3139
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1
(* Title: Nominal2_Base
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2
Authors: Christian Urban, Brian Huffman, Cezary Kaliszyk
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 4
Basic definitions and lemma infrastructure for
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 5
Nominal Isabelle.
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 6
*)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 7
theory Nominal2_Base
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 8
imports Main
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 9
"~~/src/HOL/Library/Infinite_Set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 10
"~~/src/HOL/Quotient_Examples/FSet"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 11
"GPerm"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 12
keywords
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 13
"atom_decl" "equivariance" :: thy_decl
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 14
uses ("nominal_basics.ML")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 15
("nominal_thmdecls.ML")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 16
("nominal_permeq.ML")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 17
("nominal_library.ML")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 18
("nominal_atoms.ML")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 19
("nominal_eqvt.ML")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 20
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 21
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 22
section {* Atoms and Sorts *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 23
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 24
text {* A simple implementation for atom_sorts is strings. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 25
(* types atom_sort = string *)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 26
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 27
text {* To deal with Church-like binding we use trees of
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 28
strings as sorts. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 29
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 30
datatype atom_sort = Sort "string" "atom_sort list"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 31
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 32
datatype atom = Atom atom_sort nat
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 33
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 34
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 35
text {* Basic projection function. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 36
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 37
primrec
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 38
sort_of :: "atom \<Rightarrow> atom_sort"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 39
where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 40
"sort_of (Atom s n) = s"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 41
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 42
primrec
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 43
nat_of :: "atom \<Rightarrow> nat"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 44
where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 45
"nat_of (Atom s n) = n"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 46
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 47
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 48
text {* There are infinitely many atoms of each sort. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 49
lemma INFM_sort_of_eq:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 50
shows "INFM a. sort_of a = s"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 51
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 52
have "INFM i. sort_of (Atom s i) = s" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 53
moreover have "inj (Atom s)" by (simp add: inj_on_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 54
ultimately show "INFM a. sort_of a = s" by (rule INFM_inj)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 55
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 56
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 57
lemma infinite_sort_of_eq:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 58
shows "infinite {a. sort_of a = s}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 59
using INFM_sort_of_eq unfolding INFM_iff_infinite .
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 60
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 61
lemma atom_infinite [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 62
shows "infinite (UNIV :: atom set)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 63
using subset_UNIV infinite_sort_of_eq
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 64
by (rule infinite_super)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 65
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 66
lemma obtain_atom:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 67
fixes X :: "atom set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 68
assumes X: "finite X"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 69
obtains a where "a \<notin> X" "sort_of a = s"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 70
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 71
from X have "MOST a. a \<notin> X"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 72
unfolding MOST_iff_cofinite by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 73
with INFM_sort_of_eq
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 74
have "INFM a. sort_of a = s \<and> a \<notin> X"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 75
by (rule INFM_conjI)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 76
then obtain a where "a \<notin> X" "sort_of a = s"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 77
by (auto elim: INFM_E)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 78
then show ?thesis ..
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 79
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 80
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 81
lemma atom_components_eq_iff:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 82
fixes a b :: atom
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 83
shows "a = b \<longleftrightarrow> sort_of a = sort_of b \<and> nat_of a = nat_of b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 84
by (induct a, induct b, simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 85
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 86
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 87
section {* Sort-Respecting Permutations *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 88
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 89
definition "sort_respecting p \<longleftrightarrow> (\<forall>a. sort_of (gpermute p a) = sort_of a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 90
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 91
lemma sort_respecting_0[simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 92
"sort_respecting (0\<Colon>atom gperm)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 93
by (simp add: sort_respecting_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 94
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 95
typedef (open) perm = "{p::atom gperm. sort_respecting p}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 96
by (auto intro: exI[of _ "0"])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 97
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 98
lemma perm_eq_rep:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 99
"p = q \<longleftrightarrow> Rep_perm p = Rep_perm q"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 100
by (simp add: Rep_perm_inject)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 101
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 102
definition mk_perm :: "atom gperm \<Rightarrow> perm" where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 103
"mk_perm p = Abs_perm (if sort_respecting p then p else 0)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 104
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 105
lemma sort_respecting_Rep_perm [simp, intro]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 106
"sort_respecting (Rep_perm p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 107
using Rep_perm [of p] by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 108
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 109
lemma Rep_perm_mk_perm [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 110
"Rep_perm (mk_perm p) = (if sort_respecting p then p else 0)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 111
by (simp add: mk_perm_def Abs_perm_inverse)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 112
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 113
lemma mk_perm_Rep_perm [simp, code abstype]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 114
"mk_perm (Rep_perm dxs) = dxs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 115
by (simp add: mk_perm_def Rep_perm_inverse)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 116
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Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 117
instance perm :: size ..
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 118
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 119
instantiation perm :: group_add
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 120
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 121
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Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 122
definition "(0 :: perm) = mk_perm 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 123
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 124
definition "uminus p = mk_perm (uminus (Rep_perm p))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 125
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 126
definition "p + q = mk_perm ((Rep_perm p) + (Rep_perm q))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 127
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 128
definition "(p :: perm) - q = p + - q"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 129
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 130
lemma [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 131
"sort_respecting x \<Longrightarrow> sort_respecting y \<Longrightarrow> sort_respecting (x + y)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 132
unfolding sort_respecting_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 133
by descending (simp add: perm_add_apply)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 134
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 135
lemma [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 136
"sort_respecting y \<Longrightarrow> sort_respecting (- y)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 137
unfolding sort_respecting_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 138
by partiality_descending
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 139
(auto, metis perm_apply_minus)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 140
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 141
lemma Rep_perm_0 [simp, code abstract]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 142
"Rep_perm 0 = 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 143
by (simp add: zero_perm_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 144
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 145
lemma Rep_perm_uminus [simp, code abstract]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 146
"Rep_perm (- p) = - (Rep_perm p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 147
by (simp add: uminus_perm_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 148
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 149
lemma Rep_perm_add [simp, code abstract]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 150
"Rep_perm (p + q) = (Rep_perm p) + (Rep_perm q)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 151
by (simp add: plus_perm_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 152
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 153
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 154
by default (auto simp add: perm_eq_rep add_assoc minus_perm_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 155
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 156
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 157
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 158
definition swap :: "atom \<Rightarrow> atom \<Rightarrow> perm" ("'(_ \<rightleftharpoons> _')")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 159
where "swap a b = (if sort_of a = sort_of b then mk_perm (gswap a b) else 0)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 160
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 161
lemma sort_respecting_swap [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 162
"sort_of a = sort_of b \<Longrightarrow> sort_respecting (gswap a b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 163
unfolding sort_respecting_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 164
by descending auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 165
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 166
lemma Rep_swap [simp, code abstract]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 167
"Rep_perm (swap a b) = (if sort_of a = sort_of b then gswap a b else 0)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 168
by (simp add: swap_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 169
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 170
lemma swap_different_sorts [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 171
"sort_of a \<noteq> sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) = 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 172
by (simp add: perm_eq_rep)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 173
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 174
lemma swap_cancel:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 175
shows "(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 176
and "(a \<rightleftharpoons> b) + (b \<rightleftharpoons> a) = 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 177
by (simp_all add: perm_eq_rep)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 178
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 179
lemma swap_self [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 180
"(a \<rightleftharpoons> a) = 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 181
by (simp add: perm_eq_rep)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 182
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 183
lemma minus_swap [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 184
"- (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 185
by (simp add: perm_eq_rep)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 186
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 187
lemma swap_commute:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 188
"(a \<rightleftharpoons> b) = (b \<rightleftharpoons> a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 189
by (simp add: perm_eq_rep swap_commute)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 190
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 191
lemma swap_triple:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 192
assumes "a \<noteq> b" and "c \<noteq> b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 193
assumes "sort_of a = sort_of b" "sort_of b = sort_of c"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 194
shows "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 195
using assms by (simp add: perm_eq_rep swap_triple)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 196
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 197
section {* Permutation Types *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 198
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 199
text {*
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 200
Infix syntax for @{text permute} has higher precedence than
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 201
addition, but lower than unary minus.
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 202
*}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 203
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 204
class pt =
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 205
fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a" ("_ \<bullet> _" [76, 75] 75)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 206
assumes permute_zero [simp]: "0 \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 207
assumes permute_plus [simp]: "(p + q) \<bullet> x = p \<bullet> (q \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 208
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 209
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 210
lemma permute_diff [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 211
shows "(p - q) \<bullet> x = p \<bullet> - q \<bullet> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 212
unfolding diff_minus by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 213
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 214
lemma permute_minus_cancel [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 215
shows "p \<bullet> - p \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 216
and "- p \<bullet> p \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 217
unfolding permute_plus [symmetric] by simp_all
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 218
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 219
lemma permute_swap_cancel [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 220
shows "(a \<rightleftharpoons> b) \<bullet> (a \<rightleftharpoons> b) \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 221
unfolding permute_plus [symmetric]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 222
by (simp add: swap_cancel)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 223
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 224
lemma permute_swap_cancel2 [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 225
shows "(a \<rightleftharpoons> b) \<bullet> (b \<rightleftharpoons> a) \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 226
unfolding permute_plus [symmetric]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 227
by (simp add: swap_commute)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 228
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 229
lemma inj_permute [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 230
shows "inj (permute p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 231
by (rule inj_on_inverseI)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 232
(rule permute_minus_cancel)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 233
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 234
lemma surj_permute [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 235
shows "surj (permute p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 236
by (rule surjI, rule permute_minus_cancel)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 237
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 238
lemma bij_permute [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 239
shows "bij (permute p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 240
by (rule bijI [OF inj_permute surj_permute])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 241
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 242
lemma inv_permute:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 243
shows "inv (permute p) = permute (- p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 244
by (rule inv_equality) (simp_all)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 245
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 246
lemma permute_minus:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 247
shows "permute (- p) = inv (permute p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 248
by (simp add: inv_permute)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 249
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 250
lemma permute_eq_iff [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 251
shows "p \<bullet> x = p \<bullet> y \<longleftrightarrow> x = y"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 252
by (rule inj_permute [THEN inj_eq])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 253
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 254
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 255
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 256
subsection {* Permutations for atoms *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 257
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 258
instantiation atom :: pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 259
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 260
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 261
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 262
"p \<bullet> a = gpermute (Rep_perm p) a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 263
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 264
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 265
by default (simp_all add: permute_atom_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 266
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 267
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 268
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 269
lemma sort_of_permute [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 270
shows "sort_of (p \<bullet> a) = sort_of a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 271
by (metis sort_respecting_Rep_perm sort_respecting_def permute_atom_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 272
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 273
lemma swap_atom:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 274
shows "(a \<rightleftharpoons> b) \<bullet> c =
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 275
(if sort_of a = sort_of b
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 276
then (if c = a then b else if c = b then a else c) else c)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 277
by (auto simp add: permute_atom_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 278
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 279
lemma swap_atom_simps [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 280
"sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> a = b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 281
"sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> b = a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 282
"c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> c = c"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 283
unfolding swap_atom by simp_all
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 284
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 285
lemma perm_eq_iff:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 286
fixes p q :: "perm"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 287
shows "p = q \<longleftrightarrow> (\<forall>a::atom. p \<bullet> a = q \<bullet> a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 288
unfolding permute_atom_def perm_eq_rep
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 289
by (simp add: gperm_eq)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 290
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 291
subsection {* Permutations for permutations *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 292
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 293
instantiation perm :: pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 294
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 295
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 296
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 297
"p \<bullet> q = p + q - p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 298
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 299
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 300
by default
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 301
(simp_all add: permute_perm_def diff_minus minus_add add_assoc)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 302
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 303
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 304
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 305
lemma permute_self:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 306
shows "p \<bullet> p = p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 307
unfolding permute_perm_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 308
by (simp add: diff_minus add_assoc)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 309
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 310
lemma pemute_minus_self:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 311
shows "- p \<bullet> p = p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 312
unfolding permute_perm_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 313
by (simp add: diff_minus add_assoc)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 314
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 315
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 316
subsection {* Permutations for functions *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 317
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 318
instantiation "fun" :: (pt, pt) pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 319
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 320
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 321
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 322
"p \<bullet> f = (\<lambda>x. p \<bullet> (f (- p \<bullet> x)))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 323
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 324
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 325
by default
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 326
(simp_all add: permute_fun_def minus_add)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 327
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 328
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 329
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 330
lemma permute_fun_app_eq:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 331
shows "p \<bullet> (f x) = (p \<bullet> f) (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 332
unfolding permute_fun_def by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 333
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 334
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 335
subsection {* Permutations for booleans *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 336
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 337
instantiation bool :: pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 338
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 339
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 340
definition "p \<bullet> (b::bool) = b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 341
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 342
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 343
by (default)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 344
(simp_all add: permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 345
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 346
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 347
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 348
lemma permute_boolE:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 349
fixes P::"bool"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 350
shows "p \<bullet> P \<Longrightarrow> P"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 351
by (simp add: permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 352
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 353
lemma permute_boolI:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 354
fixes P::"bool"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 355
shows "P \<Longrightarrow> p \<bullet> P"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 356
by(simp add: permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 357
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 358
subsection {* Permutations for sets *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 359
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 360
instantiation "set" :: (pt) pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 361
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 362
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 363
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 364
"p \<bullet> X = {p \<bullet> x | x. x \<in> X}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 365
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 366
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 367
apply default
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 368
apply (auto simp add: permute_set_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 369
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 370
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 371
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 372
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 373
lemma permute_set_eq:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 374
shows "p \<bullet> X = {x. - p \<bullet> x \<in> X}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 375
unfolding permute_set_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 376
by (auto) (metis permute_minus_cancel(1))
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 377
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 378
lemma permute_set_eq_image:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 379
shows "p \<bullet> X = permute p ` X"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 380
unfolding permute_set_def by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 381
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 382
lemma permute_set_eq_vimage:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 383
shows "p \<bullet> X = permute (- p) -` X"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 384
unfolding permute_set_eq vimage_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 385
by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 386
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 387
lemma permute_finite [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 388
shows "finite (p \<bullet> X) = finite X"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 389
unfolding permute_set_eq_vimage
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 390
using bij_permute by (rule finite_vimage_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 391
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 392
lemma swap_set_not_in:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 393
assumes a: "a \<notin> S" "b \<notin> S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 394
shows "(a \<rightleftharpoons> b) \<bullet> S = S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 395
unfolding permute_set_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 396
using a by (auto simp add: swap_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 397
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 398
lemma swap_set_in:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 399
assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 400
shows "(a \<rightleftharpoons> b) \<bullet> S \<noteq> S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 401
unfolding permute_set_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 402
using a by (auto simp add: swap_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 403
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 404
lemma swap_set_in_eq:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 405
assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 406
shows "(a \<rightleftharpoons> b) \<bullet> S = (S - {a}) \<union> {b}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 407
unfolding permute_set_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 408
using a by (auto simp add: swap_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 409
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 410
lemma swap_set_both_in:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 411
assumes a: "a \<in> S" "b \<in> S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 412
shows "(a \<rightleftharpoons> b) \<bullet> S = S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 413
unfolding permute_set_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 414
using a by (auto simp add: swap_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 415
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 416
lemma mem_permute_iff:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 417
shows "(p \<bullet> x) \<in> (p \<bullet> X) \<longleftrightarrow> x \<in> X"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 418
unfolding permute_set_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 419
by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 420
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 421
lemma empty_eqvt:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 422
shows "p \<bullet> {} = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 423
unfolding permute_set_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 424
by (simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 425
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 426
lemma insert_eqvt:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 427
shows "p \<bullet> (insert x A) = insert (p \<bullet> x) (p \<bullet> A)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 428
unfolding permute_set_eq_image image_insert ..
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 429
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 430
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 431
subsection {* Permutations for @{typ unit} *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 432
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 433
instantiation unit :: pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 434
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 435
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 436
definition "p \<bullet> (u::unit) = u"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 437
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 438
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 439
by (default) (simp_all add: permute_unit_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 440
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 441
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 442
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 443
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 444
subsection {* Permutations for products *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 445
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 446
instantiation prod :: (pt, pt) pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 447
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 448
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 449
primrec
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 450
permute_prod
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 451
where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 452
Pair_eqvt: "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 453
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 454
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 455
by default auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 456
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 457
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 458
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 459
subsection {* Permutations for sums *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 460
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 461
instantiation sum :: (pt, pt) pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 462
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 463
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 464
primrec
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 465
permute_sum
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 466
where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 467
Inl_eqvt: "p \<bullet> (Inl x) = Inl (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 468
| Inr_eqvt: "p \<bullet> (Inr y) = Inr (p \<bullet> y)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 469
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 470
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 471
by (default) (case_tac [!] x, simp_all)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 472
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 473
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 474
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 475
subsection {* Permutations for @{typ "'a list"} *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 476
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 477
instantiation list :: (pt) pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 478
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 479
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 480
primrec
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 481
permute_list
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 482
where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 483
Nil_eqvt: "p \<bullet> [] = []"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 484
| Cons_eqvt: "p \<bullet> (x # xs) = p \<bullet> x # p \<bullet> xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 485
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 486
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 487
by (default) (induct_tac [!] x, simp_all)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 488
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 489
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 490
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 491
lemma set_eqvt:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 492
shows "p \<bullet> (set xs) = set (p \<bullet> xs)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 493
by (induct xs) (simp_all add: empty_eqvt insert_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 494
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 495
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 496
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 497
subsection {* Permutations for @{typ "'a option"} *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 498
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 499
instantiation option :: (pt) pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 500
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 501
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 502
primrec
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 503
permute_option
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 504
where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 505
None_eqvt: "p \<bullet> None = None"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 506
| Some_eqvt: "p \<bullet> (Some x) = Some (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 507
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 508
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 509
by (default) (induct_tac [!] x, simp_all)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 510
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 511
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 512
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 513
subsection {* Permutations for @{typ "'a multiset"} *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 514
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 515
instantiation multiset :: (pt) pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 516
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 517
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 518
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 519
"p \<bullet> M = {# p \<bullet> x. x :# M #}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 520
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 521
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 522
proof
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 523
fix M :: "'a multiset" and p q :: "perm"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 524
show "0 \<bullet> M = M"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 525
unfolding permute_multiset_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 526
by (induct_tac M) (simp_all)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 527
show "(p + q) \<bullet> M = p \<bullet> q \<bullet> M"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 528
unfolding permute_multiset_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 529
by (induct_tac M) (simp_all)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 530
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 531
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 532
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 533
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 534
lemma permute_multiset [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 535
fixes M N::"('a::pt) multiset"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 536
shows "(p \<bullet> {#}) = ({#} ::('a::pt) multiset)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 537
and "(p \<bullet> {# x #}) = {# p \<bullet> x #}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 538
and "(p \<bullet> (M + N)) = (p \<bullet> M) + (p \<bullet> N)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 539
unfolding permute_multiset_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 540
by (simp_all)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 541
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 542
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 543
subsection {* Permutations for @{typ "'a fset"} *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 544
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 545
lemma permute_fset_rsp[quot_respect]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 546
shows "(op = ===> list_eq ===> list_eq) permute permute"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 547
unfolding fun_rel_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 548
by (simp add: set_eqvt[symmetric])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 549
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 550
instantiation fset :: (pt) pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 551
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 552
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 553
quotient_definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 554
"permute_fset :: perm \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 555
is
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 556
"permute :: perm \<Rightarrow> 'a list \<Rightarrow> 'a list"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 557
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 558
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 559
proof
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 560
fix x :: "'a fset" and p q :: "perm"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 561
have lst: "\<And>l :: 'a list. 0 \<bullet> l = l" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 562
show "0 \<bullet> x = x" by (lifting lst)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 563
have lst: "\<And>p q :: perm. \<And>x :: 'a list. (p + q) \<bullet> x = p \<bullet> q \<bullet> x" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 564
show "(p + q) \<bullet> x = p \<bullet> q \<bullet> x" by (lifting lst)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 565
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 566
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 567
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 568
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 569
lemma permute_fset [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 570
fixes S::"('a::pt) fset"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 571
shows "(p \<bullet> {||}) = ({||} ::('a::pt) fset)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 572
and "(p \<bullet> insert_fset x S) = insert_fset (p \<bullet> x) (p \<bullet> S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 573
by (lifting permute_list.simps)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 574
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 575
lemma fset_eqvt:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 576
shows "p \<bullet> (fset S) = fset (p \<bullet> S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 577
by (lifting set_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 578
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 579
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 580
subsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 581
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 582
instantiation char :: pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 583
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 584
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 585
definition "p \<bullet> (c::char) = c"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 586
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 587
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 588
by (default) (simp_all add: permute_char_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 589
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 590
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 591
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 592
instantiation nat :: pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 593
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 594
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 595
definition "p \<bullet> (n::nat) = n"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 596
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 597
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 598
by (default) (simp_all add: permute_nat_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 599
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 600
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 601
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 602
instantiation int :: pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 603
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 604
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 605
definition "p \<bullet> (i::int) = i"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 606
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 607
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 608
by (default) (simp_all add: permute_int_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 609
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 610
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 611
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 612
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 613
section {* Pure types *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 614
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 615
text {* Pure types will have always empty support. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 616
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 617
class pure = pt +
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 618
assumes permute_pure: "p \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 619
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 620
text {* Types @{typ unit} and @{typ bool} are pure. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 621
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 622
instance unit :: pure
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 623
proof qed (rule permute_unit_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 624
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 625
instance bool :: pure
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 626
proof qed (rule permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 627
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 628
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 629
text {* Other type constructors preserve purity. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 630
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 631
instance "fun" :: (pure, pure) pure
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 632
by default (simp add: permute_fun_def permute_pure)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 633
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 634
instance set :: (pure) pure
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 635
by default (simp add: permute_set_def permute_pure)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 636
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 637
instance prod :: (pure, pure) pure
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 638
by default (induct_tac x, simp add: permute_pure)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 639
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 640
instance sum :: (pure, pure) pure
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 641
by default (induct_tac x, simp_all add: permute_pure)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 642
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 643
instance list :: (pure) pure
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 644
by default (induct_tac x, simp_all add: permute_pure)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 645
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 646
instance option :: (pure) pure
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 647
by default (induct_tac x, simp_all add: permute_pure)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 648
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 649
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 650
subsection {* Types @{typ char}, @{typ nat}, and @{typ int} *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 651
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 652
instance char :: pure
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 653
proof qed (rule permute_char_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 654
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 655
instance nat :: pure
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 656
proof qed (rule permute_nat_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 657
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 658
instance int :: pure
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 659
proof qed (rule permute_int_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 660
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 661
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 662
section {* Infrastructure for Equivariance and Perm_simp *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 663
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 664
subsection {* Basic functions about permutations *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 665
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 666
use "nominal_basics.ML"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 667
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 668
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 669
subsection {* Eqvt infrastructure *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 670
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 671
text {* Setup of the theorem attributes @{text eqvt} and @{text eqvt_raw} *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 672
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 673
use "nominal_thmdecls.ML"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 674
setup "Nominal_ThmDecls.setup"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 675
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 676
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 677
lemmas [eqvt] =
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 678
(* pt types *)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 679
permute_prod.simps
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 680
permute_list.simps
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 681
permute_option.simps
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 682
permute_sum.simps
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 683
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 684
(* sets *)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 685
empty_eqvt insert_eqvt set_eqvt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 686
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 687
(* fsets *)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 688
permute_fset fset_eqvt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 689
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 690
(* multisets *)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 691
permute_multiset
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 692
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 693
subsection {* perm_simp infrastructure *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 694
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 695
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 696
"unpermute p = permute (- p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 697
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 698
lemma eqvt_apply:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 699
fixes f :: "'a::pt \<Rightarrow> 'b::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 700
and x :: "'a::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 701
shows "p \<bullet> (f x) \<equiv> (p \<bullet> f) (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 702
unfolding permute_fun_def by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 703
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 704
lemma eqvt_lambda:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 705
fixes f :: "'a::pt \<Rightarrow> 'b::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 706
shows "p \<bullet> f \<equiv> (\<lambda>x. p \<bullet> (f (unpermute p x)))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 707
unfolding permute_fun_def unpermute_def by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 708
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 709
lemma eqvt_bound:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 710
shows "p \<bullet> unpermute p x \<equiv> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 711
unfolding unpermute_def by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 712
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 713
text {* provides perm_simp methods *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 714
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 715
use "nominal_permeq.ML"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 716
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 717
method_setup perm_simp =
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 718
{* Nominal_Permeq.args_parser >> Nominal_Permeq.perm_simp_meth *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 719
{* pushes permutations inside. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 720
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 721
method_setup perm_strict_simp =
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 722
{* Nominal_Permeq.args_parser >> Nominal_Permeq.perm_strict_simp_meth *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 723
{* pushes permutations inside, raises an error if it cannot solve all permutations. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 724
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 725
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 726
subsubsection {* Equivariance for permutations and swapping *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 727
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 728
lemma permute_eqvt:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 729
shows "p \<bullet> (q \<bullet> x) = (p \<bullet> q) \<bullet> (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 730
unfolding permute_perm_def by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 731
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 732
(* the normal version of this lemma would cause loops *)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 733
lemma permute_eqvt_raw [eqvt_raw]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 734
shows "p \<bullet> permute \<equiv> permute"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 735
apply(simp add: fun_eq_iff permute_fun_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 736
apply(subst permute_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 737
apply(simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 738
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 739
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 740
lemma zero_perm_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 741
shows "p \<bullet> (0::perm) = 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 742
unfolding permute_perm_def by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 743
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 744
lemma add_perm_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 745
fixes p p1 p2 :: perm
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 746
shows "p \<bullet> (p1 + p2) = p \<bullet> p1 + p \<bullet> p2"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 747
unfolding permute_perm_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 748
by (simp add: perm_eq_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 749
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 750
lemma swap_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 751
shows "p \<bullet> (a \<rightleftharpoons> b) = (p \<bullet> a \<rightleftharpoons> p \<bullet> b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 752
unfolding permute_perm_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 753
by (auto simp add: swap_atom perm_eq_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 754
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 755
lemma uminus_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 756
fixes p q::"perm"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 757
shows "p \<bullet> (- q) = - (p \<bullet> q)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 758
unfolding permute_perm_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 759
by (simp add: diff_minus minus_add add_assoc)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 760
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 761
subsubsection {* Equivariance of Logical Operators *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 762
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 763
lemma eq_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 764
shows "p \<bullet> (x = y) \<longleftrightarrow> (p \<bullet> x) = (p \<bullet> y)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 765
unfolding permute_eq_iff permute_bool_def ..
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 766
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 767
lemma Not_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 768
shows "p \<bullet> (\<not> A) \<longleftrightarrow> \<not> (p \<bullet> A)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 769
by (simp add: permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 770
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 771
lemma conj_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 772
shows "p \<bullet> (A \<and> B) \<longleftrightarrow> (p \<bullet> A) \<and> (p \<bullet> B)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 773
by (simp add: permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 774
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 775
lemma imp_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 776
shows "p \<bullet> (A \<longrightarrow> B) \<longleftrightarrow> (p \<bullet> A) \<longrightarrow> (p \<bullet> B)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 777
by (simp add: permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 778
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 779
declare imp_eqvt[folded induct_implies_def, eqvt]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 780
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 781
lemma all_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 782
shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. (p \<bullet> P) x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 783
unfolding All_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 784
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 785
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 786
declare all_eqvt[folded induct_forall_def, eqvt]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 787
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 788
lemma ex_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 789
shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. (p \<bullet> P) x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 790
unfolding Ex_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 791
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 792
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 793
lemma ex1_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 794
shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. (p \<bullet> P) x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 795
unfolding Ex1_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 796
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 797
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 798
lemma if_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 799
shows "p \<bullet> (if b then x else y) = (if p \<bullet> b then p \<bullet> x else p \<bullet> y)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 800
by (simp add: permute_fun_def permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 801
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 802
lemma True_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 803
shows "p \<bullet> True = True"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 804
unfolding permute_bool_def ..
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 805
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 806
lemma False_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 807
shows "p \<bullet> False = False"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 808
unfolding permute_bool_def ..
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 809
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 810
lemma disj_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 811
shows "p \<bullet> (A \<or> B) \<longleftrightarrow> (p \<bullet> A) \<or> (p \<bullet> B)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 812
by (simp add: permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 813
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 814
lemma all_eqvt2:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 815
shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. p \<bullet> P (- p \<bullet> x))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 816
by (perm_simp add: permute_minus_cancel) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 817
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 818
lemma ex_eqvt2:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 819
shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. p \<bullet> P (- p \<bullet> x))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 820
by (perm_simp add: permute_minus_cancel) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 821
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 822
lemma ex1_eqvt2:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 823
shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. p \<bullet> P (- p \<bullet> x))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 824
by (perm_simp add: permute_minus_cancel) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 825
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 826
lemma the_eqvt:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 827
assumes unique: "\<exists>!x. P x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 828
shows "(p \<bullet> (THE x. P x)) = (THE x. (p \<bullet> P) x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 829
apply(rule the1_equality [symmetric])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 830
apply(rule_tac p="-p" in permute_boolE)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 831
apply(perm_simp add: permute_minus_cancel)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 832
apply(rule unique)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 833
apply(rule_tac p="-p" in permute_boolE)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 834
apply(perm_simp add: permute_minus_cancel)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 835
apply(rule theI'[OF unique])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 836
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 837
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 838
lemma the_eqvt2:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 839
assumes unique: "\<exists>!x. P x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 840
shows "(p \<bullet> (THE x. P x)) = (THE x. p \<bullet> P (- p \<bullet> x))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 841
apply(rule the1_equality [symmetric])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 842
apply(simp add: ex1_eqvt2[symmetric])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 843
apply(simp add: permute_bool_def unique)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 844
apply(simp add: permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 845
apply(rule theI'[OF unique])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 846
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 847
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 848
subsubsection {* Equivariance of Set operators *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 849
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 850
lemma mem_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 851
shows "p \<bullet> (x \<in> A) \<longleftrightarrow> (p \<bullet> x) \<in> (p \<bullet> A)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 852
unfolding permute_bool_def permute_set_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 853
by (auto)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 854
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 855
lemma Collect_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 856
shows "p \<bullet> {x. P x} = {x. (p \<bullet> P) x}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 857
unfolding permute_set_eq permute_fun_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 858
by (auto simp add: permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 859
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 860
lemma inter_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 861
shows "p \<bullet> (A \<inter> B) = (p \<bullet> A) \<inter> (p \<bullet> B)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 862
unfolding Int_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 863
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 864
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 865
lemma Bex_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 866
shows "p \<bullet> (\<exists>x \<in> S. P x) = (\<exists>x \<in> (p \<bullet> S). (p \<bullet> P) x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 867
unfolding Bex_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 868
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 869
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 870
lemma Ball_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 871
shows "p \<bullet> (\<forall>x \<in> S. P x) = (\<forall>x \<in> (p \<bullet> S). (p \<bullet> P) x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 872
unfolding Ball_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 873
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 874
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 875
lemma image_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 876
shows "p \<bullet> (f ` A) = (p \<bullet> f) ` (p \<bullet> A)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 877
unfolding image_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 878
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 879
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 880
lemma Image_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 881
shows "p \<bullet> (R `` A) = (p \<bullet> R) `` (p \<bullet> A)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 882
unfolding Image_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 883
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 884
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 885
lemma UNIV_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 886
shows "p \<bullet> UNIV = UNIV"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 887
unfolding UNIV_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 888
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 889
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 890
lemma union_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 891
shows "p \<bullet> (A \<union> B) = (p \<bullet> A) \<union> (p \<bullet> B)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 892
unfolding Un_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 893
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 894
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 895
lemma Diff_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 896
fixes A B :: "'a::pt set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 897
shows "p \<bullet> (A - B) = (p \<bullet> A) - (p \<bullet> B)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 898
unfolding set_diff_eq
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 899
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 900
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 901
lemma Compl_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 902
fixes A :: "'a::pt set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 903
shows "p \<bullet> (- A) = - (p \<bullet> A)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 904
unfolding Compl_eq_Diff_UNIV
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 905
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 906
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 907
lemma subset_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 908
shows "p \<bullet> (S \<subseteq> T) \<longleftrightarrow> (p \<bullet> S) \<subseteq> (p \<bullet> T)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 909
unfolding subset_eq
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 910
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 911
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 912
lemma psubset_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 913
shows "p \<bullet> (S \<subset> T) \<longleftrightarrow> (p \<bullet> S) \<subset> (p \<bullet> T)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 914
unfolding psubset_eq
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 915
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 916
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 917
lemma vimage_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 918
shows "p \<bullet> (f -` A) = (p \<bullet> f) -` (p \<bullet> A)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 919
unfolding vimage_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 920
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 921
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 922
lemma Union_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 923
shows "p \<bullet> (\<Union> S) = \<Union> (p \<bullet> S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 924
unfolding Union_eq
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 925
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 926
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 927
lemma Inter_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 928
shows "p \<bullet> (\<Inter> S) = \<Inter> (p \<bullet> S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 929
unfolding Inter_eq
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 930
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 931
3147
+ − 932
lemma foldr_eqvt[eqvt]:
+ − 933
"p \<bullet> foldr a b c = foldr (p \<bullet> a) (p \<bullet> b) (p \<bullet> c)"
+ − 934
apply (induct b)
+ − 935
apply simp_all
+ − 936
apply (perm_simp)
+ − 937
apply simp
+ − 938
done
+ − 939
3139
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 940
(* FIXME: eqvt attribute *)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 941
lemma Sigma_eqvt:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 942
shows "(p \<bullet> (X \<times> Y)) = (p \<bullet> X) \<times> (p \<bullet> Y)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 943
unfolding Sigma_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 944
unfolding SUP_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 945
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 946
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 947
text {*
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 948
In order to prove that lfp is equivariant we need two
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 949
auxiliary classes which specify that (op <=) and
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 950
Inf are equivariant. Instances for bool and fun are
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 951
given.
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 952
*}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 953
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 954
class le_eqvt = order +
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 955
assumes le_eqvt [eqvt]: "p \<bullet> (x \<le> y) = ((p \<bullet> x) \<le> (p \<bullet> (y::('a::{pt, order}))))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 956
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 957
class inf_eqvt = complete_lattice +
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 958
assumes inf_eqvt [eqvt]: "p \<bullet> (Inf X) = Inf (p \<bullet> (X::('a::{pt, Inf}) set))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 959
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 960
instantiation bool :: le_eqvt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 961
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 962
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 963
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 964
apply(default)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 965
apply perm_simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 966
apply(rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 967
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 968
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 969
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 970
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 971
instantiation "fun" :: (pt, le_eqvt) le_eqvt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 972
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 973
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 974
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 975
apply(default)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 976
unfolding le_fun_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 977
apply(perm_simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 978
apply(rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 979
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 980
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 981
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 982
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 983
instantiation bool :: inf_eqvt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 984
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 985
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 986
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 987
apply(default)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 988
apply(perm_simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 989
apply(rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 990
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 991
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 992
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 993
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 994
instantiation "fun" :: (pt, inf_eqvt) inf_eqvt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 995
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 996
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 997
instance
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 998
apply(default)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 999
unfolding Inf_fun_def INF_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1000
apply(perm_simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1001
apply(rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1002
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1003
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1004
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1005
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1006
lemma lfp_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1007
fixes F::"('a \<Rightarrow> 'b) \<Rightarrow> ('a::pt \<Rightarrow> 'b::{inf_eqvt, le_eqvt})"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1008
shows "p \<bullet> (lfp F) = lfp (p \<bullet> F)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1009
unfolding lfp_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1010
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1011
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1012
lemma finite_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1013
shows "p \<bullet> finite A = finite (p \<bullet> A)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1014
unfolding finite_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1015
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1016
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1017
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1018
subsubsection {* Equivariance for product operations *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1019
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1020
lemma fst_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1021
shows "p \<bullet> (fst x) = fst (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1022
by (cases x) simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1023
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1024
lemma snd_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1025
shows "p \<bullet> (snd x) = snd (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1026
by (cases x) simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1027
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1028
lemma split_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1029
shows "p \<bullet> (split P x) = split (p \<bullet> P) (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1030
unfolding split_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1031
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1032
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1033
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1034
subsubsection {* Equivariance for list operations *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1035
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1036
lemma append_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1037
shows "p \<bullet> (xs @ ys) = (p \<bullet> xs) @ (p \<bullet> ys)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1038
by (induct xs) auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1039
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1040
lemma rev_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1041
shows "p \<bullet> (rev xs) = rev (p \<bullet> xs)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1042
by (induct xs) (simp_all add: append_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1043
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1044
lemma map_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1045
shows "p \<bullet> (map f xs) = map (p \<bullet> f) (p \<bullet> xs)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1046
by (induct xs) (simp_all, simp only: permute_fun_app_eq)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1047
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1048
lemma removeAll_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1049
shows "p \<bullet> (removeAll x xs) = removeAll (p \<bullet> x) (p \<bullet> xs)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1050
by (induct xs) (auto)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1051
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1052
lemma filter_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1053
shows "p \<bullet> (filter f xs) = filter (p \<bullet> f) (p \<bullet> xs)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1054
apply(induct xs)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1055
apply(simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1056
apply(simp only: filter.simps permute_list.simps if_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1057
apply(simp only: permute_fun_app_eq)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1058
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1059
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1060
lemma distinct_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1061
shows "p \<bullet> (distinct xs) = distinct (p \<bullet> xs)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1062
apply(induct xs)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1063
apply(simp add: permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1064
apply(simp add: conj_eqvt Not_eqvt mem_eqvt set_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1065
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1066
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1067
lemma length_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1068
shows "p \<bullet> (length xs) = length (p \<bullet> xs)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1069
by (induct xs) (simp_all add: permute_pure)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1070
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1071
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1072
subsubsection {* Equivariance for @{typ "'a option"} *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1073
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1074
lemma option_map_eqvt[eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1075
shows "p \<bullet> (Option.map f x) = Option.map (p \<bullet> f) (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1076
by (cases x) (simp_all, simp add: permute_fun_app_eq)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1077
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1078
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1079
subsubsection {* Equivariance for @{typ "'a fset"} *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1080
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1081
lemma in_fset_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1082
shows "(p \<bullet> (x |\<in>| S)) = ((p \<bullet> x) |\<in>| (p \<bullet> S))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1083
unfolding in_fset
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1084
by (perm_simp) (simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1085
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1086
lemma union_fset_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1087
shows "(p \<bullet> (S |\<union>| T)) = ((p \<bullet> S) |\<union>| (p \<bullet> T))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1088
by (induct S) (simp_all)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1089
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1090
lemma inter_list_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1091
shows "p \<bullet> (inter_list S T) = inter_list (p \<bullet> S) (p \<bullet> T)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1092
unfolding list_eq_def inter_list_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1093
by perm_simp simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1094
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1095
lemma inter_fset_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1096
shows "(p \<bullet> (S |\<inter>| T)) = ((p \<bullet> S) |\<inter>| (p \<bullet> T))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1097
by (lifting inter_list_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1098
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1099
lemma sub_list_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1100
shows "p \<bullet> (sub_list S T) = sub_list (p \<bullet> S) (p \<bullet> T)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1101
unfolding sub_list_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1102
by perm_simp simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1103
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1104
lemma subset_fset_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1105
shows "(p \<bullet> (S |\<subseteq>| T)) = ((p \<bullet> S) |\<subseteq>| (p \<bullet> T))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1106
by (lifting sub_list_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1107
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1108
lemma map_fset_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1109
shows "p \<bullet> (map_fset f S) = map_fset (p \<bullet> f) (p \<bullet> S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1110
by (lifting map_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1111
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1112
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1113
section {* Supp, Freshness and Supports *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1114
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1115
context pt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1116
begin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1117
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1118
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1119
supp :: "'a \<Rightarrow> atom set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1120
where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1121
"supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1122
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1123
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1124
fresh :: "atom \<Rightarrow> 'a \<Rightarrow> bool" ("_ \<sharp> _" [55, 55] 55)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1125
where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1126
"a \<sharp> x \<equiv> a \<notin> supp x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1127
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1128
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1129
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1130
lemma supp_conv_fresh:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1131
shows "supp x = {a. \<not> a \<sharp> x}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1132
unfolding fresh_def by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1133
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1134
lemma swap_rel_trans:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1135
assumes "sort_of a = sort_of b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1136
assumes "sort_of b = sort_of c"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1137
assumes "(a \<rightleftharpoons> c) \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1138
assumes "(b \<rightleftharpoons> c) \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1139
shows "(a \<rightleftharpoons> b) \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1140
proof (cases)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1141
assume "a = b \<or> c = b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1142
with assms show "(a \<rightleftharpoons> b) \<bullet> x = x" by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1143
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1144
assume *: "\<not> (a = b \<or> c = b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1145
have "((a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c)) \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1146
using assms by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1147
also have "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1148
using assms * by (simp add: swap_triple)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1149
finally show "(a \<rightleftharpoons> b) \<bullet> x = x" .
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1150
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1151
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1152
lemma swap_fresh_fresh:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1153
assumes a: "a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1154
and b: "b \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1155
shows "(a \<rightleftharpoons> b) \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1156
proof (cases)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1157
assume asm: "sort_of a = sort_of b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1158
have "finite {c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x}" "finite {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1159
using a b unfolding fresh_def supp_def by simp_all
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1160
then have "finite ({c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x} \<union> {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x})" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1161
then obtain c
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1162
where "(a \<rightleftharpoons> c) \<bullet> x = x" "(b \<rightleftharpoons> c) \<bullet> x = x" "sort_of c = sort_of b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1163
by (rule obtain_atom) (auto)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1164
then show "(a \<rightleftharpoons> b) \<bullet> x = x" using asm by (rule_tac swap_rel_trans) (simp_all)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1165
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1166
assume "sort_of a \<noteq> sort_of b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1167
then show "(a \<rightleftharpoons> b) \<bullet> x = x" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1168
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1169
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1170
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1171
subsection {* supp and fresh are equivariant *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1172
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1173
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1174
lemma supp_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1175
shows "p \<bullet> (supp x) = supp (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1176
unfolding supp_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1177
by (perm_simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1178
(simp only: permute_eqvt[symmetric])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1179
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1180
lemma fresh_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1181
shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1182
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1183
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1184
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1185
lemma fresh_permute_iff:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1186
shows "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1187
by (simp only: fresh_eqvt[symmetric] permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1188
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1189
lemma fresh_permute_left:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1190
shows "a \<sharp> p \<bullet> x \<longleftrightarrow> - p \<bullet> a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1191
proof
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1192
assume "a \<sharp> p \<bullet> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1193
then have "- p \<bullet> a \<sharp> - p \<bullet> p \<bullet> x" by (simp only: fresh_permute_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1194
then show "- p \<bullet> a \<sharp> x" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1195
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1196
assume "- p \<bullet> a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1197
then have "p \<bullet> - p \<bullet> a \<sharp> p \<bullet> x" by (simp only: fresh_permute_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1198
then show "a \<sharp> p \<bullet> x" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1199
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1200
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1201
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1202
section {* supports *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1203
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1204
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1205
supports :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" (infixl "supports" 80)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1206
where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1207
"S supports x \<equiv> \<forall>a b. (a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1208
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1209
lemma supp_is_subset:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1210
fixes S :: "atom set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1211
and x :: "'a::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1212
assumes a1: "S supports x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1213
and a2: "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1214
shows "(supp x) \<subseteq> S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1215
proof (rule ccontr)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1216
assume "\<not> (supp x \<subseteq> S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1217
then obtain a where b1: "a \<in> supp x" and b2: "a \<notin> S" by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1218
from a1 b2 have "\<forall>b. b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x" unfolding supports_def by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1219
then have "{b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x} \<subseteq> S" by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1220
with a2 have "finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}" by (simp add: finite_subset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1221
then have "a \<notin> (supp x)" unfolding supp_def by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1222
with b1 show False by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1223
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1224
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1225
lemma supports_finite:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1226
fixes S :: "atom set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1227
and x :: "'a::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1228
assumes a1: "S supports x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1229
and a2: "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1230
shows "finite (supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1231
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1232
have "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1233
then show "finite (supp x)" using a2 by (simp add: finite_subset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1234
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1235
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1236
lemma supp_supports:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1237
fixes x :: "'a::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1238
shows "(supp x) supports x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1239
unfolding supports_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1240
proof (intro strip)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1241
fix a b
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1242
assume "a \<notin> (supp x) \<and> b \<notin> (supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1243
then have "a \<sharp> x" and "b \<sharp> x" by (simp_all add: fresh_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1244
then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1245
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1246
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1247
lemma supports_fresh:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1248
fixes x :: "'a::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1249
assumes a1: "S supports x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1250
and a2: "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1251
and a3: "a \<notin> S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1252
shows "a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1253
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1254
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1255
have "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1256
then show "a \<notin> (supp x)" using a3 by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1257
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1258
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1259
lemma supp_is_least_supports:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1260
fixes S :: "atom set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1261
and x :: "'a::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1262
assumes a1: "S supports x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1263
and a2: "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1264
and a3: "\<And>S'. finite S' \<Longrightarrow> (S' supports x) \<Longrightarrow> S \<subseteq> S'"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1265
shows "(supp x) = S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1266
proof (rule equalityI)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1267
show "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1268
with a2 have fin: "finite (supp x)" by (rule rev_finite_subset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1269
have "(supp x) supports x" by (rule supp_supports)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1270
with fin a3 show "S \<subseteq> supp x" by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1271
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1272
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1273
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1274
lemma subsetCI:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1275
shows "(\<And>x. x \<in> A \<Longrightarrow> x \<notin> B \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> B"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1276
by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1277
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1278
lemma finite_supp_unique:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1279
assumes a1: "S supports x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1280
assumes a2: "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1281
assumes a3: "\<And>a b. \<lbrakk>a \<in> S; b \<notin> S; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1282
shows "(supp x) = S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1283
using a1 a2
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1284
proof (rule supp_is_least_supports)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1285
fix S'
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1286
assume "finite S'" and "S' supports x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1287
show "S \<subseteq> S'"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1288
proof (rule subsetCI)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1289
fix a
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1290
assume "a \<in> S" and "a \<notin> S'"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1291
have "finite (S \<union> S')"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1292
using `finite S` `finite S'` by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1293
then obtain b where "b \<notin> S \<union> S'" and "sort_of b = sort_of a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1294
by (rule obtain_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1295
then have "b \<notin> S" and "b \<notin> S'" and "sort_of a = sort_of b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1296
by simp_all
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1297
then have "(a \<rightleftharpoons> b) \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1298
using `a \<notin> S'` `S' supports x` by (simp add: supports_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1299
moreover have "(a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1300
using `a \<in> S` `b \<notin> S` `sort_of a = sort_of b`
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1301
by (rule a3)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1302
ultimately show "False" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1303
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1304
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1305
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1306
section {* Support w.r.t. relations *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1307
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1308
text {*
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1309
This definition is used for unquotient types, where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1310
alpha-equivalence does not coincide with equality.
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1311
*}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1312
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1313
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1314
"supp_rel R x = {a. infinite {b. \<not>(R ((a \<rightleftharpoons> b) \<bullet> x) x)}}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1315
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1316
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1317
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1318
section {* Finitely-supported types *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1319
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1320
class fs = pt +
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1321
assumes finite_supp: "finite (supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1322
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1323
lemma pure_supp:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1324
fixes x::"'a::pure"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1325
shows "supp x = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1326
unfolding supp_def by (simp add: permute_pure)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1327
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1328
lemma pure_fresh:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1329
fixes x::"'a::pure"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1330
shows "a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1331
unfolding fresh_def by (simp add: pure_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1332
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1333
instance pure < fs
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1334
by default (simp add: pure_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1335
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1336
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1337
subsection {* Type @{typ atom} is finitely-supported. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1338
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1339
lemma supp_atom:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1340
shows "supp a = {a}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1341
by (rule finite_supp_unique)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1342
(auto simp add: supports_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1343
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1344
lemma fresh_atom:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1345
shows "a \<sharp> b \<longleftrightarrow> a \<noteq> b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1346
unfolding fresh_def supp_atom by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1347
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1348
instance atom :: fs
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1349
by default (simp add: supp_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1350
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1351
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1352
section {* Type @{typ perm} is finitely-supported. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1353
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1354
lemma perm_swap_eq:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1355
shows "(a \<rightleftharpoons> b) \<bullet> p = p \<longleftrightarrow> (p \<bullet> (a \<rightleftharpoons> b)) = (a \<rightleftharpoons> b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1356
unfolding permute_perm_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1357
by (metis add_diff_cancel minus_perm_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1358
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1359
lemma supports_perm:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1360
shows "{a. p \<bullet> a \<noteq> a} supports p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1361
unfolding supports_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1362
unfolding perm_swap_eq
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1363
by (simp add: swap_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1364
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1365
lemma finite_perm_lemma:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1366
shows "finite {a::atom. p \<bullet> a \<noteq> a}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1367
unfolding permute_atom_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1368
using finite_gpermute_neq .
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1369
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1370
lemma supp_perm:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1371
shows "supp p = {a. p \<bullet> a \<noteq> a}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1372
apply (rule finite_supp_unique)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1373
apply (simp_all add: perm_swap_eq swap_eqvt supports_perm finite_perm_lemma)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1374
apply (auto simp add: perm_eq_iff swap_atom perm_swap_eq swap_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1375
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1376
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1377
lemma fresh_perm:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1378
shows "a \<sharp> p \<longleftrightarrow> p \<bullet> a = a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1379
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1380
by (simp add: supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1381
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1382
lemma supp_swap:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1383
shows "supp (a \<rightleftharpoons> b) = (if a = b \<or> sort_of a \<noteq> sort_of b then {} else {a, b})"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1384
by (auto simp add: supp_perm swap_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1385
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1386
lemma fresh_zero_perm:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1387
shows "a \<sharp> (0::perm)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1388
unfolding fresh_perm by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1389
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1390
lemma supp_zero_perm:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1391
shows "supp (0::perm) = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1392
unfolding supp_perm by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1393
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1394
lemma fresh_plus_perm:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1395
fixes p q::perm
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1396
assumes "a \<sharp> p" "a \<sharp> q"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1397
shows "a \<sharp> (p + q)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1398
using assms
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1399
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1400
by (auto simp add: supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1401
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1402
lemma supp_plus_perm:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1403
fixes p q::perm
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1404
shows "supp (p + q) \<subseteq> supp p \<union> supp q"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1405
by (auto simp add: supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1406
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1407
lemma fresh_minus_perm:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1408
fixes p::perm
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1409
shows "a \<sharp> (- p) \<longleftrightarrow> a \<sharp> p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1410
unfolding fresh_def supp_perm
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1411
by (simp) (metis permute_minus_cancel(1))
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1412
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1413
lemma supp_minus_perm:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1414
fixes p::perm
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1415
shows "supp (- p) = supp p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1416
unfolding supp_conv_fresh
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1417
by (simp add: fresh_minus_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1418
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1419
lemma plus_perm_eq:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1420
fixes p q::"perm"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1421
assumes asm: "supp p \<inter> supp q = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1422
shows "p + q = q + p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1423
unfolding perm_eq_iff
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1424
proof
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1425
fix a::"atom"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1426
show "(p + q) \<bullet> a = (q + p) \<bullet> a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1427
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1428
{ assume "a \<notin> supp p" "a \<notin> supp q"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1429
then have "(p + q) \<bullet> a = (q + p) \<bullet> a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1430
by (simp add: supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1431
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1432
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1433
{ assume a: "a \<in> supp p" "a \<notin> supp q"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1434
then have "p \<bullet> a \<in> supp p" by (simp add: supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1435
then have "p \<bullet> a \<notin> supp q" using asm by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1436
with a have "(p + q) \<bullet> a = (q + p) \<bullet> a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1437
by (simp add: supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1438
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1439
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1440
{ assume a: "a \<notin> supp p" "a \<in> supp q"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1441
then have "q \<bullet> a \<in> supp q" by (simp add: supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1442
then have "q \<bullet> a \<notin> supp p" using asm by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1443
with a have "(p + q) \<bullet> a = (q + p) \<bullet> a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1444
by (simp add: supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1445
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1446
ultimately show "(p + q) \<bullet> a = (q + p) \<bullet> a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1447
using asm by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1448
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1449
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1450
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1451
lemma supp_plus_perm_eq:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1452
fixes p q::perm
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1453
assumes asm: "supp p \<inter> supp q = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1454
shows "supp (p + q) = supp p \<union> supp q"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1455
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1456
{ fix a::"atom"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1457
assume "a \<in> supp p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1458
then have "a \<notin> supp q" using asm by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1459
then have "a \<in> supp (p + q)" using `a \<in> supp p`
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1460
by (simp add: supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1461
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1462
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1463
{ fix a::"atom"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1464
assume "a \<in> supp q"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1465
then have "a \<notin> supp p" using asm by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1466
then have "a \<in> supp (q + p)" using `a \<in> supp q`
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1467
by (simp add: supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1468
then have "a \<in> supp (p + q)" using asm plus_perm_eq
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1469
by metis
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1470
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1471
ultimately have "supp p \<union> supp q \<subseteq> supp (p + q)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1472
by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1473
then show "supp (p + q) = supp p \<union> supp q" using supp_plus_perm
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1474
by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1475
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1476
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1477
instance perm :: fs
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1478
by default (simp add: supp_perm finite_perm_lemma)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1479
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1480
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1481
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1482
section {* Finite Support instances for other types *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1483
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1484
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1485
subsection {* Type @{typ "'a \<times> 'b"} is finitely-supported. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1486
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1487
lemma supp_Pair:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1488
shows "supp (x, y) = supp x \<union> supp y"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1489
by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1490
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1491
lemma fresh_Pair:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1492
shows "a \<sharp> (x, y) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1493
by (simp add: fresh_def supp_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1494
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1495
lemma supp_Unit:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1496
shows "supp () = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1497
by (simp add: supp_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1498
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1499
lemma fresh_Unit:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1500
shows "a \<sharp> ()"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1501
by (simp add: fresh_def supp_Unit)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1502
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1503
instance prod :: (fs, fs) fs
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1504
by default (auto simp add: supp_Pair finite_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1505
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1506
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1507
subsection {* Type @{typ "'a + 'b"} is finitely supported *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1508
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1509
lemma supp_Inl:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1510
shows "supp (Inl x) = supp x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1511
by (simp add: supp_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1512
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1513
lemma supp_Inr:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1514
shows "supp (Inr x) = supp x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1515
by (simp add: supp_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1516
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1517
lemma fresh_Inl:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1518
shows "a \<sharp> Inl x \<longleftrightarrow> a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1519
by (simp add: fresh_def supp_Inl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1520
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1521
lemma fresh_Inr:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1522
shows "a \<sharp> Inr y \<longleftrightarrow> a \<sharp> y"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1523
by (simp add: fresh_def supp_Inr)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1524
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1525
instance sum :: (fs, fs) fs
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1526
apply default
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1527
apply (case_tac x)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1528
apply (simp_all add: supp_Inl supp_Inr finite_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1529
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1530
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1531
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1532
subsection {* Type @{typ "'a option"} is finitely supported *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1533
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1534
lemma supp_None:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1535
shows "supp None = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1536
by (simp add: supp_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1537
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1538
lemma supp_Some:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1539
shows "supp (Some x) = supp x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1540
by (simp add: supp_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1541
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1542
lemma fresh_None:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1543
shows "a \<sharp> None"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1544
by (simp add: fresh_def supp_None)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1545
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1546
lemma fresh_Some:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1547
shows "a \<sharp> Some x \<longleftrightarrow> a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1548
by (simp add: fresh_def supp_Some)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1549
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1550
instance option :: (fs) fs
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1551
apply default
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1552
apply (induct_tac x)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1553
apply (simp_all add: supp_None supp_Some finite_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1554
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1555
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1556
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1557
subsubsection {* Type @{typ "'a list"} is finitely supported *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1558
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1559
lemma supp_Nil:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1560
shows "supp [] = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1561
by (simp add: supp_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1562
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1563
lemma fresh_Nil:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1564
shows "a \<sharp> []"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1565
by (simp add: fresh_def supp_Nil)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1566
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1567
lemma supp_Cons:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1568
shows "supp (x # xs) = supp x \<union> supp xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1569
by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1570
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1571
lemma fresh_Cons:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1572
shows "a \<sharp> (x # xs) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1573
by (simp add: fresh_def supp_Cons)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1574
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1575
lemma supp_append:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1576
shows "supp (xs @ ys) = supp xs \<union> supp ys"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1577
by (induct xs) (auto simp add: supp_Nil supp_Cons)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1578
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1579
lemma fresh_append:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1580
shows "a \<sharp> (xs @ ys) \<longleftrightarrow> a \<sharp> xs \<and> a \<sharp> ys"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1581
by (induct xs) (simp_all add: fresh_Nil fresh_Cons)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1582
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1583
lemma supp_rev:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1584
shows "supp (rev xs) = supp xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1585
by (induct xs) (auto simp add: supp_append supp_Cons supp_Nil)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1586
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1587
lemma fresh_rev:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1588
shows "a \<sharp> rev xs \<longleftrightarrow> a \<sharp> xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1589
by (induct xs) (auto simp add: fresh_append fresh_Cons fresh_Nil)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1590
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1591
lemma supp_removeAll:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1592
fixes x::"atom"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1593
shows "supp (removeAll x xs) = supp xs - {x}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1594
by (induct xs)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1595
(auto simp add: supp_Nil supp_Cons supp_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1596
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1597
lemma supp_of_atom_list:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1598
fixes as::"atom list"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1599
shows "supp as = set as"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1600
by (induct as)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1601
(simp_all add: supp_Nil supp_Cons supp_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1602
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1603
instance list :: (fs) fs
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1604
apply default
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1605
apply (induct_tac x)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1606
apply (simp_all add: supp_Nil supp_Cons finite_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1607
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1608
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1609
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1610
section {* Support and Freshness for Applications *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1611
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1612
lemma fresh_conv_MOST:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1613
shows "a \<sharp> x \<longleftrightarrow> (MOST b. (a \<rightleftharpoons> b) \<bullet> x = x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1614
unfolding fresh_def supp_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1615
unfolding MOST_iff_cofinite by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1616
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1617
lemma fresh_fun_app:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1618
assumes "a \<sharp> f" and "a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1619
shows "a \<sharp> f x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1620
using assms
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1621
unfolding fresh_conv_MOST
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1622
unfolding permute_fun_app_eq
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1623
by (elim MOST_rev_mp) (simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1624
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1625
lemma supp_fun_app:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1626
shows "supp (f x) \<subseteq> (supp f) \<union> (supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1627
using fresh_fun_app
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1628
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1629
by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1630
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1631
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1632
subsection {* Equivariance Predicate @{text eqvt} and @{text eqvt_at}*}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1633
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1634
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1635
"eqvt f \<equiv> \<forall>p. p \<bullet> f = f"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1636
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1637
lemma eqvt_boolI:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1638
fixes f::"bool"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1639
shows "eqvt f"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1640
unfolding eqvt_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1641
by (simp add: permute_bool_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1642
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1643
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1644
text {* equivariance of a function at a given argument *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1645
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1646
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1647
"eqvt_at f x \<equiv> \<forall>p. p \<bullet> (f x) = f (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1648
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1649
lemma eqvtI:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1650
shows "(\<And>p. p \<bullet> f \<equiv> f) \<Longrightarrow> eqvt f"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1651
unfolding eqvt_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1652
by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1653
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1654
lemma eqvt_at_perm:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1655
assumes "eqvt_at f x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1656
shows "eqvt_at f (q \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1657
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1658
{ fix p::"perm"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1659
have "p \<bullet> (f (q \<bullet> x)) = p \<bullet> q \<bullet> (f x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1660
using assms by (simp add: eqvt_at_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1661
also have "\<dots> = (p + q) \<bullet> (f x)" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1662
also have "\<dots> = f ((p + q) \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1663
using assms by (simp add: eqvt_at_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1664
finally have "p \<bullet> (f (q \<bullet> x)) = f (p \<bullet> q \<bullet> x)" by simp }
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1665
then show "eqvt_at f (q \<bullet> x)" unfolding eqvt_at_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1666
by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1667
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1668
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1669
lemma supp_fun_eqvt:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1670
assumes a: "eqvt f"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1671
shows "supp f = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1672
using a
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1673
unfolding eqvt_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1674
unfolding supp_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1675
by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1676
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1677
lemma fresh_fun_eqvt_app:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1678
assumes a: "eqvt f"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1679
shows "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1680
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1681
from a have "supp f = {}" by (simp add: supp_fun_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1682
then show "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1683
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1684
using supp_fun_app by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1685
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1686
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1687
lemma supp_fun_app_eqvt:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1688
assumes a: "eqvt f"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1689
shows "supp (f x) \<subseteq> supp x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1690
using fresh_fun_eqvt_app[OF a]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1691
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1692
by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1693
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1694
lemma supp_eqvt_at:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1695
assumes asm: "eqvt_at f x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1696
and fin: "finite (supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1697
shows "supp (f x) \<subseteq> supp x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1698
apply(rule supp_is_subset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1699
unfolding supports_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1700
unfolding fresh_def[symmetric]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1701
using asm
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1702
apply(simp add: eqvt_at_def swap_fresh_fresh)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1703
apply(rule fin)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1704
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1705
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1706
lemma finite_supp_eqvt_at:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1707
assumes asm: "eqvt_at f x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1708
and fin: "finite (supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1709
shows "finite (supp (f x))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1710
apply(rule finite_subset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1711
apply(rule supp_eqvt_at[OF asm fin])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1712
apply(rule fin)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1713
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1714
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1715
lemma fresh_eqvt_at:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1716
assumes asm: "eqvt_at f x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1717
and fin: "finite (supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1718
and fresh: "a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1719
shows "a \<sharp> f x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1720
using fresh
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1721
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1722
using supp_eqvt_at[OF asm fin]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1723
by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1724
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1725
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1726
subsection {* helper functions for nominal_functions *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1727
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1728
lemma THE_defaultI2:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1729
assumes "\<exists>!x. P x" "\<And>x. P x \<Longrightarrow> Q x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1730
shows "Q (THE_default d P)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1731
by (iprover intro: assms THE_defaultI')
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1732
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1733
lemma the_default_eqvt:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1734
assumes unique: "\<exists>!x. P x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1735
shows "(p \<bullet> (THE_default d P)) = (THE_default (p \<bullet> d) (p \<bullet> P))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1736
apply(rule THE_default1_equality [symmetric])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1737
apply(rule_tac p="-p" in permute_boolE)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1738
apply(simp add: ex1_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1739
apply(rule unique)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1740
apply(rule_tac p="-p" in permute_boolE)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1741
apply(rule subst[OF permute_fun_app_eq])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1742
apply(simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1743
apply(rule THE_defaultI'[OF unique])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1744
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1745
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1746
lemma fundef_ex1_eqvt:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1747
fixes x::"'a::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1748
assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (G x))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1749
assumes eqvt: "eqvt G"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1750
assumes ex1: "\<exists>!y. G x y"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1751
shows "(p \<bullet> (f x)) = f (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1752
apply(simp only: f_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1753
apply(subst the_default_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1754
apply(rule ex1)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1755
apply(rule THE_default1_equality [symmetric])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1756
apply(rule_tac p="-p" in permute_boolE)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1757
apply(perm_simp add: permute_minus_cancel)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1758
using eqvt[simplified eqvt_def]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1759
apply(simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1760
apply(rule ex1)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1761
apply(rule THE_defaultI2)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1762
apply(rule_tac p="-p" in permute_boolE)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1763
apply(perm_simp add: permute_minus_cancel)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1764
apply(rule ex1)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1765
apply(perm_simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1766
using eqvt[simplified eqvt_def]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1767
apply(simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1768
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1769
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1770
lemma fundef_ex1_eqvt_at:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1771
fixes x::"'a::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1772
assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (G x))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1773
assumes eqvt: "eqvt G"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1774
assumes ex1: "\<exists>!y. G x y"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1775
shows "eqvt_at f x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1776
unfolding eqvt_at_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1777
using assms
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1778
by (auto intro: fundef_ex1_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1779
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1780
lemma fundef_ex1_prop:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1781
fixes x::"'a::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1782
assumes f_def: "f \<equiv> (\<lambda>x::'a. THE_default (d x) (G x))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1783
assumes P_all: "\<And>x y. G x y \<Longrightarrow> P x y"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1784
assumes ex1: "\<exists>!y. G x y"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1785
shows "P x (f x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1786
unfolding f_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1787
using ex1
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1788
apply(erule_tac ex1E)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1789
apply(rule THE_defaultI2)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1790
apply(blast)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1791
apply(rule P_all)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1792
apply(assumption)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1793
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1794
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1795
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1796
section {* Support of Finite Sets of Finitely Supported Elements *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1797
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1798
text {* support and freshness for atom sets *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1799
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1800
lemma supp_finite_atom_set:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1801
fixes S::"atom set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1802
assumes "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1803
shows "supp S = S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1804
apply(rule finite_supp_unique)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1805
apply(simp add: supports_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1806
apply(simp add: swap_set_not_in)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1807
apply(rule assms)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1808
apply(simp add: swap_set_in)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1809
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1810
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1811
lemma supp_cofinite_atom_set:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1812
fixes S::"atom set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1813
assumes "finite (UNIV - S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1814
shows "supp S = (UNIV - S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1815
apply(rule finite_supp_unique)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1816
apply(simp add: supports_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1817
apply(simp add: swap_set_both_in)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1818
apply(rule assms)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1819
apply(subst swap_commute)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1820
apply(simp add: swap_set_in)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1821
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1822
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1823
lemma fresh_finite_atom_set:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1824
fixes S::"atom set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1825
assumes "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1826
shows "a \<sharp> S \<longleftrightarrow> a \<notin> S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1827
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1828
by (simp add: supp_finite_atom_set[OF assms])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1829
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1830
lemma fresh_minus_atom_set:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1831
fixes S::"atom set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1832
assumes "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1833
shows "a \<sharp> S - T \<longleftrightarrow> (a \<notin> T \<longrightarrow> a \<sharp> S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1834
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1835
by (auto simp add: supp_finite_atom_set assms)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1836
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1837
lemma Union_supports_set:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1838
shows "(\<Union>x \<in> S. supp x) supports S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1839
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1840
{ fix a b
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1841
have "\<forall>x \<in> S. (a \<rightleftharpoons> b) \<bullet> x = x \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> S = S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1842
unfolding permute_set_def by force
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1843
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1844
then show "(\<Union>x \<in> S. supp x) supports S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1845
unfolding supports_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1846
by (simp add: fresh_def[symmetric] swap_fresh_fresh)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1847
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1848
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1849
lemma Union_of_finite_supp_sets:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1850
fixes S::"('a::fs set)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1851
assumes fin: "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1852
shows "finite (\<Union>x\<in>S. supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1853
using fin by (induct) (auto simp add: finite_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1854
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1855
lemma Union_included_in_supp:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1856
fixes S::"('a::fs set)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1857
assumes fin: "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1858
shows "(\<Union>x\<in>S. supp x) \<subseteq> supp S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1859
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1860
have eqvt: "eqvt (\<lambda>S. \<Union> supp ` S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1861
unfolding eqvt_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1862
by (perm_simp) (simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1863
have "(\<Union>x\<in>S. supp x) = supp (\<Union>x\<in>S. supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1864
by (rule supp_finite_atom_set[symmetric]) (rule Union_of_finite_supp_sets[OF fin])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1865
also have "\<dots> = supp ((\<lambda>S. \<Union> supp ` S) S)" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1866
also have "\<dots> \<subseteq> supp S" using eqvt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1867
by (rule supp_fun_app_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1868
finally show "(\<Union>x\<in>S. supp x) \<subseteq> supp S" .
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1869
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1870
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1871
lemma supp_of_finite_sets:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1872
fixes S::"('a::fs set)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1873
assumes fin: "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1874
shows "(supp S) = (\<Union>x\<in>S. supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1875
apply(rule subset_antisym)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1876
apply(rule supp_is_subset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1877
apply(rule Union_supports_set)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1878
apply(rule Union_of_finite_supp_sets[OF fin])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1879
apply(rule Union_included_in_supp[OF fin])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1880
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1881
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1882
lemma finite_sets_supp:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1883
fixes S::"('a::fs set)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1884
assumes "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1885
shows "finite (supp S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1886
using assms
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1887
by (simp only: supp_of_finite_sets Union_of_finite_supp_sets)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1888
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1889
lemma supp_of_finite_union:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1890
fixes S T::"('a::fs) set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1891
assumes fin1: "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1892
and fin2: "finite T"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1893
shows "supp (S \<union> T) = supp S \<union> supp T"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1894
using fin1 fin2
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1895
by (simp add: supp_of_finite_sets)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1896
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1897
lemma supp_of_finite_insert:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1898
fixes S::"('a::fs) set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1899
assumes fin: "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1900
shows "supp (insert x S) = supp x \<union> supp S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1901
using fin
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1902
by (simp add: supp_of_finite_sets)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1903
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1904
lemma fresh_finite_insert:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1905
fixes S::"('a::fs) set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1906
assumes fin: "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1907
shows "a \<sharp> (insert x S) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1908
using fin unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1909
by (simp add: supp_of_finite_insert)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1910
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1911
lemma supp_set_empty:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1912
shows "supp {} = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1913
unfolding supp_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1914
by (simp add: empty_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1915
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1916
lemma fresh_set_empty:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1917
shows "a \<sharp> {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1918
by (simp add: fresh_def supp_set_empty)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1919
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1920
lemma supp_set:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1921
fixes xs :: "('a::fs) list"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1922
shows "supp (set xs) = supp xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1923
apply(induct xs)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1924
apply(simp add: supp_set_empty supp_Nil)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1925
apply(simp add: supp_Cons supp_of_finite_insert)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1926
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1927
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1928
lemma fresh_set:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1929
fixes xs :: "('a::fs) list"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1930
shows "a \<sharp> (set xs) \<longleftrightarrow> a \<sharp> xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1931
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1932
by (simp add: supp_set)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1933
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1934
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1935
subsection {* Type @{typ "'a multiset"} is finitely supported *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1936
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1937
lemma set_of_eqvt[eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1938
shows "p \<bullet> (set_of M) = set_of (p \<bullet> M)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1939
by (induct M) (simp_all add: insert_eqvt empty_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1940
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1941
lemma supp_set_of:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1942
shows "supp (set_of M) \<subseteq> supp M"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1943
apply (rule supp_fun_app_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1944
unfolding eqvt_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1945
apply(perm_simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1946
apply(simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1947
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1948
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1949
lemma Union_finite_multiset:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1950
fixes M::"'a::fs multiset"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1951
shows "finite (\<Union>{supp x | x. x \<in># M})"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1952
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1953
have "finite (\<Union>(supp ` {x. x \<in># M}))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1954
by (induct M) (simp_all add: Collect_imp_eq Collect_neg_eq finite_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1955
then show "finite (\<Union>{supp x | x. x \<in># M})"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1956
by (simp only: image_Collect)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1957
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1958
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1959
lemma Union_supports_multiset:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1960
shows "\<Union>{supp x | x. x :# M} supports M"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1961
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1962
have sw: "\<And>a b. ((\<And>x. x :# M \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x) \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> M = M)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1963
unfolding permute_multiset_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1964
apply(induct M)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1965
apply(simp_all)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1966
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1967
show "(\<Union>{supp x | x. x :# M}) supports M"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1968
unfolding supports_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1969
apply(clarify)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1970
apply(rule sw)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1971
apply(rule swap_fresh_fresh)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1972
apply(simp_all only: fresh_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1973
apply(auto)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1974
apply(metis neq0_conv)+
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1975
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1976
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1977
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1978
lemma Union_included_multiset:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1979
fixes M::"('a::fs multiset)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1980
shows "(\<Union>{supp x | x. x \<in># M}) \<subseteq> supp M"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1981
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1982
have "(\<Union>{supp x | x. x \<in># M}) = (\<Union>{supp x | x. x \<in> set_of M})" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1983
also have "... \<subseteq> (\<Union>x \<in> set_of M. supp x)" by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1984
also have "... = supp (set_of M)" by (simp add: subst supp_of_finite_sets)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1985
also have " ... \<subseteq> supp M" by (rule supp_set_of)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1986
finally show "(\<Union>{supp x | x. x \<in># M}) \<subseteq> supp M" .
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1987
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1988
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1989
lemma supp_of_multisets:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1990
fixes M::"('a::fs multiset)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1991
shows "(supp M) = (\<Union>{supp x | x. x :# M})"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1992
apply(rule subset_antisym)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1993
apply(rule supp_is_subset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1994
apply(rule Union_supports_multiset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1995
apply(rule Union_finite_multiset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1996
apply(rule Union_included_multiset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1997
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1998
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 1999
lemma multisets_supp_finite:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2000
fixes M::"('a::fs multiset)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2001
shows "finite (supp M)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2002
by (simp only: supp_of_multisets Union_finite_multiset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2003
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2004
lemma supp_of_multiset_union:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2005
fixes M N::"('a::fs) multiset"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2006
shows "supp (M + N) = supp M \<union> supp N"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2007
by (auto simp add: supp_of_multisets)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2008
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2009
lemma supp_empty_mset [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2010
shows "supp {#} = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2011
unfolding supp_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2012
by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2013
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2014
instance multiset :: (fs) fs
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2015
apply (default)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2016
apply (rule multisets_supp_finite)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2017
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2018
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2019
subsection {* Type @{typ "'a fset"} is finitely supported *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2020
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2021
lemma supp_fset [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2022
shows "supp (fset S) = supp S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2023
unfolding supp_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2024
by (simp add: fset_eqvt fset_cong)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2025
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2026
lemma supp_empty_fset [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2027
shows "supp {||} = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2028
unfolding supp_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2029
by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2030
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2031
lemma fresh_empty_fset:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2032
shows "a \<sharp> {||}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2033
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2034
by (simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2035
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2036
lemma supp_insert_fset [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2037
fixes x::"'a::fs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2038
and S::"'a fset"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2039
shows "supp (insert_fset x S) = supp x \<union> supp S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2040
apply(subst supp_fset[symmetric])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2041
apply(simp add: supp_of_finite_insert)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2042
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2043
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2044
lemma fresh_insert_fset:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2045
fixes x::"'a::fs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2046
and S::"'a fset"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2047
shows "a \<sharp> insert_fset x S \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2048
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2049
by (simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2050
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2051
lemma fset_finite_supp:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2052
fixes S::"('a::fs) fset"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2053
shows "finite (supp S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2054
by (induct S) (simp_all add: finite_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2055
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2056
lemma supp_union_fset:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2057
fixes S T::"'a::fs fset"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2058
shows "supp (S |\<union>| T) = supp S \<union> supp T"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2059
by (induct S) (auto)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2060
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2061
lemma fresh_union_fset:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2062
fixes S T::"'a::fs fset"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2063
shows "a \<sharp> S |\<union>| T \<longleftrightarrow> a \<sharp> S \<and> a \<sharp> T"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2064
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2065
by (simp add: supp_union_fset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2066
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2067
instance fset :: (fs) fs
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2068
apply (default)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2069
apply (rule fset_finite_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2070
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2071
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2072
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2073
section {* Freshness and Fresh-Star *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2074
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2075
lemma fresh_Unit_elim:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2076
shows "(a \<sharp> () \<Longrightarrow> PROP C) \<equiv> PROP C"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2077
by (simp add: fresh_Unit)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2078
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2079
lemma fresh_Pair_elim:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2080
shows "(a \<sharp> (x, y) \<Longrightarrow> PROP C) \<equiv> (a \<sharp> x \<Longrightarrow> a \<sharp> y \<Longrightarrow> PROP C)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2081
by rule (simp_all add: fresh_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2082
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2083
(* this rule needs to be added before the fresh_prodD is *)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2084
(* added to the simplifier with mksimps *)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2085
lemma [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2086
shows "a \<sharp> x1 \<Longrightarrow> a \<sharp> x2 \<Longrightarrow> a \<sharp> (x1, x2)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2087
by (simp add: fresh_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2088
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2089
lemma fresh_PairD:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2090
shows "a \<sharp> (x, y) \<Longrightarrow> a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2091
and "a \<sharp> (x, y) \<Longrightarrow> a \<sharp> y"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2092
by (simp_all add: fresh_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2093
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2094
declaration {* fn _ =>
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2095
let
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2096
val mksimps_pairs = (@{const_name Nominal2_Base.fresh}, @{thms fresh_PairD}) :: mksimps_pairs
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2097
in
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2098
Simplifier.map_ss (fn ss => Simplifier.set_mksimps (mksimps mksimps_pairs) ss)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2099
end
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2100
*}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2101
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2102
text {* The fresh-star generalisation of fresh is used in strong
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2103
induction principles. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2104
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2105
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2106
fresh_star :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp>* _" [80,80] 80)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2107
where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2108
"as \<sharp>* x \<equiv> \<forall>a \<in> as. a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2109
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2110
lemma fresh_star_supp_conv:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2111
shows "supp x \<sharp>* y \<Longrightarrow> supp y \<sharp>* x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2112
by (auto simp add: fresh_star_def fresh_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2113
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2114
lemma fresh_star_perm_set_conv:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2115
fixes p::"perm"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2116
assumes fresh: "as \<sharp>* p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2117
and fin: "finite as"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2118
shows "supp p \<sharp>* as"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2119
apply(rule fresh_star_supp_conv)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2120
apply(simp add: supp_finite_atom_set fin fresh)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2121
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2122
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2123
lemma fresh_star_atom_set_conv:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2124
assumes fresh: "as \<sharp>* bs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2125
and fin: "finite as" "finite bs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2126
shows "bs \<sharp>* as"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2127
using fresh
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2128
unfolding fresh_star_def fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2129
by (auto simp add: supp_finite_atom_set fin)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2130
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2131
lemma atom_fresh_star_disjoint:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2132
assumes fin: "finite bs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2133
shows "as \<sharp>* bs \<longleftrightarrow> (as \<inter> bs = {})"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2134
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2135
unfolding fresh_star_def fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2136
by (auto simp add: supp_finite_atom_set fin)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2137
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2138
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2139
lemma fresh_star_Pair:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2140
shows "as \<sharp>* (x, y) = (as \<sharp>* x \<and> as \<sharp>* y)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2141
by (auto simp add: fresh_star_def fresh_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2142
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2143
lemma fresh_star_list:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2144
shows "as \<sharp>* (xs @ ys) \<longleftrightarrow> as \<sharp>* xs \<and> as \<sharp>* ys"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2145
and "as \<sharp>* (x # xs) \<longleftrightarrow> as \<sharp>* x \<and> as \<sharp>* xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2146
and "as \<sharp>* []"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2147
by (auto simp add: fresh_star_def fresh_Nil fresh_Cons fresh_append)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2148
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2149
lemma fresh_star_set:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2150
fixes xs::"('a::fs) list"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2151
shows "as \<sharp>* set xs \<longleftrightarrow> as \<sharp>* xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2152
unfolding fresh_star_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2153
by (simp add: fresh_set)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2154
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2155
lemma fresh_star_singleton:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2156
fixes a::"atom"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2157
shows "as \<sharp>* {a} \<longleftrightarrow> as \<sharp>* a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2158
by (simp add: fresh_star_def fresh_finite_insert fresh_set_empty)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2159
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2160
lemma fresh_star_fset:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2161
fixes xs::"('a::fs) list"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2162
shows "as \<sharp>* fset S \<longleftrightarrow> as \<sharp>* S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2163
by (simp add: fresh_star_def fresh_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2164
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2165
lemma fresh_star_Un:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2166
shows "(as \<union> bs) \<sharp>* x = (as \<sharp>* x \<and> bs \<sharp>* x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2167
by (auto simp add: fresh_star_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2168
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2169
lemma fresh_star_insert:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2170
shows "(insert a as) \<sharp>* x = (a \<sharp> x \<and> as \<sharp>* x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2171
by (auto simp add: fresh_star_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2172
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2173
lemma fresh_star_Un_elim:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2174
"((as \<union> bs) \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (as \<sharp>* x \<Longrightarrow> bs \<sharp>* x \<Longrightarrow> PROP C)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2175
unfolding fresh_star_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2176
apply(rule)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2177
apply(erule meta_mp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2178
apply(auto)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2179
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2180
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2181
lemma fresh_star_insert_elim:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2182
"(insert a as \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (a \<sharp> x \<Longrightarrow> as \<sharp>* x \<Longrightarrow> PROP C)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2183
unfolding fresh_star_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2184
by rule (simp_all add: fresh_star_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2185
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2186
lemma fresh_star_empty_elim:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2187
"({} \<sharp>* x \<Longrightarrow> PROP C) \<equiv> PROP C"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2188
by (simp add: fresh_star_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2189
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2190
lemma fresh_star_Unit_elim:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2191
shows "(a \<sharp>* () \<Longrightarrow> PROP C) \<equiv> PROP C"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2192
by (simp add: fresh_star_def fresh_Unit)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2193
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2194
lemma fresh_star_Pair_elim:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2195
shows "(a \<sharp>* (x, y) \<Longrightarrow> PROP C) \<equiv> (a \<sharp>* x \<Longrightarrow> a \<sharp>* y \<Longrightarrow> PROP C)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2196
by (rule, simp_all add: fresh_star_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2197
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2198
lemma fresh_star_zero:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2199
shows "as \<sharp>* (0::perm)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2200
unfolding fresh_star_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2201
by (simp add: fresh_zero_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2202
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2203
lemma fresh_star_plus:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2204
fixes p q::perm
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2205
shows "\<lbrakk>a \<sharp>* p; a \<sharp>* q\<rbrakk> \<Longrightarrow> a \<sharp>* (p + q)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2206
unfolding fresh_star_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2207
by (simp add: fresh_plus_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2208
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2209
lemma fresh_star_permute_iff:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2210
shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) \<longleftrightarrow> a \<sharp>* x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2211
unfolding fresh_star_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2212
by (metis mem_permute_iff permute_minus_cancel(1) fresh_permute_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2213
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2214
lemma fresh_star_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2215
shows "p \<bullet> (as \<sharp>* x) \<longleftrightarrow> (p \<bullet> as) \<sharp>* (p \<bullet> x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2216
unfolding fresh_star_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2217
by (perm_simp) (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2218
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2219
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2220
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2221
section {* Induction principle for permutations *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2222
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2223
lemma smaller_supp:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2224
assumes a: "a \<in> supp p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2225
shows "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<subset> supp p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2226
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2227
have "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<subseteq> supp p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2228
unfolding supp_perm by (auto simp add: swap_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2229
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2230
have "a \<notin> supp ((p \<bullet> a \<rightleftharpoons> a) + p)" by (simp add: supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2231
then have "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<noteq> supp p" using a by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2232
ultimately
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2233
show "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<subset> supp p" by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2234
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2235
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2236
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2237
lemma perm_struct_induct[consumes 1, case_names zero swap]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2238
assumes S: "supp p \<subseteq> S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2239
and zero: "P 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2240
and swap: "\<And>p a b. \<lbrakk>P p; supp p \<subseteq> S; a \<in> S; b \<in> S; a \<noteq> b; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> P ((a \<rightleftharpoons> b) + p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2241
shows "P p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2242
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2243
have "finite (supp p)" by (simp add: finite_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2244
then show "P p" using S
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2245
proof(induct A\<equiv>"supp p" arbitrary: p rule: finite_psubset_induct)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2246
case (psubset p)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2247
then have ih: "\<And>q. supp q \<subset> supp p \<Longrightarrow> P q" by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2248
have as: "supp p \<subseteq> S" by fact
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2249
{ assume "supp p = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2250
then have "p = 0" by (simp add: supp_perm perm_eq_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2251
then have "P p" using zero by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2252
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2253
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2254
{ assume "supp p \<noteq> {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2255
then obtain a where a0: "a \<in> supp p" by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2256
then have a1: "p \<bullet> a \<in> S" "a \<in> S" "sort_of (p \<bullet> a) = sort_of a" "p \<bullet> a \<noteq> a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2257
using as by (auto simp add: supp_atom supp_perm swap_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2258
let ?q = "(p \<bullet> a \<rightleftharpoons> a) + p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2259
have a2: "supp ?q \<subset> supp p" using a0 smaller_supp by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2260
then have "P ?q" using ih by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2261
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2262
have "supp ?q \<subseteq> S" using as a2 by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2263
ultimately have "P ((p \<bullet> a \<rightleftharpoons> a) + ?q)" using as a1 swap by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2264
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2265
have "p = (p \<bullet> a \<rightleftharpoons> a) + ?q" by (simp add: perm_eq_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2266
ultimately have "P p" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2267
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2268
ultimately show "P p" by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2269
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2270
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2271
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2272
lemma perm_simple_struct_induct[case_names zero swap]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2273
assumes zero: "P 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2274
and swap: "\<And>p a b. \<lbrakk>P p; a \<noteq> b; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> P ((a \<rightleftharpoons> b) + p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2275
shows "P p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2276
by (rule_tac S="supp p" in perm_struct_induct)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2277
(auto intro: zero swap)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2278
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2279
lemma perm_struct_induct2[consumes 1, case_names zero swap plus]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2280
assumes S: "supp p \<subseteq> S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2281
assumes zero: "P 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2282
assumes swap: "\<And>a b. \<lbrakk>sort_of a = sort_of b; a \<noteq> b; a \<in> S; b \<in> S\<rbrakk> \<Longrightarrow> P (a \<rightleftharpoons> b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2283
assumes plus: "\<And>p1 p2. \<lbrakk>P p1; P p2; supp p1 \<subseteq> S; supp p2 \<subseteq> S\<rbrakk> \<Longrightarrow> P (p1 + p2)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2284
shows "P p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2285
using S
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2286
by (induct p rule: perm_struct_induct)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2287
(auto intro: zero plus swap simp add: supp_swap)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2288
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2289
lemma perm_simple_struct_induct2[case_names zero swap plus]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2290
assumes zero: "P 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2291
assumes swap: "\<And>a b. \<lbrakk>sort_of a = sort_of b; a \<noteq> b\<rbrakk> \<Longrightarrow> P (a \<rightleftharpoons> b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2292
assumes plus: "\<And>p1 p2. \<lbrakk>P p1; P p2\<rbrakk> \<Longrightarrow> P (p1 + p2)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2293
shows "P p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2294
by (rule_tac S="supp p" in perm_struct_induct2)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2295
(auto intro: zero swap plus)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2296
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2297
lemma supp_perm_singleton:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2298
fixes p::"perm"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2299
shows "supp p \<subseteq> {b} \<longleftrightarrow> p = 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2300
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2301
{ assume "supp p \<subseteq> {b}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2302
then have "p = 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2303
by (induct p rule: perm_struct_induct) (simp_all)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2304
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2305
then show "supp p \<subseteq> {b} \<longleftrightarrow> p = 0" by (auto simp add: supp_zero_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2306
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2307
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2308
lemma supp_perm_pair:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2309
fixes p::"perm"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2310
shows "supp p \<subseteq> {a, b} \<longleftrightarrow> p = 0 \<or> p = (b \<rightleftharpoons> a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2311
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2312
{ assume "supp p \<subseteq> {a, b}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2313
then have "p = 0 \<or> p = (b \<rightleftharpoons> a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2314
apply (induct p rule: perm_struct_induct)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2315
apply (auto simp add: swap_cancel supp_zero_perm supp_swap)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2316
apply (simp add: swap_commute)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2317
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2318
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2319
then show "supp p \<subseteq> {a, b} \<longleftrightarrow> p = 0 \<or> p = (b \<rightleftharpoons> a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2320
by (auto simp add: supp_zero_perm supp_swap split: if_splits)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2321
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2322
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2323
lemma supp_perm_eq:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2324
assumes "(supp x) \<sharp>* p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2325
shows "p \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2326
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2327
from assms have "supp p \<subseteq> {a. a \<sharp> x}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2328
unfolding supp_perm fresh_star_def fresh_def by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2329
then show "p \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2330
proof (induct p rule: perm_struct_induct)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2331
case zero
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2332
show "0 \<bullet> x = x" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2333
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2334
case (swap p a b)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2335
then have "a \<sharp> x" "b \<sharp> x" "p \<bullet> x = x" by simp_all
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2336
then show "((a \<rightleftharpoons> b) + p) \<bullet> x = x" by (simp add: swap_fresh_fresh)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2337
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2338
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2339
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2340
text {* same lemma as above, but proved with a different induction principle *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2341
lemma supp_perm_eq_test:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2342
assumes "(supp x) \<sharp>* p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2343
shows "p \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2344
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2345
from assms have "supp p \<subseteq> {a. a \<sharp> x}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2346
unfolding supp_perm fresh_star_def fresh_def by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2347
then show "p \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2348
proof (induct p rule: perm_struct_induct2)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2349
case zero
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2350
show "0 \<bullet> x = x" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2351
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2352
case (swap a b)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2353
then have "a \<sharp> x" "b \<sharp> x" by simp_all
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2354
then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2355
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2356
case (plus p1 p2)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2357
have "p1 \<bullet> x = x" "p2 \<bullet> x = x" by fact+
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2358
then show "(p1 + p2) \<bullet> x = x" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2359
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2360
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2361
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2362
lemma perm_supp_eq:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2363
assumes a: "(supp p) \<sharp>* x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2364
shows "p \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2365
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2366
from assms have "supp p \<subseteq> {a. a \<sharp> x}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2367
unfolding supp_perm fresh_star_def fresh_def by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2368
then show "p \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2369
proof (induct p rule: perm_struct_induct2)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2370
case zero
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2371
show "0 \<bullet> x = x" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2372
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2373
case (swap a b)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2374
then have "a \<sharp> x" "b \<sharp> x" by simp_all
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2375
then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2376
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2377
case (plus p1 p2)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2378
have "p1 \<bullet> x = x" "p2 \<bullet> x = x" by fact+
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2379
then show "(p1 + p2) \<bullet> x = x" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2380
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2381
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2382
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2383
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2384
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2385
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2386
lemma supp_perm_perm_eq:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2387
assumes a: "\<forall>a \<in> supp x. p \<bullet> a = q \<bullet> a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2388
shows "p \<bullet> x = q \<bullet> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2389
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2390
from a have "\<forall>a \<in> supp x. (-q + p) \<bullet> a = a" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2391
then have "\<forall>a \<in> supp x. a \<notin> supp (-q + p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2392
unfolding supp_perm by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2393
then have "supp x \<sharp>* (-q + p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2394
unfolding fresh_star_def fresh_def by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2395
then have "(-q + p) \<bullet> x = x" by (simp only: supp_perm_eq)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2396
then show "p \<bullet> x = q \<bullet> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2397
by (metis permute_minus_cancel(1) permute_plus)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2398
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2399
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2400
text {* disagreement set *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2401
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2402
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2403
dset :: "perm \<Rightarrow> perm \<Rightarrow> atom set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2404
where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2405
"dset p q = {a::atom. p \<bullet> a \<noteq> q \<bullet> a}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2406
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2407
lemma ds_fresh:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2408
assumes "dset p q \<sharp>* x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2409
shows "p \<bullet> x = q \<bullet> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2410
using assms
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2411
unfolding dset_def fresh_star_def fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2412
by (auto intro: supp_perm_perm_eq)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2413
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2414
lemma atom_set_perm_eq:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2415
assumes a: "as \<sharp>* p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2416
shows "p \<bullet> as = as"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2417
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2418
from a have "supp p \<subseteq> {a. a \<notin> as}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2419
unfolding supp_perm fresh_star_def fresh_def by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2420
then show "p \<bullet> as = as"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2421
proof (induct p rule: perm_struct_induct)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2422
case zero
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2423
show "0 \<bullet> as = as" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2424
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2425
case (swap p a b)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2426
then have "a \<notin> as" "b \<notin> as" "p \<bullet> as = as" by simp_all
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2427
then show "((a \<rightleftharpoons> b) + p) \<bullet> as = as" by (simp add: swap_set_not_in)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2428
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2429
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2430
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2431
section {* Avoiding of atom sets *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2432
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2433
text {*
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2434
For every set of atoms, there is another set of atoms
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2435
avoiding a finitely supported c and there is a permutation
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2436
which 'translates' between both sets.
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2437
*}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2438
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2439
lemma at_set_avoiding_aux:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2440
fixes Xs::"atom set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2441
and As::"atom set"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2442
assumes b: "Xs \<subseteq> As"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2443
and c: "finite As"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2444
shows "\<exists>p. (p \<bullet> Xs) \<inter> As = {} \<and> (supp p) = (Xs \<union> (p \<bullet> Xs))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2445
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2446
from b c have "finite Xs" by (rule finite_subset)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2447
then show ?thesis using b
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2448
proof (induct rule: finite_subset_induct)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2449
case empty
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2450
have "0 \<bullet> {} \<inter> As = {}" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2451
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2452
have "supp (0::perm) = {} \<union> 0 \<bullet> {}" by (simp add: supp_zero_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2453
ultimately show ?case by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2454
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2455
case (insert x Xs)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2456
then obtain p where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2457
p1: "(p \<bullet> Xs) \<inter> As = {}" and
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2458
p2: "supp p = (Xs \<union> (p \<bullet> Xs))" by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2459
from `x \<in> As` p1 have "x \<notin> p \<bullet> Xs" by fast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2460
with `x \<notin> Xs` p2 have "x \<notin> supp p" by fast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2461
hence px: "p \<bullet> x = x" unfolding supp_perm by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2462
have "finite (As \<union> p \<bullet> Xs \<union> supp p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2463
using `finite As` `finite Xs`
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2464
by (simp add: permute_set_eq_image finite_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2465
then obtain y where "y \<notin> (As \<union> p \<bullet> Xs \<union> supp p)" "sort_of y = sort_of x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2466
by (rule obtain_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2467
hence y: "y \<notin> As" "y \<notin> p \<bullet> Xs" "y \<notin> supp p" "sort_of y = sort_of x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2468
by simp_all
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2469
hence py: "p \<bullet> y = y" "x \<noteq> y" using `x \<in> As`
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2470
by (auto simp add: supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2471
let ?q = "(x \<rightleftharpoons> y) + p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2472
have q: "?q \<bullet> insert x Xs = insert y (p \<bullet> Xs)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2473
unfolding insert_eqvt
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2474
using `p \<bullet> x = x` `sort_of y = sort_of x`
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2475
using `x \<notin> p \<bullet> Xs` `y \<notin> p \<bullet> Xs`
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2476
by (simp add: swap_atom swap_set_not_in)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2477
have "?q \<bullet> insert x Xs \<inter> As = {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2478
using `y \<notin> As` `p \<bullet> Xs \<inter> As = {}`
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2479
unfolding q by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2480
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2481
have "supp (x \<rightleftharpoons> y) \<inter> supp p = {}" using px py `sort_of y = sort_of x`
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2482
unfolding supp_swap by (simp add: supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2483
then have "supp ?q = (supp (x \<rightleftharpoons> y) \<union> supp p)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2484
by (simp add: supp_plus_perm_eq)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2485
then have "supp ?q = insert x Xs \<union> ?q \<bullet> insert x Xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2486
using p2 `sort_of y = sort_of x` `x \<noteq> y` unfolding q supp_swap
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2487
by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2488
ultimately show ?case by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2489
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2490
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2491
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2492
lemma at_set_avoiding:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2493
assumes a: "finite Xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2494
and b: "finite (supp c)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2495
obtains p::"perm" where "(p \<bullet> Xs)\<sharp>*c" and "(supp p) = (Xs \<union> (p \<bullet> Xs))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2496
using a b at_set_avoiding_aux [where Xs="Xs" and As="Xs \<union> supp c"]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2497
unfolding fresh_star_def fresh_def by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2498
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2499
lemma at_set_avoiding1:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2500
assumes "finite xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2501
and "finite (supp c)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2502
shows "\<exists>p. (p \<bullet> xs) \<sharp>* c"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2503
using assms
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2504
apply(erule_tac c="c" in at_set_avoiding)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2505
apply(auto)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2506
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2507
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2508
lemma at_set_avoiding2:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2509
assumes "finite xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2510
and "finite (supp c)" "finite (supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2511
and "xs \<sharp>* x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2512
shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2513
using assms
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2514
apply(erule_tac c="(c, x)" in at_set_avoiding)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2515
apply(simp add: supp_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2516
apply(rule_tac x="p" in exI)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2517
apply(simp add: fresh_star_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2518
apply(rule fresh_star_supp_conv)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2519
apply(auto simp add: fresh_star_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2520
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2521
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2522
lemma at_set_avoiding3:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2523
assumes "finite xs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2524
and "finite (supp c)" "finite (supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2525
and "xs \<sharp>* x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2526
shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p \<and> supp p = xs \<union> (p \<bullet> xs)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2527
using assms
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2528
apply(erule_tac c="(c, x)" in at_set_avoiding)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2529
apply(simp add: supp_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2530
apply(rule_tac x="p" in exI)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2531
apply(simp add: fresh_star_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2532
apply(rule fresh_star_supp_conv)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2533
apply(auto simp add: fresh_star_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2534
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2535
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2536
lemma at_set_avoiding2_atom:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2537
assumes "finite (supp c)" "finite (supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2538
and b: "a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2539
shows "\<exists>p. (p \<bullet> a) \<sharp> c \<and> supp x \<sharp>* p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2540
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2541
have a: "{a} \<sharp>* x" unfolding fresh_star_def by (simp add: b)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2542
obtain p where p1: "(p \<bullet> {a}) \<sharp>* c" and p2: "supp x \<sharp>* p"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2543
using at_set_avoiding2[of "{a}" "c" "x"] assms a by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2544
have c: "(p \<bullet> a) \<sharp> c" using p1
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2545
unfolding fresh_star_def Ball_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2546
by(erule_tac x="p \<bullet> a" in allE) (simp add: permute_set_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2547
hence "p \<bullet> a \<sharp> c \<and> supp x \<sharp>* p" using p2 by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2548
then show "\<exists>p. (p \<bullet> a) \<sharp> c \<and> supp x \<sharp>* p" by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2549
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2550
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2551
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2552
section {* Renaming permutations *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2553
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2554
lemma set_renaming_perm:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2555
assumes b: "finite bs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2556
shows "\<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2557
using b
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2558
proof (induct)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2559
case empty
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2560
have "(\<forall>b \<in> {}. 0 \<bullet> b = p \<bullet> b) \<and> supp (0::perm) \<subseteq> {} \<union> p \<bullet> {}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2561
by (simp add: permute_set_def supp_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2562
then show "\<exists>q. (\<forall>b \<in> {}. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> {} \<union> p \<bullet> {}" by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2563
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2564
case (insert a bs)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2565
then have " \<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> p \<bullet> bs" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2566
then obtain q where *: "\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b" and **: "supp q \<subseteq> bs \<union> p \<bullet> bs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2567
by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2568
{ assume 1: "q \<bullet> a = p \<bullet> a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2569
have "\<forall>b \<in> (insert a bs). q \<bullet> b = p \<bullet> b" using 1 * by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2570
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2571
have "supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2572
using ** by (auto simp add: insert_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2573
ultimately
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2574
have "\<exists>q. (\<forall>b \<in> insert a bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2575
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2576
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2577
{ assume 2: "q \<bullet> a \<noteq> p \<bullet> a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2578
def q' \<equiv> "((q \<bullet> a) \<rightleftharpoons> (p \<bullet> a)) + q"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2579
have "\<forall>b \<in> insert a bs. q' \<bullet> b = p \<bullet> b" using 2 * `a \<notin> bs` unfolding q'_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2580
by (auto simp add: swap_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2581
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2582
{ have "{q \<bullet> a, p \<bullet> a} \<subseteq> insert a bs \<union> p \<bullet> insert a bs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2583
using **
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2584
apply (auto simp add: supp_perm insert_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2585
apply (subgoal_tac "q \<bullet> a \<in> bs \<union> p \<bullet> bs")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2586
apply(auto)[1]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2587
apply(subgoal_tac "q \<bullet> a \<in> {a. q \<bullet> a \<noteq> a}")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2588
apply(blast)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2589
apply(simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2590
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2591
then have "supp (q \<bullet> a \<rightleftharpoons> p \<bullet> a) \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by (simp add: supp_swap)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2592
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2593
have "supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2594
using ** by (auto simp add: insert_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2595
ultimately
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2596
have "supp q' \<subseteq> insert a bs \<union> p \<bullet> insert a bs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2597
unfolding q'_def using supp_plus_perm by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2598
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2599
ultimately
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2600
have "\<exists>q. (\<forall>b \<in> insert a bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2601
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2602
ultimately show "\<exists>q. (\<forall>b \<in> insert a bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2603
by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2604
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2605
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2606
lemma set_renaming_perm2:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2607
shows "\<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2608
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2609
have "finite (bs \<inter> supp p)" by (simp add: finite_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2610
then obtain q
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2611
where *: "\<forall>b \<in> bs \<inter> supp p. q \<bullet> b = p \<bullet> b" and **: "supp q \<subseteq> (bs \<inter> supp p) \<union> (p \<bullet> (bs \<inter> supp p))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2612
using set_renaming_perm by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2613
from ** have "supp q \<subseteq> bs \<union> (p \<bullet> bs)" by (auto simp add: inter_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2614
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2615
have "\<forall>b \<in> bs - supp p. q \<bullet> b = p \<bullet> b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2616
apply(auto)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2617
apply(subgoal_tac "b \<notin> supp q")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2618
apply(simp add: fresh_def[symmetric])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2619
apply(simp add: fresh_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2620
apply(clarify)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2621
apply(rotate_tac 2)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2622
apply(drule subsetD[OF **])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2623
apply(simp add: inter_eqvt supp_eqvt permute_self)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2624
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2625
ultimately have "(\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)" using * by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2626
then show "\<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)" by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2627
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2628
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2629
lemma list_renaming_perm:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2630
shows "\<exists>q. (\<forall>b \<in> set bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set bs \<union> (p \<bullet> set bs)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2631
proof (induct bs)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2632
case (Cons a bs)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2633
then have " \<exists>q. (\<forall>b \<in> set bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set bs \<union> p \<bullet> (set bs)" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2634
then obtain q where *: "\<forall>b \<in> set bs. q \<bullet> b = p \<bullet> b" and **: "supp q \<subseteq> set bs \<union> p \<bullet> (set bs)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2635
by (blast)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2636
{ assume 1: "a \<in> set bs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2637
have "q \<bullet> a = p \<bullet> a" using * 1 by (induct bs) (auto)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2638
then have "\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b" using * by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2639
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2640
have "supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" using ** by (auto simp add: insert_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2641
ultimately
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2642
have "\<exists>q. (\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2643
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2644
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2645
{ assume 2: "a \<notin> set bs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2646
def q' \<equiv> "((q \<bullet> a) \<rightleftharpoons> (p \<bullet> a)) + q"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2647
have "\<forall>b \<in> set (a # bs). q' \<bullet> b = p \<bullet> b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2648
unfolding q'_def using 2 * `a \<notin> set bs` by (auto simp add: swap_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2649
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2650
{ have "{q \<bullet> a, p \<bullet> a} \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2651
using **
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2652
apply (auto simp add: supp_perm insert_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2653
apply (subgoal_tac "q \<bullet> a \<in> set bs \<union> p \<bullet> set bs")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2654
apply(auto)[1]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2655
apply(subgoal_tac "q \<bullet> a \<in> {a. q \<bullet> a \<noteq> a}")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2656
apply(blast)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2657
apply(simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2658
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2659
then have "supp (q \<bullet> a \<rightleftharpoons> p \<bullet> a) \<subseteq> set (a # bs) \<union> p \<bullet> set (a # bs)" by (simp add: supp_swap)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2660
moreover
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2661
have "supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2662
using ** by (auto simp add: insert_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2663
ultimately
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2664
have "supp q' \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2665
unfolding q'_def using supp_plus_perm by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2666
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2667
ultimately
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2668
have "\<exists>q. (\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2669
}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2670
ultimately show "\<exists>q. (\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2671
by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2672
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2673
case Nil
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2674
have "(\<forall>b \<in> set []. 0 \<bullet> b = p \<bullet> b) \<and> supp (0::perm) \<subseteq> set [] \<union> p \<bullet> set []"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2675
by (simp add: supp_zero_perm)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2676
then show "\<exists>q. (\<forall>b \<in> set []. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set [] \<union> p \<bullet> (set [])" by blast
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2677
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2678
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2679
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2680
section {* Concrete Atoms Types *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2681
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2682
text {*
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2683
Class @{text at_base} allows types containing multiple sorts of atoms.
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2684
Class @{text at} only allows types with a single sort.
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2685
*}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2686
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2687
class at_base = pt +
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2688
fixes atom :: "'a \<Rightarrow> atom"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2689
assumes atom_eq_iff [simp]: "atom a = atom b \<longleftrightarrow> a = b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2690
assumes atom_eqvt: "p \<bullet> (atom a) = atom (p \<bullet> a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2691
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2692
declare atom_eqvt[eqvt]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2693
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2694
class at = at_base +
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2695
assumes sort_of_atom_eq [simp]: "sort_of (atom a) = sort_of (atom b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2696
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2697
lemma sort_ineq [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2698
assumes "sort_of (atom a) \<noteq> sort_of (atom b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2699
shows "atom a \<noteq> atom b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2700
using assms by metis
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2701
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2702
lemma supp_at_base:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2703
fixes a::"'a::at_base"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2704
shows "supp a = {atom a}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2705
by (simp add: supp_atom [symmetric] supp_def atom_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2706
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2707
lemma fresh_at_base:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2708
shows "sort_of a \<noteq> sort_of (atom b) \<Longrightarrow> a \<sharp> b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2709
and "a \<sharp> b \<longleftrightarrow> a \<noteq> atom b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2710
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2711
apply(simp_all add: supp_at_base)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2712
apply(metis)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2713
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2714
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2715
lemma fresh_atom_at_base:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2716
fixes b::"'a::at_base"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2717
shows "a \<sharp> atom b \<longleftrightarrow> a \<sharp> b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2718
by (simp add: fresh_def supp_at_base supp_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2719
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2720
lemma fresh_star_atom_at_base:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2721
fixes b::"'a::at_base"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2722
shows "as \<sharp>* atom b \<longleftrightarrow> as \<sharp>* b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2723
by (simp add: fresh_star_def fresh_atom_at_base)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2724
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2725
instance at_base < fs
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2726
proof qed (simp add: supp_at_base)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2727
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2728
lemma at_base_infinite [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2729
shows "infinite (UNIV :: 'a::at_base set)" (is "infinite ?U")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2730
proof
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2731
obtain a :: 'a where "True" by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2732
assume "finite ?U"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2733
hence "finite (atom ` ?U)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2734
by (rule finite_imageI)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2735
then obtain b where b: "b \<notin> atom ` ?U" "sort_of b = sort_of (atom a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2736
by (rule obtain_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2737
from b(2) have "b = atom ((atom a \<rightleftharpoons> b) \<bullet> a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2738
unfolding atom_eqvt [symmetric]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2739
by (simp add: swap_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2740
hence "b \<in> atom ` ?U" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2741
with b(1) show "False" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2742
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2743
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2744
lemma swap_at_base_simps [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2745
fixes x y::"'a::at_base"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2746
shows "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> x = y"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2747
and "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> y = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2748
and "atom x \<noteq> a \<Longrightarrow> atom x \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2749
unfolding atom_eq_iff [symmetric]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2750
unfolding atom_eqvt [symmetric]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2751
by simp_all
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2752
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2753
lemma obtain_at_base:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2754
assumes X: "finite X"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2755
obtains a::"'a::at_base" where "atom a \<notin> X"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2756
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2757
have "inj (atom :: 'a \<Rightarrow> atom)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2758
by (simp add: inj_on_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2759
with X have "finite (atom -` X :: 'a set)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2760
by (rule finite_vimageI)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2761
with at_base_infinite have "atom -` X \<noteq> (UNIV :: 'a set)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2762
by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2763
then obtain a :: 'a where "atom a \<notin> X"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2764
by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2765
thus ?thesis ..
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2766
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2767
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2768
lemma obtain_fresh':
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2769
assumes fin: "finite (supp x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2770
obtains a::"'a::at_base" where "atom a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2771
using obtain_at_base[where X="supp x"]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2772
by (auto simp add: fresh_def fin)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2773
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2774
lemma obtain_fresh:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2775
fixes x::"'b::fs"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2776
obtains a::"'a::at_base" where "atom a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2777
by (rule obtain_fresh') (auto simp add: finite_supp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2778
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2779
lemma supp_finite_set_at_base:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2780
assumes a: "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2781
shows "supp S = atom ` S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2782
apply(simp add: supp_of_finite_sets[OF a])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2783
apply(simp add: supp_at_base)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2784
apply(auto)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2785
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2786
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2787
(* FIXME
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2788
lemma supp_cofinite_set_at_base:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2789
assumes a: "finite (UNIV - S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2790
shows "supp S = atom ` (UNIV - S)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2791
apply(rule finite_supp_unique)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2792
*)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2793
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2794
lemma fresh_finite_set_at_base:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2795
fixes a::"'a::at_base"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2796
assumes a: "finite S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2797
shows "atom a \<sharp> S \<longleftrightarrow> a \<notin> S"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2798
unfolding fresh_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2799
apply(simp add: supp_finite_set_at_base[OF a])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2800
apply(subst inj_image_mem_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2801
apply(simp add: inj_on_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2802
apply(simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2803
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2804
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2805
lemma fresh_at_base_permute_iff [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2806
fixes a::"'a::at_base"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2807
shows "atom (p \<bullet> a) \<sharp> p \<bullet> x \<longleftrightarrow> atom a \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2808
unfolding atom_eqvt[symmetric]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2809
by (simp add: fresh_permute_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2810
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2811
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2812
section {* Infrastructure for concrete atom types *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2813
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2814
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2815
flip :: "'a::at_base \<Rightarrow> 'a \<Rightarrow> perm" ("'(_ \<leftrightarrow> _')")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2816
where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2817
"(a \<leftrightarrow> b) = (atom a \<rightleftharpoons> atom b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2818
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2819
lemma flip_self [simp]: "(a \<leftrightarrow> a) = 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2820
unfolding flip_def by (rule swap_self)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2821
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2822
lemma flip_commute: "(a \<leftrightarrow> b) = (b \<leftrightarrow> a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2823
unfolding flip_def by (rule swap_commute)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2824
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2825
lemma minus_flip [simp]: "- (a \<leftrightarrow> b) = (a \<leftrightarrow> b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2826
unfolding flip_def by (rule minus_swap)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2827
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2828
lemma add_flip_cancel: "(a \<leftrightarrow> b) + (a \<leftrightarrow> b) = 0"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2829
unfolding flip_def by (rule swap_cancel)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2830
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2831
lemma permute_flip_cancel [simp]: "(a \<leftrightarrow> b) \<bullet> (a \<leftrightarrow> b) \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2832
unfolding permute_plus [symmetric] add_flip_cancel by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2833
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2834
lemma permute_flip_cancel2 [simp]: "(a \<leftrightarrow> b) \<bullet> (b \<leftrightarrow> a) \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2835
by (simp add: flip_commute)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2836
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2837
lemma flip_eqvt [eqvt]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2838
fixes a b c::"'a::at_base"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2839
shows "p \<bullet> (a \<leftrightarrow> b) = (p \<bullet> a \<leftrightarrow> p \<bullet> b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2840
unfolding flip_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2841
by (simp add: swap_eqvt atom_eqvt)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2842
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2843
lemma flip_at_base_simps [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2844
shows "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> a = b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2845
and "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> b = a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2846
and "\<lbrakk>a \<noteq> c; b \<noteq> c\<rbrakk> \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> c = c"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2847
and "sort_of (atom a) \<noteq> sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2848
unfolding flip_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2849
unfolding atom_eq_iff [symmetric]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2850
unfolding atom_eqvt [symmetric]
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2851
by simp_all
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2852
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2853
text {* the following two lemmas do not hold for at_base,
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2854
only for single sort atoms from at *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2855
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2856
lemma permute_flip_at:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2857
fixes a b c::"'a::at"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2858
shows "(a \<leftrightarrow> b) \<bullet> c = (if c = a then b else if c = b then a else c)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2859
unfolding flip_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2860
apply (rule atom_eq_iff [THEN iffD1])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2861
apply (subst atom_eqvt [symmetric])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2862
apply (simp add: swap_atom)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2863
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2864
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2865
lemma flip_at_simps [simp]:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2866
fixes a b::"'a::at"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2867
shows "(a \<leftrightarrow> b) \<bullet> a = b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2868
and "(a \<leftrightarrow> b) \<bullet> b = a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2869
unfolding permute_flip_at by simp_all
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2870
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2871
lemma flip_fresh_fresh:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2872
fixes a b::"'a::at_base"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2873
assumes "atom a \<sharp> x" "atom b \<sharp> x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2874
shows "(a \<leftrightarrow> b) \<bullet> x = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2875
using assms
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2876
by (simp add: flip_def swap_fresh_fresh)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2877
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2878
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2879
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2880
subsection {* Syntax for coercing at-elements to the atom-type *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2881
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2882
syntax
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2883
"_atom_constrain" :: "logic \<Rightarrow> type \<Rightarrow> logic" ("_:::_" [4, 0] 3)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2884
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2885
translations
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2886
"_atom_constrain a t" => "CONST atom (_constrain a t)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2887
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2888
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2889
subsection {* A lemma for proving instances of class @{text at}. *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2890
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2891
setup {* Sign.add_const_constraint (@{const_name "permute"}, NONE) *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2892
setup {* Sign.add_const_constraint (@{const_name "atom"}, NONE) *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2893
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2894
text {*
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2895
New atom types are defined as subtypes of @{typ atom}.
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2896
*}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2897
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2898
lemma exists_eq_simple_sort:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2899
shows "\<exists>a. a \<in> {a. sort_of a = s}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2900
by (rule_tac x="Atom s 0" in exI, simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2901
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2902
lemma exists_eq_sort:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2903
shows "\<exists>a. a \<in> {a. sort_of a \<in> range sort_fun}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2904
by (rule_tac x="Atom (sort_fun x) y" in exI, simp)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2905
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2906
lemma at_base_class:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2907
fixes sort_fun :: "'b \<Rightarrow> atom_sort"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2908
fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2909
assumes type: "type_definition Rep Abs {a. sort_of a \<in> range sort_fun}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2910
assumes atom_def: "\<And>a. atom a = Rep a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2911
assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2912
shows "OFCLASS('a, at_base_class)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2913
proof
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2914
interpret type_definition Rep Abs "{a. sort_of a \<in> range sort_fun}" by (rule type)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2915
have sort_of_Rep: "\<And>a. sort_of (Rep a) \<in> range sort_fun" using Rep by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2916
fix a b :: 'a and p p1 p2 :: perm
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2917
show "0 \<bullet> a = a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2918
unfolding permute_def by (simp add: Rep_inverse)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2919
show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2920
unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2921
show "atom a = atom b \<longleftrightarrow> a = b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2922
unfolding atom_def by (simp add: Rep_inject)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2923
show "p \<bullet> atom a = atom (p \<bullet> a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2924
unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2925
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2926
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2927
(*
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2928
lemma at_class:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2929
fixes s :: atom_sort
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2930
fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2931
assumes type: "type_definition Rep Abs {a. sort_of a \<in> range (\<lambda>x::unit. s)}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2932
assumes atom_def: "\<And>a. atom a = Rep a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2933
assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2934
shows "OFCLASS('a, at_class)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2935
proof
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2936
interpret type_definition Rep Abs "{a. sort_of a \<in> range (\<lambda>x::unit. s)}" by (rule type)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2937
have sort_of_Rep: "\<And>a. sort_of (Rep a) = s" using Rep by (simp add: image_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2938
fix a b :: 'a and p p1 p2 :: perm
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2939
show "0 \<bullet> a = a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2940
unfolding permute_def by (simp add: Rep_inverse)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2941
show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2942
unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2943
show "sort_of (atom a) = sort_of (atom b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2944
unfolding atom_def by (simp add: sort_of_Rep)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2945
show "atom a = atom b \<longleftrightarrow> a = b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2946
unfolding atom_def by (simp add: Rep_inject)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2947
show "p \<bullet> atom a = atom (p \<bullet> a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2948
unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2949
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2950
*)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2951
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2952
lemma at_class:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2953
fixes s :: atom_sort
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2954
fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2955
assumes type: "type_definition Rep Abs {a. sort_of a = s}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2956
assumes atom_def: "\<And>a. atom a = Rep a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2957
assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2958
shows "OFCLASS('a, at_class)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2959
proof
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2960
interpret type_definition Rep Abs "{a. sort_of a = s}" by (rule type)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2961
have sort_of_Rep: "\<And>a. sort_of (Rep a) = s" using Rep by (simp add: image_def)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2962
fix a b :: 'a and p p1 p2 :: perm
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2963
show "0 \<bullet> a = a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2964
unfolding permute_def by (simp add: Rep_inverse)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2965
show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2966
unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2967
show "sort_of (atom a) = sort_of (atom b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2968
unfolding atom_def by (simp add: sort_of_Rep)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2969
show "atom a = atom b \<longleftrightarrow> a = b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2970
unfolding atom_def by (simp add: Rep_inject)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2971
show "p \<bullet> atom a = atom (p \<bullet> a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2972
unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2973
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2974
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2975
lemma at_class_sort:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2976
fixes s :: atom_sort
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2977
fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2978
fixes a::"'a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2979
assumes type: "type_definition Rep Abs {a. sort_of a = s}"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2980
assumes atom_def: "\<And>a. atom a = Rep a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2981
shows "sort_of (atom a) = s"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2982
using atom_def type
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2983
unfolding type_definition_def by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2984
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2985
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2986
setup {* Sign.add_const_constraint
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2987
(@{const_name "permute"}, SOME @{typ "perm \<Rightarrow> 'a::pt \<Rightarrow> 'a"}) *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2988
setup {* Sign.add_const_constraint
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2989
(@{const_name "atom"}, SOME @{typ "'a::at_base \<Rightarrow> atom"}) *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2990
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2991
section {* The freshness lemma according to Andy Pitts *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2992
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2993
lemma freshness_lemma:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2994
fixes h :: "'a::at \<Rightarrow> 'b::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2995
assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2996
shows "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2997
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2998
from a obtain b where a1: "atom b \<sharp> h" and a2: "atom b \<sharp> h b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 2999
by (auto simp add: fresh_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3000
show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3001
proof (intro exI allI impI)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3002
fix a :: 'a
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3003
assume a3: "atom a \<sharp> h"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3004
show "h a = h b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3005
proof (cases "a = b")
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3006
assume "a = b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3007
thus "h a = h b" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3008
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3009
assume "a \<noteq> b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3010
hence "atom a \<sharp> b" by (simp add: fresh_at_base)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3011
with a3 have "atom a \<sharp> h b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3012
by (rule fresh_fun_app)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3013
with a2 have d1: "(atom b \<rightleftharpoons> atom a) \<bullet> (h b) = (h b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3014
by (rule swap_fresh_fresh)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3015
from a1 a3 have d2: "(atom b \<rightleftharpoons> atom a) \<bullet> h = h"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3016
by (rule swap_fresh_fresh)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3017
from d1 have "h b = (atom b \<rightleftharpoons> atom a) \<bullet> (h b)" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3018
also have "\<dots> = ((atom b \<rightleftharpoons> atom a) \<bullet> h) ((atom b \<rightleftharpoons> atom a) \<bullet> b)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3019
by (rule permute_fun_app_eq)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3020
also have "\<dots> = h a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3021
using d2 by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3022
finally show "h a = h b" by simp
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3023
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3024
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3025
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3026
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3027
lemma freshness_lemma_unique:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3028
fixes h :: "'a::at \<Rightarrow> 'b::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3029
assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3030
shows "\<exists>!x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3031
proof (rule ex_ex1I)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3032
from a show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3033
by (rule freshness_lemma)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3034
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3035
fix x y
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3036
assume x: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3037
assume y: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = y"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3038
from a x y show "x = y"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3039
by (auto simp add: fresh_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3040
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3041
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3042
text {* packaging the freshness lemma into a function *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3043
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3044
definition
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3045
fresh_fun :: "('a::at \<Rightarrow> 'b::pt) \<Rightarrow> 'b"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3046
where
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3047
"fresh_fun h = (THE x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3048
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3049
lemma fresh_fun_apply:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3050
fixes h :: "'a::at \<Rightarrow> 'b::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3051
assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3052
assumes b: "atom a \<sharp> h"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3053
shows "fresh_fun h = h a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3054
unfolding fresh_fun_def
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3055
proof (rule the_equality)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3056
show "\<forall>a'. atom a' \<sharp> h \<longrightarrow> h a' = h a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3057
proof (intro strip)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3058
fix a':: 'a
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3059
assume c: "atom a' \<sharp> h"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3060
from a have "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x" by (rule freshness_lemma)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3061
with b c show "h a' = h a" by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3062
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3063
next
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3064
fix fr :: 'b
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3065
assume "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = fr"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3066
with b show "fr = h a" by auto
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3067
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3068
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3069
lemma fresh_fun_apply':
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3070
fixes h :: "'a::at \<Rightarrow> 'b::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3071
assumes a: "atom a \<sharp> h" "atom a \<sharp> h a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3072
shows "fresh_fun h = h a"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3073
apply (rule fresh_fun_apply)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3074
apply (auto simp add: fresh_Pair intro: a)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3075
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3076
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3077
lemma fresh_fun_eqvt:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3078
fixes h :: "'a::at \<Rightarrow> 'b::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3079
assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3080
shows "p \<bullet> (fresh_fun h) = fresh_fun (p \<bullet> h)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3081
using a
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3082
apply (clarsimp simp add: fresh_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3083
apply (subst fresh_fun_apply', assumption+)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3084
apply (drule fresh_permute_iff [where p=p, THEN iffD2])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3085
apply (drule fresh_permute_iff [where p=p, THEN iffD2])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3086
apply (simp only: atom_eqvt permute_fun_app_eq [where f=h])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3087
apply (erule (1) fresh_fun_apply' [symmetric])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3088
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3089
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3090
lemma fresh_fun_supports:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3091
fixes h :: "'a::at \<Rightarrow> 'b::pt"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3092
assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3093
shows "(supp h) supports (fresh_fun h)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3094
apply (simp add: supports_def fresh_def [symmetric])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3095
apply (simp add: fresh_fun_eqvt [OF a] swap_fresh_fresh)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3096
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3097
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3098
notation fresh_fun (binder "FRESH " 10)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3099
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3100
lemma FRESH_f_iff:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3101
fixes P :: "'a::at \<Rightarrow> 'b::pure"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3102
fixes f :: "'b \<Rightarrow> 'c::pure"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3103
assumes P: "finite (supp P)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3104
shows "(FRESH x. f (P x)) = f (FRESH x. P x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3105
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3106
obtain a::'a where "atom a \<sharp> P" using P by (rule obtain_fresh')
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3107
show "(FRESH x. f (P x)) = f (FRESH x. P x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3108
apply (subst fresh_fun_apply' [where a=a, OF _ pure_fresh])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3109
apply (cut_tac `atom a \<sharp> P`)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3110
apply (simp add: fresh_conv_MOST)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3111
apply (elim MOST_rev_mp, rule MOST_I, clarify)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3112
apply (simp add: permute_fun_def permute_pure fun_eq_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3113
apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> P` pure_fresh])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3114
apply (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3115
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3116
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3117
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3118
lemma FRESH_binop_iff:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3119
fixes P :: "'a::at \<Rightarrow> 'b::pure"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3120
fixes Q :: "'a::at \<Rightarrow> 'c::pure"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3121
fixes binop :: "'b \<Rightarrow> 'c \<Rightarrow> 'd::pure"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3122
assumes P: "finite (supp P)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3123
and Q: "finite (supp Q)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3124
shows "(FRESH x. binop (P x) (Q x)) = binop (FRESH x. P x) (FRESH x. Q x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3125
proof -
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3126
from assms have "finite (supp (P, Q))" by (simp add: supp_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3127
then obtain a::'a where "atom a \<sharp> (P, Q)" by (rule obtain_fresh')
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3128
then have "atom a \<sharp> P" and "atom a \<sharp> Q" by (simp_all add: fresh_Pair)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3129
show ?thesis
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3130
apply (subst fresh_fun_apply' [where a=a, OF _ pure_fresh])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3131
apply (cut_tac `atom a \<sharp> P` `atom a \<sharp> Q`)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3132
apply (simp add: fresh_conv_MOST)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3133
apply (elim MOST_rev_mp, rule MOST_I, clarify)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3134
apply (simp add: permute_fun_def permute_pure fun_eq_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3135
apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> P` pure_fresh])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3136
apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> Q` pure_fresh])
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3137
apply (rule refl)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3138
done
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3139
qed
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3140
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3141
lemma FRESH_conj_iff:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3142
fixes P Q :: "'a::at \<Rightarrow> bool"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3143
assumes P: "finite (supp P)" and Q: "finite (supp Q)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3144
shows "(FRESH x. P x \<and> Q x) \<longleftrightarrow> (FRESH x. P x) \<and> (FRESH x. Q x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3145
using P Q by (rule FRESH_binop_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3146
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3147
lemma FRESH_disj_iff:
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3148
fixes P Q :: "'a::at \<Rightarrow> bool"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3149
assumes P: "finite (supp P)" and Q: "finite (supp Q)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3150
shows "(FRESH x. P x \<or> Q x) \<longleftrightarrow> (FRESH x. P x) \<or> (FRESH x. Q x)"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3151
using P Q by (rule FRESH_binop_iff)
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3152
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3153
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3154
section {* Library functions for the nominal infrastructure *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3155
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3156
use "nominal_library.ML"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3157
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3158
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3159
section {* Automation for creating concrete atom types *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3160
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3161
text {* at the moment only single-sort concrete atoms are supported *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3162
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3163
use "nominal_atoms.ML"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3164
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3165
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3166
section {* automatic equivariance procedure for inductive definitions *}
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3167
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3168
use "nominal_eqvt.ML"
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3169
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3170
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3171
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3172
e05c033d69c1
Alternate version of Nominal_Base: Executable version.
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents:
diff
changeset
+ − 3173
end