Nominal/Ex/Lambda.thy
author Christian Urban <urbanc@in.tum.de>
Mon, 07 Jun 2010 11:43:01 +0200
changeset 2311 4da5c5c29009
parent 2173 477293d841e8
child 2424 621ebd8b13c4
permissions -rw-r--r--
work on transitivity proof
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
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theory Lambda
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imports "../NewParser" Quotient_Option
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begin
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atom_decl name
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declare [[STEPS = 9]]
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nominal_datatype lam =
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  Var "name"
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| App "lam" "lam"
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| Lam x::"name" l::"lam"  bind x in l
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b7423c6b5564 deleted offending [eqvt]-attribute in Abs; Lambda works again, but there is now a problem in CoreHaskell
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thm lam.fv
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thm lam.supp
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lemmas supp_fn' = lam.fv[simplified lam.supp]
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declare lam.perm[eqvt]
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section {* Strong Induction Principles*}
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3842464ee03b Move 2 more to NewParser
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(*
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  Old way of establishing strong induction
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
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  principles by chosing a fresh name.
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*)
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lemma
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  fixes c::"'a::fs"
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78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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  assumes a1: "\<And>name c. P c (Var name)"
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  and     a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"
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  and     a3: "\<And>name lam c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lam\<rbrakk> \<Longrightarrow> P c (Lam name lam)"
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  shows "P c lam"
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proof -
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  have "\<And>p. P c (p \<bullet> lam)"
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    apply(induct lam arbitrary: c rule: lam.induct)
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    apply(perm_simp)
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    apply(rule a1)
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    apply(perm_simp)
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    apply(rule a2)
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fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
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    apply(assumption)
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    apply(assumption)
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    apply(subgoal_tac "\<exists>new::name. (atom new) \<sharp> (c, Lam (p \<bullet> name) (p \<bullet> lam))")
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    defer
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    apply(simp add: fresh_def)
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    apply(rule_tac X="supp (c, Lam (p \<bullet> name) (p \<bullet> lam))" in obtain_at_base)
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    apply(simp add: supp_Pair finite_supp)
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    apply(blast)
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    apply(erule exE)
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    apply(rule_tac t="p \<bullet> Lam name lam" and 
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                   s="(((p \<bullet> name) \<leftrightarrow> new) + p) \<bullet> Lam name lam" in subst)
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    apply(simp del: lam.perm)
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    apply(subst lam.perm)
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    apply(subst (2) lam.perm)
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    apply(rule flip_fresh_fresh)
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    apply(simp add: fresh_def)
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    apply(simp only: supp_fn')
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    apply(simp)
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    apply(simp add: fresh_Pair)
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    apply(simp)
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    apply(rule a3)
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    apply(simp add: fresh_Pair)
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    apply(drule_tac x="((p \<bullet> name) \<leftrightarrow> new) + p" in meta_spec)
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    apply(simp)
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    done
1800
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parents: 1797
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  then have "P c (0 \<bullet> lam)" by blast
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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parents: 1797
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    65
  then show "P c lam" by simp
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qed
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1797
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
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parents: 1773
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    68
(* 
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
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parents: 1773
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    69
  New way of establishing strong induction
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
parents: 1773
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    70
  principles by using a appropriate permutation.
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
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parents: 1773
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    71
*)
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    72
lemma
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  fixes c::"'a::fs"
1800
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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parents: 1797
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    74
  assumes a1: "\<And>name c. P c (Var name)"
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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    75
  and     a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"
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    76
  and     a3: "\<And>name lam c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lam\<rbrakk> \<Longrightarrow> P c (Lam name lam)"
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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parents: 1797
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    77
  shows "P c lam"
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    78
proof -
1800
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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parents: 1797
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    79
  have "\<And>p. P c (p \<bullet> lam)"
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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    80
    apply(induct lam arbitrary: c rule: lam.induct)
1805
f187f20f0a79 a few tests
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    81
    apply(perm_simp)
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    82
    apply(rule a1)
1805
f187f20f0a79 a few tests
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parents: 1800
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    83
    apply(perm_simp)
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parents:
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    84
    apply(rule a2)
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    85
    apply(assumption)
1797
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
parents: 1773
diff changeset
    86
    apply(assumption)
1800
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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parents: 1797
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    87
    apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom name}) \<sharp>* c \<and> supp (p \<bullet> Lam name lam) \<sharp>* q")
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892fcdb96c96 Move LamEx out of Test.
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parents:
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    88
    apply(erule exE)
1800
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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parents: 1797
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    89
    apply(rule_tac t="p \<bullet> Lam name lam" and 
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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    90
                   s="q \<bullet> p \<bullet> Lam name lam" in subst)
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    91
    defer
1805
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parents: 1800
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    92
    apply(simp)
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parents:
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    93
    apply(rule a3)
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parents:
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    94
    apply(simp add: eqvts fresh_star_def)
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parents:
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    95
    apply(drule_tac x="q + p" in meta_spec)
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parents:
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    96
    apply(simp)
1797
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
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parents: 1773
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    97
    apply(rule at_set_avoiding2)
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
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parents: 1773
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    98
    apply(simp add: finite_supp)
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
parents: 1773
diff changeset
    99
    apply(simp add: finite_supp)
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
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parents: 1773
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   100
    apply(simp add: finite_supp)
1805
f187f20f0a79 a few tests
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parents: 1800
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   101
    apply(perm_simp)
1797
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
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parents: 1773
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   102
    apply(simp add: fresh_star_def fresh_def supp_fn')
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
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parents: 1773
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   103
    apply(rule supp_perm_eq)
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
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parents: 1773
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   104
    apply(simp)
fddb470720f1 renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
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parents: 1773
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   105
    done
1800
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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parents: 1797
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   106
  then have "P c (0 \<bullet> lam)" by blast
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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parents: 1797
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   107
  then show "P c lam" by simp
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parents:
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   108
qed
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parents:
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   109
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   110
section {* Typing *}
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   111
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   112
nominal_datatype ty =
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   113
  TVar string
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parents: 1805
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   114
| TFun ty ty
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   115
894930834ca8 fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
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   116
notation
894930834ca8 fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
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   117
 TFun ("_ \<rightarrow> _") 
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   118
894930834ca8 fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
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   119
declare ty.perm[eqvt]
1805
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inductive
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   122
  valid :: "(name \<times> ty) list \<Rightarrow> bool"
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   123
where
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   124
  "valid []"
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| "\<lbrakk>atom x \<sharp> Gamma; valid Gamma\<rbrakk> \<Longrightarrow> valid ((x, T)#Gamma)"
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   126
1828
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   127
inductive
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   128
  typing :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool" ("_ \<turnstile> _ : _" [60,60,60] 60) 
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   129
where
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   130
    t_Var[intro]: "\<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Var x : T"
1947
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   131
  | t_App[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : T2"
1828
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   132
  | t_Lam[intro]: "\<lbrakk>atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam x t : T1 \<rightarrow> T2"
f374ffd50c7c preliminary tests
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   133
1831
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equivariance valid
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   135
equivariance typing
1816
56cebe7f8e24 some small tunings (incompleted work in Lambda.thy)
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   136
1831
16653e702d89 first working version of the automatic equivariance procedure
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   137
thm valid.eqvt
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   138
thm typing.eqvt
1811
ae176476b525 implemented in thmdecls the case where eqvt-lemmas are of the form _ ==> _
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   139
thm eqvts
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   140
thm eqvts_raw
ae176476b525 implemented in thmdecls the case where eqvt-lemmas are of the form _ ==> _
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   141
1949
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   142
thm typing.induct[of "\<Gamma>" "t" "T", no_vars]
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diff changeset
   143
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   144
lemma
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   145
  fixes c::"'a::fs"
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diff changeset
   146
  assumes a: "\<Gamma> \<turnstile> t : T" 
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   147
  and a1: "\<And>\<Gamma> x T c. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P c \<Gamma> (Var x) T"
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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   148
  and a2: "\<And>\<Gamma> t1 T1 T2 t2 c. \<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<And>d. P d \<Gamma> t1 T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1; \<And>d. P d \<Gamma> t2 T1\<rbrakk> 
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   149
           \<Longrightarrow> P c \<Gamma> (App t1 t2) T2"
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   150
  and a3: "\<And>x \<Gamma> T1 t T2 c. \<lbrakk>atom x \<sharp> c; atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2; \<And>d. P d ((x, T1) # \<Gamma>) t T2\<rbrakk> 
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diff changeset
   151
           \<Longrightarrow> P c \<Gamma> (Lam x t) T1 \<rightarrow> T2"
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   152
  shows "P c \<Gamma> t T"
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   153
proof -
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   154
  from a have "\<And>p c. P c (p \<bullet> \<Gamma>) (p \<bullet> t) (p \<bullet> T)"
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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   155
  proof (induct)
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   156
    case (t_Var \<Gamma> x T p c)
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   157
    then show ?case
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   158
      apply -
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   159
      apply(perm_strict_simp)
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   160
      apply(rule a1)
1954
23480003f9c5 some changes to the paper
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parents: 1950
diff changeset
   161
      apply(drule_tac p="p" in permute_boolI)
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   162
      apply(perm_strict_simp add: permute_minus_cancel)
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   163
      apply(assumption)
1954
23480003f9c5 some changes to the paper
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diff changeset
   164
      apply(rotate_tac 1)
23480003f9c5 some changes to the paper
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diff changeset
   165
      apply(drule_tac p="p" in permute_boolI)
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   166
      apply(perm_strict_simp add: permute_minus_cancel)
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diff changeset
   167
      apply(assumption)
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   168
      done
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   169
  next
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   170
    case (t_App \<Gamma> t1 T1 T2 t2 p c)
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   171
    then show ?case
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   172
      apply -
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   173
      apply(perm_strict_simp)
1954
23480003f9c5 some changes to the paper
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parents: 1950
diff changeset
   174
      apply(rule a2)
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diff changeset
   175
      apply(drule_tac p="p" in permute_boolI)
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   176
      apply(perm_strict_simp add: permute_minus_cancel)
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   177
      apply(assumption)
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   178
      apply(assumption)
1954
23480003f9c5 some changes to the paper
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parents: 1950
diff changeset
   179
      apply(rotate_tac 2)
23480003f9c5 some changes to the paper
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parents: 1950
diff changeset
   180
      apply(drule_tac p="p" in permute_boolI)
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   181
      apply(perm_strict_simp add: permute_minus_cancel)
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diff changeset
   182
      apply(assumption)
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diff changeset
   183
      apply(assumption)
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   184
      done
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   185
  next
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   186
    case (t_Lam x \<Gamma> T1 t T2 p c)
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diff changeset
   187
    then show ?case
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diff changeset
   188
      apply -
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diff changeset
   189
      apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom x}) \<sharp>* c \<and> 
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   190
        supp (p \<bullet> \<Gamma>, p \<bullet> Lam x t, p \<bullet> (T1 \<rightarrow> T2)) \<sharp>* q")
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   191
      apply(erule exE)
1954
23480003f9c5 some changes to the paper
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diff changeset
   192
      apply(rule_tac t="p \<bullet> \<Gamma>" and  s="(q + p) \<bullet> \<Gamma>" in subst)
23480003f9c5 some changes to the paper
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diff changeset
   193
      apply(simp only: permute_plus)
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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   194
      apply(rule supp_perm_eq)
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   195
      apply(simp add: supp_Pair fresh_star_union)
1954
23480003f9c5 some changes to the paper
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diff changeset
   196
      apply(rule_tac t="p \<bullet> Lam x t" and  s="(q + p) \<bullet> Lam x t" in subst)
23480003f9c5 some changes to the paper
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diff changeset
   197
      apply(simp only: permute_plus)
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   198
      apply(rule supp_perm_eq)
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diff changeset
   199
      apply(simp add: supp_Pair fresh_star_union)
1954
23480003f9c5 some changes to the paper
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diff changeset
   200
      apply(rule_tac t="p \<bullet> (T1 \<rightarrow> T2)" and  s="(q + p) \<bullet> (T1 \<rightarrow> T2)" in subst)
23480003f9c5 some changes to the paper
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parents: 1950
diff changeset
   201
      apply(simp only: permute_plus)
23480003f9c5 some changes to the paper
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diff changeset
   202
      apply(rule supp_perm_eq)
23480003f9c5 some changes to the paper
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diff changeset
   203
      apply(simp add: supp_Pair fresh_star_union)
23480003f9c5 some changes to the paper
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diff changeset
   204
      apply(simp (no_asm) only: eqvts)
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   205
      apply(rule a3)
1954
23480003f9c5 some changes to the paper
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parents: 1950
diff changeset
   206
      apply(simp only: eqvts permute_plus)
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   207
      apply(simp add: fresh_star_def)
1954
23480003f9c5 some changes to the paper
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parents: 1950
diff changeset
   208
      apply(drule_tac p="q + p" in permute_boolI)
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   209
      apply(perm_strict_simp add: permute_minus_cancel)
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diff changeset
   210
      apply(assumption)
1954
23480003f9c5 some changes to the paper
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parents: 1950
diff changeset
   211
      apply(rotate_tac 1)
23480003f9c5 some changes to the paper
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parents: 1950
diff changeset
   212
      apply(drule_tac p="q + p" in permute_boolI)
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   213
      apply(perm_strict_simp add: permute_minus_cancel)
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parents: 1947
diff changeset
   214
      apply(assumption)
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diff changeset
   215
      apply(drule_tac x="d" in meta_spec)
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   216
      apply(drule_tac x="q + p" in meta_spec)
1954
23480003f9c5 some changes to the paper
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parents: 1950
diff changeset
   217
      apply(perm_strict_simp add: permute_minus_cancel)
23480003f9c5 some changes to the paper
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parents: 1950
diff changeset
   218
      apply(assumption)
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   219
      apply(rule at_set_avoiding2)
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   220
      apply(simp add: finite_supp)
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   221
      apply(simp add: finite_supp)
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   222
      apply(simp add: finite_supp)
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   223
      apply(rule_tac p="-p" in permute_boolE)
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   224
      apply(perm_strict_simp add: permute_minus_cancel)
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   225
	(* supplied by the user *)
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parents: 1947
diff changeset
   226
      apply(simp add: fresh_star_prod)
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   227
      apply(simp add: fresh_star_def)
1954
23480003f9c5 some changes to the paper
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parents: 1950
diff changeset
   228
      sorry
23480003f9c5 some changes to the paper
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diff changeset
   229
  qed
23480003f9c5 some changes to the paper
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diff changeset
   230
  then have "P c (0 \<bullet> \<Gamma>) (0 \<bullet> t) (0 \<bullet> T)" .
23480003f9c5 some changes to the paper
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parents: 1950
diff changeset
   231
  then show "P c \<Gamma> t T" by simp
23480003f9c5 some changes to the paper
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diff changeset
   232
qed
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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diff changeset
   233
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   234
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   235
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   236
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   237
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
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parents: 1947
diff changeset
   238
1833
2050b5723c04 added a library for basic nominal functions; separated nominal_eqvt file
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parents: 1831
diff changeset
   239
2050b5723c04 added a library for basic nominal functions; separated nominal_eqvt file
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parents: 1831
diff changeset
   240
inductive
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diff changeset
   241
  tt :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"  
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diff changeset
   242
  for r :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"
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diff changeset
   243
where
1835
636de31888a6 tuned and removed dead code
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parents: 1833
diff changeset
   244
    aa: "tt r a a"
636de31888a6 tuned and removed dead code
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parents: 1833
diff changeset
   245
  | bb: "tt r a b ==> tt r a c"
1833
2050b5723c04 added a library for basic nominal functions; separated nominal_eqvt file
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parents: 1831
diff changeset
   246
1835
636de31888a6 tuned and removed dead code
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parents: 1833
diff changeset
   247
(* PROBLEM: derived eqvt-theorem does not conform with [eqvt]
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equivariance tt
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*)
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inductive
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 alpha_lam_raw'
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where
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  a1: "name = namea \<Longrightarrow> alpha_lam_raw' (Var_raw name) (Var_raw namea)"
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| a2: "\<lbrakk>alpha_lam_raw' lam_raw1 lam_raw1a; alpha_lam_raw' lam_raw2 lam_raw2a\<rbrakk> \<Longrightarrow>
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   alpha_lam_raw' (App_raw lam_raw1 lam_raw2) (App_raw lam_raw1a lam_raw2a)"
1865
b71b838b0a75 Finished proof in Lambda.thy
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| a3: "\<exists>pi. ({atom name}, lam_raw) \<approx>gen alpha_lam_raw' fv_lam_raw pi ({atom namea}, lam_rawa) \<Longrightarrow>
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   alpha_lam_raw' (Lam_raw name lam_raw) (Lam_raw namea lam_rawa)"
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1866
6d4e4bf9bce6 automatic proofs for equivariance of alphas
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equivariance alpha_lam_raw'
1947
51f411b1197d tuned and cleaned
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1861
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thm eqvts_raw
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1800
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section {* size function *}
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lemma size_eqvt_raw:
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  fixes t::"lam_raw"
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  shows "size (pi \<bullet> t)  = size t"
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  apply (induct rule: lam_raw.inducts)
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  apply simp_all
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  done
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instantiation lam :: size 
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begin
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quotient_definition
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  "size_lam :: lam \<Rightarrow> nat"
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is
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  "size :: lam_raw \<Rightarrow> nat"
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lemma size_rsp:
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  "alpha_lam_raw x y \<Longrightarrow> size x = size y"
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  apply (induct rule: alpha_lam_raw.inducts)
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  apply (simp_all only: lam_raw.size)
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  apply (simp_all only: alphas)
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  apply clarify
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  apply (simp_all only: size_eqvt_raw)
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  done
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lemma [quot_respect]:
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  "(alpha_lam_raw ===> op =) size size"
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  by (simp_all add: size_rsp)
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lemma [quot_preserve]:
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  "(rep_lam ---> id) size = size"
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  by (simp_all add: size_lam_def)
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instance
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  by default
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end
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lemmas size_lam[simp] = 
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  lam_raw.size(4)[quot_lifted]
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  lam_raw.size(5)[quot_lifted]
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  lam_raw.size(6)[quot_lifted]
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(* is this needed? *)
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lemma [measure_function]: 
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  "is_measure (size::lam\<Rightarrow>nat)" 
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  by (rule is_measure_trivial)
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section {* Matching *}
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definition
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  MATCH :: "('c::pt \<Rightarrow> (bool * 'a::pt * 'b::pt)) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"
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where
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  "MATCH M d x \<equiv> if (\<exists>!r. \<exists>q. M q = (True, x, r)) then (THE r. \<exists>q. M q = (True, x, r)) else d"
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78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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(*
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lemma MATCH_eqvt:
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  shows "p \<bullet> (MATCH M d x) = MATCH (p \<bullet> M) (p \<bullet> d) (p \<bullet> x)"
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unfolding MATCH_def
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apply(perm_simp the_eqvt)
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apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
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apply(simp)
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thm eqvts_raw 
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apply(subst if_eqvt)
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apply(subst ex1_eqvt)
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apply(subst permute_fun_def)
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apply(subst ex_eqvt)
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apply(subst permute_fun_def)
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apply(subst eq_eqvt)
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apply(subst permute_fun_app_eq[where f="M"])
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   336
apply(simp only: permute_minus_cancel)
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   337
apply(subst permute_prod.simps)
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apply(subst permute_prod.simps)
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apply(simp only: permute_minus_cancel)
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apply(simp only: permute_bool_def)
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apply(simp)
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apply(subst ex1_eqvt)
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apply(subst permute_fun_def)
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apply(subst ex_eqvt)
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apply(subst permute_fun_def)
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apply(subst eq_eqvt)
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78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
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apply(simp only: eqvts)
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apply(simp)
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apply(subgoal_tac "(p \<bullet> (\<exists>!r. \<exists>q. M q = (True, x, r))) = (\<exists>!r. \<exists>q. (p \<bullet> M) q = (True, p \<bullet> x, r))")
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apply(drule sym)
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apply(simp)
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apply(rule impI)
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apply(simp add: perm_bool)
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   355
apply(rule trans)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   356
apply(rule pt_the_eqvt[OF pta at])
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   357
apply(assumption)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   358
apply(simp add: pt_ex_eqvt[OF pt at])
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   359
apply(simp add: pt_eq_eqvt[OF ptb at])
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   360
apply(rule cheat)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   361
apply(rule trans)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   362
apply(rule pt_ex1_eqvt)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   363
apply(rule pta)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   364
apply(rule at)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   365
apply(simp add: pt_ex_eqvt[OF pt at])
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   366
apply(simp add: pt_eq_eqvt[OF ptb at])
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   367
apply(subst pt_pi_rev[OF pta at])
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   368
apply(subst pt_fun_app_eq[OF pt at])
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   369
apply(subst pt_pi_rev[OF pt at])
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   370
apply(simp)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   371
done
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   372
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   373
lemma MATCH_cng:
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   374
  assumes a: "M1 = M2" "d1 = d2"
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   375
  shows "MATCH M1 d1 x = MATCH M2 d2 x"
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   376
using a by simp
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   377
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   378
lemma MATCH_eq:
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   379
  assumes a: "t = l x" "G x" "\<And>x'. t = l x' \<Longrightarrow> G x' \<Longrightarrow> r x' = r x"
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   380
  shows "MATCH (\<lambda>x. (G x, l x, r x)) d t = r x"
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   381
using a
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   382
unfolding MATCH_def
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   383
apply(subst if_P)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   384
apply(rule_tac a="r x" in ex1I)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   385
apply(rule_tac x="x" in exI)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   386
apply(blast)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   387
apply(erule exE)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   388
apply(drule_tac x="q" in meta_spec)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   389
apply(auto)[1]
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   390
apply(rule the_equality)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   391
apply(blast)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   392
apply(erule exE)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   393
apply(drule_tac x="q" in meta_spec)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   394
apply(auto)[1]
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   395
done
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   396
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   397
lemma MATCH_eq2:
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   398
  assumes a: "t = l x1 x2" "G x1 x2" "\<And>x1' x2'. t = l x1' x2' \<Longrightarrow> G x1' x2' \<Longrightarrow> r x1' x2' = r x1 x2"
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   399
  shows "MATCH (\<lambda>(x1,x2). (G x1 x2, l x1 x2, r x1 x2)) d t = r x1 x2"
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   400
sorry
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   401
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   402
lemma MATCH_neq:
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   403
  assumes a: "\<And>x. t = l x \<Longrightarrow> G x \<Longrightarrow> False"
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   404
  shows "MATCH (\<lambda>x. (G x, l x, r x)) d t = d"
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   405
using a
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   406
unfolding MATCH_def
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   407
apply(subst if_not_P)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   408
apply(blast)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   409
apply(rule refl)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   410
done
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   411
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   412
lemma MATCH_neq2:
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   413
  assumes a: "\<And>x1 x2. t = l x1 x2 \<Longrightarrow> G x1 x2 \<Longrightarrow> False"
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   414
  shows "MATCH (\<lambda>(x1,x2). (G x1 x2, l x1 x2, r x1 x2)) d t = d"
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   415
using a
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   416
unfolding MATCH_def
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   417
apply(subst if_not_P)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   418
apply(auto)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   419
done
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   420
*)
78fdc6b36a1c changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
parents: 1797
diff changeset
   421
1954
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   422
ML {*
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   423
fun mk_avoids ctxt params name set =
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   424
  let
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   425
    val (_, ctxt') = ProofContext.add_fixes
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   426
      (map (fn (s, T) => (Binding.name s, SOME T, NoSyn)) params) ctxt;
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   427
    fun mk s =
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   428
      let
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   429
        val t = Syntax.read_term ctxt' s;
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   430
        val t' = list_abs_free (params, t) |>
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   431
          funpow (length params) (fn Abs (_, _, t) => t)
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   432
      in (t', HOLogic.dest_setT (fastype_of t)) end
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   433
      handle TERM _ =>
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   434
        error ("Expression " ^ quote s ^ " to be avoided in case " ^
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   435
          quote name ^ " is not a set type");
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   436
    fun add_set p [] = [p]
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   437
      | add_set (t, T) ((u, U) :: ps) =
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   438
          if T = U then
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   439
            let val S = HOLogic.mk_setT T
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   440
            in (Const (@{const_name sup}, S --> S --> S) $ u $ t, T) :: ps
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   441
            end
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   442
          else (u, U) :: add_set (t, T) ps
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   443
  in
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   444
    (mk #> add_set) set 
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   445
  end;
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   446
*}
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   447
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   448
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   449
ML {* 
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   450
  writeln (commas (map (Syntax.string_of_term @{context} o fst) 
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   451
    (mk_avoids @{context} [] "t_Var" "{x}" []))) 
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   452
*}
23480003f9c5 some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents: 1950
diff changeset
   453
1947
51f411b1197d tuned and cleaned
Christian Urban <urbanc@in.tum.de>
parents: 1866
diff changeset
   454
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   455
ML {*
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   456
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   457
fun prove_strong_ind (pred_name, avoids) ctxt = 
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   458
  Proof.theorem NONE (K I) [] ctxt
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   459
2169
61a89e41c55b Renamings.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2162
diff changeset
   460
local structure P = Parse and K = Keyword in
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   461
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   462
val _ =
2169
61a89e41c55b Renamings.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2162
diff changeset
   463
  Outer_Syntax.local_theory_to_proof "nominal_inductive"
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   464
    "proves strong induction theorem for inductive predicate involving nominal datatypes" K.thy_goal
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   465
      (P.xname -- (Scan.optional (P.$$$ "avoids" |-- P.enum1 "|" (P.name --
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   466
        (P.$$$ ":" |-- P.and_list1 P.term))) []) >>  prove_strong_ind)
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   467
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   468
end;
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   469
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   470
*}
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   471
1950
7de54c9f81ac eliminated command so that all compiles
Christian Urban <urbanc@in.tum.de>
parents: 1949
diff changeset
   472
(*
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   473
nominal_inductive typing
1950
7de54c9f81ac eliminated command so that all compiles
Christian Urban <urbanc@in.tum.de>
parents: 1949
diff changeset
   474
*)
1949
0b692f37a771 changed theorem_i to theorem....requires new Isabelle
Christian Urban <urbanc@in.tum.de>
parents: 1947
diff changeset
   475
2157
a1d27083e688 subst experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   476
(* Substitution *)
2159
ce00205e07ab Single variable substitution
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2158
diff changeset
   477
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   478
primrec match_Var_raw where
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   479
  "match_Var_raw (Var_raw x) = Some x"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   480
| "match_Var_raw (App_raw x y) = None"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   481
| "match_Var_raw (Lam_raw n t) = None"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   482
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   483
quotient_definition
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   484
  "match_Var :: lam \<Rightarrow> name option"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   485
is match_Var_raw
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   486
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   487
lemma [quot_respect]: "(alpha_lam_raw ===> op =) match_Var_raw match_Var_raw"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   488
  apply rule
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   489
  apply (induct_tac a b rule: alpha_lam_raw.induct)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   490
  apply simp_all
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   491
  done
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   492
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   493
lemmas match_Var_simps = match_Var_raw.simps[quot_lifted]
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   494
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   495
primrec match_App_raw where
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   496
  "match_App_raw (Var_raw x) = None"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   497
| "match_App_raw (App_raw x y) = Some (x, y)"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   498
| "match_App_raw (Lam_raw n t) = None"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   499
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   500
quotient_definition
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   501
  "match_App :: lam \<Rightarrow> (lam \<times> lam) option"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   502
is match_App_raw
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   503
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   504
lemma [quot_respect]:
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   505
  "(alpha_lam_raw ===> option_rel (prod_rel alpha_lam_raw alpha_lam_raw)) match_App_raw match_App_raw"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   506
  apply (intro fun_relI)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   507
  apply (induct_tac a b rule: alpha_lam_raw.induct)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   508
  apply simp_all
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   509
  done
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   510
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   511
lemmas match_App_simps = match_App_raw.simps[quot_lifted]
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   512
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   513
definition new where
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   514
  "new (s :: 'a :: fs) = (THE x. \<forall>a \<in> supp s. atom x \<noteq> a)"
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   515
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   516
definition
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   517
  "match_Lam (S :: 'a :: fs) t = (if (\<exists>n s. (t = Lam n s)) then
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   518
    (let z = new (S, t) in Some (z, THE s. t = Lam z s)) else None)"
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   519
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   520
lemma lam_half_inj: "(Lam z s = Lam z sa) = (s = sa)"
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   521
  apply auto
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   522
  apply (simp only: lam.eq_iff alphas)
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   523
  apply clarify
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   524
  apply (simp add: eqvts)
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   525
  sorry
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   526
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   527
lemma match_Lam_simps:
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   528
  "match_Lam S (Var n) = None"
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   529
  "match_Lam S (App l r) = None"
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   530
  "z = new (S, (Lam z s)) \<Longrightarrow> match_Lam S (Lam z s) = Some (z, s)"
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   531
  apply (simp_all add: match_Lam_def)
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   532
  apply (simp add: lam_half_inj)
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   533
  apply auto
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   534
  done
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   535
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   536
(*
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   537
lemma match_Lam_simps2:
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   538
  "atom n \<sharp> ((S :: 'a :: fs), Lam n s) \<Longrightarrow> match_Lam S (Lam n s) = Some (n, s)"
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   539
  apply (rule_tac t="Lam n s"
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   540
              and s="Lam (new (S, (Lam n s))) ((n \<leftrightarrow> (new (S, (Lam n s)))) \<bullet> s)" in subst)
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   541
  defer
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   542
  apply (subst match_Lam_simps(3))
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   543
  defer
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   544
  apply simp
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   545
*)
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   546
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   547
(*primrec match_Lam_raw where
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   548
  "match_Lam_raw (S :: atom set) (Var_raw x) = None"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   549
| "match_Lam_raw S (App_raw x y) = None"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   550
| "match_Lam_raw S (Lam_raw n t) = (let z = new (S \<union> (fv_lam_raw t - {atom n})) in Some (z, (n \<leftrightarrow> z) \<bullet> t))"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   551
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   552
quotient_definition
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   553
  "match_Lam :: (atom set) \<Rightarrow> lam \<Rightarrow> (name \<times> lam) option"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   554
is match_Lam_raw
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   555
2172
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   556
lemma swap_fresh:
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   557
  assumes a: "fv_lam_raw t \<sharp>* p"
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   558
  shows "alpha_lam_raw (p \<bullet> t) t"
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   559
  using a apply (induct t)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   560
  apply (simp add: supp_at_base fresh_star_def)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   561
  apply (rule alpha_lam_raw.intros)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   562
  apply (metis Rep_name_inverse atom_eqvt atom_name_def fresh_perm)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   563
  apply (simp)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   564
  apply (simp only: fresh_star_union)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   565
  apply clarify
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   566
  apply (rule alpha_lam_raw.intros)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   567
  apply simp
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   568
  apply simp
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   569
  apply simp
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   570
  apply (rule alpha_lam_raw.intros)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   571
  sorry
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   572
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   573
lemma [quot_respect]:
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   574
  "(op = ===> alpha_lam_raw ===> option_rel (prod_rel op = alpha_lam_raw)) match_Lam_raw match_Lam_raw"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   575
  proof (intro fun_relI, clarify)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   576
    fix S t s
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   577
    assume a: "alpha_lam_raw t s"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   578
    show "option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S t) (match_Lam_raw S s)"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   579
      using a proof (induct t s rule: alpha_lam_raw.induct)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   580
      case goal1 show ?case by simp
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   581
    next
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   582
      case goal2 show ?case by simp
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   583
    next
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   584
      case (goal3 x t y s)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   585
      then obtain p where "({atom x}, t) \<approx>gen (\<lambda>x1 x2. alpha_lam_raw x1 x2 \<and>
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   586
                              option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S x1)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   587
                               (match_Lam_raw S x2)) fv_lam_raw p ({atom y}, s)" ..
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   588
      then have
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   589
        c: "fv_lam_raw t - {atom x} = fv_lam_raw s - {atom y}" and
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   590
        d: "(fv_lam_raw t - {atom x}) \<sharp>* p" and
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   591
        e: "alpha_lam_raw (p \<bullet> t) s" and
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   592
        f: "option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S (p \<bullet> t)) (match_Lam_raw S s)" and
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   593
        g: "p \<bullet> {atom x} = {atom y}" unfolding alphas(1) by - (elim conjE, assumption)+
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   594
      let ?z = "new (S \<union> (fv_lam_raw t - {atom x}))"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   595
      have h: "?z = new (S \<union> (fv_lam_raw s - {atom y}))" using c by simp
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   596
      show ?case
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   597
        unfolding match_Lam_raw.simps Let_def option_rel.simps prod_rel.simps split_conv
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   598
      proof
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   599
        show "?z = new (S \<union> (fv_lam_raw s - {atom y}))" by (fact h)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   600
      next
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   601
        have "atom y \<sharp> p" sorry
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   602
        have "fv_lam_raw t \<sharp>* ((x \<leftrightarrow> y) \<bullet> p)" sorry
2172
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   603
        then have "alpha_lam_raw (((x \<leftrightarrow> y) \<bullet> p) \<bullet> t) t" using swap_fresh by auto
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   604
        then have "alpha_lam_raw (p \<bullet> t) ((x \<leftrightarrow> y) \<bullet> t)" sorry
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   605
        have "alpha_lam_raw t ((x \<leftrightarrow> y) \<bullet> s)" sorry
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   606
        then have "alpha_lam_raw ((x \<leftrightarrow> ?z) \<bullet> t) ((y \<leftrightarrow> ?z) \<bullet> s)" using eqvts(15) sorry
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   607
        then show "alpha_lam_raw ((x \<leftrightarrow> new (S \<union> (fv_lam_raw t - {atom x}))) \<bullet> t)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   608
                  ((y \<leftrightarrow> new (S \<union> (fv_lam_raw s - {atom y}))) \<bullet> s)" unfolding h .
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   609
      qed
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   610
    qed
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   611
  qed
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   612
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   613
lemmas match_Lam_simps = match_Lam_raw.simps[quot_lifted]
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   614
*)
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   615
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   616
lemma app_some: "match_App x = Some (a, b) \<Longrightarrow> x = App a b"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   617
by (induct x rule: lam.induct) (simp_all add: match_App_simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   618
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   619
lemma lam_some: "match_Lam S x = Some (z, s) \<Longrightarrow> x = Lam z s \<and> atom z \<sharp> S"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   620
  apply (induct x rule: lam.induct)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   621
  apply (simp_all add: match_Lam_simps)
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   622
  apply (thin_tac "match_Lam S lam = Some (z, s) \<Longrightarrow> lam = Lam z s \<and> atom z \<sharp> S")
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   623
  apply (simp add: match_Lam_def)
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   624
  apply (subgoal_tac "\<exists>n s. Lam name lam = Lam n s")
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   625
  prefer 2
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   626
  apply auto[1]
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   627
  apply (simp add: Let_def)
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   628
  apply (thin_tac "\<exists>n s. Lam name lam = Lam n s")
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   629
  apply clarify
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   630
  apply (rule conjI)
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   631
  apply (rule_tac t="THE s. Lam name lam = Lam (new (S, Lam name lam)) s" and
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   632
                  s="(name \<leftrightarrow> (new (S, Lam name lam))) \<bullet> lam" in subst)
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   633
  defer
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   634
  apply (simp add: lam.eq_iff)
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   635
  apply (rule_tac x="(name \<leftrightarrow> (new (S, Lam name lam)))" in exI)
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   636
  apply (simp add: alphas)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   637
  apply (simp add: eqvts)
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   638
  apply (rule conjI)
2172
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   639
  sorry
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   640
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   641
function subst where
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   642
"subst v s t = (
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   643
  case match_Var t of Some n \<Rightarrow> if n = v then s else Var n | None \<Rightarrow>
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   644
  case match_App t of Some (l, r) \<Rightarrow> App (subst v s l) (subst v s r) | None \<Rightarrow>
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   645
  case match_Lam (v,s) t of Some (n, t) \<Rightarrow> Lam n (subst v s t) | None \<Rightarrow> undefined)"
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   646
by pat_completeness auto
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   647
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   648
termination apply (relation "measure (\<lambda>(_, _, t). size t)")
2172
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   649
  apply auto[1]
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   650
  apply (case_tac a) apply simp
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   651
  apply (frule lam_some) apply simp
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   652
  apply (case_tac a) apply simp
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   653
  apply (frule app_some) apply simp
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   654
  apply (case_tac a) apply simp
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   655
  apply (frule app_some) apply simp
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   656
done
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   657
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   658
lemmas lam_exhaust = lam_raw.exhaust[quot_lifted]
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   659
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   660
lemma subst_eqvt:
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   661
  "p \<bullet> (subst v s t) = subst (p \<bullet> v) (p \<bullet> s) (p \<bullet> t)"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   662
  proof (induct v s t rule: subst.induct)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   663
    case (1 v s t)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   664
    show ?case proof (cases t rule: lam_exhaust)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   665
      fix n
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   666
      assume "t = Var n"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   667
      then show ?thesis by (simp add: match_Var_simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   668
    next
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   669
      fix l r
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   670
      assume "t = App l r"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   671
      then show ?thesis
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   672
        apply (simp only:)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   673
        apply (subst subst.simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   674
        apply (subst match_Var_simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   675
        apply (simp only: option.cases)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   676
        apply (subst match_App_simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   677
        apply (simp only: option.cases)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   678
        apply (simp only: prod.cases)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   679
        apply (simp only: lam.perm)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   680
        apply (subst (3) subst.simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   681
        apply (subst match_Var_simps)
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   682
        apply (simp only: option.cases)
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   683
        apply (subst match_App_simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   684
        apply (simp only: option.cases)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   685
        apply (simp only: prod.cases)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   686
        apply (subst 1(2)[of "(l, r)" "l" "r"])
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   687
        apply (simp add: match_Var_simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   688
        apply (simp add: match_App_simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   689
        apply (rule refl)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   690
        apply (subst 1(3)[of "(l, r)" "l" "r"])
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   691
        apply (simp add: match_Var_simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   692
        apply (simp add: match_App_simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   693
        apply (rule refl)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   694
        apply (rule refl)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   695
        done
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   696
    next
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   697
      fix n t'
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   698
      assume "t = Lam n t'"
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   699
      then show ?thesis
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   700
        apply (simp only: )
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   701
        apply (simp only: lam.perm)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   702
        apply (subst subst.simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   703
        apply (subst match_Var_simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   704
        apply (simp only: option.cases)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   705
        apply (subst match_App_simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   706
        apply (simp only: option.cases)
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   707
        apply (rule_tac t="Lam n t'" and s="Lam (new ((v, s), Lam n t')) ((n \<leftrightarrow> new ((v, s), Lam n t')) \<bullet> t')" in subst)
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   708
        defer
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   709
        apply (subst match_Lam_simps)
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   710
        defer
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   711
        apply (simp only: option.cases)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   712
        apply (simp only: prod.cases)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   713
        apply (subst (2) subst.simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   714
        apply (subst match_Var_simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   715
        apply (simp only: option.cases)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   716
        apply (subst match_App_simps)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   717
        apply (simp only: option.cases)
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   718
        apply (rule_tac t="Lam (p \<bullet> n) (p \<bullet> t')" and s="Lam (new ((p \<bullet> v, p \<bullet> s), Lam (p \<bullet> n) (p \<bullet> t'))) (((p \<bullet> n) \<leftrightarrow> new ((p \<bullet> v, p \<bullet> s), Lam (p \<bullet> n) (p \<bullet> t'))) \<bullet> t')" in subst)
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   719
        defer
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   720
        apply (subst match_Lam_simps)
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   721
        defer
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   722
        apply (simp only: option.cases)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   723
        apply (simp only: prod.cases)
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   724
        apply (simp only: lam.perm)
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   725
        thm 1(1)
2165
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   726
        sorry
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   727
    qed
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   728
  qed
e838f7d90f81 Function experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   729
2172
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   730
lemma subst_proper_eqs:
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   731
  "subst y s (Var x) = (if x = y then s else (Var x))"
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   732
  "subst y s (App l r) = App (subst y s l) (subst y s r)"
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   733
  "atom x \<sharp> (t, s) \<Longrightarrow> subst y s (Lam x t) = Lam x (subst y s t)"
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   734
  apply (subst subst.simps)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   735
  apply (simp only: match_Var_simps)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   736
  apply (simp only: option.simps)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   737
  apply (subst subst.simps)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   738
  apply (simp only: match_App_simps)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   739
  apply (simp only: option.simps)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   740
  apply (simp only: prod.simps)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   741
  apply (simp only: match_Var_simps)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   742
  apply (simp only: option.simps)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   743
  apply (subst subst.simps)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   744
  apply (simp only: match_Var_simps)
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   745
  apply (simp only: option.simps)
2172
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   746
  apply (simp only: match_App_simps)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   747
  apply (simp only: option.simps)
2173
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   748
  apply (rule_tac t="Lam x t" and s="Lam (new ((y, s), Lam x t)) ((x \<leftrightarrow> new ((y, s), Lam x t)) \<bullet> t)" in subst)
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   749
  defer
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   750
  apply (subst match_Lam_simps)
477293d841e8 Match_Lam defined on Quotient Level.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2172
diff changeset
   751
  defer
2172
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   752
  apply (simp only: option.simps)
fd5eec72c3f5 More on Function-defined subst.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2170
diff changeset
   753
  apply (simp only: prod.simps)
2157
a1d27083e688 subst experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   754
  sorry
a1d27083e688 subst experiments
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 2120
diff changeset
   755
1594
892fcdb96c96 Move LamEx out of Test.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   756
end
892fcdb96c96 Move LamEx out of Test.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   757
892fcdb96c96 Move LamEx out of Test.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   758
892fcdb96c96 Move LamEx out of Test.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   759