1797
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1 theory Lambda
2165
+ − 2 imports "../NewParser" Quotient_Option
1594
+ − 3 begin
+ − 4
+ − 5 atom_decl name
2311
+ − 6 declare [[STEPS = 9]]
1594
+ − 7
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 8 nominal_datatype lam =
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 9 Var "name"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 10 | App "lam" "lam"
2311
+ − 11 | Lam x::"name" l::"lam" bind x in l
1594
+ − 12
1845
b7423c6b5564
deleted offending [eqvt]-attribute in Abs; Lambda works again, but there is now a problem in CoreHaskell
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 13 thm lam.fv
b7423c6b5564
deleted offending [eqvt]-attribute in Abs; Lambda works again, but there is now a problem in CoreHaskell
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 14 thm lam.supp
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 15 lemmas supp_fn' = lam.fv[simplified lam.supp]
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 16
1805
+ − 17 declare lam.perm[eqvt]
+ − 18
2082
0854af516f14
cleaned up a bit the examples; added equivariance to all examples
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 19
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 20 section {* Strong Induction Principles*}
1594
+ − 21
2041
+ − 22 (*
1797
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 23 Old 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>
diff
changeset
+ − 24 principles by chosing a fresh name.
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 25 *)
1594
+ − 26 lemma
+ − 27 fixes c::"'a::fs"
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 28 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
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 29 and a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 30 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
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 31 shows "P c lam"
1594
+ − 32 proof -
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 33 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
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 34 apply(induct lam arbitrary: c rule: lam.induct)
1805
+ − 35 apply(perm_simp)
1594
+ − 36 apply(rule a1)
1805
+ − 37 apply(perm_simp)
1594
+ − 38 apply(rule a2)
1797
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 39 apply(assumption)
1594
+ − 40 apply(assumption)
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 41 apply(subgoal_tac "\<exists>new::name. (atom new) \<sharp> (c, Lam (p \<bullet> name) (p \<bullet> lam))")
1594
+ − 42 defer
+ − 43 apply(simp add: fresh_def)
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 44 apply(rule_tac X="supp (c, Lam (p \<bullet> name) (p \<bullet> lam))" in obtain_at_base)
1594
+ − 45 apply(simp add: supp_Pair finite_supp)
+ − 46 apply(blast)
+ − 47 apply(erule exE)
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 48 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
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 49 s="(((p \<bullet> name) \<leftrightarrow> new) + p) \<bullet> Lam name lam" in subst)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 50 apply(simp del: lam.perm)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 51 apply(subst lam.perm)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 52 apply(subst (2) lam.perm)
1594
+ − 53 apply(rule flip_fresh_fresh)
+ − 54 apply(simp add: fresh_def)
+ − 55 apply(simp only: supp_fn')
+ − 56 apply(simp)
+ − 57 apply(simp add: fresh_Pair)
+ − 58 apply(simp)
+ − 59 apply(rule a3)
+ − 60 apply(simp add: fresh_Pair)
+ − 61 apply(drule_tac x="((p \<bullet> name) \<leftrightarrow> new) + p" in meta_spec)
+ − 62 apply(simp)
+ − 63 done
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 64 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
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 65 then show "P c lam" by simp
1594
+ − 66 qed
+ − 67
1797
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 68 (*
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 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>
diff
changeset
+ − 70 principles by using a appropriate permutation.
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 71 *)
1594
+ − 72 lemma
+ − 73 fixes c::"'a::fs"
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 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
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 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)"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 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
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 77 shows "P c lam"
1594
+ − 78 proof -
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 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
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 80 apply(induct lam arbitrary: c rule: lam.induct)
1805
+ − 81 apply(perm_simp)
1594
+ − 82 apply(rule a1)
1805
+ − 83 apply(perm_simp)
1594
+ − 84 apply(rule a2)
+ − 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>
diff
changeset
+ − 86 apply(assumption)
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 87 apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom name}) \<sharp>* c \<and> supp (p \<bullet> Lam name lam) \<sharp>* q")
1594
+ − 88 apply(erule exE)
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 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
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 90 s="q \<bullet> p \<bullet> Lam name lam" in subst)
1594
+ − 91 defer
1805
+ − 92 apply(simp)
1594
+ − 93 apply(rule a3)
+ − 94 apply(simp add: eqvts fresh_star_def)
+ − 95 apply(drule_tac x="q + p" in meta_spec)
+ − 96 apply(simp)
1797
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 97 apply(rule at_set_avoiding2)
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 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>
diff
changeset
+ − 99 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>
diff
changeset
+ − 100 apply(simp add: finite_supp)
1805
+ − 101 apply(perm_simp)
1797
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 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
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 103 apply(rule supp_perm_eq)
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 104 apply(simp)
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 105 done
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 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
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 107 then show "P c lam" by simp
1594
+ − 108 qed
+ − 109
1805
+ − 110 section {* Typing *}
+ − 111
+ − 112 nominal_datatype ty =
+ − 113 TVar string
1810
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 114 | TFun ty ty
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 115
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 116 notation
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 117 TFun ("_ \<rightarrow> _")
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 118
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 119 declare ty.perm[eqvt]
1805
+ − 120
+ − 121 inductive
+ − 122 valid :: "(name \<times> ty) list \<Rightarrow> bool"
+ − 123 where
+ − 124 "valid []"
+ − 125 | "\<lbrakk>atom x \<sharp> Gamma; valid Gamma\<rbrakk> \<Longrightarrow> valid ((x, T)#Gamma)"
+ − 126
1828
+ − 127 inductive
+ − 128 typing :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool" ("_ \<turnstile> _ : _" [60,60,60] 60)
+ − 129 where
+ − 130 t_Var[intro]: "\<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Var x : T"
1947
+ − 131 | t_App[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : T2"
1828
+ − 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"
+ − 133
1831
+ − 134 equivariance valid
+ − 135 equivariance typing
1816
+ − 136
1831
+ − 137 thm valid.eqvt
+ − 138 thm typing.eqvt
1811
ae176476b525
implemented in thmdecls the case where eqvt-lemmas are of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 139 thm eqvts
ae176476b525
implemented in thmdecls the case where eqvt-lemmas are of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 140 thm eqvts_raw
ae176476b525
implemented in thmdecls the case where eqvt-lemmas are of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 141
1949
+ − 142 thm typing.induct[of "\<Gamma>" "t" "T", no_vars]
+ − 143
+ − 144 lemma
+ − 145 fixes c::"'a::fs"
+ − 146 assumes a: "\<Gamma> \<turnstile> t : T"
+ − 147 and a1: "\<And>\<Gamma> x T c. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P c \<Gamma> (Var x) T"
+ − 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>
+ − 149 \<Longrightarrow> P c \<Gamma> (App t1 t2) T2"
+ − 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>
+ − 151 \<Longrightarrow> P c \<Gamma> (Lam x t) T1 \<rightarrow> T2"
+ − 152 shows "P c \<Gamma> t T"
+ − 153 proof -
+ − 154 from a have "\<And>p c. P c (p \<bullet> \<Gamma>) (p \<bullet> t) (p \<bullet> T)"
+ − 155 proof (induct)
+ − 156 case (t_Var \<Gamma> x T p c)
+ − 157 then show ?case
+ − 158 apply -
+ − 159 apply(perm_strict_simp)
+ − 160 apply(rule a1)
1954
+ − 161 apply(drule_tac p="p" in permute_boolI)
1949
+ − 162 apply(perm_strict_simp add: permute_minus_cancel)
+ − 163 apply(assumption)
1954
+ − 164 apply(rotate_tac 1)
+ − 165 apply(drule_tac p="p" in permute_boolI)
1949
+ − 166 apply(perm_strict_simp add: permute_minus_cancel)
+ − 167 apply(assumption)
+ − 168 done
+ − 169 next
+ − 170 case (t_App \<Gamma> t1 T1 T2 t2 p c)
+ − 171 then show ?case
+ − 172 apply -
+ − 173 apply(perm_strict_simp)
1954
+ − 174 apply(rule a2)
+ − 175 apply(drule_tac p="p" in permute_boolI)
1949
+ − 176 apply(perm_strict_simp add: permute_minus_cancel)
+ − 177 apply(assumption)
+ − 178 apply(assumption)
1954
+ − 179 apply(rotate_tac 2)
+ − 180 apply(drule_tac p="p" in permute_boolI)
1949
+ − 181 apply(perm_strict_simp add: permute_minus_cancel)
+ − 182 apply(assumption)
+ − 183 apply(assumption)
+ − 184 done
+ − 185 next
+ − 186 case (t_Lam x \<Gamma> T1 t T2 p c)
+ − 187 then show ?case
+ − 188 apply -
+ − 189 apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom x}) \<sharp>* c \<and>
+ − 190 supp (p \<bullet> \<Gamma>, p \<bullet> Lam x t, p \<bullet> (T1 \<rightarrow> T2)) \<sharp>* q")
+ − 191 apply(erule exE)
1954
+ − 192 apply(rule_tac t="p \<bullet> \<Gamma>" and s="(q + p) \<bullet> \<Gamma>" in subst)
+ − 193 apply(simp only: permute_plus)
1949
+ − 194 apply(rule supp_perm_eq)
+ − 195 apply(simp add: supp_Pair fresh_star_union)
1954
+ − 196 apply(rule_tac t="p \<bullet> Lam x t" and s="(q + p) \<bullet> Lam x t" in subst)
+ − 197 apply(simp only: permute_plus)
1949
+ − 198 apply(rule supp_perm_eq)
+ − 199 apply(simp add: supp_Pair fresh_star_union)
1954
+ − 200 apply(rule_tac t="p \<bullet> (T1 \<rightarrow> T2)" and s="(q + p) \<bullet> (T1 \<rightarrow> T2)" in subst)
+ − 201 apply(simp only: permute_plus)
+ − 202 apply(rule supp_perm_eq)
+ − 203 apply(simp add: supp_Pair fresh_star_union)
+ − 204 apply(simp (no_asm) only: eqvts)
1949
+ − 205 apply(rule a3)
1954
+ − 206 apply(simp only: eqvts permute_plus)
1949
+ − 207 apply(simp add: fresh_star_def)
1954
+ − 208 apply(drule_tac p="q + p" in permute_boolI)
1949
+ − 209 apply(perm_strict_simp add: permute_minus_cancel)
+ − 210 apply(assumption)
1954
+ − 211 apply(rotate_tac 1)
+ − 212 apply(drule_tac p="q + p" in permute_boolI)
1949
+ − 213 apply(perm_strict_simp add: permute_minus_cancel)
+ − 214 apply(assumption)
+ − 215 apply(drule_tac x="d" in meta_spec)
+ − 216 apply(drule_tac x="q + p" in meta_spec)
1954
+ − 217 apply(perm_strict_simp add: permute_minus_cancel)
+ − 218 apply(assumption)
1949
+ − 219 apply(rule at_set_avoiding2)
+ − 220 apply(simp add: finite_supp)
+ − 221 apply(simp add: finite_supp)
+ − 222 apply(simp add: finite_supp)
+ − 223 apply(rule_tac p="-p" in permute_boolE)
+ − 224 apply(perm_strict_simp add: permute_minus_cancel)
+ − 225 (* supplied by the user *)
+ − 226 apply(simp add: fresh_star_prod)
+ − 227 apply(simp add: fresh_star_def)
1954
+ − 228 sorry
+ − 229 qed
+ − 230 then have "P c (0 \<bullet> \<Gamma>) (0 \<bullet> t) (0 \<bullet> T)" .
+ − 231 then show "P c \<Gamma> t T" by simp
+ − 232 qed
1949
+ − 233
+ − 234
+ − 235
+ − 236
+ − 237
+ − 238
1833
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 239
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 240 inductive
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 241 tt :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 242 for r :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 243 where
1835
+ − 244 aa: "tt r a a"
+ − 245 | bb: "tt r a b ==> tt r a c"
1833
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 246
1835
+ − 247 (* PROBLEM: derived eqvt-theorem does not conform with [eqvt]
1833
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 248 equivariance tt
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 249 *)
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 250
1810
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 251
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 252 inductive
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 253 alpha_lam_raw'
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 254 where
1861
+ − 255 a1: "name = namea \<Longrightarrow> alpha_lam_raw' (Var_raw name) (Var_raw namea)"
+ − 256 | a2: "\<lbrakk>alpha_lam_raw' lam_raw1 lam_raw1a; alpha_lam_raw' lam_raw2 lam_raw2a\<rbrakk> \<Longrightarrow>
1810
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 257 alpha_lam_raw' (App_raw lam_raw1 lam_raw2) (App_raw lam_raw1a lam_raw2a)"
1865
+ − 258 | a3: "\<exists>pi. ({atom name}, lam_raw) \<approx>gen alpha_lam_raw' fv_lam_raw pi ({atom namea}, lam_rawa) \<Longrightarrow>
1810
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 259 alpha_lam_raw' (Lam_raw name lam_raw) (Lam_raw namea lam_rawa)"
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 260
1866
+ − 261 equivariance alpha_lam_raw'
1947
+ − 262
1861
+ − 263 thm eqvts_raw
+ − 264
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 265 section {* size function *}
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 266
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 267 lemma size_eqvt_raw:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 268 fixes t::"lam_raw"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 269 shows "size (pi \<bullet> t) = size t"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 270 apply (induct rule: lam_raw.inducts)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 271 apply simp_all
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 272 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>
diff
changeset
+ − 273
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 274 instantiation lam :: size
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 275 begin
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 276
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 277 quotient_definition
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 278 "size_lam :: lam \<Rightarrow> nat"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 279 is
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 280 "size :: lam_raw \<Rightarrow> nat"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 281
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 282 lemma size_rsp:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 283 "alpha_lam_raw x y \<Longrightarrow> size x = size y"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 284 apply (induct rule: alpha_lam_raw.inducts)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 285 apply (simp_all only: lam_raw.size)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 286 apply (simp_all only: alphas)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 287 apply clarify
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 288 apply (simp_all only: size_eqvt_raw)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 289 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>
diff
changeset
+ − 290
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 291 lemma [quot_respect]:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 292 "(alpha_lam_raw ===> op =) size size"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 293 by (simp_all add: size_rsp)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 294
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 295 lemma [quot_preserve]:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 296 "(rep_lam ---> id) size = size"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 297 by (simp_all add: size_lam_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>
diff
changeset
+ − 298
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 299 instance
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 300 by default
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 301
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 302 end
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 303
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 304 lemmas size_lam[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>
diff
changeset
+ − 305 lam_raw.size(4)[quot_lifted]
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 306 lam_raw.size(5)[quot_lifted]
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 307 lam_raw.size(6)[quot_lifted]
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 308
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 309 (* is this needed? *)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 310 lemma [measure_function]:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 311 "is_measure (size::lam\<Rightarrow>nat)"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 312 by (rule is_measure_trivial)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 313
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 314 section {* Matching *}
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 315
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 316 definition
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 317 MATCH :: "('c::pt \<Rightarrow> (bool * 'a::pt * 'b::pt)) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 318 where
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 319 "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"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 320
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 321 (*
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 322 lemma MATCH_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>
diff
changeset
+ − 323 shows "p \<bullet> (MATCH M d x) = MATCH (p \<bullet> M) (p \<bullet> d) (p \<bullet> 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>
diff
changeset
+ − 324 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>
diff
changeset
+ − 325 apply(perm_simp the_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>
diff
changeset
+ − 326 apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 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>
diff
changeset
+ − 327 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>
diff
changeset
+ − 328 thm eqvts_raw
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 329 apply(subst if_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>
diff
changeset
+ − 330 apply(subst 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>
diff
changeset
+ − 331 apply(subst permute_fun_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>
diff
changeset
+ − 332 apply(subst ex_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>
diff
changeset
+ − 333 apply(subst permute_fun_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>
diff
changeset
+ − 334 apply(subst eq_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>
diff
changeset
+ − 335 apply(subst permute_fun_app_eq[where f="M"])
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 336 apply(simp only: permute_minus_cancel)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 337 apply(subst permute_prod.simps)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 338 apply(subst permute_prod.simps)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 339 apply(simp only: permute_minus_cancel)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 340 apply(simp only: permute_bool_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>
diff
changeset
+ − 341 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>
diff
changeset
+ − 342 apply(subst 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>
diff
changeset
+ − 343 apply(subst permute_fun_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>
diff
changeset
+ − 344 apply(subst ex_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>
diff
changeset
+ − 345 apply(subst permute_fun_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>
diff
changeset
+ − 346 apply(subst eq_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>
diff
changeset
+ − 347
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 348 apply(simp only: eqvts)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 349 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>
diff
changeset
+ − 350 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))")
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 351 apply(drule sym)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 352 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>
diff
changeset
+ − 353 apply(rule impI)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 354 apply(simp add: perm_bool)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
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>
diff
changeset
+ − 421
1954
+ − 422 ML {*
+ − 423 fun mk_avoids ctxt params name set =
+ − 424 let
+ − 425 val (_, ctxt') = ProofContext.add_fixes
+ − 426 (map (fn (s, T) => (Binding.name s, SOME T, NoSyn)) params) ctxt;
+ − 427 fun mk s =
+ − 428 let
+ − 429 val t = Syntax.read_term ctxt' s;
+ − 430 val t' = list_abs_free (params, t) |>
+ − 431 funpow (length params) (fn Abs (_, _, t) => t)
+ − 432 in (t', HOLogic.dest_setT (fastype_of t)) end
+ − 433 handle TERM _ =>
+ − 434 error ("Expression " ^ quote s ^ " to be avoided in case " ^
+ − 435 quote name ^ " is not a set type");
+ − 436 fun add_set p [] = [p]
+ − 437 | add_set (t, T) ((u, U) :: ps) =
+ − 438 if T = U then
+ − 439 let val S = HOLogic.mk_setT T
+ − 440 in (Const (@{const_name sup}, S --> S --> S) $ u $ t, T) :: ps
+ − 441 end
+ − 442 else (u, U) :: add_set (t, T) ps
+ − 443 in
+ − 444 (mk #> add_set) set
+ − 445 end;
+ − 446 *}
+ − 447
+ − 448
+ − 449 ML {*
+ − 450 writeln (commas (map (Syntax.string_of_term @{context} o fst)
+ − 451 (mk_avoids @{context} [] "t_Var" "{x}" [])))
+ − 452 *}
+ − 453
1947
+ − 454
1949
+ − 455 ML {*
+ − 456
+ − 457 fun prove_strong_ind (pred_name, avoids) ctxt =
+ − 458 Proof.theorem NONE (K I) [] ctxt
+ − 459
2169
+ − 460 local structure P = Parse and K = Keyword in
1949
+ − 461
+ − 462 val _ =
2169
+ − 463 Outer_Syntax.local_theory_to_proof "nominal_inductive"
1949
+ − 464 "proves strong induction theorem for inductive predicate involving nominal datatypes" K.thy_goal
+ − 465 (P.xname -- (Scan.optional (P.$$$ "avoids" |-- P.enum1 "|" (P.name --
+ − 466 (P.$$$ ":" |-- P.and_list1 P.term))) []) >> prove_strong_ind)
+ − 467
+ − 468 end;
+ − 469
+ − 470 *}
+ − 471
1950
+ − 472 (*
1949
+ − 473 nominal_inductive typing
1950
+ − 474 *)
1949
+ − 475
2157
+ − 476 (* Substitution *)
2159
+ − 477
2165
+ − 478 primrec match_Var_raw where
+ − 479 "match_Var_raw (Var_raw x) = Some x"
+ − 480 | "match_Var_raw (App_raw x y) = None"
+ − 481 | "match_Var_raw (Lam_raw n t) = None"
+ − 482
+ − 483 quotient_definition
+ − 484 "match_Var :: lam \<Rightarrow> name option"
+ − 485 is match_Var_raw
+ − 486
+ − 487 lemma [quot_respect]: "(alpha_lam_raw ===> op =) match_Var_raw match_Var_raw"
+ − 488 apply rule
+ − 489 apply (induct_tac a b rule: alpha_lam_raw.induct)
+ − 490 apply simp_all
+ − 491 done
+ − 492
+ − 493 lemmas match_Var_simps = match_Var_raw.simps[quot_lifted]
+ − 494
+ − 495 primrec match_App_raw where
+ − 496 "match_App_raw (Var_raw x) = None"
+ − 497 | "match_App_raw (App_raw x y) = Some (x, y)"
+ − 498 | "match_App_raw (Lam_raw n t) = None"
+ − 499
+ − 500 quotient_definition
+ − 501 "match_App :: lam \<Rightarrow> (lam \<times> lam) option"
+ − 502 is match_App_raw
+ − 503
+ − 504 lemma [quot_respect]:
+ − 505 "(alpha_lam_raw ===> option_rel (prod_rel alpha_lam_raw alpha_lam_raw)) match_App_raw match_App_raw"
+ − 506 apply (intro fun_relI)
+ − 507 apply (induct_tac a b rule: alpha_lam_raw.induct)
+ − 508 apply simp_all
+ − 509 done
+ − 510
+ − 511 lemmas match_App_simps = match_App_raw.simps[quot_lifted]
+ − 512
2173
+ − 513 definition new where
+ − 514 "new (s :: 'a :: fs) = (THE x. \<forall>a \<in> supp s. atom x \<noteq> a)"
+ − 515
+ − 516 definition
+ − 517 "match_Lam (S :: 'a :: fs) t = (if (\<exists>n s. (t = Lam n s)) then
+ − 518 (let z = new (S, t) in Some (z, THE s. t = Lam z s)) else None)"
+ − 519
+ − 520 lemma lam_half_inj: "(Lam z s = Lam z sa) = (s = sa)"
+ − 521 apply auto
+ − 522 apply (simp only: lam.eq_iff alphas)
+ − 523 apply clarify
+ − 524 apply (simp add: eqvts)
+ − 525 sorry
+ − 526
+ − 527 lemma match_Lam_simps:
+ − 528 "match_Lam S (Var n) = None"
+ − 529 "match_Lam S (App l r) = None"
+ − 530 "z = new (S, (Lam z s)) \<Longrightarrow> match_Lam S (Lam z s) = Some (z, s)"
+ − 531 apply (simp_all add: match_Lam_def)
+ − 532 apply (simp add: lam_half_inj)
+ − 533 apply auto
+ − 534 done
+ − 535
+ − 536 (*
+ − 537 lemma match_Lam_simps2:
+ − 538 "atom n \<sharp> ((S :: 'a :: fs), Lam n s) \<Longrightarrow> match_Lam S (Lam n s) = Some (n, s)"
+ − 539 apply (rule_tac t="Lam n s"
+ − 540 and s="Lam (new (S, (Lam n s))) ((n \<leftrightarrow> (new (S, (Lam n s)))) \<bullet> s)" in subst)
+ − 541 defer
+ − 542 apply (subst match_Lam_simps(3))
+ − 543 defer
+ − 544 apply simp
+ − 545 *)
+ − 546
+ − 547 (*primrec match_Lam_raw where
2165
+ − 548 "match_Lam_raw (S :: atom set) (Var_raw x) = None"
+ − 549 | "match_Lam_raw S (App_raw x y) = None"
+ − 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))"
+ − 551
+ − 552 quotient_definition
+ − 553 "match_Lam :: (atom set) \<Rightarrow> lam \<Rightarrow> (name \<times> lam) option"
+ − 554 is match_Lam_raw
+ − 555
2172
+ − 556 lemma swap_fresh:
+ − 557 assumes a: "fv_lam_raw t \<sharp>* p"
+ − 558 shows "alpha_lam_raw (p \<bullet> t) t"
+ − 559 using a apply (induct t)
+ − 560 apply (simp add: supp_at_base fresh_star_def)
+ − 561 apply (rule alpha_lam_raw.intros)
+ − 562 apply (metis Rep_name_inverse atom_eqvt atom_name_def fresh_perm)
+ − 563 apply (simp)
+ − 564 apply (simp only: fresh_star_union)
+ − 565 apply clarify
+ − 566 apply (rule alpha_lam_raw.intros)
+ − 567 apply simp
+ − 568 apply simp
+ − 569 apply simp
+ − 570 apply (rule alpha_lam_raw.intros)
+ − 571 sorry
+ − 572
2165
+ − 573 lemma [quot_respect]:
+ − 574 "(op = ===> alpha_lam_raw ===> option_rel (prod_rel op = alpha_lam_raw)) match_Lam_raw match_Lam_raw"
+ − 575 proof (intro fun_relI, clarify)
+ − 576 fix S t s
+ − 577 assume a: "alpha_lam_raw t s"
+ − 578 show "option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S t) (match_Lam_raw S s)"
+ − 579 using a proof (induct t s rule: alpha_lam_raw.induct)
+ − 580 case goal1 show ?case by simp
+ − 581 next
+ − 582 case goal2 show ?case by simp
+ − 583 next
+ − 584 case (goal3 x t y s)
+ − 585 then obtain p where "({atom x}, t) \<approx>gen (\<lambda>x1 x2. alpha_lam_raw x1 x2 \<and>
+ − 586 option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S x1)
+ − 587 (match_Lam_raw S x2)) fv_lam_raw p ({atom y}, s)" ..
+ − 588 then have
+ − 589 c: "fv_lam_raw t - {atom x} = fv_lam_raw s - {atom y}" and
+ − 590 d: "(fv_lam_raw t - {atom x}) \<sharp>* p" and
+ − 591 e: "alpha_lam_raw (p \<bullet> t) s" and
+ − 592 f: "option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S (p \<bullet> t)) (match_Lam_raw S s)" and
+ − 593 g: "p \<bullet> {atom x} = {atom y}" unfolding alphas(1) by - (elim conjE, assumption)+
+ − 594 let ?z = "new (S \<union> (fv_lam_raw t - {atom x}))"
+ − 595 have h: "?z = new (S \<union> (fv_lam_raw s - {atom y}))" using c by simp
+ − 596 show ?case
+ − 597 unfolding match_Lam_raw.simps Let_def option_rel.simps prod_rel.simps split_conv
+ − 598 proof
+ − 599 show "?z = new (S \<union> (fv_lam_raw s - {atom y}))" by (fact h)
+ − 600 next
+ − 601 have "atom y \<sharp> p" sorry
+ − 602 have "fv_lam_raw t \<sharp>* ((x \<leftrightarrow> y) \<bullet> p)" sorry
2172
+ − 603 then have "alpha_lam_raw (((x \<leftrightarrow> y) \<bullet> p) \<bullet> t) t" using swap_fresh by auto
+ − 604 then have "alpha_lam_raw (p \<bullet> t) ((x \<leftrightarrow> y) \<bullet> t)" sorry
2165
+ − 605 have "alpha_lam_raw t ((x \<leftrightarrow> y) \<bullet> s)" sorry
+ − 606 then have "alpha_lam_raw ((x \<leftrightarrow> ?z) \<bullet> t) ((y \<leftrightarrow> ?z) \<bullet> s)" using eqvts(15) sorry
+ − 607 then show "alpha_lam_raw ((x \<leftrightarrow> new (S \<union> (fv_lam_raw t - {atom x}))) \<bullet> t)
+ − 608 ((y \<leftrightarrow> new (S \<union> (fv_lam_raw s - {atom y}))) \<bullet> s)" unfolding h .
+ − 609 qed
+ − 610 qed
+ − 611 qed
+ − 612
+ − 613 lemmas match_Lam_simps = match_Lam_raw.simps[quot_lifted]
2173
+ − 614 *)
2165
+ − 615
+ − 616 lemma app_some: "match_App x = Some (a, b) \<Longrightarrow> x = App a b"
+ − 617 by (induct x rule: lam.induct) (simp_all add: match_App_simps)
+ − 618
+ − 619 lemma lam_some: "match_Lam S x = Some (z, s) \<Longrightarrow> x = Lam z s \<and> atom z \<sharp> S"
+ − 620 apply (induct x rule: lam.induct)
+ − 621 apply (simp_all add: match_Lam_simps)
2173
+ − 622 apply (thin_tac "match_Lam S lam = Some (z, s) \<Longrightarrow> lam = Lam z s \<and> atom z \<sharp> S")
+ − 623 apply (simp add: match_Lam_def)
+ − 624 apply (subgoal_tac "\<exists>n s. Lam name lam = Lam n s")
+ − 625 prefer 2
+ − 626 apply auto[1]
2165
+ − 627 apply (simp add: Let_def)
2173
+ − 628 apply (thin_tac "\<exists>n s. Lam name lam = Lam n s")
+ − 629 apply clarify
2165
+ − 630 apply (rule conjI)
2173
+ − 631 apply (rule_tac t="THE s. Lam name lam = Lam (new (S, Lam name lam)) s" and
+ − 632 s="(name \<leftrightarrow> (new (S, Lam name lam))) \<bullet> lam" in subst)
+ − 633 defer
2165
+ − 634 apply (simp add: lam.eq_iff)
2173
+ − 635 apply (rule_tac x="(name \<leftrightarrow> (new (S, Lam name lam)))" in exI)
2165
+ − 636 apply (simp add: alphas)
+ − 637 apply (simp add: eqvts)
2173
+ − 638 apply (rule conjI)
2172
+ − 639 sorry
2165
+ − 640
+ − 641 function subst where
+ − 642 "subst v s t = (
+ − 643 case match_Var t of Some n \<Rightarrow> if n = v then s else Var n | None \<Rightarrow>
+ − 644 case match_App t of Some (l, r) \<Rightarrow> App (subst v s l) (subst v s r) | None \<Rightarrow>
2173
+ − 645 case match_Lam (v,s) t of Some (n, t) \<Rightarrow> Lam n (subst v s t) | None \<Rightarrow> undefined)"
2165
+ − 646 by pat_completeness auto
+ − 647
+ − 648 termination apply (relation "measure (\<lambda>(_, _, t). size t)")
2172
+ − 649 apply auto[1]
+ − 650 apply (case_tac a) apply simp
+ − 651 apply (frule lam_some) apply simp
+ − 652 apply (case_tac a) apply simp
+ − 653 apply (frule app_some) apply simp
+ − 654 apply (case_tac a) apply simp
+ − 655 apply (frule app_some) apply simp
2165
+ − 656 done
+ − 657
+ − 658 lemmas lam_exhaust = lam_raw.exhaust[quot_lifted]
+ − 659
+ − 660 lemma subst_eqvt:
+ − 661 "p \<bullet> (subst v s t) = subst (p \<bullet> v) (p \<bullet> s) (p \<bullet> t)"
+ − 662 proof (induct v s t rule: subst.induct)
+ − 663 case (1 v s t)
+ − 664 show ?case proof (cases t rule: lam_exhaust)
+ − 665 fix n
+ − 666 assume "t = Var n"
+ − 667 then show ?thesis by (simp add: match_Var_simps)
+ − 668 next
+ − 669 fix l r
+ − 670 assume "t = App l r"
+ − 671 then show ?thesis
+ − 672 apply (simp only:)
+ − 673 apply (subst subst.simps)
+ − 674 apply (subst match_Var_simps)
+ − 675 apply (simp only: option.cases)
+ − 676 apply (subst match_App_simps)
+ − 677 apply (simp only: option.cases)
+ − 678 apply (simp only: prod.cases)
+ − 679 apply (simp only: lam.perm)
+ − 680 apply (subst (3) subst.simps)
+ − 681 apply (subst match_Var_simps)
2173
+ − 682 apply (simp only: option.cases)
2165
+ − 683 apply (subst match_App_simps)
+ − 684 apply (simp only: option.cases)
+ − 685 apply (simp only: prod.cases)
+ − 686 apply (subst 1(2)[of "(l, r)" "l" "r"])
+ − 687 apply (simp add: match_Var_simps)
+ − 688 apply (simp add: match_App_simps)
+ − 689 apply (rule refl)
+ − 690 apply (subst 1(3)[of "(l, r)" "l" "r"])
+ − 691 apply (simp add: match_Var_simps)
+ − 692 apply (simp add: match_App_simps)
+ − 693 apply (rule refl)
+ − 694 apply (rule refl)
+ − 695 done
+ − 696 next
+ − 697 fix n t'
+ − 698 assume "t = Lam n t'"
+ − 699 then show ?thesis
+ − 700 apply (simp only: )
+ − 701 apply (simp only: lam.perm)
+ − 702 apply (subst subst.simps)
+ − 703 apply (subst match_Var_simps)
+ − 704 apply (simp only: option.cases)
+ − 705 apply (subst match_App_simps)
+ − 706 apply (simp only: option.cases)
2173
+ − 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)
+ − 708 defer
2165
+ − 709 apply (subst match_Lam_simps)
2173
+ − 710 defer
2165
+ − 711 apply (simp only: option.cases)
+ − 712 apply (simp only: prod.cases)
+ − 713 apply (subst (2) subst.simps)
+ − 714 apply (subst match_Var_simps)
+ − 715 apply (simp only: option.cases)
+ − 716 apply (subst match_App_simps)
+ − 717 apply (simp only: option.cases)
2173
+ − 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)
+ − 719 defer
2165
+ − 720 apply (subst match_Lam_simps)
2173
+ − 721 defer
2165
+ − 722 apply (simp only: option.cases)
+ − 723 apply (simp only: prod.cases)
+ − 724 apply (simp only: lam.perm)
2173
+ − 725 thm 1(1)
2165
+ − 726 sorry
+ − 727 qed
+ − 728 qed
+ − 729
2172
+ − 730 lemma subst_proper_eqs:
+ − 731 "subst y s (Var x) = (if x = y then s else (Var x))"
+ − 732 "subst y s (App l r) = App (subst y s l) (subst y s r)"
+ − 733 "atom x \<sharp> (t, s) \<Longrightarrow> subst y s (Lam x t) = Lam x (subst y s t)"
+ − 734 apply (subst subst.simps)
+ − 735 apply (simp only: match_Var_simps)
+ − 736 apply (simp only: option.simps)
+ − 737 apply (subst subst.simps)
+ − 738 apply (simp only: match_App_simps)
+ − 739 apply (simp only: option.simps)
+ − 740 apply (simp only: prod.simps)
+ − 741 apply (simp only: match_Var_simps)
+ − 742 apply (simp only: option.simps)
+ − 743 apply (subst subst.simps)
+ − 744 apply (simp only: match_Var_simps)
2173
+ − 745 apply (simp only: option.simps)
2172
+ − 746 apply (simp only: match_App_simps)
+ − 747 apply (simp only: option.simps)
2173
+ − 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)
+ − 749 defer
+ − 750 apply (subst match_Lam_simps)
+ − 751 defer
2172
+ − 752 apply (simp only: option.simps)
+ − 753 apply (simp only: prod.simps)
2157
+ − 754 sorry
+ − 755
1594
+ − 756 end
+ − 757
+ − 758
+ − 759