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
2454
+ − 2 imports "../Nominal2"
1594
+ − 3 begin
+ − 4
+ − 5 atom_decl name
2440
0a36825b16c1
"isabelle make test" makes all major examples....they work up to supp theorems (excluding)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 6 declare [[STEPS = 100]]
1594
+ − 7
2436
+ − 8 nominal_datatype lam =
1800
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diff
changeset
+ − 9 Var "name"
2436
+ − 10 | App "lam" "lam"
+ − 11 | Lam x::"name" l::"lam" bind x in l
2431
331873ebc5cd
can now deal with type variables in nominal datatype definitions
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diff
changeset
+ − 12
2436
+ − 13 thm lam.distinct
+ − 14 thm lam.induct
+ − 15 thm lam.exhaust
+ − 16 thm lam.fv_defs
+ − 17 thm lam.bn_defs
+ − 18 thm lam.perm_simps
+ − 19 thm lam.eq_iff
2442
+ − 20 thm lam.eq_iff[folded alphas]
2436
+ − 21 thm lam.fv_bn_eqvt
+ − 22 thm lam.size_eqvt
2431
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Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 23
2424
621ebd8b13c4
changed parser so that the binding mode is indicated as "bind (list)", "bind (set)" or "bind (res)"; if only "bind" is given, then bind (list) is assumed as default
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 24
2497
+ − 25 section {* Strong Exhausts Lemma and Strong Induction Lemma via Induct_schema *}
+ − 26
+ − 27 lemma strong_exhaust:
+ − 28 fixes c::"'a::fs"
+ − 29 assumes "\<And>name ca. \<lbrakk>c = ca; y = Var name\<rbrakk> \<Longrightarrow> P"
+ − 30 and "\<And>lam1 lam2 ca. \<lbrakk>c = ca; y = App lam1 lam2\<rbrakk> \<Longrightarrow> P"
+ − 31 and "\<And>name lam ca. \<lbrakk>c = ca; atom name \<sharp> ca; y = Lam name lam\<rbrakk> \<Longrightarrow> P"
+ − 32 shows "P"
+ − 33 apply(rule_tac y="y" in lam.exhaust)
+ − 34 apply(rule assms(1))
+ − 35 apply(auto)[2]
+ − 36 apply(rule assms(2))
+ − 37 apply(auto)[2]
+ − 38 apply(subgoal_tac "\<exists>q. (q \<bullet> (atom name)) \<sharp> c \<and> supp (Lam name lam) \<sharp>* q")
+ − 39 apply(erule exE)
+ − 40 apply(erule conjE)
+ − 41 apply(rule assms(3))
+ − 42 apply(rule refl)
+ − 43 apply(simp add: atom_eqvt)
+ − 44 apply(clarify)
+ − 45 apply(drule supp_perm_eq[symmetric])
+ − 46 apply(simp)
+ − 47 apply(rule at_set_avoiding2_atom)
+ − 48 apply(simp add: finite_supp)
+ − 49 apply(simp add: finite_supp)
+ − 50 apply(simp add: lam.fresh)
+ − 51 done
+ − 52
+ − 53
+ − 54 lemma strong_induct:
+ − 55 fixes c::"'a::fs"
+ − 56 assumes a1: "\<And>name c. P c (Var name)"
+ − 57 and a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"
+ − 58 and a3: "\<And>name lam c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lam\<rbrakk> \<Longrightarrow> P c (Lam name lam)"
+ − 59 shows "P c lam"
+ − 60 using assms
+ − 61 apply(induction_schema)
+ − 62 apply(rule_tac y="lam" in strong_exhaust)
+ − 63 apply(blast)
+ − 64 apply(blast)
+ − 65 apply(blast)
+ − 66 apply(relation "measure (\<lambda>(x,y). size y)")
+ − 67 apply(auto)[1]
+ − 68 apply(simp add: lam.size)
+ − 69 apply(simp add: lam.size)
+ − 70 apply(simp add: lam.size)
+ − 71 done
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0854af516f14
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Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 72
1800
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diff
changeset
+ − 73 section {* Strong Induction Principles*}
1594
+ − 74
2041
+ − 75 (*
1797
fddb470720f1
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Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 76 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
+ − 77 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
+ − 78 *)
2436
+ − 79 (*
1594
+ − 80 lemma
+ − 81 fixes c::"'a::fs"
1800
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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
+ − 82 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
+ − 83 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
+ − 84 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
+ − 85 shows "P c lam"
1594
+ − 86 proof -
1800
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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 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
+ − 88 apply(induct lam arbitrary: c rule: lam.induct)
1805
+ − 89 apply(perm_simp)
1594
+ − 90 apply(rule a1)
1805
+ − 91 apply(perm_simp)
1594
+ − 92 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
+ − 93 apply(assumption)
1594
+ − 94 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
+ − 95 apply(subgoal_tac "\<exists>new::name. (atom new) \<sharp> (c, Lam (p \<bullet> name) (p \<bullet> lam))")
1594
+ − 96 defer
+ − 97 apply(simp add: fresh_def)
1800
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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
+ − 98 apply(rule_tac X="supp (c, Lam (p \<bullet> name) (p \<bullet> lam))" in obtain_at_base)
1594
+ − 99 apply(simp add: supp_Pair finite_supp)
+ − 100 apply(blast)
+ − 101 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
+ − 102 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
+ − 103 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
+ − 104 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
+ − 105 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
+ − 106 apply(subst (2) lam.perm)
1594
+ − 107 apply(rule flip_fresh_fresh)
+ − 108 apply(simp add: fresh_def)
+ − 109 apply(simp only: supp_fn')
+ − 110 apply(simp)
+ − 111 apply(simp add: fresh_Pair)
+ − 112 apply(simp)
+ − 113 apply(rule a3)
+ − 114 apply(simp add: fresh_Pair)
+ − 115 apply(drule_tac x="((p \<bullet> name) \<leftrightarrow> new) + p" in meta_spec)
+ − 116 apply(simp)
+ − 117 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
+ − 118 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
+ − 119 then show "P c lam" by simp
1594
+ − 120 qed
2436
+ − 121 *)
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
+ − 122 (*
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 123 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
+ − 124 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
+ − 125 *)
2436
+ − 126 (*
1594
+ − 127 lemma
+ − 128 fixes c::"'a::fs"
1800
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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
+ − 129 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
+ − 130 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
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Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 131 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
+ − 132 shows "P c lam"
1594
+ − 133 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
+ − 134 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
+ − 135 apply(induct lam arbitrary: c rule: lam.induct)
1805
+ − 136 apply(perm_simp)
1594
+ − 137 apply(rule a1)
1805
+ − 138 apply(perm_simp)
1594
+ − 139 apply(rule a2)
+ − 140 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
+ − 141 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
+ − 142 apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom name}) \<sharp>* c \<and> supp (p \<bullet> Lam name lam) \<sharp>* q")
1594
+ − 143 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
+ − 144 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
+ − 145 s="q \<bullet> p \<bullet> Lam name lam" in subst)
1594
+ − 146 defer
1805
+ − 147 apply(simp)
1594
+ − 148 apply(rule a3)
+ − 149 apply(simp add: eqvts fresh_star_def)
+ − 150 apply(drule_tac x="q + p" in meta_spec)
+ − 151 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
+ − 152 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
+ − 153 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
+ − 154 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
+ − 155 apply(simp add: finite_supp)
1805
+ − 156 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
+ − 157 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
+ − 158 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
+ − 159 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
+ − 160 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
+ − 161 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
+ − 162 then show "P c lam" by simp
1594
+ − 163 qed
2436
+ − 164 *)
1594
+ − 165
1805
+ − 166 section {* Typing *}
+ − 167
+ − 168 nominal_datatype ty =
+ − 169 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
+ − 170 | TFun ty ty
894930834ca8
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Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 171
894930834ca8
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Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 172 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
+ − 173 TFun ("_ \<rightarrow> _")
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fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 174
2436
+ − 175 (*
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
+ − 176 declare ty.perm[eqvt]
1805
+ − 177
+ − 178 inductive
+ − 179 valid :: "(name \<times> ty) list \<Rightarrow> bool"
+ − 180 where
+ − 181 "valid []"
+ − 182 | "\<lbrakk>atom x \<sharp> Gamma; valid Gamma\<rbrakk> \<Longrightarrow> valid ((x, T)#Gamma)"
+ − 183
1828
+ − 184 inductive
+ − 185 typing :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool" ("_ \<turnstile> _ : _" [60,60,60] 60)
+ − 186 where
+ − 187 t_Var[intro]: "\<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Var x : T"
1947
+ − 188 | t_App[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : T2"
1828
+ − 189 | t_Lam[intro]: "\<lbrakk>atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam x t : T1 \<rightarrow> T2"
+ − 190
1831
+ − 191 equivariance valid
+ − 192 equivariance typing
1816
+ − 193
1831
+ − 194 thm valid.eqvt
+ − 195 thm typing.eqvt
1811
ae176476b525
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Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 196 thm eqvts
ae176476b525
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Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 197 thm eqvts_raw
ae176476b525
implemented in thmdecls the case where eqvt-lemmas are of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 198
1949
+ − 199 thm typing.induct[of "\<Gamma>" "t" "T", no_vars]
+ − 200
+ − 201 lemma
+ − 202 fixes c::"'a::fs"
+ − 203 assumes a: "\<Gamma> \<turnstile> t : T"
+ − 204 and a1: "\<And>\<Gamma> x T c. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P c \<Gamma> (Var x) T"
+ − 205 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>
+ − 206 \<Longrightarrow> P c \<Gamma> (App t1 t2) T2"
+ − 207 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>
+ − 208 \<Longrightarrow> P c \<Gamma> (Lam x t) T1 \<rightarrow> T2"
+ − 209 shows "P c \<Gamma> t T"
+ − 210 proof -
+ − 211 from a have "\<And>p c. P c (p \<bullet> \<Gamma>) (p \<bullet> t) (p \<bullet> T)"
+ − 212 proof (induct)
+ − 213 case (t_Var \<Gamma> x T p c)
+ − 214 then show ?case
+ − 215 apply -
+ − 216 apply(perm_strict_simp)
+ − 217 apply(rule a1)
1954
+ − 218 apply(drule_tac p="p" in permute_boolI)
1949
+ − 219 apply(perm_strict_simp add: permute_minus_cancel)
+ − 220 apply(assumption)
1954
+ − 221 apply(rotate_tac 1)
+ − 222 apply(drule_tac p="p" in permute_boolI)
1949
+ − 223 apply(perm_strict_simp add: permute_minus_cancel)
+ − 224 apply(assumption)
+ − 225 done
+ − 226 next
+ − 227 case (t_App \<Gamma> t1 T1 T2 t2 p c)
+ − 228 then show ?case
+ − 229 apply -
+ − 230 apply(perm_strict_simp)
1954
+ − 231 apply(rule a2)
+ − 232 apply(drule_tac p="p" in permute_boolI)
1949
+ − 233 apply(perm_strict_simp add: permute_minus_cancel)
+ − 234 apply(assumption)
+ − 235 apply(assumption)
1954
+ − 236 apply(rotate_tac 2)
+ − 237 apply(drule_tac p="p" in permute_boolI)
1949
+ − 238 apply(perm_strict_simp add: permute_minus_cancel)
+ − 239 apply(assumption)
+ − 240 apply(assumption)
+ − 241 done
+ − 242 next
+ − 243 case (t_Lam x \<Gamma> T1 t T2 p c)
+ − 244 then show ?case
+ − 245 apply -
+ − 246 apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom x}) \<sharp>* c \<and>
+ − 247 supp (p \<bullet> \<Gamma>, p \<bullet> Lam x t, p \<bullet> (T1 \<rightarrow> T2)) \<sharp>* q")
+ − 248 apply(erule exE)
1954
+ − 249 apply(rule_tac t="p \<bullet> \<Gamma>" and s="(q + p) \<bullet> \<Gamma>" in subst)
+ − 250 apply(simp only: permute_plus)
1949
+ − 251 apply(rule supp_perm_eq)
+ − 252 apply(simp add: supp_Pair fresh_star_union)
1954
+ − 253 apply(rule_tac t="p \<bullet> Lam x t" and s="(q + p) \<bullet> Lam x t" in subst)
+ − 254 apply(simp only: permute_plus)
1949
+ − 255 apply(rule supp_perm_eq)
+ − 256 apply(simp add: supp_Pair fresh_star_union)
1954
+ − 257 apply(rule_tac t="p \<bullet> (T1 \<rightarrow> T2)" and s="(q + p) \<bullet> (T1 \<rightarrow> T2)" in subst)
+ − 258 apply(simp only: permute_plus)
+ − 259 apply(rule supp_perm_eq)
+ − 260 apply(simp add: supp_Pair fresh_star_union)
+ − 261 apply(simp (no_asm) only: eqvts)
1949
+ − 262 apply(rule a3)
1954
+ − 263 apply(simp only: eqvts permute_plus)
1949
+ − 264 apply(simp add: fresh_star_def)
1954
+ − 265 apply(drule_tac p="q + p" in permute_boolI)
1949
+ − 266 apply(perm_strict_simp add: permute_minus_cancel)
+ − 267 apply(assumption)
1954
+ − 268 apply(rotate_tac 1)
+ − 269 apply(drule_tac p="q + p" in permute_boolI)
1949
+ − 270 apply(perm_strict_simp add: permute_minus_cancel)
+ − 271 apply(assumption)
+ − 272 apply(drule_tac x="d" in meta_spec)
+ − 273 apply(drule_tac x="q + p" in meta_spec)
1954
+ − 274 apply(perm_strict_simp add: permute_minus_cancel)
+ − 275 apply(assumption)
1949
+ − 276 apply(rule at_set_avoiding2)
+ − 277 apply(simp add: finite_supp)
+ − 278 apply(simp add: finite_supp)
+ − 279 apply(simp add: finite_supp)
+ − 280 apply(rule_tac p="-p" in permute_boolE)
+ − 281 apply(perm_strict_simp add: permute_minus_cancel)
2436
+ − 282 --"supplied by the user"
1949
+ − 283 apply(simp add: fresh_star_prod)
+ − 284 apply(simp add: fresh_star_def)
1954
+ − 285 sorry
+ − 286 qed
+ − 287 then have "P c (0 \<bullet> \<Gamma>) (0 \<bullet> t) (0 \<bullet> T)" .
+ − 288 then show "P c \<Gamma> t T" by simp
+ − 289 qed
1949
+ − 290
1833
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 291 *)
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 292
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
+ − 293
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
+ − 294 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
+ − 295
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 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
+ − 297 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
+ − 298 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
+ − 299 "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
+ − 300
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 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
+ − 303 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
+ − 304 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
+ − 305 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
+ − 306 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
+ − 307 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
+ − 308 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
+ − 309 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
+ − 310 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
+ − 311 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
+ − 312 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
+ − 313 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
+ − 314 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
+ − 315 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
+ − 316 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
+ − 317 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
+ − 318 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
+ − 319 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
+ − 320 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
+ − 321 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
+ − 322 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
+ − 323 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
+ − 324 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
+ − 325 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
+ − 326 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
+ − 327
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 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
+ − 329 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
+ − 330 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
+ − 331 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
+ − 332 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
+ − 333 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
+ − 334 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
+ − 335 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
+ − 336 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
+ − 337 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
+ − 338 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
+ − 339 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
+ − 340 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
+ − 341 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
+ − 342 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
+ − 343 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
+ − 344 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
+ − 345 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
+ − 346 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
+ − 347 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
+ − 348 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
+ − 349 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
+ − 350 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
+ − 351 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
+ − 352
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 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
+ − 354 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
+ − 355 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
+ − 356 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
+ − 357
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 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
+ − 359 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
+ − 360 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
+ − 361 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
+ − 362 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
+ − 363 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
+ − 364 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
+ − 365 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
+ − 366 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
+ − 367 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
+ − 368 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
+ − 369 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
+ − 370 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
+ − 371 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
+ − 372 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
+ − 373 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
+ − 374 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
+ − 375 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
+ − 376
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 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
+ − 378 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
+ − 379 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
+ − 380 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
+ − 381
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 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
+ − 383 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
+ − 384 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
+ − 385 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
+ − 386 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
+ − 387 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
+ − 388 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
+ − 389 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
+ − 390 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
+ − 391
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 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
+ − 393 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
+ − 394 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
+ − 395 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
+ − 396 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
+ − 397 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
+ − 398 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
+ − 399 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
+ − 400 *)
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
1954
+ − 402 ML {*
+ − 403 fun mk_avoids ctxt params name set =
+ − 404 let
+ − 405 val (_, ctxt') = ProofContext.add_fixes
+ − 406 (map (fn (s, T) => (Binding.name s, SOME T, NoSyn)) params) ctxt;
+ − 407 fun mk s =
+ − 408 let
+ − 409 val t = Syntax.read_term ctxt' s;
+ − 410 val t' = list_abs_free (params, t) |>
+ − 411 funpow (length params) (fn Abs (_, _, t) => t)
+ − 412 in (t', HOLogic.dest_setT (fastype_of t)) end
+ − 413 handle TERM _ =>
+ − 414 error ("Expression " ^ quote s ^ " to be avoided in case " ^
+ − 415 quote name ^ " is not a set type");
+ − 416 fun add_set p [] = [p]
+ − 417 | add_set (t, T) ((u, U) :: ps) =
+ − 418 if T = U then
+ − 419 let val S = HOLogic.mk_setT T
+ − 420 in (Const (@{const_name sup}, S --> S --> S) $ u $ t, T) :: ps
+ − 421 end
+ − 422 else (u, U) :: add_set (t, T) ps
+ − 423 in
+ − 424 (mk #> add_set) set
+ − 425 end;
+ − 426 *}
+ − 427
+ − 428
+ − 429 ML {*
+ − 430 writeln (commas (map (Syntax.string_of_term @{context} o fst)
+ − 431 (mk_avoids @{context} [] "t_Var" "{x}" [])))
+ − 432 *}
+ − 433
1947
+ − 434
1949
+ − 435 ML {*
+ − 436
+ − 437 fun prove_strong_ind (pred_name, avoids) ctxt =
+ − 438 Proof.theorem NONE (K I) [] ctxt
+ − 439
2169
+ − 440 local structure P = Parse and K = Keyword in
1949
+ − 441
+ − 442 val _ =
2169
+ − 443 Outer_Syntax.local_theory_to_proof "nominal_inductive"
1949
+ − 444 "proves strong induction theorem for inductive predicate involving nominal datatypes" K.thy_goal
+ − 445 (P.xname -- (Scan.optional (P.$$$ "avoids" |-- P.enum1 "|" (P.name --
+ − 446 (P.$$$ ":" |-- P.and_list1 P.term))) []) >> prove_strong_ind)
+ − 447
+ − 448 end;
+ − 449
+ − 450 *}
+ − 451
1950
+ − 452 (*
1949
+ − 453 nominal_inductive typing
1950
+ − 454 *)
1949
+ − 455
2157
+ − 456 (* Substitution *)
2159
+ − 457
2165
+ − 458 primrec match_Var_raw where
+ − 459 "match_Var_raw (Var_raw x) = Some x"
+ − 460 | "match_Var_raw (App_raw x y) = None"
+ − 461 | "match_Var_raw (Lam_raw n t) = None"
+ − 462
+ − 463 quotient_definition
+ − 464 "match_Var :: lam \<Rightarrow> name option"
+ − 465 is match_Var_raw
+ − 466
+ − 467 lemma [quot_respect]: "(alpha_lam_raw ===> op =) match_Var_raw match_Var_raw"
+ − 468 apply rule
+ − 469 apply (induct_tac a b rule: alpha_lam_raw.induct)
+ − 470 apply simp_all
+ − 471 done
+ − 472
+ − 473 lemmas match_Var_simps = match_Var_raw.simps[quot_lifted]
+ − 474
+ − 475 primrec match_App_raw where
+ − 476 "match_App_raw (Var_raw x) = None"
+ − 477 | "match_App_raw (App_raw x y) = Some (x, y)"
+ − 478 | "match_App_raw (Lam_raw n t) = None"
+ − 479
2436
+ − 480 (*
2165
+ − 481 quotient_definition
+ − 482 "match_App :: lam \<Rightarrow> (lam \<times> lam) option"
+ − 483 is match_App_raw
+ − 484
+ − 485 lemma [quot_respect]:
+ − 486 "(alpha_lam_raw ===> option_rel (prod_rel alpha_lam_raw alpha_lam_raw)) match_App_raw match_App_raw"
+ − 487 apply (intro fun_relI)
+ − 488 apply (induct_tac a b rule: alpha_lam_raw.induct)
+ − 489 apply simp_all
+ − 490 done
+ − 491
+ − 492 lemmas match_App_simps = match_App_raw.simps[quot_lifted]
+ − 493
2173
+ − 494 definition new where
+ − 495 "new (s :: 'a :: fs) = (THE x. \<forall>a \<in> supp s. atom x \<noteq> a)"
+ − 496
+ − 497 definition
+ − 498 "match_Lam (S :: 'a :: fs) t = (if (\<exists>n s. (t = Lam n s)) then
+ − 499 (let z = new (S, t) in Some (z, THE s. t = Lam z s)) else None)"
+ − 500
+ − 501 lemma lam_half_inj: "(Lam z s = Lam z sa) = (s = sa)"
+ − 502 apply auto
+ − 503 apply (simp only: lam.eq_iff alphas)
+ − 504 apply clarify
+ − 505 apply (simp add: eqvts)
+ − 506 sorry
+ − 507
+ − 508 lemma match_Lam_simps:
+ − 509 "match_Lam S (Var n) = None"
+ − 510 "match_Lam S (App l r) = None"
+ − 511 "z = new (S, (Lam z s)) \<Longrightarrow> match_Lam S (Lam z s) = Some (z, s)"
+ − 512 apply (simp_all add: match_Lam_def)
+ − 513 apply (simp add: lam_half_inj)
+ − 514 apply auto
+ − 515 done
2436
+ − 516 *)
2173
+ − 517 (*
+ − 518 lemma match_Lam_simps2:
+ − 519 "atom n \<sharp> ((S :: 'a :: fs), Lam n s) \<Longrightarrow> match_Lam S (Lam n s) = Some (n, s)"
+ − 520 apply (rule_tac t="Lam n s"
+ − 521 and s="Lam (new (S, (Lam n s))) ((n \<leftrightarrow> (new (S, (Lam n s)))) \<bullet> s)" in subst)
+ − 522 defer
+ − 523 apply (subst match_Lam_simps(3))
+ − 524 defer
+ − 525 apply simp
+ − 526 *)
+ − 527
+ − 528 (*primrec match_Lam_raw where
2165
+ − 529 "match_Lam_raw (S :: atom set) (Var_raw x) = None"
+ − 530 | "match_Lam_raw S (App_raw x y) = None"
+ − 531 | "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))"
+ − 532
+ − 533 quotient_definition
+ − 534 "match_Lam :: (atom set) \<Rightarrow> lam \<Rightarrow> (name \<times> lam) option"
+ − 535 is match_Lam_raw
+ − 536
2172
+ − 537 lemma swap_fresh:
+ − 538 assumes a: "fv_lam_raw t \<sharp>* p"
+ − 539 shows "alpha_lam_raw (p \<bullet> t) t"
+ − 540 using a apply (induct t)
+ − 541 apply (simp add: supp_at_base fresh_star_def)
+ − 542 apply (rule alpha_lam_raw.intros)
+ − 543 apply (metis Rep_name_inverse atom_eqvt atom_name_def fresh_perm)
+ − 544 apply (simp)
+ − 545 apply (simp only: fresh_star_union)
+ − 546 apply clarify
+ − 547 apply (rule alpha_lam_raw.intros)
+ − 548 apply simp
+ − 549 apply simp
+ − 550 apply simp
+ − 551 apply (rule alpha_lam_raw.intros)
+ − 552 sorry
+ − 553
2165
+ − 554 lemma [quot_respect]:
+ − 555 "(op = ===> alpha_lam_raw ===> option_rel (prod_rel op = alpha_lam_raw)) match_Lam_raw match_Lam_raw"
+ − 556 proof (intro fun_relI, clarify)
+ − 557 fix S t s
+ − 558 assume a: "alpha_lam_raw t s"
+ − 559 show "option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S t) (match_Lam_raw S s)"
+ − 560 using a proof (induct t s rule: alpha_lam_raw.induct)
+ − 561 case goal1 show ?case by simp
+ − 562 next
+ − 563 case goal2 show ?case by simp
+ − 564 next
+ − 565 case (goal3 x t y s)
+ − 566 then obtain p where "({atom x}, t) \<approx>gen (\<lambda>x1 x2. alpha_lam_raw x1 x2 \<and>
+ − 567 option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S x1)
+ − 568 (match_Lam_raw S x2)) fv_lam_raw p ({atom y}, s)" ..
+ − 569 then have
+ − 570 c: "fv_lam_raw t - {atom x} = fv_lam_raw s - {atom y}" and
+ − 571 d: "(fv_lam_raw t - {atom x}) \<sharp>* p" and
+ − 572 e: "alpha_lam_raw (p \<bullet> t) s" and
+ − 573 f: "option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S (p \<bullet> t)) (match_Lam_raw S s)" and
+ − 574 g: "p \<bullet> {atom x} = {atom y}" unfolding alphas(1) by - (elim conjE, assumption)+
+ − 575 let ?z = "new (S \<union> (fv_lam_raw t - {atom x}))"
+ − 576 have h: "?z = new (S \<union> (fv_lam_raw s - {atom y}))" using c by simp
+ − 577 show ?case
+ − 578 unfolding match_Lam_raw.simps Let_def option_rel.simps prod_rel.simps split_conv
+ − 579 proof
+ − 580 show "?z = new (S \<union> (fv_lam_raw s - {atom y}))" by (fact h)
+ − 581 next
+ − 582 have "atom y \<sharp> p" sorry
+ − 583 have "fv_lam_raw t \<sharp>* ((x \<leftrightarrow> y) \<bullet> p)" sorry
2172
+ − 584 then have "alpha_lam_raw (((x \<leftrightarrow> y) \<bullet> p) \<bullet> t) t" using swap_fresh by auto
+ − 585 then have "alpha_lam_raw (p \<bullet> t) ((x \<leftrightarrow> y) \<bullet> t)" sorry
2165
+ − 586 have "alpha_lam_raw t ((x \<leftrightarrow> y) \<bullet> s)" sorry
+ − 587 then have "alpha_lam_raw ((x \<leftrightarrow> ?z) \<bullet> t) ((y \<leftrightarrow> ?z) \<bullet> s)" using eqvts(15) sorry
+ − 588 then show "alpha_lam_raw ((x \<leftrightarrow> new (S \<union> (fv_lam_raw t - {atom x}))) \<bullet> t)
+ − 589 ((y \<leftrightarrow> new (S \<union> (fv_lam_raw s - {atom y}))) \<bullet> s)" unfolding h .
+ − 590 qed
+ − 591 qed
+ − 592 qed
+ − 593
+ − 594 lemmas match_Lam_simps = match_Lam_raw.simps[quot_lifted]
2173
+ − 595 *)
2436
+ − 596 (*
2165
+ − 597 lemma app_some: "match_App x = Some (a, b) \<Longrightarrow> x = App a b"
+ − 598 by (induct x rule: lam.induct) (simp_all add: match_App_simps)
+ − 599
+ − 600 lemma lam_some: "match_Lam S x = Some (z, s) \<Longrightarrow> x = Lam z s \<and> atom z \<sharp> S"
+ − 601 apply (induct x rule: lam.induct)
+ − 602 apply (simp_all add: match_Lam_simps)
2173
+ − 603 apply (thin_tac "match_Lam S lam = Some (z, s) \<Longrightarrow> lam = Lam z s \<and> atom z \<sharp> S")
+ − 604 apply (simp add: match_Lam_def)
+ − 605 apply (subgoal_tac "\<exists>n s. Lam name lam = Lam n s")
+ − 606 prefer 2
+ − 607 apply auto[1]
2165
+ − 608 apply (simp add: Let_def)
2173
+ − 609 apply (thin_tac "\<exists>n s. Lam name lam = Lam n s")
+ − 610 apply clarify
2165
+ − 611 apply (rule conjI)
2173
+ − 612 apply (rule_tac t="THE s. Lam name lam = Lam (new (S, Lam name lam)) s" and
+ − 613 s="(name \<leftrightarrow> (new (S, Lam name lam))) \<bullet> lam" in subst)
+ − 614 defer
2165
+ − 615 apply (simp add: lam.eq_iff)
2173
+ − 616 apply (rule_tac x="(name \<leftrightarrow> (new (S, Lam name lam)))" in exI)
2165
+ − 617 apply (simp add: alphas)
+ − 618 apply (simp add: eqvts)
2173
+ − 619 apply (rule conjI)
2172
+ − 620 sorry
2165
+ − 621
+ − 622 function subst where
+ − 623 "subst v s t = (
+ − 624 case match_Var t of Some n \<Rightarrow> if n = v then s else Var n | None \<Rightarrow>
+ − 625 case match_App t of Some (l, r) \<Rightarrow> App (subst v s l) (subst v s r) | None \<Rightarrow>
2173
+ − 626 case match_Lam (v,s) t of Some (n, t) \<Rightarrow> Lam n (subst v s t) | None \<Rightarrow> undefined)"
2165
+ − 627 by pat_completeness auto
+ − 628
+ − 629 termination apply (relation "measure (\<lambda>(_, _, t). size t)")
2172
+ − 630 apply auto[1]
+ − 631 apply (case_tac a) apply simp
+ − 632 apply (frule lam_some) apply simp
+ − 633 apply (case_tac a) apply simp
+ − 634 apply (frule app_some) apply simp
+ − 635 apply (case_tac a) apply simp
+ − 636 apply (frule app_some) apply simp
2165
+ − 637 done
+ − 638
+ − 639 lemmas lam_exhaust = lam_raw.exhaust[quot_lifted]
+ − 640
+ − 641 lemma subst_eqvt:
+ − 642 "p \<bullet> (subst v s t) = subst (p \<bullet> v) (p \<bullet> s) (p \<bullet> t)"
+ − 643 proof (induct v s t rule: subst.induct)
+ − 644 case (1 v s t)
+ − 645 show ?case proof (cases t rule: lam_exhaust)
+ − 646 fix n
+ − 647 assume "t = Var n"
+ − 648 then show ?thesis by (simp add: match_Var_simps)
+ − 649 next
+ − 650 fix l r
+ − 651 assume "t = App l r"
+ − 652 then show ?thesis
+ − 653 apply (simp only:)
+ − 654 apply (subst subst.simps)
+ − 655 apply (subst match_Var_simps)
+ − 656 apply (simp only: option.cases)
+ − 657 apply (subst match_App_simps)
+ − 658 apply (simp only: option.cases)
+ − 659 apply (simp only: prod.cases)
+ − 660 apply (simp only: lam.perm)
+ − 661 apply (subst (3) subst.simps)
+ − 662 apply (subst match_Var_simps)
2173
+ − 663 apply (simp only: option.cases)
2165
+ − 664 apply (subst match_App_simps)
+ − 665 apply (simp only: option.cases)
+ − 666 apply (simp only: prod.cases)
+ − 667 apply (subst 1(2)[of "(l, r)" "l" "r"])
+ − 668 apply (simp add: match_Var_simps)
+ − 669 apply (simp add: match_App_simps)
+ − 670 apply (rule refl)
+ − 671 apply (subst 1(3)[of "(l, r)" "l" "r"])
+ − 672 apply (simp add: match_Var_simps)
+ − 673 apply (simp add: match_App_simps)
+ − 674 apply (rule refl)
+ − 675 apply (rule refl)
+ − 676 done
+ − 677 next
+ − 678 fix n t'
+ − 679 assume "t = Lam n t'"
+ − 680 then show ?thesis
+ − 681 apply (simp only: )
+ − 682 apply (simp only: lam.perm)
+ − 683 apply (subst subst.simps)
+ − 684 apply (subst match_Var_simps)
+ − 685 apply (simp only: option.cases)
+ − 686 apply (subst match_App_simps)
+ − 687 apply (simp only: option.cases)
2173
+ − 688 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)
+ − 689 defer
2165
+ − 690 apply (subst match_Lam_simps)
2173
+ − 691 defer
2165
+ − 692 apply (simp only: option.cases)
+ − 693 apply (simp only: prod.cases)
+ − 694 apply (subst (2) subst.simps)
+ − 695 apply (subst match_Var_simps)
+ − 696 apply (simp only: option.cases)
+ − 697 apply (subst match_App_simps)
+ − 698 apply (simp only: option.cases)
2173
+ − 699 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)
+ − 700 defer
2165
+ − 701 apply (subst match_Lam_simps)
2173
+ − 702 defer
2165
+ − 703 apply (simp only: option.cases)
+ − 704 apply (simp only: prod.cases)
+ − 705 apply (simp only: lam.perm)
2173
+ − 706 thm 1(1)
2165
+ − 707 sorry
+ − 708 qed
+ − 709 qed
+ − 710
2172
+ − 711 lemma subst_proper_eqs:
+ − 712 "subst y s (Var x) = (if x = y then s else (Var x))"
+ − 713 "subst y s (App l r) = App (subst y s l) (subst y s r)"
+ − 714 "atom x \<sharp> (t, s) \<Longrightarrow> subst y s (Lam x t) = Lam x (subst y s t)"
+ − 715 apply (subst subst.simps)
+ − 716 apply (simp only: match_Var_simps)
+ − 717 apply (simp only: option.simps)
+ − 718 apply (subst subst.simps)
+ − 719 apply (simp only: match_App_simps)
+ − 720 apply (simp only: option.simps)
+ − 721 apply (simp only: prod.simps)
+ − 722 apply (simp only: match_Var_simps)
+ − 723 apply (simp only: option.simps)
+ − 724 apply (subst subst.simps)
+ − 725 apply (simp only: match_Var_simps)
2173
+ − 726 apply (simp only: option.simps)
2172
+ − 727 apply (simp only: match_App_simps)
+ − 728 apply (simp only: option.simps)
2173
+ − 729 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)
+ − 730 defer
+ − 731 apply (subst match_Lam_simps)
+ − 732 defer
2172
+ − 733 apply (simp only: option.simps)
+ − 734 apply (simp only: prod.simps)
2157
+ − 735 sorry
2436
+ − 736 *)
1594
+ − 737 end
+ − 738
+ − 739
+ − 740