Nominal/Ex/SingleLet.thy
author Christian Urban <urbanc@in.tum.de>
Sat, 04 Sep 2010 06:23:31 +0800
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moved a proof to Abs
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theory SingleLet
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imports "../Nominal2"
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begin
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atom_decl name
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declare [[STEPS = 100]]
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nominal_datatype single_let: trm  =
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  Var "name"
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| App "trm" "trm"
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| Lam x::"name" t::"trm"  bind x in t
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| Let a::"assg" t::"trm"  bind "bn a" in t
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| Foo x::"name" y::"name" t::"trm" t1::"trm" t2::"trm" bind (set) x in y t t1 t2
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| Bar x::"name" y::"name" t::"trm" bind y x in t x y
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| Baz x::"name" t1::"trm" t2::"trm" bind x in t1, bind x in t2 
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and assg =
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  As "name" x::"name" t::"trm" bind x in t
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binder
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  bn::"assg \<Rightarrow> atom list"
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where
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  "bn (As x y t) = [atom x]"
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thm single_let.distinct
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thm single_let.induct
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thm single_let.exhaust
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thm single_let.fv_defs
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thm single_let.bn_defs
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thm single_let.perm_simps
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thm single_let.eq_iff
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thm single_let.fv_bn_eqvt
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thm single_let.size_eqvt
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thm single_let.supports
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thm single_let.fsupp
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lemma test2:
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  assumes "fv_trm t = supp t" 
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  shows "\<forall>p. fv_trm (p \<bullet> t) = supp (p \<bullet> t)"
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apply(rule allI)
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apply(rule_tac p="-p" in permute_boolE)
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apply(perm_simp add: single_let.fv_bn_eqvt permute_minus_cancel)
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apply(rule assms)
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done
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lemma supp_fv:
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  "fv_trm t = supp t \<and> fv_assg as = supp as \<and> fv_bn as = {a. infinite {b. \<not>alpha_bn ((a \<rightleftharpoons> b) \<bullet> as) as}}"
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apply(rule single_let.induct)
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apply(simp_all (no_asm_use) only: single_let.fv_defs)[2]
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apply(simp_all (no_asm_use) only: supp_def)[2]
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apply(simp_all (no_asm_use) only: single_let.perm_simps)[2]
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apply(simp_all (no_asm_use) only: single_let.eq_iff)[2]
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apply(simp_all (no_asm_use) only: de_Morgan_conj)[2]
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apply(simp_all (no_asm_use) only: Collect_disj_eq)[2]
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apply(simp_all (no_asm_use) only: finite_Un)[2]
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apply(simp_all (no_asm_use) only: de_Morgan_conj)[2]
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apply(simp_all (no_asm_use) only: Collect_disj_eq)[2]
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apply(simp)
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--" 1 "
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apply(simp only: single_let.fv_defs supp_Pair[symmetric])
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apply(simp only: supp_abs(3)[symmetric])
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apply(simp (no_asm) only: supp_def)
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apply(perm_simp add: single_let.perm_simps single_let.fv_bn_eqvt)
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apply(simp (no_asm) only: single_let.eq_iff Abs_eq_iff)
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apply(simp only: alphas prod_alpha_def prod_fv.simps prod_rel.simps permute_prod_def prod.recs prod.cases prod.inject)
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apply(drule test2)
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apply(simp only:)
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-- " 2 "
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apply(erule conjE)+
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apply(simp only: single_let.fv_defs supp_Pair[symmetric])
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apply(simp only: supp_abs(3)[symmetric])
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apply(simp (no_asm) only: supp_def)
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apply(perm_simp add: single_let.perm_simps single_let.fv_bn_eqvt)
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apply(simp (no_asm) only: single_let.eq_iff Abs_eq_iff)
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apply(simp only: de_Morgan_conj Collect_disj_eq finite_Un)
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apply(simp only: alphas prod_alpha_def prod_fv.simps prod_rel.simps permute_prod_def prod.recs prod.cases prod.inject)
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apply(drule test2)
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apply(simp only:)
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-- " 3 "
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apply(simp only: single_let.fv_defs supp_Pair[symmetric])
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apply(simp only: supp_abs(1)[symmetric])
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apply(simp (no_asm) only: supp_def)
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apply(perm_simp add: single_let.perm_simps single_let.fv_bn_eqvt)
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apply(simp (no_asm) only: single_let.eq_iff Abs_eq_iff)
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apply(simp only: alphas prod_alpha_def prod_fv.simps prod_rel.simps permute_prod_def prod.recs prod.cases prod.inject)
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apply(drule test2)+
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apply(simp only: supp_Pair Un_assoc conj_assoc)
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-- " Bar "
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apply(simp only: single_let.fv_defs supp_Pair[symmetric])
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apply(simp only: supp_abs(3)[symmetric])
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apply(simp (no_asm) only: supp_def)
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apply(perm_simp add: single_let.perm_simps single_let.fv_bn_eqvt)
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apply(simp (no_asm) only: single_let.eq_iff Abs_eq_iff)
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apply(simp only: alphas prod_alpha_def prod_fv.simps prod_rel.simps permute_prod_def prod.recs prod.cases prod.inject)
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apply(drule test2)+
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apply(simp only: supp_Pair Un_assoc conj_assoc)
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-- " Baz "
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apply(simp only: single_let.fv_defs supp_Pair[symmetric])
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apply(simp only: supp_abs(3)[symmetric])
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apply(simp (no_asm) only: supp_def)
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apply(perm_simp add: single_let.perm_simps single_let.fv_bn_eqvt)
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apply(simp (no_asm) only: single_let.eq_iff Abs_eq_iff)
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apply(simp only: alphas prod_alpha_def prod_fv.simps prod_rel.simps permute_prod_def prod.recs prod.cases prod.inject)
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apply(simp only: de_Morgan_conj Collect_disj_eq finite_Un)
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apply(drule test2)+
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apply(simp only: supp_Pair Un_assoc conj_assoc)
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-- "last"
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apply(rule conjI)
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apply(simp only: single_let.fv_defs supp_Pair[symmetric])
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apply(simp only: supp_abs(3)[symmetric])
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apply(simp (no_asm) only: supp_def)
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apply(perm_simp add: single_let.perm_simps single_let.fv_bn_eqvt)
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apply(simp (no_asm) only: single_let.eq_iff Abs_eq_iff)
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apply(simp only: alphas prod_alpha_def prod_fv.simps prod_rel.simps permute_prod_def prod.recs prod.cases prod.inject)
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apply(simp only: de_Morgan_conj Collect_disj_eq finite_Un)
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apply(drule test2)+
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apply(simp only: supp_Pair Un_assoc conj_assoc)
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-- "other case"
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apply(simp only: single_let.fv_defs supp_Pair[symmetric])
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apply(simp only: supp_abs(3)[symmetric])
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apply(simp (no_asm) only: supp_def)
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apply(perm_simp add: single_let.perm_simps single_let.fv_bn_eqvt)
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apply(simp (no_asm) only: single_let.eq_iff Abs_eq_iff)
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apply(simp only: alphas prod_alpha_def prod_fv.simps prod_rel.simps permute_prod_def prod.recs prod.cases prod.inject)
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apply(simp only: de_Morgan_conj Collect_disj_eq finite_Un)?
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apply(drule test2)+
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apply(simp only: supp_Pair Un_assoc conj_assoc)
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done
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86028b2016bd some experiments with support
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86028b2016bd some experiments with support
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86028b2016bd some experiments with support
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86028b2016bd some experiments with support
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86028b2016bd some experiments with support
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text {* *}
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(*
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consts perm_bn_trm :: "perm \<Rightarrow> trm \<Rightarrow> trm"
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consts perm_bn_assg :: "perm \<Rightarrow> assg \<Rightarrow> assg"
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lemma y:
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  "perm_bn_trm p (Var x) = (Var x)"
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  "perm_bn_trm p (App t1 t2) = (App t1 t2)"
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  "perm_bn_trm p ("
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typ trm
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typ assg
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thm trm_assg.fv
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thm trm_assg.supp
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thm trm_assg.eq_iff
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thm trm_assg.bn
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thm trm_assg.perm
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thm trm_assg.induct
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thm trm_assg.inducts
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thm trm_assg.distinct
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ML {* Sign.of_sort @{theory} (@{typ trm}, @{sort fs}) *}
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*)
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end
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