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theory LocalWeakening(* Formalisation of weakening using locally nameless *) (* terms; the nominal infrastructure can also derive *) (* strong induction principles for such representations *) (* *) (* Authors: Christian Urban and Randy Pollack *) theory LocalWeakening imports "../Nominal" begin atom_decl name text {* Curry-style lambda terms in locally nameless representation without any binders *} nominal_datatype llam = lPar "name" | lVar "nat" | lApp "llam" "llam" | lLam "llam" text {* definition of vsub - at the moment a bit cumbersome *} lemma llam_cases: "(∃a. t = lPar a) ∨ (∃n. t = lVar n) ∨ (∃t1 t2. t = lApp t1 t2) ∨ (∃t1. t = lLam t1)" by (induct t rule: llam.induct) (auto simp add: llam.inject) nominal_primrec llam_size :: "llam => nat" where "llam_size (lPar a) = 1" | "llam_size (lVar n) = 1" | "llam_size (lApp t1 t2) = 1 + (llam_size t1) + (llam_size t2)" | "llam_size (lLam t) = 1 + (llam_size t)" by (rule TrueI)+ function vsub :: "llam => nat => llam => llam" where vsub_lPar: "vsub (lPar p) x s = lPar p" | vsub_lVar: "vsub (lVar n) x s = (if n < x then (lVar n) else (if n = x then s else (lVar (n - 1))))" | vsub_lApp: "vsub (lApp t u) x s = (lApp (vsub t x s) (vsub u x s))" | vsub_lLam: "vsub (lLam t) x s = (lLam (vsub t (x + 1) s))" using llam_cases apply(auto simp add: llam.inject) apply(rotate_tac 4) apply(drule_tac x="a" in meta_spec) apply(blast) done termination by (relation "measure (llam_size o fst)") auto lemma vsub_eqvt[eqvt]: fixes pi::"name prm" shows "pi•(vsub t n s) = vsub (pi•t) (pi•n) (pi•s)" by (induct t arbitrary: n rule: llam.induct) (simp_all add: perm_nat_def) constdefs freshen :: "llam => name => llam" "freshen t p ≡ vsub t 0 (lPar p)" lemma freshen_eqvt[eqvt]: fixes pi::"name prm" shows "pi•(freshen t p) = freshen (pi•t) (pi•p)" by (simp add: vsub_eqvt freshen_def perm_nat_def) text {* types *} nominal_datatype ty = TVar "nat" | TArr "ty" "ty" (infix "->" 200) lemma ty_fresh[simp]: fixes x::"name" and T::"ty" shows "x\<sharp>T" by (induct T rule: ty.induct) (simp_all add: fresh_nat) text {* valid contexts *} types cxt = "(name×ty) list" inductive valid :: "cxt => bool" where v1[intro]: "valid []" | v2[intro]: "[|valid Γ;a\<sharp>Γ|]==> valid ((a,T)#Γ)" equivariance valid lemma v2_elim: assumes a: "valid ((a,T)#Γ)" shows "a\<sharp>Γ ∧ valid Γ" using a by (cases) (auto) text {* "weak" typing relation *} inductive typing :: "cxt=>llam=>ty=>bool" (" _ \<turnstile> _ : _ " [60,60,60] 60) where t_lPar[intro]: "[|valid Γ; (x,T)∈set Γ|]==> Γ \<turnstile> lPar x : T" | t_lApp[intro]: "[|Γ \<turnstile> t1 : T1->T2; Γ \<turnstile> t2 : T1|]==> Γ \<turnstile> lApp t1 t2 : T2" | t_lLam[intro]: "[|x\<sharp>t; (x,T1)#Γ \<turnstile> freshen t x : T2|]==> Γ \<turnstile> lLam t : T1->T2" equivariance typing lemma typing_implies_valid: assumes a: "Γ \<turnstile> t : T" shows "valid Γ" using a by (induct) (auto dest: v2_elim) text {* we have to explicitly state that nominal_inductive should strengthen over the variable x (since x is not a binder) *} nominal_inductive typing avoids t_lLam: x by (auto simp add: fresh_prod dest: v2_elim typing_implies_valid) text {* strong induction principle for typing *} thm typing.strong_induct text {* sub-contexts *} abbreviation "sub_context" :: "cxt => cxt => bool" ("_ ⊆ _" [60,60] 60) where "Γ1 ⊆ Γ2 ≡ ∀x T. (x,T)∈set Γ1 --> (x,T)∈set Γ2" lemma weakening_almost_automatic: fixes Γ1 Γ2 :: cxt assumes a: "Γ1 \<turnstile> t : T" and b: "Γ1 ⊆ Γ2" and c: "valid Γ2" shows "Γ2 \<turnstile> t : T" using a b c apply(nominal_induct Γ1 t T avoiding: Γ2 rule: typing.strong_induct) apply(auto) apply(drule_tac x="(x,T1)#Γ2" in meta_spec) apply(auto intro!: t_lLam) done lemma weakening_in_more_detail: fixes Γ1 Γ2 :: cxt assumes a: "Γ1 \<turnstile> t : T" and b: "Γ1 ⊆ Γ2" and c: "valid Γ2" shows "Γ2 \<turnstile> t : T" using a b c proof(nominal_induct Γ1 t T avoiding: Γ2 rule: typing.strong_induct) case (t_lPar Γ1 x T Γ2) (* variable case *) have "Γ1 ⊆ Γ2" by fact moreover have "valid Γ2" by fact moreover have "(x,T)∈ set Γ1" by fact ultimately show "Γ2 \<turnstile> lPar x : T" by auto next case (t_lLam x t T1 Γ1 T2 Γ2) (* lambda case *) have vc: "x\<sharp>Γ2" "x\<sharp>t" by fact+ (* variable convention *) have ih: "[|(x,T1)#Γ1 ⊆ (x,T1)#Γ2; valid ((x,T1)#Γ2)|] ==> (x,T1)#Γ2 \<turnstile> freshen t x : T2" by fact have "Γ1 ⊆ Γ2" by fact then have "(x,T1)#Γ1 ⊆ (x,T1)#Γ2" by simp moreover have "valid Γ2" by fact then have "valid ((x,T1)#Γ2)" using vc by (simp add: v2) ultimately have "(x,T1)#Γ2 \<turnstile> freshen t x : T2" using ih by simp with vc show "Γ2 \<turnstile> lLam t : T1->T2" by auto next case (t_lApp Γ1 t1 T1 T2 t2 Γ2) then show "Γ2 \<turnstile> lApp t1 t2 : T2" by auto qed end