(* Title: ZF/Resid/Substitution.thy ID: $Id: Substitution.thy,v 1.13 2005/06/17 14:15:11 haftmann Exp $ Author: Ole Rasmussen Copyright 1995 University of Cambridge Logic Image: ZF *) theory Substitution imports Redex begin consts lift_aux :: "i=>i" lift :: "i=>i" subst_aux :: "i=>i" "'/" :: "[i,i]=>i" (infixl 70) (*subst*) constdefs lift_rec :: "[i,i]=> i" "lift_rec(r,k) == lift_aux(r)`k" subst_rec :: "[i,i,i]=> i" (**NOTE THE ARGUMENT ORDER BELOW**) "subst_rec(u,r,k) == subst_aux(r)`u`k" translations "lift(r)" == "lift_rec(r,0)" "u/v" == "subst_rec(u,v,0)" (** The clumsy _aux functions are required because other arguments vary in the recursive calls ***) primrec "lift_aux(Var(i)) = (λk ∈ nat. if i<k then Var(i) else Var(succ(i)))" "lift_aux(Fun(t)) = (λk ∈ nat. Fun(lift_aux(t) ` succ(k)))" "lift_aux(App(b,f,a)) = (λk ∈ nat. App(b, lift_aux(f)`k, lift_aux(a)`k))" primrec "subst_aux(Var(i)) = (λr ∈ redexes. λk ∈ nat. if k<i then Var(i #- 1) else if k=i then r else Var(i))" "subst_aux(Fun(t)) = (λr ∈ redexes. λk ∈ nat. Fun(subst_aux(t) ` lift(r) ` succ(k)))" "subst_aux(App(b,f,a)) = (λr ∈ redexes. λk ∈ nat. App(b, subst_aux(f)`r`k, subst_aux(a)`r`k))" (* ------------------------------------------------------------------------- *) (* Arithmetic extensions *) (* ------------------------------------------------------------------------- *) lemma gt_not_eq: "p < n ==> n≠p" by blast lemma succ_pred [rule_format, simp]: "j ∈ nat ==> i < j --> succ(j #- 1) = j" by (induct_tac "j", auto) lemma lt_pred: "[|succ(x)<n; n ∈ nat|] ==> x < n #- 1 " apply (rule succ_leE) apply (simp add: succ_pred) done lemma gt_pred: "[|n < succ(x); p<n; n ∈ nat|] ==> n #- 1 < x " apply (rule succ_leE) apply (simp add: succ_pred) done declare not_lt_iff_le [simp] if_P [simp] if_not_P [simp] (* ------------------------------------------------------------------------- *) (* lift_rec equality rules *) (* ------------------------------------------------------------------------- *) lemma lift_rec_Var: "n ∈ nat ==> lift_rec(Var(i),n) = (if i<n then Var(i) else Var(succ(i)))" by (simp add: lift_rec_def) lemma lift_rec_le [simp]: "[|i ∈ nat; k≤i|] ==> lift_rec(Var(i),k) = Var(succ(i))" by (simp add: lift_rec_def le_in_nat) lemma lift_rec_gt [simp]: "[| k ∈ nat; i<k |] ==> lift_rec(Var(i),k) = Var(i)" by (simp add: lift_rec_def) lemma lift_rec_Fun [simp]: "k ∈ nat ==> lift_rec(Fun(t),k) = Fun(lift_rec(t,succ(k)))" by (simp add: lift_rec_def) lemma lift_rec_App [simp]: "k ∈ nat ==> lift_rec(App(b,f,a),k) = App(b,lift_rec(f,k),lift_rec(a,k))" by (simp add: lift_rec_def) (* ------------------------------------------------------------------------- *) (* substitution quality rules *) (* ------------------------------------------------------------------------- *) lemma subst_Var: "[|k ∈ nat; u ∈ redexes|] ==> subst_rec(u,Var(i),k) = (if k<i then Var(i #- 1) else if k=i then u else Var(i))" by (simp add: subst_rec_def gt_not_eq leI) lemma subst_eq [simp]: "[|n ∈ nat; u ∈ redexes|] ==> subst_rec(u,Var(n),n) = u" by (simp add: subst_rec_def) lemma subst_gt [simp]: "[|u ∈ redexes; p ∈ nat; p<n|] ==> subst_rec(u,Var(n),p) = Var(n #- 1)" by (simp add: subst_rec_def) lemma subst_lt [simp]: "[|u ∈ redexes; p ∈ nat; n<p|] ==> subst_rec(u,Var(n),p) = Var(n)" by (simp add: subst_rec_def gt_not_eq leI lt_nat_in_nat) lemma subst_Fun [simp]: "[|p ∈ nat; u ∈ redexes|] ==> subst_rec(u,Fun(t),p) = Fun(subst_rec(lift(u),t,succ(p))) " by (simp add: subst_rec_def) lemma subst_App [simp]: "[|p ∈ nat; u ∈ redexes|] ==> subst_rec(u,App(b,f,a),p) = App(b,subst_rec(u,f,p),subst_rec(u,a,p))" by (simp add: subst_rec_def) lemma lift_rec_type [rule_format, simp]: "u ∈ redexes ==> ∀k ∈ nat. lift_rec(u,k) ∈ redexes" apply (erule redexes.induct) apply (simp_all add: lift_rec_Var lift_rec_Fun lift_rec_App) done lemma subst_type [rule_format, simp]: "v ∈ redexes ==> ∀n ∈ nat. ∀u ∈ redexes. subst_rec(u,v,n) ∈ redexes" apply (erule redexes.induct) apply (simp_all add: subst_Var lift_rec_type) done (* ------------------------------------------------------------------------- *) (* lift and substitution proofs *) (* ------------------------------------------------------------------------- *) (*The i∈nat is redundant*) lemma lift_lift_rec [rule_format]: "u ∈ redexes ==> ∀n ∈ nat. ∀i ∈ nat. i≤n --> (lift_rec(lift_rec(u,i),succ(n)) = lift_rec(lift_rec(u,n),i))" apply (erule redexes.induct, auto) apply (case_tac "n < i") apply (frule lt_trans2, assumption) apply (simp_all add: lift_rec_Var leI) done lemma lift_lift: "[|u ∈ redexes; n ∈ nat|] ==> lift_rec(lift(u),succ(n)) = lift(lift_rec(u,n))" by (simp add: lift_lift_rec) lemma lt_not_m1_lt: "[|m < n; n ∈ nat; m ∈ nat|]==> ~ n #- 1 < m" by (erule natE, auto) lemma lift_rec_subst_rec [rule_format]: "v ∈ redexes ==> ∀n ∈ nat. ∀m ∈ nat. ∀u ∈ redexes. n≤m--> lift_rec(subst_rec(u,v,n),m) = subst_rec(lift_rec(u,m),lift_rec(v,succ(m)),n)" apply (erule redexes.induct, simp_all (no_asm_simp) add: lift_lift) apply safe apply (rename_tac n n' m u) apply (case_tac "n < n'") apply (frule_tac j = n' in lt_trans2, assumption) apply (simp add: leI, simp) apply (erule_tac j=n in leE) apply (auto simp add: lift_rec_Var subst_Var leI lt_not_m1_lt) done lemma lift_subst: "[|v ∈ redexes; u ∈ redexes; n ∈ nat|] ==> lift_rec(u/v,n) = lift_rec(u,n)/lift_rec(v,succ(n))" by (simp add: lift_rec_subst_rec) lemma lift_rec_subst_rec_lt [rule_format]: "v ∈ redexes ==> ∀n ∈ nat. ∀m ∈ nat. ∀u ∈ redexes. m≤n--> lift_rec(subst_rec(u,v,n),m) = subst_rec(lift_rec(u,m),lift_rec(v,m),succ(n))" apply (erule redexes.induct, simp_all (no_asm_simp) add: lift_lift) apply safe apply (rename_tac n n' m u) apply (case_tac "n < n'") apply (case_tac "n < m") apply (simp_all add: leI) apply (erule_tac i=n' in leE) apply (frule lt_trans1, assumption) apply (simp_all add: succ_pred leI gt_pred) done lemma subst_rec_lift_rec [rule_format]: "u ∈ redexes ==> ∀n ∈ nat. ∀v ∈ redexes. subst_rec(v,lift_rec(u,n),n) = u" apply (erule redexes.induct, auto) apply (case_tac "n < na", auto) done lemma subst_rec_subst_rec [rule_format]: "v ∈ redexes ==> ∀m ∈ nat. ∀n ∈ nat. ∀u ∈ redexes. ∀w ∈ redexes. m≤n --> subst_rec(subst_rec(w,u,n),subst_rec(lift_rec(w,m),v,succ(n)),m) = subst_rec(w,subst_rec(u,v,m),n)" apply (erule redexes.induct) apply (simp_all add: lift_lift [symmetric] lift_rec_subst_rec_lt, safe) apply (rename_tac n' u w) apply (case_tac "n ≤ succ(n') ") apply (erule_tac i = n in leE) apply (simp_all add: succ_pred subst_rec_lift_rec leI) apply (case_tac "n < m") apply (frule lt_trans2, assumption, simp add: gt_pred) apply simp apply (erule_tac j = n in leE, simp add: gt_pred) apply (simp add: subst_rec_lift_rec) (*final case*) apply (frule nat_into_Ord [THEN le_refl, THEN lt_trans], assumption) apply (erule leE) apply (frule succ_leI [THEN lt_trans], assumption) apply (frule_tac i = m in nat_into_Ord [THEN le_refl, THEN lt_trans], assumption) apply (simp_all add: succ_pred lt_pred) done lemma substitution: "[|v ∈ redexes; u ∈ redexes; w ∈ redexes; n ∈ nat|] ==> subst_rec(w,u,n)/subst_rec(lift(w),v,succ(n)) = subst_rec(w,u/v,n)" by (simp add: subst_rec_subst_rec) (* ------------------------------------------------------------------------- *) (* Preservation lemmas *) (* Substitution preserves comp and regular *) (* ------------------------------------------------------------------------- *) lemma lift_rec_preserve_comp [rule_format, simp]: "u ~ v ==> ∀m ∈ nat. lift_rec(u,m) ~ lift_rec(v,m)" by (erule Scomp.induct, simp_all add: comp_refl) lemma subst_rec_preserve_comp [rule_format, simp]: "u2 ~ v2 ==> ∀m ∈ nat. ∀u1 ∈ redexes. ∀v1 ∈ redexes. u1 ~ v1--> subst_rec(u1,u2,m) ~ subst_rec(v1,v2,m)" by (erule Scomp.induct, simp_all add: subst_Var lift_rec_preserve_comp comp_refl) lemma lift_rec_preserve_reg [simp]: "regular(u) ==> ∀m ∈ nat. regular(lift_rec(u,m))" by (erule Sreg.induct, simp_all add: lift_rec_Var) lemma subst_rec_preserve_reg [simp]: "regular(v) ==> ∀m ∈ nat. ∀u ∈ redexes. regular(u)-->regular(subst_rec(u,v,m))" by (erule Sreg.induct, simp_all add: subst_Var lift_rec_preserve_reg) end
lemma gt_not_eq:
p < n ==> n ≠ p
lemma succ_pred:
[| j ∈ nat; i < j |] ==> succ(j #- 1) = j
lemma lt_pred:
[| succ(x) < n; n ∈ nat |] ==> x < n #- 1
lemma gt_pred:
[| n ≤ x; p < n; n ∈ nat |] ==> n #- 1 < x
lemma lift_rec_Var:
n ∈ nat ==> lift_rec(Var(i), n) = (if i < n then Var(i) else Var(succ(i)))
lemma lift_rec_le:
[| i ∈ nat; k ≤ i |] ==> lift_rec(Var(i), k) = Var(succ(i))
lemma lift_rec_gt:
[| k ∈ nat; i < k |] ==> lift_rec(Var(i), k) = Var(i)
lemma lift_rec_Fun:
k ∈ nat ==> lift_rec(Fun(t), k) = Fun(lift_rec(t, succ(k)))
lemma lift_rec_App:
k ∈ nat ==> lift_rec(App(b, f, a), k) = App(b, lift_rec(f, k), lift_rec(a, k))
lemma subst_Var:
[| k ∈ nat; u ∈ redexes |] ==> subst_rec(u, Var(i), k) = (if k < i then Var(i #- 1) else if k = i then u else Var(i))
lemma subst_eq:
[| n ∈ nat; u ∈ redexes |] ==> subst_rec(u, Var(n), n) = u
lemma subst_gt:
[| u ∈ redexes; p ∈ nat; p < n |] ==> subst_rec(u, Var(n), p) = Var(n #- 1)
lemma subst_lt:
[| u ∈ redexes; p ∈ nat; n < p |] ==> subst_rec(u, Var(n), p) = Var(n)
lemma subst_Fun:
[| p ∈ nat; u ∈ redexes |] ==> subst_rec(u, Fun(t), p) = Fun(subst_rec(lift(u), t, succ(p)))
lemma subst_App:
[| p ∈ nat; u ∈ redexes |] ==> subst_rec(u, App(b, f, a), p) = App(b, subst_rec(u, f, p), subst_rec(u, a, p))
lemma lift_rec_type:
[| u ∈ redexes; k ∈ nat |] ==> lift_rec(u, k) ∈ redexes
lemma subst_type:
[| v ∈ redexes; n ∈ nat; u ∈ redexes |] ==> subst_rec(u, v, n) ∈ redexes
lemma lift_lift_rec:
[| u ∈ redexes; n ∈ nat; i ∈ nat; i ≤ n |] ==> lift_rec(lift_rec(u, i), succ(n)) = lift_rec(lift_rec(u, n), i)
lemma lift_lift:
[| u ∈ redexes; n ∈ nat |] ==> lift_rec(lift(u), succ(n)) = lift(lift_rec(u, n))
lemma lt_not_m1_lt:
[| m < n; n ∈ nat; m ∈ nat |] ==> ¬ n #- 1 < m
lemma lift_rec_subst_rec:
[| v ∈ redexes; n ∈ nat; m ∈ nat; u ∈ redexes; n ≤ m |] ==> lift_rec(subst_rec(u, v, n), m) = subst_rec(lift_rec(u, m), lift_rec(v, succ(m)), n)
lemma lift_subst:
[| v ∈ redexes; u ∈ redexes; n ∈ nat |] ==> lift_rec(u / v, n) = lift_rec(u, n) / lift_rec(v, succ(n))
lemma lift_rec_subst_rec_lt:
[| v ∈ redexes; n ∈ nat; m ∈ nat; u ∈ redexes; m ≤ n |] ==> lift_rec(subst_rec(u, v, n), m) = subst_rec(lift_rec(u, m), lift_rec(v, m), succ(n))
lemma subst_rec_lift_rec:
[| u ∈ redexes; n ∈ nat; v ∈ redexes |] ==> subst_rec(v, lift_rec(u, n), n) = u
lemma subst_rec_subst_rec:
[| v ∈ redexes; m ∈ nat; n ∈ nat; u ∈ redexes; w ∈ redexes; m ≤ n |] ==> subst_rec(subst_rec(w, u, n), subst_rec(lift_rec(w, m), v, succ(n)), m) = subst_rec(w, subst_rec(u, v, m), n)
lemma substitution:
[| v ∈ redexes; u ∈ redexes; w ∈ redexes; n ∈ nat |] ==> subst_rec(w, u, n) / subst_rec(lift(w), v, succ(n)) = subst_rec(w, u / v, n)
lemma lift_rec_preserve_comp:
[| u ~ v; m ∈ nat |] ==> lift_rec(u, m) ~ lift_rec(v, m)
lemma subst_rec_preserve_comp:
[| u2.0 ~ v2.0; m ∈ nat; u1.0 ∈ redexes; v1.0 ∈ redexes; u1.0 ~ v1.0 |] ==> subst_rec(u1.0, u2.0, m) ~ subst_rec(v1.0, v2.0, m)
lemma lift_rec_preserve_reg:
regular(u) ==> ∀m∈nat. regular(lift_rec(u, m))
lemma subst_rec_preserve_reg:
regular(v) ==> ∀m∈nat. ∀u∈redexes. regular(u) --> regular(subst_rec(u, v, m))