(* Title: Substitution.ML
ID: $Id$
Author: Ole Rasmussen
Copyright 1995 University of Cambridge
Logic Image: ZF
*)
open Substitution;
(* ------------------------------------------------------------------------- *)
(* Arithmetic extensions *)
(* ------------------------------------------------------------------------- *)
goal Arith.thy
"!!m.[| p < n; n:nat|]==> n~=p";
by (fast_tac lt_cs 1);
val gt_not_eq = result();
val succ_pred = prove_goal Arith.thy
"!!i.[|j:nat; i:nat|]==> i < j --> succ(j #- 1) = j"
(fn prems =>[(etac nat_induct 1),
(fast_tac lt_cs 1),
(asm_simp_tac arith_ss 1)]);
goal Arith.thy
"!!i.[|succ(x)<n; n:nat; x:nat|]==> x < n#-1 ";
by (resolve_tac [succ_leE] 1);
by (forward_tac [nat_into_Ord RS le_refl RS lt_trans] 1 THEN assume_tac 1);
by (asm_simp_tac (arith_ss addsimps [succ_pred]) 1);
val lt_pred = result();
goal Arith.thy
"!!i.[|n < succ(x); p<n; p:nat; n:nat; x:nat|]==> n#-1 < x ";
by (resolve_tac [succ_leE] 1);
by (asm_simp_tac (arith_ss addsimps [succ_pred]) 1);
val gt_pred = result();
val lift_ss = (union_ss addsimps
[add_0_right,add_succ_right,nat_into_Ord,
not_lt_iff_le,if_P,if_not_P]);
(* ------------------------------------------------------------------------- *)
(* lift_rec equality rules *)
(* ------------------------------------------------------------------------- *)
goalw Substitution.thy [lift_rec_def]
"!!n.[|n:nat; i:nat|]==> lift_rec(Var(i),n) =if(i<n,Var(i),Var(succ(i)))";
by (asm_full_simp_tac lift_ss 1);
val lift_rec_Var = result();
goalw Substitution.thy [lift_rec_def]
"!!n.[|n:nat; i:nat; k:nat; k le i|]==> lift_rec(Var(i),k) = Var(succ(i))";
by (asm_full_simp_tac lift_ss 1);
val lift_rec_le = result();
goalw Substitution.thy [lift_rec_def]
"!!n.[|i:nat; k:nat; i<k |]==> lift_rec(Var(i),k) = Var(i)";
by (asm_full_simp_tac lift_ss 1);
val lift_rec_gt = result();
goalw Substitution.thy [lift_rec_def]
"!!n.[|n:nat; k:nat|]==> \
\ lift_rec(Fun(t),k) = Fun(lift_rec(t,succ(k)))";
by (asm_full_simp_tac lift_ss 1);
val lift_rec_Fun = result();
goalw Substitution.thy [lift_rec_def]
"!!n.[|n:nat; k:nat|]==> \
\ lift_rec(App(b,f,a),k) = App(b,lift_rec(f,k),lift_rec(a,k))";
by (asm_full_simp_tac lift_ss 1);
val lift_rec_App = result();
(* ------------------------------------------------------------------------- *)
(* substitution quality rules *)
(* ------------------------------------------------------------------------- *)
goalw Substitution.thy [subst_rec_def]
"!!n.[|i:nat; k:nat; u:redexes|]==> \
\ subst_rec(u,Var(i),k) = if(k<i,Var(i#-1),if(k=i,u,Var(i)))";
by (asm_full_simp_tac (lift_ss addsimps [gt_not_eq,leI]) 1);
val subst_Var = result();
goalw Substitution.thy [subst_rec_def]
"!!n.[|n:nat; u:redexes|]==> subst_rec(u,Var(n),n) = u";
by (asm_full_simp_tac (lift_ss) 1);
val subst_eq = result();
goalw Substitution.thy [subst_rec_def]
"!!n.[|n:nat; u:redexes; p:nat; p<n|]==> \
\ subst_rec(u,Var(n),p) = Var(n#-1)";
by (asm_full_simp_tac (lift_ss) 1);
val subst_gt = result();
goalw Substitution.thy [subst_rec_def]
"!!n.[|n:nat; u:redexes; p:nat; n<p|]==> \
\ subst_rec(u,Var(n),p) = Var(n)";
by (asm_full_simp_tac (lift_ss addsimps [gt_not_eq,leI]) 1);
val subst_lt = result();
goalw Substitution.thy [subst_rec_def]
"!!n.[|p:nat; u:redexes|]==> \
\ subst_rec(u,Fun(t),p) = Fun(subst_rec(lift(u),t,succ(p))) ";
by (asm_full_simp_tac (lift_ss) 1);
val subst_Fun = result();
goalw Substitution.thy [subst_rec_def]
"!!n.[|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 (asm_full_simp_tac (lift_ss) 1);
val subst_App = result();
fun addsplit ss = (ss setloop (split_tac [expand_if])
addsimps [lift_rec_Var,subst_Var]);
goal Substitution.thy
"!!n.u:redexes ==> ALL k:nat.lift_rec(u,k):redexes";
by (eresolve_tac [redexes.induct] 1);
by (ALLGOALS(asm_full_simp_tac
((addsplit lift_ss) addsimps [lift_rec_Fun,lift_rec_App])));
val lift_rec_type_a = result();
val lift_rec_type = lift_rec_type_a RS bspec;
goalw Substitution.thy []
"!!n.v:redexes ==> ALL n:nat.ALL u:redexes.subst_rec(u,v,n):redexes";
by (eresolve_tac [redexes.induct] 1);
by (ALLGOALS(asm_full_simp_tac
((addsplit lift_ss) addsimps [subst_Fun,subst_App,
lift_rec_type])));
val subst_type_a = result();
val subst_type = subst_type_a RS bspec RS bspec;
val subst_ss = (lift_ss addsimps
[subst_eq,subst_gt,subst_lt,subst_Fun,subst_App,subst_type,
lift_rec_le,lift_rec_gt,lift_rec_Fun,lift_rec_App,
lift_rec_type]);
(* ------------------------------------------------------------------------- *)
(* lift and substitution proofs *)
(* ------------------------------------------------------------------------- *)
goalw Substitution.thy []
"!!n.u:redexes ==> ALL n:nat.ALL i:nat.i le n --> \
\ (lift_rec(lift_rec(u,i),succ(n)) = lift_rec(lift_rec(u,n),i))";
by ((eresolve_tac [redexes.induct] 1)
THEN (ALLGOALS(asm_simp_tac subst_ss)));
by (step_tac ZF_cs 1);
by (excluded_middle_tac "na < xa" 1);
by ((forward_tac [lt_trans2] 2) THEN (assume_tac 2));
by (ALLGOALS(asm_full_simp_tac
((addsplit subst_ss) addsimps [leI])));
val lift_lift_rec = result();
goalw Substitution.thy []
"!!n.[|u:redexes; n:nat|]==> \
\ lift_rec(lift(u),succ(n)) = lift(lift_rec(u,n))";
by (asm_simp_tac (lift_ss addsimps [lift_lift_rec]) 1);
val lift_lift = result();
goal Substitution.thy
"!!n.v:redexes ==> \
\ ALL n:nat.ALL m:nat.ALL u:redexes.n le m-->\
\ lift_rec(subst_rec(u,v,n),m) = \
\ subst_rec(lift_rec(u,m),lift_rec(v,succ(m)),n)";
by ((eresolve_tac [redexes.induct] 1)
THEN (ALLGOALS(asm_simp_tac (subst_ss addsimps [lift_lift]))));
by (step_tac ZF_cs 1);
by (excluded_middle_tac "na < x" 1);
by (asm_full_simp_tac (subst_ss) 1);
by (eres_inst_tac [("j","na")] leE 1);
by (asm_full_simp_tac ((addsplit subst_ss)
addsimps [leI,gt_pred,succ_pred]) 1);
by (hyp_subst_tac 1);
by (asm_full_simp_tac (subst_ss) 1);
by (forw_inst_tac [("j","x")] lt_trans2 1);
by (assume_tac 1);
by (asm_full_simp_tac (subst_ss addsimps [leI]) 1);
val lift_rec_subst_rec = result();
goalw Substitution.thy []
"!!n.[|v:redexes; u:redexes; n:nat|]==> \
\ lift_rec(u/v,n) = lift_rec(u,n)/lift_rec(v,succ(n))";
by (asm_full_simp_tac (subst_ss addsimps [lift_rec_subst_rec]) 1);
val lift_subst = result();
goalw Substitution.thy []
"!!n.v:redexes ==> \
\ ALL n:nat.ALL m:nat.ALL u:redexes.m le n-->\
\ lift_rec(subst_rec(u,v,n),m) = \
\ subst_rec(lift_rec(u,m),lift_rec(v,m),succ(n))";
by ((eresolve_tac [redexes.induct] 1)
THEN (ALLGOALS(asm_simp_tac (subst_ss addsimps [lift_lift]))));
by (step_tac ZF_cs 1);
by (excluded_middle_tac "na < x" 1);
by (asm_full_simp_tac (subst_ss) 1);
by (eres_inst_tac [("i","x")] leE 1);
by (forward_tac [lt_trans1] 1 THEN assume_tac 1);
by (ALLGOALS(asm_full_simp_tac
(subst_ss addsimps [succ_pred,leI,gt_pred])));
by (hyp_subst_tac 1);
by (asm_full_simp_tac (subst_ss addsimps [leI]) 1);
by (excluded_middle_tac "na < xa" 1);
by (asm_full_simp_tac (subst_ss) 1);
by (asm_full_simp_tac (subst_ss addsimps [leI]) 1);
val lift_rec_subst_rec_lt = result();
goalw Substitution.thy []
"!!n.u:redexes ==> \
\ ALL n:nat.ALL v:redexes.subst_rec(v,lift_rec(u,n),n) = u";
by ((eresolve_tac [redexes.induct] 1)
THEN (ALLGOALS(asm_simp_tac subst_ss)));
by (step_tac ZF_cs 1);
by (excluded_middle_tac "na < x" 1);
(* x <= na *)
by (asm_full_simp_tac (subst_ss) 1);
by (asm_full_simp_tac (subst_ss) 1);
val subst_rec_lift_rec = result();
goal Substitution.thy
"!!n.v:redexes ==> \
\ ALL m:nat.ALL n:nat.ALL u:redexes.ALL w:redexes.m le 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)";
by ((eresolve_tac [redexes.induct] 1) THEN
(ALLGOALS(asm_simp_tac (subst_ss addsimps
[lift_lift RS sym,lift_rec_subst_rec_lt]))));
by (step_tac ZF_cs 1);
by (excluded_middle_tac "na le succ(xa)" 1);
by (asm_full_simp_tac (subst_ss) 1);
by (forward_tac [nat_into_Ord RS le_refl RS lt_trans] 1 THEN assume_tac 1);
by (eresolve_tac [leE] 1);
by (asm_simp_tac (subst_ss addsimps [succ_pred,lt_pred]) 2);
by (forward_tac [succ_leI RS lt_trans] 1 THEN assume_tac 1);
by (forw_inst_tac [("i","x")]
(nat_into_Ord RS le_refl RS lt_trans) 1 THEN assume_tac 1);
by (asm_simp_tac (subst_ss addsimps [succ_pred,lt_pred]) 1);
by (eres_inst_tac [("i","na")] leE 1);
by (asm_full_simp_tac
(subst_ss addsimps [succ_pred,subst_rec_lift_rec,leI]) 2);
by (excluded_middle_tac "na < x" 1);
by (asm_full_simp_tac (subst_ss) 1);
by (eres_inst_tac [("j","na")] leE 1);
by (asm_simp_tac (subst_ss addsimps [gt_pred]) 1);
by (asm_simp_tac (subst_ss addsimps [subst_rec_lift_rec]) 1);
by (forward_tac [lt_trans2] 1 THEN assume_tac 1);
by (asm_simp_tac (subst_ss addsimps [gt_pred]) 1);
val subst_rec_subst_rec = result();
goalw Substitution.thy []
"!!n.[|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 (asm_simp_tac (subst_ss addsimps [subst_rec_subst_rec]) 1);
val substitution = result();
(* ------------------------------------------------------------------------- *)
(* Preservation lemmas *)
(* Substitution preserves comp and regular *)
(* ------------------------------------------------------------------------- *)
goal Substitution.thy
"!!n.[|n:nat; u ~ v|]==> ALL m:nat.lift_rec(u,m) ~ lift_rec(v,m)";
by (eresolve_tac [comp_induct] 1);
by (ALLGOALS(asm_simp_tac (subst_ss addsimps [comp_refl])));
val lift_rec_preserve_comp = result();
goal Substitution.thy
"!!n.u2 ~ v2 ==> ALL m:nat.ALL u1:redexes.ALL v1:redexes.\
\ u1 ~ v1--> subst_rec(u1,u2,m) ~ subst_rec(v1,v2,m)";
by (eresolve_tac [comp_induct] 1);
by (ALLGOALS(asm_full_simp_tac ((addsplit subst_ss) addsimps
([lift_rec_preserve_comp,comp_refl]))));
val subst_rec_preserve_comp = result();
goal Substitution.thy
"!!n.regular(u) ==> ALL m:nat.regular(lift_rec(u,m))";
by (eresolve_tac [Sreg.induct] 1);
by (ALLGOALS(asm_full_simp_tac (addsplit subst_ss)));
val lift_rec_preserve_reg = result();
goal Substitution.thy
"!!n.regular(v) ==> \
\ ALL m:nat.ALL u:redexes.regular(u)-->regular(subst_rec(u,v,m))";
by (eresolve_tac [Sreg.induct] 1);
by (ALLGOALS(asm_full_simp_tac ((addsplit subst_ss) addsimps
[lift_rec_preserve_reg])));
val subst_rec_preserve_reg = result();