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