--- a/Subst/Subst.ML Tue Oct 24 14:59:17 1995 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,185 +0,0 @@
-(* Title: Substitutions/subst.ML
- Author: Martin Coen, Cambridge University Computer Laboratory
- Copyright 1993 University of Cambridge
-
-For subst.thy.
-*)
-
-open Subst;
-
-(***********)
-
-val subst_defs = [subst_def,comp_def,sdom_def];
-
-val raw_subst_ss = utlemmas_ss addsimps al_rews;
-
-local fun mk_thm s = prove_goalw Subst.thy subst_defs s
- (fn _ => [simp_tac raw_subst_ss 1])
-in val subst_rews = map mk_thm
-["Const(c) <| al = Const(c)",
- "Comb(t,u) <| al = Comb(t <| al, u <| al)",
- "[] <> bl = bl",
- "<a,b>#al <> bl = <a,b <| bl> # (al <> bl)",
- "sdom([]) = {}",
- "sdom(<a,b>#al) = if(Var(a)=b,sdom(al) Int Compl({a}),sdom(al) Un {a})"
-];
- (* This rewrite isn't always desired *)
- val Var_subst = mk_thm "Var(x) <| al = assoc(x,Var(x),al)";
-end;
-
-val subst_ss = raw_subst_ss addsimps subst_rews;
-
-(**** Substitutions ****)
-
-goal Subst.thy "t <| [] = t";
-by (uterm_ind_tac "t" 1);
-by (ALLGOALS (asm_simp_tac (subst_ss addsimps [Var_subst])));
-qed "subst_Nil";
-
-goal Subst.thy "t <: u --> t <| s <: u <| s";
-by (uterm_ind_tac "u" 1);
-by (ALLGOALS (asm_simp_tac subst_ss));
-val subst_mono = store_thm("subst_mono", result() RS mp);
-
-goal Subst.thy "~ (Var(v) <: t) --> t <| <v,t <| s>#s = t <| s";
-by (imp_excluded_middle_tac "t = Var(v)" 1);
-by (res_inst_tac [("P",
- "%x.~x=Var(v) --> ~(Var(v) <: x) --> x <| <v,t<|s>#s=x<|s")]
- uterm_induct 2);
-by (ALLGOALS (simp_tac (subst_ss addsimps [Var_subst])));
-by (fast_tac HOL_cs 1);
-val Var_not_occs = store_thm("Var_not_occs", result() RS mp);
-
-goal Subst.thy
- "(t <|r = t <|s) = (! v.v : vars_of(t) --> Var(v) <|r = Var(v) <|s)";
-by (uterm_ind_tac "t" 1);
-by (REPEAT (etac rev_mp 3));
-by (ALLGOALS (asm_simp_tac subst_ss));
-by (ALLGOALS (fast_tac HOL_cs));
-qed "agreement";
-
-goal Subst.thy "~ v: vars_of(t) --> t <| <v,u>#s = t <| s";
-by(simp_tac(subst_ss addsimps [agreement,Var_subst]
- setloop (split_tac [expand_if])) 1);
-val repl_invariance = store_thm("repl_invariance", result() RS mp);
-
-val asms = goal Subst.thy
- "v : vars_of(t) --> w : vars_of(t <| <v,Var(w)>#s)";
-by (uterm_ind_tac "t" 1);
-by (ALLGOALS (asm_simp_tac (subst_ss addsimps [Var_subst])));
-val Var_in_subst = store_thm("Var_in_subst", result() RS mp);
-
-(**** Equality between Substitutions ****)
-
-goalw Subst.thy [subst_eq_def] "r =s= s = (! t.t <| r = t <| s)";
-by (simp_tac subst_ss 1);
-qed "subst_eq_iff";
-
-local fun mk_thm s = prove_goal Subst.thy s
- (fn prems => [cut_facts_tac prems 1,
- REPEAT (etac rev_mp 1),
- simp_tac (subst_ss addsimps [subst_eq_iff]) 1])
-in
- val subst_refl = mk_thm "r = s ==> r =s= s";
- val subst_sym = mk_thm "r =s= s ==> s =s= r";
- val subst_trans = mk_thm "[| q =s= r; r =s= s |] ==> q =s= s";
-end;
-
-val eq::prems = goalw Subst.thy [subst_eq_def]
- "[| r =s= s; P(t <| r,u <| r) |] ==> P(t <| s,u <| s)";
-by (resolve_tac [eq RS spec RS subst] 1);
-by (resolve_tac (prems RL [eq RS spec RS subst]) 1);
-qed "subst_subst2";
-
-val ssubst_subst2 = subst_sym RS subst_subst2;
-
-(**** Composition of Substitutions ****)
-
-goal Subst.thy "s <> [] = s";
-by (alist_ind_tac "s" 1);
-by (ALLGOALS (asm_simp_tac (subst_ss addsimps [subst_Nil])));
-qed "comp_Nil";
-
-goal Subst.thy "(t <| r <> s) = (t <| r <| s)";
-by (uterm_ind_tac "t" 1);
-by (ALLGOALS (asm_simp_tac (subst_ss addsimps [Var_subst])));
-by (alist_ind_tac "r" 1);
-by (ALLGOALS (asm_simp_tac (subst_ss addsimps [Var_subst,subst_Nil]
- setloop (split_tac [expand_if]))));
-qed "subst_comp";
-
-goal Subst.thy "q <> r <> s =s= q <> (r <> s)";
-by (simp_tac (subst_ss addsimps [subst_eq_iff,subst_comp]) 1);
-qed "comp_assoc";
-
-goal Subst.thy "<w,Var(w) <| s>#s =s= s";
-by (rtac (allI RS (subst_eq_iff RS iffD2)) 1);
-by (uterm_ind_tac "t" 1);
-by (REPEAT (etac rev_mp 3));
-by (ALLGOALS (simp_tac (subst_ss addsimps[Var_subst]
- setloop (split_tac [expand_if]))));
-qed "Cons_trivial";
-
-val [prem] = goal Subst.thy "q <> r =s= s ==> t <| q <| r = t <| s";
-by (simp_tac (subst_ss addsimps [prem RS (subst_eq_iff RS iffD1),
- subst_comp RS sym]) 1);
-qed "comp_subst_subst";
-
-(**** Domain and range of Substitutions ****)
-
-goal Subst.thy "(v : sdom(s)) = (~ Var(v) <| s = Var(v))";
-by (alist_ind_tac "s" 1);
-by (ALLGOALS (asm_simp_tac (subst_ss addsimps [Var_subst]
- setloop (split_tac[expand_if]))));
-by (fast_tac HOL_cs 1);
-qed "sdom_iff";
-
-goalw Subst.thy [srange_def]
- "v : srange(s) = (? w.w : sdom(s) & v : vars_of(Var(w) <| s))";
-by (fast_tac set_cs 1);
-qed "srange_iff";
-
-goal Subst.thy "(t <| s = t) = (sdom(s) Int vars_of(t) = {})";
-by (uterm_ind_tac "t" 1);
-by (REPEAT (etac rev_mp 3));
-by (ALLGOALS (simp_tac (subst_ss addsimps [sdom_iff,Var_subst])));
-by (ALLGOALS (fast_tac set_cs));
-qed "invariance";
-
-goal Subst.thy "v : sdom(s) --> ~v : srange(s) --> ~v : vars_of(t <| s)";
-by (uterm_ind_tac "t" 1);
-by (imp_excluded_middle_tac "x : sdom(s)" 1);
-by (ALLGOALS (asm_simp_tac (subst_ss addsimps [sdom_iff,srange_iff])));
-by (ALLGOALS (fast_tac set_cs));
-val Var_elim = store_thm("Var_elim", result() RS mp RS mp);
-
-val asms = goal Subst.thy
- "[| v : sdom(s); v : vars_of(t <| s) |] ==> v : srange(s)";
-by (REPEAT (ares_tac (asms @ [Var_elim RS swap RS classical]) 1));
-qed "Var_elim2";
-
-goal Subst.thy "v : vars_of(t <| s) --> v : srange(s) | v : vars_of(t)";
-by (uterm_ind_tac "t" 1);
-by (REPEAT_SOME (etac rev_mp ));
-by (ALLGOALS (simp_tac (subst_ss addsimps [sdom_iff,srange_iff])));
-by (REPEAT (step_tac (set_cs addIs [vars_var_iff RS iffD1 RS sym]) 1));
-by (etac notE 1);
-by (etac subst 1);
-by (ALLGOALS (fast_tac set_cs));
-val Var_intro = store_thm("Var_intro", result() RS mp);
-
-goal Subst.thy
- "v : srange(s) --> (? w.w : sdom(s) & v : vars_of(Var(w) <| s))";
-by (simp_tac (subst_ss addsimps [srange_iff]) 1);
-val srangeE = store_thm("srangeE", make_elim (result() RS mp));
-
-val asms = goal Subst.thy
- "sdom(s) Int srange(s) = {} = (! t.sdom(s) Int vars_of(t <| s) = {})";
-by (simp_tac subst_ss 1);
-by (fast_tac (set_cs addIs [Var_elim2] addEs [srangeE]) 1);
-qed "dom_range_disjoint";
-
-val asms = goal Subst.thy "~ u <| s = u --> (? x.x : sdom(s))";
-by (simp_tac (subst_ss addsimps [invariance]) 1);
-by (fast_tac set_cs 1);
-val subst_not_empty = store_thm("subst_not_empty", result() RS mp);