(* Title: HOL/Subst/subst.ML
ID: $Id$
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 = simpset_of "UTLemmas" 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 then (sdom al) Int Compl({a}) \
\ else (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);