(* 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)",
"Nil <> bl = bl",
"<a,b>#al <> bl = <a,b <| bl> # (al <> bl)",
"sdom(Nil) = {}",
"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 <| Nil = t";
by (uterm_ind_tac "t" 1);
by (ALLGOALS (asm_simp_tac (subst_ss addsimps [Var_subst])));
val subst_Nil = result();
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 = 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 = 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));
val agreement = result();
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 = 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 = 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);
val subst_eq_iff = result();
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);
val subst_subst2 = result();
val ssubst_subst2 = subst_sym RS subst_subst2;
(**** Composition of Substitutions ****)
goal Subst.thy "s <> Nil = s";
by (alist_ind_tac "s" 1);
by (ALLGOALS (asm_simp_tac (subst_ss addsimps [subst_Nil])));
val comp_Nil = result();
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]))));
val subst_comp = result();
goal Subst.thy "q <> r <> s =s= q <> (r <> s)";
by (simp_tac (subst_ss addsimps [subst_eq_iff,subst_comp]) 1);
val comp_assoc = result();
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]))));
val Cons_trivial = result();
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);
val comp_subst_subst = result();
(**** 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);
val sdom_iff = result();
goalw Subst.thy [srange_def]
"v : srange(s) = (? w.w : sdom(s) & v : vars_of(Var(w) <| s))";
by (fast_tac set_cs 1);
val srange_iff = result();
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));
val invariance = result();
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 = 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));
val Var_elim2 = result();
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 = 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 = 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);
val dom_range_disjoint = result();
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 = result() RS mp;