diff -r 000000000000 -r 7949f97df77a Subst/Subst.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Subst/Subst.ML Thu Sep 16 12:21:07 1993 +0200 @@ -0,0 +1,185 @@ +(* 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", + "Cons(,al) <> bl = Cons(, al <> bl)", + "sdom(Nil) = {}", + "sdom(Cons(,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 <| Cons(,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 <| Cons(,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 <| Cons(,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 <| Cons(,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 "Cons(,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;