1 (* Title: Substitutions/subst.ML |
|
2 Author: Martin Coen, Cambridge University Computer Laboratory |
|
3 Copyright 1993 University of Cambridge |
|
4 |
|
5 For subst.thy. |
|
6 *) |
|
7 |
|
8 open Subst; |
|
9 |
|
10 (***********) |
|
11 |
|
12 val subst_defs = [subst_def,comp_def,sdom_def]; |
|
13 |
|
14 val raw_subst_ss = utlemmas_ss addsimps al_rews; |
|
15 |
|
16 local fun mk_thm s = prove_goalw Subst.thy subst_defs s |
|
17 (fn _ => [simp_tac raw_subst_ss 1]) |
|
18 in val subst_rews = map mk_thm |
|
19 ["Const(c) <| al = Const(c)", |
|
20 "Comb(t,u) <| al = Comb(t <| al, u <| al)", |
|
21 "[] <> bl = bl", |
|
22 "<a,b>#al <> bl = <a,b <| bl> # (al <> bl)", |
|
23 "sdom([]) = {}", |
|
24 "sdom(<a,b>#al) = if(Var(a)=b,sdom(al) Int Compl({a}),sdom(al) Un {a})" |
|
25 ]; |
|
26 (* This rewrite isn't always desired *) |
|
27 val Var_subst = mk_thm "Var(x) <| al = assoc(x,Var(x),al)"; |
|
28 end; |
|
29 |
|
30 val subst_ss = raw_subst_ss addsimps subst_rews; |
|
31 |
|
32 (**** Substitutions ****) |
|
33 |
|
34 goal Subst.thy "t <| [] = t"; |
|
35 by (uterm_ind_tac "t" 1); |
|
36 by (ALLGOALS (asm_simp_tac (subst_ss addsimps [Var_subst]))); |
|
37 qed "subst_Nil"; |
|
38 |
|
39 goal Subst.thy "t <: u --> t <| s <: u <| s"; |
|
40 by (uterm_ind_tac "u" 1); |
|
41 by (ALLGOALS (asm_simp_tac subst_ss)); |
|
42 val subst_mono = store_thm("subst_mono", result() RS mp); |
|
43 |
|
44 goal Subst.thy "~ (Var(v) <: t) --> t <| <v,t <| s>#s = t <| s"; |
|
45 by (imp_excluded_middle_tac "t = Var(v)" 1); |
|
46 by (res_inst_tac [("P", |
|
47 "%x.~x=Var(v) --> ~(Var(v) <: x) --> x <| <v,t<|s>#s=x<|s")] |
|
48 uterm_induct 2); |
|
49 by (ALLGOALS (simp_tac (subst_ss addsimps [Var_subst]))); |
|
50 by (fast_tac HOL_cs 1); |
|
51 val Var_not_occs = store_thm("Var_not_occs", result() RS mp); |
|
52 |
|
53 goal Subst.thy |
|
54 "(t <|r = t <|s) = (! v.v : vars_of(t) --> Var(v) <|r = Var(v) <|s)"; |
|
55 by (uterm_ind_tac "t" 1); |
|
56 by (REPEAT (etac rev_mp 3)); |
|
57 by (ALLGOALS (asm_simp_tac subst_ss)); |
|
58 by (ALLGOALS (fast_tac HOL_cs)); |
|
59 qed "agreement"; |
|
60 |
|
61 goal Subst.thy "~ v: vars_of(t) --> t <| <v,u>#s = t <| s"; |
|
62 by(simp_tac(subst_ss addsimps [agreement,Var_subst] |
|
63 setloop (split_tac [expand_if])) 1); |
|
64 val repl_invariance = store_thm("repl_invariance", result() RS mp); |
|
65 |
|
66 val asms = goal Subst.thy |
|
67 "v : vars_of(t) --> w : vars_of(t <| <v,Var(w)>#s)"; |
|
68 by (uterm_ind_tac "t" 1); |
|
69 by (ALLGOALS (asm_simp_tac (subst_ss addsimps [Var_subst]))); |
|
70 val Var_in_subst = store_thm("Var_in_subst", result() RS mp); |
|
71 |
|
72 (**** Equality between Substitutions ****) |
|
73 |
|
74 goalw Subst.thy [subst_eq_def] "r =s= s = (! t.t <| r = t <| s)"; |
|
75 by (simp_tac subst_ss 1); |
|
76 qed "subst_eq_iff"; |
|
77 |
|
78 local fun mk_thm s = prove_goal Subst.thy s |
|
79 (fn prems => [cut_facts_tac prems 1, |
|
80 REPEAT (etac rev_mp 1), |
|
81 simp_tac (subst_ss addsimps [subst_eq_iff]) 1]) |
|
82 in |
|
83 val subst_refl = mk_thm "r = s ==> r =s= s"; |
|
84 val subst_sym = mk_thm "r =s= s ==> s =s= r"; |
|
85 val subst_trans = mk_thm "[| q =s= r; r =s= s |] ==> q =s= s"; |
|
86 end; |
|
87 |
|
88 val eq::prems = goalw Subst.thy [subst_eq_def] |
|
89 "[| r =s= s; P(t <| r,u <| r) |] ==> P(t <| s,u <| s)"; |
|
90 by (resolve_tac [eq RS spec RS subst] 1); |
|
91 by (resolve_tac (prems RL [eq RS spec RS subst]) 1); |
|
92 qed "subst_subst2"; |
|
93 |
|
94 val ssubst_subst2 = subst_sym RS subst_subst2; |
|
95 |
|
96 (**** Composition of Substitutions ****) |
|
97 |
|
98 goal Subst.thy "s <> [] = s"; |
|
99 by (alist_ind_tac "s" 1); |
|
100 by (ALLGOALS (asm_simp_tac (subst_ss addsimps [subst_Nil]))); |
|
101 qed "comp_Nil"; |
|
102 |
|
103 goal Subst.thy "(t <| r <> s) = (t <| r <| s)"; |
|
104 by (uterm_ind_tac "t" 1); |
|
105 by (ALLGOALS (asm_simp_tac (subst_ss addsimps [Var_subst]))); |
|
106 by (alist_ind_tac "r" 1); |
|
107 by (ALLGOALS (asm_simp_tac (subst_ss addsimps [Var_subst,subst_Nil] |
|
108 setloop (split_tac [expand_if])))); |
|
109 qed "subst_comp"; |
|
110 |
|
111 goal Subst.thy "q <> r <> s =s= q <> (r <> s)"; |
|
112 by (simp_tac (subst_ss addsimps [subst_eq_iff,subst_comp]) 1); |
|
113 qed "comp_assoc"; |
|
114 |
|
115 goal Subst.thy "<w,Var(w) <| s>#s =s= s"; |
|
116 by (rtac (allI RS (subst_eq_iff RS iffD2)) 1); |
|
117 by (uterm_ind_tac "t" 1); |
|
118 by (REPEAT (etac rev_mp 3)); |
|
119 by (ALLGOALS (simp_tac (subst_ss addsimps[Var_subst] |
|
120 setloop (split_tac [expand_if])))); |
|
121 qed "Cons_trivial"; |
|
122 |
|
123 val [prem] = goal Subst.thy "q <> r =s= s ==> t <| q <| r = t <| s"; |
|
124 by (simp_tac (subst_ss addsimps [prem RS (subst_eq_iff RS iffD1), |
|
125 subst_comp RS sym]) 1); |
|
126 qed "comp_subst_subst"; |
|
127 |
|
128 (**** Domain and range of Substitutions ****) |
|
129 |
|
130 goal Subst.thy "(v : sdom(s)) = (~ Var(v) <| s = Var(v))"; |
|
131 by (alist_ind_tac "s" 1); |
|
132 by (ALLGOALS (asm_simp_tac (subst_ss addsimps [Var_subst] |
|
133 setloop (split_tac[expand_if])))); |
|
134 by (fast_tac HOL_cs 1); |
|
135 qed "sdom_iff"; |
|
136 |
|
137 goalw Subst.thy [srange_def] |
|
138 "v : srange(s) = (? w.w : sdom(s) & v : vars_of(Var(w) <| s))"; |
|
139 by (fast_tac set_cs 1); |
|
140 qed "srange_iff"; |
|
141 |
|
142 goal Subst.thy "(t <| s = t) = (sdom(s) Int vars_of(t) = {})"; |
|
143 by (uterm_ind_tac "t" 1); |
|
144 by (REPEAT (etac rev_mp 3)); |
|
145 by (ALLGOALS (simp_tac (subst_ss addsimps [sdom_iff,Var_subst]))); |
|
146 by (ALLGOALS (fast_tac set_cs)); |
|
147 qed "invariance"; |
|
148 |
|
149 goal Subst.thy "v : sdom(s) --> ~v : srange(s) --> ~v : vars_of(t <| s)"; |
|
150 by (uterm_ind_tac "t" 1); |
|
151 by (imp_excluded_middle_tac "x : sdom(s)" 1); |
|
152 by (ALLGOALS (asm_simp_tac (subst_ss addsimps [sdom_iff,srange_iff]))); |
|
153 by (ALLGOALS (fast_tac set_cs)); |
|
154 val Var_elim = store_thm("Var_elim", result() RS mp RS mp); |
|
155 |
|
156 val asms = goal Subst.thy |
|
157 "[| v : sdom(s); v : vars_of(t <| s) |] ==> v : srange(s)"; |
|
158 by (REPEAT (ares_tac (asms @ [Var_elim RS swap RS classical]) 1)); |
|
159 qed "Var_elim2"; |
|
160 |
|
161 goal Subst.thy "v : vars_of(t <| s) --> v : srange(s) | v : vars_of(t)"; |
|
162 by (uterm_ind_tac "t" 1); |
|
163 by (REPEAT_SOME (etac rev_mp )); |
|
164 by (ALLGOALS (simp_tac (subst_ss addsimps [sdom_iff,srange_iff]))); |
|
165 by (REPEAT (step_tac (set_cs addIs [vars_var_iff RS iffD1 RS sym]) 1)); |
|
166 by (etac notE 1); |
|
167 by (etac subst 1); |
|
168 by (ALLGOALS (fast_tac set_cs)); |
|
169 val Var_intro = store_thm("Var_intro", result() RS mp); |
|
170 |
|
171 goal Subst.thy |
|
172 "v : srange(s) --> (? w.w : sdom(s) & v : vars_of(Var(w) <| s))"; |
|
173 by (simp_tac (subst_ss addsimps [srange_iff]) 1); |
|
174 val srangeE = store_thm("srangeE", make_elim (result() RS mp)); |
|
175 |
|
176 val asms = goal Subst.thy |
|
177 "sdom(s) Int srange(s) = {} = (! t.sdom(s) Int vars_of(t <| s) = {})"; |
|
178 by (simp_tac subst_ss 1); |
|
179 by (fast_tac (set_cs addIs [Var_elim2] addEs [srangeE]) 1); |
|
180 qed "dom_range_disjoint"; |
|
181 |
|
182 val asms = goal Subst.thy "~ u <| s = u --> (? x.x : sdom(s))"; |
|
183 by (simp_tac (subst_ss addsimps [invariance]) 1); |
|
184 by (fast_tac set_cs 1); |
|
185 val subst_not_empty = store_thm("subst_not_empty", result() RS mp); |
|