author  wenzelm 
Mon, 24 Oct 2016 15:00:13 +0200  
changeset 64375  74a2af7c5145 
parent 63505  42e1dece537a 
child 64980  7dc25cf5793e 
permissions  rwrr 
17441  1 
(* Title: CTT/CTT.thy 
0  2 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
3 
Copyright 1993 University of Cambridge 

4 
*) 

5 

60770  6 
section \<open>Constructive Type Theory\<close> 
0  7 

17441  8 
theory CTT 
9 
imports Pure 

10 
begin 

11 

48891  12 
ML_file "~~/src/Provers/typedsimp.ML" 
39557
fe5722fce758
renamed structure PureThy to Pure_Thy and moved most content to Global_Theory, to emphasize that this is globalonly;
wenzelm
parents:
35762
diff
changeset

13 
setup Pure_Thy.old_appl_syntax_setup 
26956
1309a6a0a29f
setup PureThy.old_appl_syntax_setup  theory Pure provides regular application syntax by default;
wenzelm
parents:
26391
diff
changeset

14 

17441  15 
typedecl i 
16 
typedecl t 

17 
typedecl o 

0  18 

19 
consts 

63505  20 
\<comment> \<open>Types\<close> 
17441  21 
F :: "t" 
63505  22 
T :: "t" \<comment> \<open>\<open>F\<close> is empty, \<open>T\<close> contains one element\<close> 
58977  23 
contr :: "i\<Rightarrow>i" 
0  24 
tt :: "i" 
63505  25 
\<comment> \<open>Natural numbers\<close> 
0  26 
N :: "t" 
58977  27 
succ :: "i\<Rightarrow>i" 
28 
rec :: "[i, i, [i,i]\<Rightarrow>i] \<Rightarrow> i" 

63505  29 
\<comment> \<open>Unions\<close> 
58977  30 
inl :: "i\<Rightarrow>i" 
31 
inr :: "i\<Rightarrow>i" 

60555
51a6997b1384
support 'when' statement, which corresponds to 'presume';
wenzelm
parents:
59780
diff
changeset

32 
"when" :: "[i, i\<Rightarrow>i, i\<Rightarrow>i]\<Rightarrow>i" 
63505  33 
\<comment> \<open>General Sum and Binary Product\<close> 
58977  34 
Sum :: "[t, i\<Rightarrow>t]\<Rightarrow>t" 
35 
fst :: "i\<Rightarrow>i" 

36 
snd :: "i\<Rightarrow>i" 

37 
split :: "[i, [i,i]\<Rightarrow>i] \<Rightarrow>i" 

63505  38 
\<comment> \<open>General Product and Function Space\<close> 
58977  39 
Prod :: "[t, i\<Rightarrow>t]\<Rightarrow>t" 
63505  40 
\<comment> \<open>Types\<close> 
58977  41 
Plus :: "[t,t]\<Rightarrow>t" (infixr "+" 40) 
63505  42 
\<comment> \<open>Equality type\<close> 
58977  43 
Eq :: "[t,i,i]\<Rightarrow>t" 
0  44 
eq :: "i" 
63505  45 
\<comment> \<open>Judgements\<close> 
58977  46 
Type :: "t \<Rightarrow> prop" ("(_ type)" [10] 5) 
47 
Eqtype :: "[t,t]\<Rightarrow>prop" ("(_ =/ _)" [10,10] 5) 

48 
Elem :: "[i, t]\<Rightarrow>prop" ("(_ /: _)" [10,10] 5) 

49 
Eqelem :: "[i,i,t]\<Rightarrow>prop" ("(_ =/ _ :/ _)" [10,10,10] 5) 

50 
Reduce :: "[i,i]\<Rightarrow>prop" ("Reduce[_,_]") 

14765  51 

63505  52 
\<comment> \<open>Types\<close> 
53 

54 
\<comment> \<open>Functions\<close> 

61391  55 
lambda :: "(i \<Rightarrow> i) \<Rightarrow> i" (binder "\<^bold>\<lambda>" 10) 
58977  56 
app :: "[i,i]\<Rightarrow>i" (infixl "`" 60) 
63505  57 
\<comment> \<open>Natural numbers\<close> 
41310  58 
Zero :: "i" ("0") 
63505  59 
\<comment> \<open>Pairing\<close> 
58977  60 
pair :: "[i,i]\<Rightarrow>i" ("(1<_,/_>)") 
0  61 

14765  62 
syntax 
61391  63 
"_PROD" :: "[idt,t,t]\<Rightarrow>t" ("(3\<Prod>_:_./ _)" 10) 
64 
"_SUM" :: "[idt,t,t]\<Rightarrow>t" ("(3\<Sum>_:_./ _)" 10) 

0  65 
translations 
61391  66 
"\<Prod>x:A. B" \<rightleftharpoons> "CONST Prod(A, \<lambda>x. B)" 
67 
"\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum(A, \<lambda>x. B)" 

19761  68 

63505  69 
abbreviation Arrow :: "[t,t]\<Rightarrow>t" (infixr "\<longrightarrow>" 30) 
70 
where "A \<longrightarrow> B \<equiv> \<Prod>_:A. B" 

71 

72 
abbreviation Times :: "[t,t]\<Rightarrow>t" (infixr "\<times>" 50) 

73 
where "A \<times> B \<equiv> \<Sum>_:A. B" 

10467
e6e7205e9e91
xsymbol support for Pi, Sigma, >, : (membership)
paulson
parents:
3837
diff
changeset

74 

63505  75 
text \<open> 
76 
Reduction: a weaker notion than equality; a hack for simplification. 

77 
\<open>Reduce[a,b]\<close> means either that \<open>a = b : A\<close> for some \<open>A\<close> or else 

78 
that \<open>a\<close> and \<open>b\<close> are textually identical. 

0  79 

63505  80 
Does not verify \<open>a:A\<close>! Sound because only \<open>trans_red\<close> uses a \<open>Reduce\<close> 
81 
premise. No new theorems can be proved about the standard judgements. 

82 
\<close> 

83 
axiomatization 

84 
where 

51308  85 
refl_red: "\<And>a. Reduce[a,a]" and 
58977  86 
red_if_equal: "\<And>a b A. a = b : A \<Longrightarrow> Reduce[a,b]" and 
87 
trans_red: "\<And>a b c A. \<lbrakk>a = b : A; Reduce[b,c]\<rbrakk> \<Longrightarrow> a = c : A" and 

0  88 

63505  89 
\<comment> \<open>Reflexivity\<close> 
0  90 

58977  91 
refl_type: "\<And>A. A type \<Longrightarrow> A = A" and 
92 
refl_elem: "\<And>a A. a : A \<Longrightarrow> a = a : A" and 

0  93 

63505  94 
\<comment> \<open>Symmetry\<close> 
0  95 

58977  96 
sym_type: "\<And>A B. A = B \<Longrightarrow> B = A" and 
97 
sym_elem: "\<And>a b A. a = b : A \<Longrightarrow> b = a : A" and 

0  98 

63505  99 
\<comment> \<open>Transitivity\<close> 
0  100 

58977  101 
trans_type: "\<And>A B C. \<lbrakk>A = B; B = C\<rbrakk> \<Longrightarrow> A = C" and 
102 
trans_elem: "\<And>a b c A. \<lbrakk>a = b : A; b = c : A\<rbrakk> \<Longrightarrow> a = c : A" and 

0  103 

58977  104 
equal_types: "\<And>a A B. \<lbrakk>a : A; A = B\<rbrakk> \<Longrightarrow> a : B" and 
105 
equal_typesL: "\<And>a b A B. \<lbrakk>a = b : A; A = B\<rbrakk> \<Longrightarrow> a = b : B" and 

0  106 

63505  107 
\<comment> \<open>Substitution\<close> 
0  108 

58977  109 
subst_type: "\<And>a A B. \<lbrakk>a : A; \<And>z. z:A \<Longrightarrow> B(z) type\<rbrakk> \<Longrightarrow> B(a) type" and 
110 
subst_typeL: "\<And>a c A B D. \<lbrakk>a = c : A; \<And>z. z:A \<Longrightarrow> B(z) = D(z)\<rbrakk> \<Longrightarrow> B(a) = D(c)" and 

0  111 

58977  112 
subst_elem: "\<And>a b A B. \<lbrakk>a : A; \<And>z. z:A \<Longrightarrow> b(z):B(z)\<rbrakk> \<Longrightarrow> b(a):B(a)" and 
17441  113 
subst_elemL: 
58977  114 
"\<And>a b c d A B. \<lbrakk>a = c : A; \<And>z. z:A \<Longrightarrow> b(z)=d(z) : B(z)\<rbrakk> \<Longrightarrow> b(a)=d(c) : B(a)" and 
0  115 

116 

63505  117 
\<comment> \<open>The type \<open>N\<close>  natural numbers\<close> 
0  118 

51308  119 
NF: "N type" and 
120 
NI0: "0 : N" and 

58977  121 
NI_succ: "\<And>a. a : N \<Longrightarrow> succ(a) : N" and 
122 
NI_succL: "\<And>a b. a = b : N \<Longrightarrow> succ(a) = succ(b) : N" and 

0  123 

17441  124 
NE: 
58977  125 
"\<And>p a b C. \<lbrakk>p: N; a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk> 
126 
\<Longrightarrow> rec(p, a, \<lambda>u v. b(u,v)) : C(p)" and 

0  127 

17441  128 
NEL: 
58977  129 
"\<And>p q a b c d C. \<lbrakk>p = q : N; a = c : C(0); 
130 
\<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v) = d(u,v): C(succ(u))\<rbrakk> 

131 
\<Longrightarrow> rec(p, a, \<lambda>u v. b(u,v)) = rec(q,c,d) : C(p)" and 

0  132 

17441  133 
NC0: 
58977  134 
"\<And>a b C. \<lbrakk>a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk> 
135 
\<Longrightarrow> rec(0, a, \<lambda>u v. b(u,v)) = a : C(0)" and 

0  136 

17441  137 
NC_succ: 
58977  138 
"\<And>p a b C. \<lbrakk>p: N; a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk> \<Longrightarrow> 
139 
rec(succ(p), a, \<lambda>u v. b(u,v)) = b(p, rec(p, a, \<lambda>u v. b(u,v))) : C(succ(p))" and 

0  140 

63505  141 
\<comment> \<open>The fourth Peano axiom. See page 91 of MartinLÃ¶f's book.\<close> 
58977  142 
zero_ne_succ: "\<And>a. \<lbrakk>a: N; 0 = succ(a) : N\<rbrakk> \<Longrightarrow> 0: F" and 
0  143 

144 

63505  145 
\<comment> \<open>The Product of a family of types\<close> 
0  146 

61391  147 
ProdF: "\<And>A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> B(x) type\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) type" and 
0  148 

17441  149 
ProdFL: 
61391  150 
"\<And>A B C D. \<lbrakk>A = C; \<And>x. x:A \<Longrightarrow> B(x) = D(x)\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) = \<Prod>x:C. D(x)" and 
0  151 

17441  152 
ProdI: 
61391  153 
"\<And>b A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> b(x):B(x)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x) : \<Prod>x:A. B(x)" and 
0  154 

58977  155 
ProdIL: "\<And>b c A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> b(x) = c(x) : B(x)\<rbrakk> \<Longrightarrow> 
61391  156 
\<^bold>\<lambda>x. b(x) = \<^bold>\<lambda>x. c(x) : \<Prod>x:A. B(x)" and 
0  157 

61391  158 
ProdE: "\<And>p a A B. \<lbrakk>p : \<Prod>x:A. B(x); a : A\<rbrakk> \<Longrightarrow> p`a : B(a)" and 
159 
ProdEL: "\<And>p q a b A B. \<lbrakk>p = q: \<Prod>x:A. B(x); a = b : A\<rbrakk> \<Longrightarrow> p`a = q`b : B(a)" and 

0  160 

61391  161 
ProdC: "\<And>a b A B. \<lbrakk>a : A; \<And>x. x:A \<Longrightarrow> b(x) : B(x)\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b(x)) ` a = b(a) : B(a)" and 
0  162 

61391  163 
ProdC2: "\<And>p A B. p : \<Prod>x:A. B(x) \<Longrightarrow> (\<^bold>\<lambda>x. p`x) = p : \<Prod>x:A. B(x)" and 
0  164 

165 

63505  166 
\<comment> \<open>The Sum of a family of types\<close> 
0  167 

61391  168 
SumF: "\<And>A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> B(x) type\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x) type" and 
169 
SumFL: "\<And>A B C D. \<lbrakk>A = C; \<And>x. x:A \<Longrightarrow> B(x) = D(x)\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x) = \<Sum>x:C. D(x)" and 

0  170 

61391  171 
SumI: "\<And>a b A B. \<lbrakk>a : A; b : B(a)\<rbrakk> \<Longrightarrow> <a,b> : \<Sum>x:A. B(x)" and 
172 
SumIL: "\<And>a b c d A B. \<lbrakk> a = c : A; b = d : B(a)\<rbrakk> \<Longrightarrow> <a,b> = <c,d> : \<Sum>x:A. B(x)" and 

0  173 

61391  174 
SumE: "\<And>p c A B C. \<lbrakk>p: \<Sum>x:A. B(x); \<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y): C(<x,y>)\<rbrakk> 
58977  175 
\<Longrightarrow> split(p, \<lambda>x y. c(x,y)) : C(p)" and 
0  176 

61391  177 
SumEL: "\<And>p q c d A B C. \<lbrakk>p = q : \<Sum>x:A. B(x); 
58977  178 
\<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y)=d(x,y): C(<x,y>)\<rbrakk> 
179 
\<Longrightarrow> split(p, \<lambda>x y. c(x,y)) = split(q, \<lambda>x y. d(x,y)) : C(p)" and 

0  180 

58977  181 
SumC: "\<And>a b c A B C. \<lbrakk>a: A; b: B(a); \<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y): C(<x,y>)\<rbrakk> 
182 
\<Longrightarrow> split(<a,b>, \<lambda>x y. c(x,y)) = c(a,b) : C(<a,b>)" and 

0  183 

61391  184 
fst_def: "\<And>a. fst(a) \<equiv> split(a, \<lambda>x y. x)" and 
185 
snd_def: "\<And>a. snd(a) \<equiv> split(a, \<lambda>x y. y)" and 

0  186 

187 

63505  188 
\<comment> \<open>The sum of two types\<close> 
0  189 

58977  190 
PlusF: "\<And>A B. \<lbrakk>A type; B type\<rbrakk> \<Longrightarrow> A+B type" and 
191 
PlusFL: "\<And>A B C D. \<lbrakk>A = C; B = D\<rbrakk> \<Longrightarrow> A+B = C+D" and 

0  192 

58977  193 
PlusI_inl: "\<And>a A B. \<lbrakk>a : A; B type\<rbrakk> \<Longrightarrow> inl(a) : A+B" and 
194 
PlusI_inlL: "\<And>a c A B. \<lbrakk>a = c : A; B type\<rbrakk> \<Longrightarrow> inl(a) = inl(c) : A+B" and 

0  195 

58977  196 
PlusI_inr: "\<And>b A B. \<lbrakk>A type; b : B\<rbrakk> \<Longrightarrow> inr(b) : A+B" and 
197 
PlusI_inrL: "\<And>b d A B. \<lbrakk>A type; b = d : B\<rbrakk> \<Longrightarrow> inr(b) = inr(d) : A+B" and 

0  198 

17441  199 
PlusE: 
58977  200 
"\<And>p c d A B C. \<lbrakk>p: A+B; 
201 
\<And>x. x:A \<Longrightarrow> c(x): C(inl(x)); 

202 
\<And>y. y:B \<Longrightarrow> d(y): C(inr(y)) \<rbrakk> \<Longrightarrow> when(p, \<lambda>x. c(x), \<lambda>y. d(y)) : C(p)" and 

0  203 

17441  204 
PlusEL: 
58977  205 
"\<And>p q c d e f A B C. \<lbrakk>p = q : A+B; 
206 
\<And>x. x: A \<Longrightarrow> c(x) = e(x) : C(inl(x)); 

207 
\<And>y. y: B \<Longrightarrow> d(y) = f(y) : C(inr(y))\<rbrakk> 

208 
\<Longrightarrow> when(p, \<lambda>x. c(x), \<lambda>y. d(y)) = when(q, \<lambda>x. e(x), \<lambda>y. f(y)) : C(p)" and 

0  209 

17441  210 
PlusC_inl: 
58977  211 
"\<And>a c d A C. \<lbrakk>a: A; 
212 
\<And>x. x:A \<Longrightarrow> c(x): C(inl(x)); 

213 
\<And>y. y:B \<Longrightarrow> d(y): C(inr(y)) \<rbrakk> 

214 
\<Longrightarrow> when(inl(a), \<lambda>x. c(x), \<lambda>y. d(y)) = c(a) : C(inl(a))" and 

0  215 

17441  216 
PlusC_inr: 
58977  217 
"\<And>b c d A B C. \<lbrakk>b: B; 
218 
\<And>x. x:A \<Longrightarrow> c(x): C(inl(x)); 

219 
\<And>y. y:B \<Longrightarrow> d(y): C(inr(y))\<rbrakk> 

220 
\<Longrightarrow> when(inr(b), \<lambda>x. c(x), \<lambda>y. d(y)) = d(b) : C(inr(b))" and 

0  221 

222 

63505  223 
\<comment> \<open>The type \<open>Eq\<close>\<close> 
0  224 

58977  225 
EqF: "\<And>a b A. \<lbrakk>A type; a : A; b : A\<rbrakk> \<Longrightarrow> Eq(A,a,b) type" and 
226 
EqFL: "\<And>a b c d A B. \<lbrakk>A = B; a = c : A; b = d : A\<rbrakk> \<Longrightarrow> Eq(A,a,b) = Eq(B,c,d)" and 

227 
EqI: "\<And>a b A. a = b : A \<Longrightarrow> eq : Eq(A,a,b)" and 

228 
EqE: "\<And>p a b A. p : Eq(A,a,b) \<Longrightarrow> a = b : A" and 

0  229 

63505  230 
\<comment> \<open>By equality of types, can prove \<open>C(p)\<close> from \<open>C(eq)\<close>, an elimination rule\<close> 
58977  231 
EqC: "\<And>p a b A. p : Eq(A,a,b) \<Longrightarrow> p = eq : Eq(A,a,b)" and 
0  232 

63505  233 

234 
\<comment> \<open>The type \<open>F\<close>\<close> 

0  235 

51308  236 
FF: "F type" and 
58977  237 
FE: "\<And>p C. \<lbrakk>p: F; C type\<rbrakk> \<Longrightarrow> contr(p) : C" and 
238 
FEL: "\<And>p q C. \<lbrakk>p = q : F; C type\<rbrakk> \<Longrightarrow> contr(p) = contr(q) : C" and 

0  239 

63505  240 

241 
\<comment> \<open>The type T\<close> 

242 
\<comment> \<open> 

243 
MartinLÃ¶f's book (page 68) discusses elimination and computation. 

244 
Elimination can be derived by computation and equality of types, 

245 
but with an extra premise \<open>C(x)\<close> type \<open>x:T\<close>. 

246 
Also computation can be derived from elimination. 

247 
\<close> 

0  248 

51308  249 
TF: "T type" and 
250 
TI: "tt : T" and 

58977  251 
TE: "\<And>p c C. \<lbrakk>p : T; c : C(tt)\<rbrakk> \<Longrightarrow> c : C(p)" and 
252 
TEL: "\<And>p q c d C. \<lbrakk>p = q : T; c = d : C(tt)\<rbrakk> \<Longrightarrow> c = d : C(p)" and 

253 
TC: "\<And>p. p : T \<Longrightarrow> p = tt : T" 

0  254 

19761  255 

256 
subsection "Tactics and derived rules for Constructive Type Theory" 

257 

63505  258 
text \<open>Formation rules.\<close> 
19761  259 
lemmas form_rls = NF ProdF SumF PlusF EqF FF TF 
260 
and formL_rls = ProdFL SumFL PlusFL EqFL 

261 

63505  262 
text \<open> 
263 
Introduction rules. OMITTED: 

264 
\<^item> \<open>EqI\<close>, because its premise is an \<open>eqelem\<close>, not an \<open>elem\<close>. 

265 
\<close> 

19761  266 
lemmas intr_rls = NI0 NI_succ ProdI SumI PlusI_inl PlusI_inr TI 
267 
and intrL_rls = NI_succL ProdIL SumIL PlusI_inlL PlusI_inrL 

268 

63505  269 
text \<open> 
270 
Elimination rules. OMITTED: 

271 
\<^item> \<open>EqE\<close>, because its conclusion is an \<open>eqelem\<close>, not an \<open>elem\<close> 

272 
\<^item> \<open>TE\<close>, because it does not involve a constructor. 

273 
\<close> 

19761  274 
lemmas elim_rls = NE ProdE SumE PlusE FE 
275 
and elimL_rls = NEL ProdEL SumEL PlusEL FEL 

276 

63505  277 
text \<open>OMITTED: \<open>eqC\<close> are \<open>TC\<close> because they make rewriting loop: \<open>p = un = un = \<dots>\<close>\<close> 
19761  278 
lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr 
279 

63505  280 
text \<open>Rules with conclusion \<open>a:A\<close>, an elem judgement.\<close> 
19761  281 
lemmas element_rls = intr_rls elim_rls 
282 

63505  283 
text \<open>Definitions are (meta)equality axioms.\<close> 
19761  284 
lemmas basic_defs = fst_def snd_def 
285 

63505  286 
text \<open>Compare with standard version: \<open>B\<close> is applied to UNSIMPLIFIED expression!\<close> 
58977  287 
lemma SumIL2: "\<lbrakk>c = a : A; d = b : B(a)\<rbrakk> \<Longrightarrow> <c,d> = <a,b> : Sum(A,B)" 
63505  288 
apply (rule sym_elem) 
289 
apply (rule SumIL) 

290 
apply (rule_tac [!] sym_elem) 

291 
apply assumption+ 

292 
done 

19761  293 

294 
lemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL 

295 

63505  296 
text \<open> 
297 
Exploit \<open>p:Prod(A,B)\<close> to create the assumption \<open>z:B(a)\<close>. 

298 
A more natural form of product elimination. 

299 
\<close> 

19761  300 
lemma subst_prodE: 
301 
assumes "p: Prod(A,B)" 

302 
and "a: A" 

58977  303 
and "\<And>z. z: B(a) \<Longrightarrow> c(z): C(z)" 
19761  304 
shows "c(p`a): C(p`a)" 
63505  305 
by (rule assms ProdE)+ 
19761  306 

307 

60770  308 
subsection \<open>Tactics for type checking\<close> 
19761  309 

60770  310 
ML \<open> 
19761  311 
local 
312 

56250  313 
fun is_rigid_elem (Const(@{const_name Elem},_) $ a $ _) = not(is_Var (head_of a)) 
314 
 is_rigid_elem (Const(@{const_name Eqelem},_) $ a $ _ $ _) = not(is_Var (head_of a)) 

315 
 is_rigid_elem (Const(@{const_name Type},_) $ a) = not(is_Var (head_of a)) 

19761  316 
 is_rigid_elem _ = false 
317 

318 
in 

319 

320 
(*Try solving a:A or a=b:A by assumption provided a is rigid!*) 

63505  321 
fun test_assume_tac ctxt = SUBGOAL (fn (prem, i) => 
322 
if is_rigid_elem (Logic.strip_assums_concl prem) 

323 
then assume_tac ctxt i else no_tac) 

19761  324 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

325 
fun ASSUME ctxt tf i = test_assume_tac ctxt i ORELSE tf i 
19761  326 

63505  327 
end 
60770  328 
\<close> 
19761  329 

63505  330 
text \<open> 
331 
For simplification: type formation and checking, 

332 
but no equalities between terms. 

333 
\<close> 

19761  334 
lemmas routine_rls = form_rls formL_rls refl_type element_rls 
335 

60770  336 
ML \<open> 
59164  337 
fun routine_tac rls ctxt prems = 
338 
ASSUME ctxt (filt_resolve_from_net_tac ctxt 4 (Tactic.build_net (prems @ rls))); 

19761  339 

340 
(*Solve all subgoals "A type" using formation rules. *) 

59164  341 
val form_net = Tactic.build_net @{thms form_rls}; 
342 
fun form_tac ctxt = 

343 
REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 form_net)); 

19761  344 

345 
(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

346 
fun typechk_tac ctxt thms = 
59164  347 
let val tac = 
348 
filt_resolve_from_net_tac ctxt 3 

349 
(Tactic.build_net (thms @ @{thms form_rls} @ @{thms element_rls})) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

350 
in REPEAT_FIRST (ASSUME ctxt tac) end 
19761  351 

352 
(*Solve a:A (a flexible, A rigid) by introduction rules. 

353 
Cannot use stringtrees (filt_resolve_tac) since 

354 
goals like ?a:SUM(A,B) have a trivial headstring *) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

355 
fun intr_tac ctxt thms = 
59164  356 
let val tac = 
357 
filt_resolve_from_net_tac ctxt 1 

358 
(Tactic.build_net (thms @ @{thms form_rls} @ @{thms intr_rls})) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

359 
in REPEAT_FIRST (ASSUME ctxt tac) end 
19761  360 

361 
(*Equality proving: solve a=b:A (where a is rigid) by long rules. *) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

362 
fun equal_tac ctxt thms = 
59164  363 
REPEAT_FIRST 
63505  364 
(ASSUME ctxt 
365 
(filt_resolve_from_net_tac ctxt 3 

366 
(Tactic.build_net (thms @ @{thms form_rls element_rls intrL_rls elimL_rls refl_elem})))) 

60770  367 
\<close> 
19761  368 

60770  369 
method_setup form = \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (form_tac ctxt))\<close> 
370 
method_setup typechk = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (typechk_tac ctxt ths))\<close> 

371 
method_setup intr = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (intr_tac ctxt ths))\<close> 

372 
method_setup equal = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (equal_tac ctxt ths))\<close> 

19761  373 

374 

60770  375 
subsection \<open>Simplification\<close> 
19761  376 

63505  377 
text \<open>To simplify the type in a goal.\<close> 
58977  378 
lemma replace_type: "\<lbrakk>B = A; a : A\<rbrakk> \<Longrightarrow> a : B" 
63505  379 
apply (rule equal_types) 
380 
apply (rule_tac [2] sym_type) 

381 
apply assumption+ 

382 
done 

19761  383 

63505  384 
text \<open>Simplify the parameter of a unary type operator.\<close> 
19761  385 
lemma subst_eqtyparg: 
23467  386 
assumes 1: "a=c : A" 
58977  387 
and 2: "\<And>z. z:A \<Longrightarrow> B(z) type" 
63505  388 
shows "B(a) = B(c)" 
389 
apply (rule subst_typeL) 

390 
apply (rule_tac [2] refl_type) 

391 
apply (rule 1) 

392 
apply (erule 2) 

393 
done 

19761  394 

63505  395 
text \<open>Simplification rules for Constructive Type Theory.\<close> 
19761  396 
lemmas reduction_rls = comp_rls [THEN trans_elem] 
397 

60770  398 
ML \<open> 
19761  399 
(*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification. 
400 
Uses other intro rules to avoid changing flexible goals.*) 

59164  401 
val eqintr_net = Tactic.build_net @{thms EqI intr_rls} 
58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

402 
fun eqintr_tac ctxt = 
59164  403 
REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 eqintr_net)) 
19761  404 

405 
(** Tactics that instantiate CTTrules. 

406 
Vars in the given terms will be incremented! 

407 
The (rtac EqE i) lets them apply to equality judgements. **) 

408 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

409 
fun NE_tac ctxt sp i = 
60754  410 
TRY (resolve_tac ctxt @{thms EqE} i) THEN 
59780  411 
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm NE} i 
19761  412 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

413 
fun SumE_tac ctxt sp i = 
60754  414 
TRY (resolve_tac ctxt @{thms EqE} i) THEN 
59780  415 
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm SumE} i 
19761  416 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

417 
fun PlusE_tac ctxt sp i = 
60754  418 
TRY (resolve_tac ctxt @{thms EqE} i) THEN 
59780  419 
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm PlusE} i 
19761  420 

421 
(** Predicate logic reasoning, WITH THINNING!! Procedures adapted from NJ. **) 

422 

423 
(*Finds f:Prod(A,B) and a:A in the assumptions, concludes there is z:B(a) *) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

424 
fun add_mp_tac ctxt i = 
60754  425 
resolve_tac ctxt @{thms subst_prodE} i THEN assume_tac ctxt i THEN assume_tac ctxt i 
19761  426 

61391  427 
(*Finds P\<longrightarrow>Q and P in the assumptions, replaces implication by Q *) 
60754  428 
fun mp_tac ctxt i = eresolve_tac ctxt @{thms subst_prodE} i THEN assume_tac ctxt i 
19761  429 

430 
(*"safe" when regarded as predicate calculus rules*) 

431 
val safe_brls = sort (make_ord lessb) 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

432 
[ (true, @{thm FE}), (true,asm_rl), 
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

433 
(false, @{thm ProdI}), (true, @{thm SumE}), (true, @{thm PlusE}) ] 
19761  434 

435 
val unsafe_brls = 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

436 
[ (false, @{thm PlusI_inl}), (false, @{thm PlusI_inr}), (false, @{thm SumI}), 
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

437 
(true, @{thm subst_prodE}) ] 
19761  438 

439 
(*0 subgoals vs 1 or more*) 

440 
val (safe0_brls, safep_brls) = 

441 
List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls 

442 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

443 
fun safestep_tac ctxt thms i = 
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

444 
form_tac ctxt ORELSE 
59498
50b60f501b05
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
wenzelm
parents:
59164
diff
changeset

445 
resolve_tac ctxt thms i ORELSE 
50b60f501b05
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
wenzelm
parents:
59164
diff
changeset

446 
biresolve_tac ctxt safe0_brls i ORELSE mp_tac ctxt i ORELSE 
50b60f501b05
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
wenzelm
parents:
59164
diff
changeset

447 
DETERM (biresolve_tac ctxt safep_brls i) 
19761  448 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

449 
fun safe_tac ctxt thms i = DEPTH_SOLVE_1 (safestep_tac ctxt thms i) 
19761  450 

59498
50b60f501b05
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
wenzelm
parents:
59164
diff
changeset

451 
fun step_tac ctxt thms = safestep_tac ctxt thms ORELSE' biresolve_tac ctxt unsafe_brls 
19761  452 

453 
(*Fails unless it solves the goal!*) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

454 
fun pc_tac ctxt thms = DEPTH_SOLVE_1 o (step_tac ctxt thms) 
60770  455 
\<close> 
19761  456 

60770  457 
method_setup eqintr = \<open>Scan.succeed (SIMPLE_METHOD o eqintr_tac)\<close> 
458 
method_setup NE = \<open> 

63120
629a4c5e953e
embedded content may be delimited via cartouches;
wenzelm
parents:
61391
diff
changeset

459 
Scan.lift Args.embedded_inner_syntax >> (fn s => fn ctxt => SIMPLE_METHOD' (NE_tac ctxt s)) 
60770  460 
\<close> 
461 
method_setup pc = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (pc_tac ctxt ths))\<close> 

462 
method_setup add_mp = \<open>Scan.succeed (SIMPLE_METHOD' o add_mp_tac)\<close> 

58972  463 

48891  464 
ML_file "rew.ML" 
60770  465 
method_setup rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (rew_tac ctxt ths))\<close> 
466 
method_setup hyp_rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_rew_tac ctxt ths))\<close> 

58972  467 

19761  468 

60770  469 
subsection \<open>The elimination rules for fst/snd\<close> 
19761  470 

58977  471 
lemma SumE_fst: "p : Sum(A,B) \<Longrightarrow> fst(p) : A" 
63505  472 
apply (unfold basic_defs) 
473 
apply (erule SumE) 

474 
apply assumption 

475 
done 

19761  476 

63505  477 
text \<open>The first premise must be \<open>p:Sum(A,B)\<close>!!.\<close> 
19761  478 
lemma SumE_snd: 
479 
assumes major: "p: Sum(A,B)" 

480 
and "A type" 

58977  481 
and "\<And>x. x:A \<Longrightarrow> B(x) type" 
19761  482 
shows "snd(p) : B(fst(p))" 
483 
apply (unfold basic_defs) 

484 
apply (rule major [THEN SumE]) 

485 
apply (rule SumC [THEN subst_eqtyparg, THEN replace_type]) 

63505  486 
apply (typechk assms) 
19761  487 
done 
488 

489 
end 