src/CTT/CTT.thy
author wenzelm
Mon Feb 08 21:28:27 2010 +0100 (2010-02-08)
changeset 35054 a5db9779b026
parent 27239 f2f42f9fa09d
child 35762 af3ff2ba4c54
permissions -rw-r--r--
modernized some syntax translations;
wenzelm@17441
     1
(*  Title:      CTT/CTT.thy
clasohm@0
     2
    ID:         $Id$
clasohm@0
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
clasohm@0
     4
    Copyright   1993  University of Cambridge
clasohm@0
     5
*)
clasohm@0
     6
wenzelm@17441
     7
header {* Constructive Type Theory *}
clasohm@0
     8
wenzelm@17441
     9
theory CTT
wenzelm@17441
    10
imports Pure
wenzelm@19761
    11
uses "~~/src/Provers/typedsimp.ML" ("rew.ML")
wenzelm@17441
    12
begin
wenzelm@17441
    13
wenzelm@26956
    14
setup PureThy.old_appl_syntax_setup
wenzelm@26956
    15
wenzelm@17441
    16
typedecl i
wenzelm@17441
    17
typedecl t
wenzelm@17441
    18
typedecl o
clasohm@0
    19
clasohm@0
    20
consts
clasohm@0
    21
  (*Types*)
wenzelm@17441
    22
  F         :: "t"
wenzelm@17441
    23
  T         :: "t"          (*F is empty, T contains one element*)
clasohm@0
    24
  contr     :: "i=>i"
clasohm@0
    25
  tt        :: "i"
clasohm@0
    26
  (*Natural numbers*)
clasohm@0
    27
  N         :: "t"
clasohm@0
    28
  succ      :: "i=>i"
clasohm@0
    29
  rec       :: "[i, i, [i,i]=>i] => i"
clasohm@0
    30
  (*Unions*)
wenzelm@17441
    31
  inl       :: "i=>i"
wenzelm@17441
    32
  inr       :: "i=>i"
clasohm@0
    33
  when      :: "[i, i=>i, i=>i]=>i"
clasohm@0
    34
  (*General Sum and Binary Product*)
clasohm@0
    35
  Sum       :: "[t, i=>t]=>t"
wenzelm@17441
    36
  fst       :: "i=>i"
wenzelm@17441
    37
  snd       :: "i=>i"
clasohm@0
    38
  split     :: "[i, [i,i]=>i] =>i"
clasohm@0
    39
  (*General Product and Function Space*)
clasohm@0
    40
  Prod      :: "[t, i=>t]=>t"
wenzelm@14765
    41
  (*Types*)
wenzelm@22808
    42
  Plus      :: "[t,t]=>t"           (infixr "+" 40)
clasohm@0
    43
  (*Equality type*)
clasohm@0
    44
  Eq        :: "[t,i,i]=>t"
clasohm@0
    45
  eq        :: "i"
clasohm@0
    46
  (*Judgements*)
clasohm@0
    47
  Type      :: "t => prop"          ("(_ type)" [10] 5)
paulson@10467
    48
  Eqtype    :: "[t,t]=>prop"        ("(_ =/ _)" [10,10] 5)
clasohm@0
    49
  Elem      :: "[i, t]=>prop"       ("(_ /: _)" [10,10] 5)
paulson@10467
    50
  Eqelem    :: "[i,i,t]=>prop"      ("(_ =/ _ :/ _)" [10,10,10] 5)
clasohm@0
    51
  Reduce    :: "[i,i]=>prop"        ("Reduce[_,_]")
clasohm@0
    52
  (*Types*)
wenzelm@14765
    53
clasohm@0
    54
  (*Functions*)
clasohm@0
    55
  lambda    :: "(i => i) => i"      (binder "lam " 10)
wenzelm@22808
    56
  app       :: "[i,i]=>i"           (infixl "`" 60)
clasohm@0
    57
  (*Natural numbers*)
clasohm@0
    58
  "0"       :: "i"                  ("0")
clasohm@0
    59
  (*Pairing*)
clasohm@0
    60
  pair      :: "[i,i]=>i"           ("(1<_,/_>)")
clasohm@0
    61
wenzelm@14765
    62
syntax
wenzelm@19761
    63
  "_PROD"   :: "[idt,t,t]=>t"       ("(3PROD _:_./ _)" 10)
wenzelm@19761
    64
  "_SUM"    :: "[idt,t,t]=>t"       ("(3SUM _:_./ _)" 10)
clasohm@0
    65
translations
wenzelm@35054
    66
  "PROD x:A. B" == "CONST Prod(A, %x. B)"
wenzelm@35054
    67
  "SUM x:A. B"  == "CONST Sum(A, %x. B)"
wenzelm@19761
    68
wenzelm@19761
    69
abbreviation
wenzelm@21404
    70
  Arrow     :: "[t,t]=>t"  (infixr "-->" 30) where
wenzelm@19761
    71
  "A --> B == PROD _:A. B"
wenzelm@21404
    72
abbreviation
wenzelm@21404
    73
  Times     :: "[t,t]=>t"  (infixr "*" 50) where
wenzelm@19761
    74
  "A * B == SUM _:A. B"
clasohm@0
    75
wenzelm@21210
    76
notation (xsymbols)
wenzelm@21524
    77
  lambda  (binder "\<lambda>\<lambda>" 10) and
wenzelm@21404
    78
  Elem  ("(_ /\<in> _)" [10,10] 5) and
wenzelm@21404
    79
  Eqelem  ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) and
wenzelm@21404
    80
  Arrow  (infixr "\<longrightarrow>" 30) and
wenzelm@19761
    81
  Times  (infixr "\<times>" 50)
wenzelm@17441
    82
wenzelm@21210
    83
notation (HTML output)
wenzelm@21524
    84
  lambda  (binder "\<lambda>\<lambda>" 10) and
wenzelm@21404
    85
  Elem  ("(_ /\<in> _)" [10,10] 5) and
wenzelm@21404
    86
  Eqelem  ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) and
wenzelm@19761
    87
  Times  (infixr "\<times>" 50)
wenzelm@17441
    88
paulson@10467
    89
syntax (xsymbols)
wenzelm@21524
    90
  "_PROD"   :: "[idt,t,t] => t"     ("(3\<Pi> _\<in>_./ _)"    10)
wenzelm@21524
    91
  "_SUM"    :: "[idt,t,t] => t"     ("(3\<Sigma> _\<in>_./ _)" 10)
paulson@10467
    92
kleing@14565
    93
syntax (HTML output)
wenzelm@21524
    94
  "_PROD"   :: "[idt,t,t] => t"     ("(3\<Pi> _\<in>_./ _)"    10)
wenzelm@21524
    95
  "_SUM"    :: "[idt,t,t] => t"     ("(3\<Sigma> _\<in>_./ _)" 10)
kleing@14565
    96
wenzelm@17441
    97
axioms
clasohm@0
    98
clasohm@0
    99
  (*Reduction: a weaker notion than equality;  a hack for simplification.
clasohm@0
   100
    Reduce[a,b] means either that  a=b:A  for some A or else that "a" and "b"
clasohm@0
   101
    are textually identical.*)
clasohm@0
   102
clasohm@0
   103
  (*does not verify a:A!  Sound because only trans_red uses a Reduce premise
clasohm@0
   104
    No new theorems can be proved about the standard judgements.*)
wenzelm@17441
   105
  refl_red: "Reduce[a,a]"
wenzelm@17441
   106
  red_if_equal: "a = b : A ==> Reduce[a,b]"
wenzelm@17441
   107
  trans_red: "[| a = b : A;  Reduce[b,c] |] ==> a = c : A"
clasohm@0
   108
clasohm@0
   109
  (*Reflexivity*)
clasohm@0
   110
wenzelm@17441
   111
  refl_type: "A type ==> A = A"
wenzelm@17441
   112
  refl_elem: "a : A ==> a = a : A"
clasohm@0
   113
clasohm@0
   114
  (*Symmetry*)
clasohm@0
   115
wenzelm@17441
   116
  sym_type:  "A = B ==> B = A"
wenzelm@17441
   117
  sym_elem:  "a = b : A ==> b = a : A"
clasohm@0
   118
clasohm@0
   119
  (*Transitivity*)
clasohm@0
   120
wenzelm@17441
   121
  trans_type:   "[| A = B;  B = C |] ==> A = C"
wenzelm@17441
   122
  trans_elem:   "[| a = b : A;  b = c : A |] ==> a = c : A"
clasohm@0
   123
wenzelm@17441
   124
  equal_types:  "[| a : A;  A = B |] ==> a : B"
wenzelm@17441
   125
  equal_typesL: "[| a = b : A;  A = B |] ==> a = b : B"
clasohm@0
   126
clasohm@0
   127
  (*Substitution*)
clasohm@0
   128
wenzelm@17441
   129
  subst_type:   "[| a : A;  !!z. z:A ==> B(z) type |] ==> B(a) type"
wenzelm@17441
   130
  subst_typeL:  "[| a = c : A;  !!z. z:A ==> B(z) = D(z) |] ==> B(a) = D(c)"
clasohm@0
   131
wenzelm@17441
   132
  subst_elem:   "[| a : A;  !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)"
wenzelm@17441
   133
  subst_elemL:
clasohm@0
   134
    "[| a=c : A;  !!z. z:A ==> b(z)=d(z) : B(z) |] ==> b(a)=d(c) : B(a)"
clasohm@0
   135
clasohm@0
   136
clasohm@0
   137
  (*The type N -- natural numbers*)
clasohm@0
   138
wenzelm@17441
   139
  NF: "N type"
wenzelm@17441
   140
  NI0: "0 : N"
wenzelm@17441
   141
  NI_succ: "a : N ==> succ(a) : N"
wenzelm@17441
   142
  NI_succL:  "a = b : N ==> succ(a) = succ(b) : N"
clasohm@0
   143
wenzelm@17441
   144
  NE:
wenzelm@17441
   145
   "[| p: N;  a: C(0);  !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]
wenzelm@3837
   146
   ==> rec(p, a, %u v. b(u,v)) : C(p)"
clasohm@0
   147
wenzelm@17441
   148
  NEL:
wenzelm@17441
   149
   "[| p = q : N;  a = c : C(0);
wenzelm@17441
   150
      !!u v. [| u: N; v: C(u) |] ==> b(u,v) = d(u,v): C(succ(u)) |]
wenzelm@3837
   151
   ==> rec(p, a, %u v. b(u,v)) = rec(q,c,d) : C(p)"
clasohm@0
   152
wenzelm@17441
   153
  NC0:
wenzelm@17441
   154
   "[| a: C(0);  !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]
wenzelm@3837
   155
   ==> rec(0, a, %u v. b(u,v)) = a : C(0)"
clasohm@0
   156
wenzelm@17441
   157
  NC_succ:
wenzelm@17441
   158
   "[| p: N;  a: C(0);
wenzelm@17441
   159
       !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] ==>
wenzelm@3837
   160
   rec(succ(p), a, %u v. b(u,v)) = b(p, rec(p, a, %u v. b(u,v))) : C(succ(p))"
clasohm@0
   161
clasohm@0
   162
  (*The fourth Peano axiom.  See page 91 of Martin-Lof's book*)
wenzelm@17441
   163
  zero_ne_succ:
clasohm@0
   164
    "[| a: N;  0 = succ(a) : N |] ==> 0: F"
clasohm@0
   165
clasohm@0
   166
clasohm@0
   167
  (*The Product of a family of types*)
clasohm@0
   168
wenzelm@17441
   169
  ProdF:  "[| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A. B(x) type"
clasohm@0
   170
wenzelm@17441
   171
  ProdFL:
wenzelm@17441
   172
   "[| A = C;  !!x. x:A ==> B(x) = D(x) |] ==>
wenzelm@3837
   173
   PROD x:A. B(x) = PROD x:C. D(x)"
clasohm@0
   174
wenzelm@17441
   175
  ProdI:
wenzelm@3837
   176
   "[| A type;  !!x. x:A ==> b(x):B(x)|] ==> lam x. b(x) : PROD x:A. B(x)"
clasohm@0
   177
wenzelm@17441
   178
  ProdIL:
wenzelm@17441
   179
   "[| A type;  !!x. x:A ==> b(x) = c(x) : B(x)|] ==>
wenzelm@3837
   180
   lam x. b(x) = lam x. c(x) : PROD x:A. B(x)"
clasohm@0
   181
wenzelm@17441
   182
  ProdE:  "[| p : PROD x:A. B(x);  a : A |] ==> p`a : B(a)"
wenzelm@17441
   183
  ProdEL: "[| p=q: PROD x:A. B(x);  a=b : A |] ==> p`a = q`b : B(a)"
clasohm@0
   184
wenzelm@17441
   185
  ProdC:
wenzelm@17441
   186
   "[| a : A;  !!x. x:A ==> b(x) : B(x)|] ==>
wenzelm@3837
   187
   (lam x. b(x)) ` a = b(a) : B(a)"
clasohm@0
   188
wenzelm@17441
   189
  ProdC2:
wenzelm@3837
   190
   "p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)"
clasohm@0
   191
clasohm@0
   192
clasohm@0
   193
  (*The Sum of a family of types*)
clasohm@0
   194
wenzelm@17441
   195
  SumF:  "[| A type;  !!x. x:A ==> B(x) type |] ==> SUM x:A. B(x) type"
wenzelm@17441
   196
  SumFL:
wenzelm@3837
   197
    "[| A = C;  !!x. x:A ==> B(x) = D(x) |] ==> SUM x:A. B(x) = SUM x:C. D(x)"
clasohm@0
   198
wenzelm@17441
   199
  SumI:  "[| a : A;  b : B(a) |] ==> <a,b> : SUM x:A. B(x)"
wenzelm@17441
   200
  SumIL: "[| a=c:A;  b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A. B(x)"
clasohm@0
   201
wenzelm@17441
   202
  SumE:
wenzelm@17441
   203
    "[| p: SUM x:A. B(x);  !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]
wenzelm@3837
   204
    ==> split(p, %x y. c(x,y)) : C(p)"
clasohm@0
   205
wenzelm@17441
   206
  SumEL:
wenzelm@17441
   207
    "[| p=q : SUM x:A. B(x);
wenzelm@17441
   208
       !!x y. [| x:A; y:B(x) |] ==> c(x,y)=d(x,y): C(<x,y>)|]
wenzelm@3837
   209
    ==> split(p, %x y. c(x,y)) = split(q, % x y. d(x,y)) : C(p)"
clasohm@0
   210
wenzelm@17441
   211
  SumC:
wenzelm@17441
   212
    "[| a: A;  b: B(a);  !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]
wenzelm@3837
   213
    ==> split(<a,b>, %x y. c(x,y)) = c(a,b) : C(<a,b>)"
clasohm@0
   214
wenzelm@17441
   215
  fst_def:   "fst(a) == split(a, %x y. x)"
wenzelm@17441
   216
  snd_def:   "snd(a) == split(a, %x y. y)"
clasohm@0
   217
clasohm@0
   218
clasohm@0
   219
  (*The sum of two types*)
clasohm@0
   220
wenzelm@17441
   221
  PlusF:   "[| A type;  B type |] ==> A+B type"
wenzelm@17441
   222
  PlusFL:  "[| A = C;  B = D |] ==> A+B = C+D"
clasohm@0
   223
wenzelm@17441
   224
  PlusI_inl:   "[| a : A;  B type |] ==> inl(a) : A+B"
wenzelm@17441
   225
  PlusI_inlL: "[| a = c : A;  B type |] ==> inl(a) = inl(c) : A+B"
clasohm@0
   226
wenzelm@17441
   227
  PlusI_inr:   "[| A type;  b : B |] ==> inr(b) : A+B"
wenzelm@17441
   228
  PlusI_inrL: "[| A type;  b = d : B |] ==> inr(b) = inr(d) : A+B"
clasohm@0
   229
wenzelm@17441
   230
  PlusE:
wenzelm@17441
   231
    "[| p: A+B;  !!x. x:A ==> c(x): C(inl(x));
wenzelm@17441
   232
                !!y. y:B ==> d(y): C(inr(y)) |]
wenzelm@3837
   233
    ==> when(p, %x. c(x), %y. d(y)) : C(p)"
clasohm@0
   234
wenzelm@17441
   235
  PlusEL:
wenzelm@17441
   236
    "[| p = q : A+B;  !!x. x: A ==> c(x) = e(x) : C(inl(x));
wenzelm@17441
   237
                     !!y. y: B ==> d(y) = f(y) : C(inr(y)) |]
wenzelm@3837
   238
    ==> when(p, %x. c(x), %y. d(y)) = when(q, %x. e(x), %y. f(y)) : C(p)"
clasohm@0
   239
wenzelm@17441
   240
  PlusC_inl:
wenzelm@17441
   241
    "[| a: A;  !!x. x:A ==> c(x): C(inl(x));
wenzelm@17441
   242
              !!y. y:B ==> d(y): C(inr(y)) |]
wenzelm@3837
   243
    ==> when(inl(a), %x. c(x), %y. d(y)) = c(a) : C(inl(a))"
clasohm@0
   244
wenzelm@17441
   245
  PlusC_inr:
wenzelm@17441
   246
    "[| b: B;  !!x. x:A ==> c(x): C(inl(x));
wenzelm@17441
   247
              !!y. y:B ==> d(y): C(inr(y)) |]
wenzelm@3837
   248
    ==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))"
clasohm@0
   249
clasohm@0
   250
clasohm@0
   251
  (*The type Eq*)
clasohm@0
   252
wenzelm@17441
   253
  EqF:    "[| A type;  a : A;  b : A |] ==> Eq(A,a,b) type"
wenzelm@17441
   254
  EqFL: "[| A=B;  a=c: A;  b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)"
wenzelm@17441
   255
  EqI: "a = b : A ==> eq : Eq(A,a,b)"
wenzelm@17441
   256
  EqE: "p : Eq(A,a,b) ==> a = b : A"
clasohm@0
   257
clasohm@0
   258
  (*By equality of types, can prove C(p) from C(eq), an elimination rule*)
wenzelm@17441
   259
  EqC: "p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)"
clasohm@0
   260
clasohm@0
   261
  (*The type F*)
clasohm@0
   262
wenzelm@17441
   263
  FF: "F type"
wenzelm@17441
   264
  FE: "[| p: F;  C type |] ==> contr(p) : C"
wenzelm@17441
   265
  FEL:  "[| p = q : F;  C type |] ==> contr(p) = contr(q) : C"
clasohm@0
   266
clasohm@0
   267
  (*The type T
clasohm@0
   268
     Martin-Lof's book (page 68) discusses elimination and computation.
clasohm@0
   269
     Elimination can be derived by computation and equality of types,
clasohm@0
   270
     but with an extra premise C(x) type x:T.
clasohm@0
   271
     Also computation can be derived from elimination. *)
clasohm@0
   272
wenzelm@17441
   273
  TF: "T type"
wenzelm@17441
   274
  TI: "tt : T"
wenzelm@17441
   275
  TE: "[| p : T;  c : C(tt) |] ==> c : C(p)"
wenzelm@17441
   276
  TEL: "[| p = q : T;  c = d : C(tt) |] ==> c = d : C(p)"
wenzelm@17441
   277
  TC: "p : T ==> p = tt : T"
clasohm@0
   278
wenzelm@19761
   279
wenzelm@19761
   280
subsection "Tactics and derived rules for Constructive Type Theory"
wenzelm@19761
   281
wenzelm@19761
   282
(*Formation rules*)
wenzelm@19761
   283
lemmas form_rls = NF ProdF SumF PlusF EqF FF TF
wenzelm@19761
   284
  and formL_rls = ProdFL SumFL PlusFL EqFL
wenzelm@19761
   285
wenzelm@19761
   286
(*Introduction rules
wenzelm@19761
   287
  OMITTED: EqI, because its premise is an eqelem, not an elem*)
wenzelm@19761
   288
lemmas intr_rls = NI0 NI_succ ProdI SumI PlusI_inl PlusI_inr TI
wenzelm@19761
   289
  and intrL_rls = NI_succL ProdIL SumIL PlusI_inlL PlusI_inrL
wenzelm@19761
   290
wenzelm@19761
   291
(*Elimination rules
wenzelm@19761
   292
  OMITTED: EqE, because its conclusion is an eqelem,  not an elem
wenzelm@19761
   293
           TE, because it does not involve a constructor *)
wenzelm@19761
   294
lemmas elim_rls = NE ProdE SumE PlusE FE
wenzelm@19761
   295
  and elimL_rls = NEL ProdEL SumEL PlusEL FEL
wenzelm@19761
   296
wenzelm@19761
   297
(*OMITTED: eqC are TC because they make rewriting loop: p = un = un = ... *)
wenzelm@19761
   298
lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr
wenzelm@19761
   299
wenzelm@19761
   300
(*rules with conclusion a:A, an elem judgement*)
wenzelm@19761
   301
lemmas element_rls = intr_rls elim_rls
wenzelm@19761
   302
wenzelm@19761
   303
(*Definitions are (meta)equality axioms*)
wenzelm@19761
   304
lemmas basic_defs = fst_def snd_def
wenzelm@19761
   305
wenzelm@19761
   306
(*Compare with standard version: B is applied to UNSIMPLIFIED expression! *)
wenzelm@19761
   307
lemma SumIL2: "[| c=a : A;  d=b : B(a) |] ==> <c,d> = <a,b> : Sum(A,B)"
wenzelm@19761
   308
apply (rule sym_elem)
wenzelm@19761
   309
apply (rule SumIL)
wenzelm@19761
   310
apply (rule_tac [!] sym_elem)
wenzelm@19761
   311
apply assumption+
wenzelm@19761
   312
done
wenzelm@19761
   313
wenzelm@19761
   314
lemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL
wenzelm@19761
   315
wenzelm@19761
   316
(*Exploit p:Prod(A,B) to create the assumption z:B(a).
wenzelm@19761
   317
  A more natural form of product elimination. *)
wenzelm@19761
   318
lemma subst_prodE:
wenzelm@19761
   319
  assumes "p: Prod(A,B)"
wenzelm@19761
   320
    and "a: A"
wenzelm@19761
   321
    and "!!z. z: B(a) ==> c(z): C(z)"
wenzelm@19761
   322
  shows "c(p`a): C(p`a)"
wenzelm@19761
   323
apply (rule prems ProdE)+
wenzelm@19761
   324
done
wenzelm@19761
   325
wenzelm@19761
   326
wenzelm@19761
   327
subsection {* Tactics for type checking *}
wenzelm@19761
   328
wenzelm@19761
   329
ML {*
wenzelm@19761
   330
wenzelm@19761
   331
local
wenzelm@19761
   332
wenzelm@19761
   333
fun is_rigid_elem (Const("CTT.Elem",_) $ a $ _) = not(is_Var (head_of a))
wenzelm@19761
   334
  | is_rigid_elem (Const("CTT.Eqelem",_) $ a $ _ $ _) = not(is_Var (head_of a))
wenzelm@19761
   335
  | is_rigid_elem (Const("CTT.Type",_) $ a) = not(is_Var (head_of a))
wenzelm@19761
   336
  | is_rigid_elem _ = false
wenzelm@19761
   337
wenzelm@19761
   338
in
wenzelm@19761
   339
wenzelm@19761
   340
(*Try solving a:A or a=b:A by assumption provided a is rigid!*)
wenzelm@19761
   341
val test_assume_tac = SUBGOAL(fn (prem,i) =>
wenzelm@19761
   342
    if is_rigid_elem (Logic.strip_assums_concl prem)
wenzelm@19761
   343
    then  assume_tac i  else  no_tac)
wenzelm@19761
   344
wenzelm@19761
   345
fun ASSUME tf i = test_assume_tac i  ORELSE  tf i
wenzelm@19761
   346
wenzelm@19761
   347
end;
wenzelm@19761
   348
wenzelm@19761
   349
*}
wenzelm@19761
   350
wenzelm@19761
   351
(*For simplification: type formation and checking,
wenzelm@19761
   352
  but no equalities between terms*)
wenzelm@19761
   353
lemmas routine_rls = form_rls formL_rls refl_type element_rls
wenzelm@19761
   354
wenzelm@19761
   355
ML {*
wenzelm@19761
   356
local
wenzelm@27208
   357
  val equal_rls = @{thms form_rls} @ @{thms element_rls} @ @{thms intrL_rls} @
wenzelm@27208
   358
    @{thms elimL_rls} @ @{thms refl_elem}
wenzelm@19761
   359
in
wenzelm@19761
   360
wenzelm@19761
   361
fun routine_tac rls prems = ASSUME (filt_resolve_tac (prems @ rls) 4);
wenzelm@19761
   362
wenzelm@19761
   363
(*Solve all subgoals "A type" using formation rules. *)
wenzelm@27208
   364
val form_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac @{thms form_rls} 1));
wenzelm@19761
   365
wenzelm@19761
   366
(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *)
wenzelm@19761
   367
fun typechk_tac thms =
wenzelm@27208
   368
  let val tac = filt_resolve_tac (thms @ @{thms form_rls} @ @{thms element_rls}) 3
wenzelm@19761
   369
  in  REPEAT_FIRST (ASSUME tac)  end
wenzelm@19761
   370
wenzelm@19761
   371
(*Solve a:A (a flexible, A rigid) by introduction rules.
wenzelm@19761
   372
  Cannot use stringtrees (filt_resolve_tac) since
wenzelm@19761
   373
  goals like ?a:SUM(A,B) have a trivial head-string *)
wenzelm@19761
   374
fun intr_tac thms =
wenzelm@27208
   375
  let val tac = filt_resolve_tac(thms @ @{thms form_rls} @ @{thms intr_rls}) 1
wenzelm@19761
   376
  in  REPEAT_FIRST (ASSUME tac)  end
wenzelm@19761
   377
wenzelm@19761
   378
(*Equality proving: solve a=b:A (where a is rigid) by long rules. *)
wenzelm@19761
   379
fun equal_tac thms =
wenzelm@19761
   380
  REPEAT_FIRST (ASSUME (filt_resolve_tac (thms @ equal_rls) 3))
clasohm@0
   381
wenzelm@17441
   382
end
wenzelm@19761
   383
wenzelm@19761
   384
*}
wenzelm@19761
   385
wenzelm@19761
   386
wenzelm@19761
   387
subsection {* Simplification *}
wenzelm@19761
   388
wenzelm@19761
   389
(*To simplify the type in a goal*)
wenzelm@19761
   390
lemma replace_type: "[| B = A;  a : A |] ==> a : B"
wenzelm@19761
   391
apply (rule equal_types)
wenzelm@19761
   392
apply (rule_tac [2] sym_type)
wenzelm@19761
   393
apply assumption+
wenzelm@19761
   394
done
wenzelm@19761
   395
wenzelm@19761
   396
(*Simplify the parameter of a unary type operator.*)
wenzelm@19761
   397
lemma subst_eqtyparg:
wenzelm@23467
   398
  assumes 1: "a=c : A"
wenzelm@23467
   399
    and 2: "!!z. z:A ==> B(z) type"
wenzelm@19761
   400
  shows "B(a)=B(c)"
wenzelm@19761
   401
apply (rule subst_typeL)
wenzelm@19761
   402
apply (rule_tac [2] refl_type)
wenzelm@23467
   403
apply (rule 1)
wenzelm@23467
   404
apply (erule 2)
wenzelm@19761
   405
done
wenzelm@19761
   406
wenzelm@19761
   407
(*Simplification rules for Constructive Type Theory*)
wenzelm@19761
   408
lemmas reduction_rls = comp_rls [THEN trans_elem]
wenzelm@19761
   409
wenzelm@19761
   410
ML {*
wenzelm@19761
   411
(*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification.
wenzelm@19761
   412
  Uses other intro rules to avoid changing flexible goals.*)
wenzelm@27208
   413
val eqintr_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac (@{thm EqI} :: @{thms intr_rls}) 1))
wenzelm@19761
   414
wenzelm@19761
   415
(** Tactics that instantiate CTT-rules.
wenzelm@19761
   416
    Vars in the given terms will be incremented!
wenzelm@19761
   417
    The (rtac EqE i) lets them apply to equality judgements. **)
wenzelm@19761
   418
wenzelm@27208
   419
fun NE_tac ctxt sp i =
wenzelm@27239
   420
  TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm NE} i
wenzelm@19761
   421
wenzelm@27208
   422
fun SumE_tac ctxt sp i =
wenzelm@27239
   423
  TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm SumE} i
wenzelm@19761
   424
wenzelm@27208
   425
fun PlusE_tac ctxt sp i =
wenzelm@27239
   426
  TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm PlusE} i
wenzelm@19761
   427
wenzelm@19761
   428
(** Predicate logic reasoning, WITH THINNING!!  Procedures adapted from NJ. **)
wenzelm@19761
   429
wenzelm@19761
   430
(*Finds f:Prod(A,B) and a:A in the assumptions, concludes there is z:B(a) *)
wenzelm@19761
   431
fun add_mp_tac i =
wenzelm@27208
   432
    rtac @{thm subst_prodE} i  THEN  assume_tac i  THEN  assume_tac i
wenzelm@19761
   433
wenzelm@19761
   434
(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
wenzelm@27208
   435
fun mp_tac i = etac @{thm subst_prodE} i  THEN  assume_tac i
wenzelm@19761
   436
wenzelm@19761
   437
(*"safe" when regarded as predicate calculus rules*)
wenzelm@19761
   438
val safe_brls = sort (make_ord lessb)
wenzelm@27208
   439
    [ (true, @{thm FE}), (true,asm_rl),
wenzelm@27208
   440
      (false, @{thm ProdI}), (true, @{thm SumE}), (true, @{thm PlusE}) ]
wenzelm@19761
   441
wenzelm@19761
   442
val unsafe_brls =
wenzelm@27208
   443
    [ (false, @{thm PlusI_inl}), (false, @{thm PlusI_inr}), (false, @{thm SumI}),
wenzelm@27208
   444
      (true, @{thm subst_prodE}) ]
wenzelm@19761
   445
wenzelm@19761
   446
(*0 subgoals vs 1 or more*)
wenzelm@19761
   447
val (safe0_brls, safep_brls) =
wenzelm@19761
   448
    List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls
wenzelm@19761
   449
wenzelm@19761
   450
fun safestep_tac thms i =
wenzelm@19761
   451
    form_tac  ORELSE
wenzelm@19761
   452
    resolve_tac thms i  ORELSE
wenzelm@19761
   453
    biresolve_tac safe0_brls i  ORELSE  mp_tac i  ORELSE
wenzelm@19761
   454
    DETERM (biresolve_tac safep_brls i)
wenzelm@19761
   455
wenzelm@19761
   456
fun safe_tac thms i = DEPTH_SOLVE_1 (safestep_tac thms i)
wenzelm@19761
   457
wenzelm@19761
   458
fun step_tac thms = safestep_tac thms  ORELSE'  biresolve_tac unsafe_brls
wenzelm@19761
   459
wenzelm@19761
   460
(*Fails unless it solves the goal!*)
wenzelm@19761
   461
fun pc_tac thms = DEPTH_SOLVE_1 o (step_tac thms)
wenzelm@19761
   462
*}
wenzelm@19761
   463
wenzelm@19761
   464
use "rew.ML"
wenzelm@19761
   465
wenzelm@19761
   466
wenzelm@19761
   467
subsection {* The elimination rules for fst/snd *}
wenzelm@19761
   468
wenzelm@19761
   469
lemma SumE_fst: "p : Sum(A,B) ==> fst(p) : A"
wenzelm@19761
   470
apply (unfold basic_defs)
wenzelm@19761
   471
apply (erule SumE)
wenzelm@19761
   472
apply assumption
wenzelm@19761
   473
done
wenzelm@19761
   474
wenzelm@19761
   475
(*The first premise must be p:Sum(A,B) !!*)
wenzelm@19761
   476
lemma SumE_snd:
wenzelm@19761
   477
  assumes major: "p: Sum(A,B)"
wenzelm@19761
   478
    and "A type"
wenzelm@19761
   479
    and "!!x. x:A ==> B(x) type"
wenzelm@19761
   480
  shows "snd(p) : B(fst(p))"
wenzelm@19761
   481
  apply (unfold basic_defs)
wenzelm@19761
   482
  apply (rule major [THEN SumE])
wenzelm@19761
   483
  apply (rule SumC [THEN subst_eqtyparg, THEN replace_type])
wenzelm@26391
   484
  apply (tactic {* typechk_tac @{thms assms} *})
wenzelm@19761
   485
  done
wenzelm@19761
   486
wenzelm@19761
   487
end