src/CTT/CTT.thy
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(*  Title:      CTT/CTT.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1993  University of Cambridge
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*)
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header {* Constructive Type Theory *}
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theory CTT
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imports Pure
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uses "~~/src/Provers/typedsimp.ML" ("rew.ML")
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begin
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setup PureThy.old_appl_syntax_setup
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typedecl i
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typedecl t
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typedecl o
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consts
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  (*Types*)
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  F         :: "t"
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  T         :: "t"          (*F is empty, T contains one element*)
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  contr     :: "i=>i"
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  tt        :: "i"
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  (*Natural numbers*)
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  N         :: "t"
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  succ      :: "i=>i"
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  rec       :: "[i, i, [i,i]=>i] => i"
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  (*Unions*)
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  inl       :: "i=>i"
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  inr       :: "i=>i"
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  when      :: "[i, i=>i, i=>i]=>i"
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  (*General Sum and Binary Product*)
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  Sum       :: "[t, i=>t]=>t"
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  fst       :: "i=>i"
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  snd       :: "i=>i"
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  split     :: "[i, [i,i]=>i] =>i"
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  (*General Product and Function Space*)
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  Prod      :: "[t, i=>t]=>t"
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  (*Types*)
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  Plus      :: "[t,t]=>t"           (infixr "+" 40)
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  (*Equality type*)
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  Eq        :: "[t,i,i]=>t"
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  eq        :: "i"
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  (*Judgements*)
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  Type      :: "t => prop"          ("(_ type)" [10] 5)
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  Eqtype    :: "[t,t]=>prop"        ("(_ =/ _)" [10,10] 5)
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  Elem      :: "[i, t]=>prop"       ("(_ /: _)" [10,10] 5)
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  Eqelem    :: "[i,i,t]=>prop"      ("(_ =/ _ :/ _)" [10,10,10] 5)
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  Reduce    :: "[i,i]=>prop"        ("Reduce[_,_]")
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  (*Types*)
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  (*Functions*)
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  lambda    :: "(i => i) => i"      (binder "lam " 10)
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  app       :: "[i,i]=>i"           (infixl "`" 60)
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  (*Natural numbers*)
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  "0"       :: "i"                  ("0")
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  (*Pairing*)
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  pair      :: "[i,i]=>i"           ("(1<_,/_>)")
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syntax
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  "_PROD"   :: "[idt,t,t]=>t"       ("(3PROD _:_./ _)" 10)
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  "_SUM"    :: "[idt,t,t]=>t"       ("(3SUM _:_./ _)" 10)
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translations
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  "PROD x:A. B" == "Prod(A, %x. B)"
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  "SUM x:A. B"  == "Sum(A, %x. B)"
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abbreviation
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  Arrow     :: "[t,t]=>t"  (infixr "-->" 30) where
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  "A --> B == PROD _:A. B"
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abbreviation
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  Times     :: "[t,t]=>t"  (infixr "*" 50) where
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  "A * B == SUM _:A. B"
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notation (xsymbols)
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  lambda  (binder "\<lambda>\<lambda>" 10) and
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  Elem  ("(_ /\<in> _)" [10,10] 5) and
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  Eqelem  ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) and
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  Arrow  (infixr "\<longrightarrow>" 30) and
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  Times  (infixr "\<times>" 50)
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notation (HTML output)
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  lambda  (binder "\<lambda>\<lambda>" 10) and
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  Elem  ("(_ /\<in> _)" [10,10] 5) and
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  Eqelem  ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) and
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  Times  (infixr "\<times>" 50)
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syntax (xsymbols)
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  "_PROD"   :: "[idt,t,t] => t"     ("(3\<Pi> _\<in>_./ _)"    10)
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  "_SUM"    :: "[idt,t,t] => t"     ("(3\<Sigma> _\<in>_./ _)" 10)
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syntax (HTML output)
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  "_PROD"   :: "[idt,t,t] => t"     ("(3\<Pi> _\<in>_./ _)"    10)
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  "_SUM"    :: "[idt,t,t] => t"     ("(3\<Sigma> _\<in>_./ _)" 10)
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axioms
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  (*Reduction: a weaker notion than equality;  a hack for simplification.
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    Reduce[a,b] means either that  a=b:A  for some A or else that "a" and "b"
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    are textually identical.*)
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  (*does not verify a:A!  Sound because only trans_red uses a Reduce premise
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    No new theorems can be proved about the standard judgements.*)
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  refl_red: "Reduce[a,a]"
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  red_if_equal: "a = b : A ==> Reduce[a,b]"
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  trans_red: "[| a = b : A;  Reduce[b,c] |] ==> a = c : A"
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  (*Reflexivity*)
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  refl_type: "A type ==> A = A"
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  refl_elem: "a : A ==> a = a : A"
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  (*Symmetry*)
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  sym_type:  "A = B ==> B = A"
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  sym_elem:  "a = b : A ==> b = a : A"
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  (*Transitivity*)
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  trans_type:   "[| A = B;  B = C |] ==> A = C"
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  trans_elem:   "[| a = b : A;  b = c : A |] ==> a = c : A"
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  equal_types:  "[| a : A;  A = B |] ==> a : B"
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  equal_typesL: "[| a = b : A;  A = B |] ==> a = b : B"
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  (*Substitution*)
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  subst_type:   "[| a : A;  !!z. z:A ==> B(z) type |] ==> B(a) type"
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  subst_typeL:  "[| a = c : A;  !!z. z:A ==> B(z) = D(z) |] ==> B(a) = D(c)"
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  subst_elem:   "[| a : A;  !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)"
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  subst_elemL:
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    "[| a=c : A;  !!z. z:A ==> b(z)=d(z) : B(z) |] ==> b(a)=d(c) : B(a)"
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  (*The type N -- natural numbers*)
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  NF: "N type"
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  NI0: "0 : N"
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  NI_succ: "a : N ==> succ(a) : N"
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  NI_succL:  "a = b : N ==> succ(a) = succ(b) : N"
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  NE:
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   "[| p: N;  a: C(0);  !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]
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   ==> rec(p, a, %u v. b(u,v)) : C(p)"
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  NEL:
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   "[| p = q : N;  a = c : C(0);
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      !!u v. [| u: N; v: C(u) |] ==> b(u,v) = d(u,v): C(succ(u)) |]
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   ==> rec(p, a, %u v. b(u,v)) = rec(q,c,d) : C(p)"
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  NC0:
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   "[| a: C(0);  !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]
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   ==> rec(0, a, %u v. b(u,v)) = a : C(0)"
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  NC_succ:
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   "[| p: N;  a: C(0);
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       !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] ==>
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   rec(succ(p), a, %u v. b(u,v)) = b(p, rec(p, a, %u v. b(u,v))) : C(succ(p))"
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  (*The fourth Peano axiom.  See page 91 of Martin-Lof's book*)
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  zero_ne_succ:
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    "[| a: N;  0 = succ(a) : N |] ==> 0: F"
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  (*The Product of a family of types*)
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  ProdF:  "[| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A. B(x) type"
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  ProdFL:
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   "[| A = C;  !!x. x:A ==> B(x) = D(x) |] ==>
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   PROD x:A. B(x) = PROD x:C. D(x)"
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  ProdI:
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   "[| A type;  !!x. x:A ==> b(x):B(x)|] ==> lam x. b(x) : PROD x:A. B(x)"
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  ProdIL:
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   "[| A type;  !!x. x:A ==> b(x) = c(x) : B(x)|] ==>
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   lam x. b(x) = lam x. c(x) : PROD x:A. B(x)"
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  ProdE:  "[| p : PROD x:A. B(x);  a : A |] ==> p`a : B(a)"
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  ProdEL: "[| p=q: PROD x:A. B(x);  a=b : A |] ==> p`a = q`b : B(a)"
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  ProdC:
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   "[| a : A;  !!x. x:A ==> b(x) : B(x)|] ==>
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   (lam x. b(x)) ` a = b(a) : B(a)"
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  ProdC2:
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   "p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)"
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  (*The Sum of a family of types*)
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  SumF:  "[| A type;  !!x. x:A ==> B(x) type |] ==> SUM x:A. B(x) type"
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  SumFL:
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    "[| A = C;  !!x. x:A ==> B(x) = D(x) |] ==> SUM x:A. B(x) = SUM x:C. D(x)"
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  SumI:  "[| a : A;  b : B(a) |] ==> <a,b> : SUM x:A. B(x)"
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  SumIL: "[| a=c:A;  b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A. B(x)"
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  SumE:
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    "[| p: SUM x:A. B(x);  !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]
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    ==> split(p, %x y. c(x,y)) : C(p)"
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  SumEL:
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    "[| p=q : SUM x:A. B(x);
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       !!x y. [| x:A; y:B(x) |] ==> c(x,y)=d(x,y): C(<x,y>)|]
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    ==> split(p, %x y. c(x,y)) = split(q, % x y. d(x,y)) : C(p)"
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  SumC:
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    "[| a: A;  b: B(a);  !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]
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    ==> split(<a,b>, %x y. c(x,y)) = c(a,b) : C(<a,b>)"
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  fst_def:   "fst(a) == split(a, %x y. x)"
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  snd_def:   "snd(a) == split(a, %x y. y)"
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  (*The sum of two types*)
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  PlusF:   "[| A type;  B type |] ==> A+B type"
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  PlusFL:  "[| A = C;  B = D |] ==> A+B = C+D"
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  PlusI_inl:   "[| a : A;  B type |] ==> inl(a) : A+B"
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  PlusI_inlL: "[| a = c : A;  B type |] ==> inl(a) = inl(c) : A+B"
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  PlusI_inr:   "[| A type;  b : B |] ==> inr(b) : A+B"
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  PlusI_inrL: "[| A type;  b = d : B |] ==> inr(b) = inr(d) : A+B"
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  PlusE:
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    "[| p: A+B;  !!x. x:A ==> c(x): C(inl(x));
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                !!y. y:B ==> d(y): C(inr(y)) |]
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    ==> when(p, %x. c(x), %y. d(y)) : C(p)"
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  PlusEL:
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    "[| p = q : A+B;  !!x. x: A ==> c(x) = e(x) : C(inl(x));
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                     !!y. y: B ==> d(y) = f(y) : C(inr(y)) |]
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    ==> when(p, %x. c(x), %y. d(y)) = when(q, %x. e(x), %y. f(y)) : C(p)"
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  PlusC_inl:
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    "[| a: A;  !!x. x:A ==> c(x): C(inl(x));
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              !!y. y:B ==> d(y): C(inr(y)) |]
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    ==> when(inl(a), %x. c(x), %y. d(y)) = c(a) : C(inl(a))"
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  PlusC_inr:
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    "[| b: B;  !!x. x:A ==> c(x): C(inl(x));
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              !!y. y:B ==> d(y): C(inr(y)) |]
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    ==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))"
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  (*The type Eq*)
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  EqF:    "[| A type;  a : A;  b : A |] ==> Eq(A,a,b) type"
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  EqFL: "[| A=B;  a=c: A;  b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)"
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  EqI: "a = b : A ==> eq : Eq(A,a,b)"
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  EqE: "p : Eq(A,a,b) ==> a = b : A"
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  (*By equality of types, can prove C(p) from C(eq), an elimination rule*)
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  EqC: "p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)"
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  (*The type F*)
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  FF: "F type"
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  FE: "[| p: F;  C type |] ==> contr(p) : C"
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  FEL:  "[| p = q : F;  C type |] ==> contr(p) = contr(q) : C"
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  (*The type T
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     Martin-Lof's book (page 68) discusses elimination and computation.
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     Elimination can be derived by computation and equality of types,
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     but with an extra premise C(x) type x:T.
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     Also computation can be derived from elimination. *)
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  TF: "T type"
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  TI: "tt : T"
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  TE: "[| p : T;  c : C(tt) |] ==> c : C(p)"
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  TEL: "[| p = q : T;  c = d : C(tt) |] ==> c = d : C(p)"
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  TC: "p : T ==> p = tt : T"
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19761
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subsection "Tactics and derived rules for Constructive Type Theory"
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(*Formation rules*)
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lemmas form_rls = NF ProdF SumF PlusF EqF FF TF
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  and formL_rls = ProdFL SumFL PlusFL EqFL
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(*Introduction rules
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  OMITTED: EqI, because its premise is an eqelem, not an elem*)
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lemmas intr_rls = NI0 NI_succ ProdI SumI PlusI_inl PlusI_inr TI
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  and intrL_rls = NI_succL ProdIL SumIL PlusI_inlL PlusI_inrL
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(*Elimination rules
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  OMITTED: EqE, because its conclusion is an eqelem,  not an elem
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           TE, because it does not involve a constructor *)
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lemmas elim_rls = NE ProdE SumE PlusE FE
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  and elimL_rls = NEL ProdEL SumEL PlusEL FEL
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(*OMITTED: eqC are TC because they make rewriting loop: p = un = un = ... *)
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lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr
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(*rules with conclusion a:A, an elem judgement*)
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lemmas element_rls = intr_rls elim_rls
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(*Definitions are (meta)equality axioms*)
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lemmas basic_defs = fst_def snd_def
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(*Compare with standard version: B is applied to UNSIMPLIFIED expression! *)
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lemma SumIL2: "[| c=a : A;  d=b : B(a) |] ==> <c,d> = <a,b> : Sum(A,B)"
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apply (rule sym_elem)
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apply (rule SumIL)
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apply (rule_tac [!] sym_elem)
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apply assumption+
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done
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lemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL
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(*Exploit p:Prod(A,B) to create the assumption z:B(a).
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  A more natural form of product elimination. *)
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lemma subst_prodE:
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  assumes "p: Prod(A,B)"
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    and "a: A"
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    and "!!z. z: B(a) ==> c(z): C(z)"
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  shows "c(p`a): C(p`a)"
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apply (rule prems ProdE)+
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done
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subsection {* Tactics for type checking *}
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ML {*
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local
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fun is_rigid_elem (Const("CTT.Elem",_) $ a $ _) = not(is_Var (head_of a))
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  | is_rigid_elem (Const("CTT.Eqelem",_) $ a $ _ $ _) = not(is_Var (head_of a))
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  | is_rigid_elem (Const("CTT.Type",_) $ a) = not(is_Var (head_of a))
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  | is_rigid_elem _ = false
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in
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(*Try solving a:A or a=b:A by assumption provided a is rigid!*)
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val test_assume_tac = SUBGOAL(fn (prem,i) =>
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    if is_rigid_elem (Logic.strip_assums_concl prem)
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    then  assume_tac i  else  no_tac)
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fun ASSUME tf i = test_assume_tac i  ORELSE  tf i
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end;
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*}
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(*For simplification: type formation and checking,
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  but no equalities between terms*)
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lemmas routine_rls = form_rls formL_rls refl_type element_rls
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ML {*
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local
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  val equal_rls = @{thms form_rls} @ @{thms element_rls} @ @{thms intrL_rls} @
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    @{thms elimL_rls} @ @{thms refl_elem}
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in
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fun routine_tac rls prems = ASSUME (filt_resolve_tac (prems @ rls) 4);
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(*Solve all subgoals "A type" using formation rules. *)
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val form_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac @{thms form_rls} 1));
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(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *)
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fun typechk_tac thms =
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  let val tac = filt_resolve_tac (thms @ @{thms form_rls} @ @{thms element_rls}) 3
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  in  REPEAT_FIRST (ASSUME tac)  end
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(*Solve a:A (a flexible, A rigid) by introduction rules.
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  Cannot use stringtrees (filt_resolve_tac) since
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  goals like ?a:SUM(A,B) have a trivial head-string *)
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fun intr_tac thms =
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  let val tac = filt_resolve_tac(thms @ @{thms form_rls} @ @{thms intr_rls}) 1
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  in  REPEAT_FIRST (ASSUME tac)  end
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(*Equality proving: solve a=b:A (where a is rigid) by long rules. *)
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fun equal_tac thms =
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  REPEAT_FIRST (ASSUME (filt_resolve_tac (thms @ equal_rls) 3))
0
a5a9c433f639 Initial revision
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   382
end
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*}
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   387
subsection {* Simplification *}
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   388
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   389
(*To simplify the type in a goal*)
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   390
lemma replace_type: "[| B = A;  a : A |] ==> a : B"
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   391
apply (rule equal_types)
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   392
apply (rule_tac [2] sym_type)
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   393
apply assumption+
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   394
done
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   395
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   396
(*Simplify the parameter of a unary type operator.*)
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   397
lemma subst_eqtyparg:
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   398
  assumes 1: "a=c : A"
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   399
    and 2: "!!z. z:A ==> B(z) type"
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  shows "B(a)=B(c)"
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   401
apply (rule subst_typeL)
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   402
apply (rule_tac [2] refl_type)
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   403
apply (rule 1)
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   404
apply (erule 2)
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   405
done
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   406
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   407
(*Simplification rules for Constructive Type Theory*)
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   408
lemmas reduction_rls = comp_rls [THEN trans_elem]
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diff changeset
   409
5cd82054c2c6 removed obsolete ML files;
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   410
ML {*
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   411
(*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification.
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  Uses other intro rules to avoid changing flexible goals.*)
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   413
val eqintr_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac (@{thm EqI} :: @{thms intr_rls}) 1))
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   414
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   415
(** Tactics that instantiate CTT-rules.
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   416
    Vars in the given terms will be incremented!
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   417
    The (rtac EqE i) lets them apply to equality judgements. **)
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   418
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   419
fun NE_tac ctxt sp i =
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   420
  TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm NE} i
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   421
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   422
fun SumE_tac ctxt sp i =
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   423
  TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm SumE} i
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   424
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   425
fun PlusE_tac ctxt sp i =
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   426
  TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm PlusE} i
19761
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diff changeset
   427
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   428
(** Predicate logic reasoning, WITH THINNING!!  Procedures adapted from NJ. **)
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diff changeset
   429
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   430
(*Finds f:Prod(A,B) and a:A in the assumptions, concludes there is z:B(a) *)
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   431
fun add_mp_tac i =
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   432
    rtac @{thm subst_prodE} i  THEN  assume_tac i  THEN  assume_tac i
19761
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diff changeset
   433
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   434
(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
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   435
fun mp_tac i = etac @{thm subst_prodE} i  THEN  assume_tac i
19761
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diff changeset
   436
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   437
(*"safe" when regarded as predicate calculus rules*)
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   438
val safe_brls = sort (make_ord lessb)
27208
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   439
    [ (true, @{thm FE}), (true,asm_rl),
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   440
      (false, @{thm ProdI}), (true, @{thm SumE}), (true, @{thm PlusE}) ]
19761
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diff changeset
   441
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   442
val unsafe_brls =
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   443
    [ (false, @{thm PlusI_inl}), (false, @{thm PlusI_inr}), (false, @{thm SumI}),
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diff changeset
   444
      (true, @{thm subst_prodE}) ]
19761
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diff changeset
   445
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   446
(*0 subgoals vs 1 or more*)
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   447
val (safe0_brls, safep_brls) =
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   448
    List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls
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diff changeset
   449
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   450
fun safestep_tac thms i =
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   451
    form_tac  ORELSE
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   452
    resolve_tac thms i  ORELSE
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   453
    biresolve_tac safe0_brls i  ORELSE  mp_tac i  ORELSE
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diff changeset
   454
    DETERM (biresolve_tac safep_brls i)
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diff changeset
   455
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   456
fun safe_tac thms i = DEPTH_SOLVE_1 (safestep_tac thms i)
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diff changeset
   457
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   458
fun step_tac thms = safestep_tac thms  ORELSE'  biresolve_tac unsafe_brls
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diff changeset
   459
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   460
(*Fails unless it solves the goal!*)
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   461
fun pc_tac thms = DEPTH_SOLVE_1 o (step_tac thms)
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   462
*}
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diff changeset
   463
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   464
use "rew.ML"
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diff changeset
   465
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diff changeset
   466
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   467
subsection {* The elimination rules for fst/snd *}
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diff changeset
   468
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diff changeset
   469
lemma SumE_fst: "p : Sum(A,B) ==> fst(p) : A"
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diff changeset
   470
apply (unfold basic_defs)
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diff changeset
   471
apply (erule SumE)
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diff changeset
   472
apply assumption
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   473
done
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diff changeset
   474
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diff changeset
   475
(*The first premise must be p:Sum(A,B) !!*)
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diff changeset
   476
lemma SumE_snd:
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diff changeset
   477
  assumes major: "p: Sum(A,B)"
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diff changeset
   478
    and "A type"
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diff changeset
   479
    and "!!x. x:A ==> B(x) type"
5cd82054c2c6 removed obsolete ML files;
wenzelm
parents: 17782
diff changeset
   480
  shows "snd(p) : B(fst(p))"
5cd82054c2c6 removed obsolete ML files;
wenzelm
parents: 17782
diff changeset
   481
  apply (unfold basic_defs)
5cd82054c2c6 removed obsolete ML files;
wenzelm
parents: 17782
diff changeset
   482
  apply (rule major [THEN SumE])
5cd82054c2c6 removed obsolete ML files;
wenzelm
parents: 17782
diff changeset
   483
  apply (rule SumC [THEN subst_eqtyparg, THEN replace_type])
26391
6e8aa5a4eb82 more antiquotations;
wenzelm
parents: 23467
diff changeset
   484
  apply (tactic {* typechk_tac @{thms assms} *})
19761
5cd82054c2c6 removed obsolete ML files;
wenzelm
parents: 17782
diff changeset
   485
  done
5cd82054c2c6 removed obsolete ML files;
wenzelm
parents: 17782
diff changeset
   486
5cd82054c2c6 removed obsolete ML files;
wenzelm
parents: 17782
diff changeset
   487
end