author | wenzelm |
Fri, 02 Jun 2006 18:15:38 +0200 | |
changeset 19761 | 5cd82054c2c6 |
parent 17782 | b3846df9d643 |
child 21210 | c17fd2df4e9e |
permissions | -rw-r--r-- |
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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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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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"+" :: "[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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"`" :: "[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) |
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"A --> B == PROD _:A. B" |
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Times :: "[t,t]=>t" (infixr "*" 50) |
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"A * B == SUM _:A. B" |
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const_syntax (xsymbols) |
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Elem ("(_ /\<in> _)" [10,10] 5) |
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Eqelem ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) |
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Arrow (infixr "\<longrightarrow>" 30) |
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Times (infixr "\<times>" 50) |
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const_syntax (HTML output) |
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Elem ("(_ /\<in> _)" [10,10] 5) |
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Eqelem ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) |
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Times (infixr "\<times>" 50) |
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syntax (xsymbols) |
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"@SUM" :: "[idt,t,t] => t" ("(3\<Sigma> _\<in>_./ _)" 10) |
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"@PROD" :: "[idt,t,t] => t" ("(3\<Pi> _\<in>_./ _)" 10) |
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"lam " :: "[idts, i] => i" ("(3\<lambda>\<lambda>_./ _)" 10) |
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syntax (HTML output) |
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"@SUM" :: "[idt,t,t] => t" ("(3\<Sigma> _\<in>_./ _)" 10) |
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"@PROD" :: "[idt,t,t] => t" ("(3\<Pi> _\<in>_./ _)" 10) |
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"lam " :: "[idts, i] => i" ("(3\<lambda>\<lambda>_./ _)" 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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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 routine_rls = thms "routine_rls"; |
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val form_rls = thms "form_rls"; |
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val intr_rls = thms "intr_rls"; |
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val element_rls = thms "element_rls"; |
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val equal_rls = form_rls @ element_rls @ thms "intrL_rls" @ |
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thms "elimL_rls" @ thms "refl_elem" |
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360 |
in |
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361 |
||
362 |
fun routine_tac rls prems = ASSUME (filt_resolve_tac (prems @ rls) 4); |
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363 |
||
364 |
(*Solve all subgoals "A type" using formation rules. *) |
|
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val form_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac(form_rls) 1)); |
|
366 |
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367 |
(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *) |
|
368 |
fun typechk_tac thms = |
|
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let val tac = filt_resolve_tac (thms @ form_rls @ element_rls) 3 |
|
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in REPEAT_FIRST (ASSUME tac) end |
|
371 |
||
372 |
(*Solve a:A (a flexible, A rigid) by introduction rules. |
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Cannot use stringtrees (filt_resolve_tac) since |
|
374 |
goals like ?a:SUM(A,B) have a trivial head-string *) |
|
375 |
fun intr_tac thms = |
|
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let val tac = filt_resolve_tac(thms@form_rls@intr_rls) 1 |
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377 |
in REPEAT_FIRST (ASSUME tac) end |
|
378 |
||
379 |
(*Equality proving: solve a=b:A (where a is rigid) by long rules. *) |
|
380 |
fun equal_tac thms = |
|
381 |
REPEAT_FIRST (ASSUME (filt_resolve_tac (thms @ equal_rls) 3)) |
|
0 | 382 |
|
17441 | 383 |
end |
19761 | 384 |
|
385 |
*} |
|
386 |
||
387 |
||
388 |
subsection {* Simplification *} |
|
389 |
||
390 |
(*To simplify the type in a goal*) |
|
391 |
lemma replace_type: "[| B = A; a : A |] ==> a : B" |
|
392 |
apply (rule equal_types) |
|
393 |
apply (rule_tac [2] sym_type) |
|
394 |
apply assumption+ |
|
395 |
done |
|
396 |
||
397 |
(*Simplify the parameter of a unary type operator.*) |
|
398 |
lemma subst_eqtyparg: |
|
399 |
assumes "a=c : A" |
|
400 |
and "!!z. z:A ==> B(z) type" |
|
401 |
shows "B(a)=B(c)" |
|
402 |
apply (rule subst_typeL) |
|
403 |
apply (rule_tac [2] refl_type) |
|
404 |
apply (rule prems) |
|
405 |
apply assumption |
|
406 |
done |
|
407 |
||
408 |
(*Simplification rules for Constructive Type Theory*) |
|
409 |
lemmas reduction_rls = comp_rls [THEN trans_elem] |
|
410 |
||
411 |
ML {* |
|
412 |
local |
|
413 |
val EqI = thm "EqI"; |
|
414 |
val EqE = thm "EqE"; |
|
415 |
val NE = thm "NE"; |
|
416 |
val FE = thm "FE"; |
|
417 |
val ProdI = thm "ProdI"; |
|
418 |
val SumI = thm "SumI"; |
|
419 |
val SumE = thm "SumE"; |
|
420 |
val PlusE = thm "PlusE"; |
|
421 |
val PlusI_inl = thm "PlusI_inl"; |
|
422 |
val PlusI_inr = thm "PlusI_inr"; |
|
423 |
val subst_prodE = thm "subst_prodE"; |
|
424 |
val intr_rls = thms "intr_rls"; |
|
425 |
in |
|
426 |
||
427 |
(*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification. |
|
428 |
Uses other intro rules to avoid changing flexible goals.*) |
|
429 |
val eqintr_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac(EqI::intr_rls) 1)) |
|
430 |
||
431 |
(** Tactics that instantiate CTT-rules. |
|
432 |
Vars in the given terms will be incremented! |
|
433 |
The (rtac EqE i) lets them apply to equality judgements. **) |
|
434 |
||
435 |
fun NE_tac (sp: string) i = |
|
436 |
TRY (rtac EqE i) THEN res_inst_tac [ ("p",sp) ] NE i |
|
437 |
||
438 |
fun SumE_tac (sp: string) i = |
|
439 |
TRY (rtac EqE i) THEN res_inst_tac [ ("p",sp) ] SumE i |
|
440 |
||
441 |
fun PlusE_tac (sp: string) i = |
|
442 |
TRY (rtac EqE i) THEN res_inst_tac [ ("p",sp) ] PlusE i |
|
443 |
||
444 |
(** Predicate logic reasoning, WITH THINNING!! Procedures adapted from NJ. **) |
|
445 |
||
446 |
(*Finds f:Prod(A,B) and a:A in the assumptions, concludes there is z:B(a) *) |
|
447 |
fun add_mp_tac i = |
|
448 |
rtac subst_prodE i THEN assume_tac i THEN assume_tac i |
|
449 |
||
450 |
(*Finds P-->Q and P in the assumptions, replaces implication by Q *) |
|
451 |
fun mp_tac i = etac subst_prodE i THEN assume_tac i |
|
452 |
||
453 |
(*"safe" when regarded as predicate calculus rules*) |
|
454 |
val safe_brls = sort (make_ord lessb) |
|
455 |
[ (true,FE), (true,asm_rl), |
|
456 |
(false,ProdI), (true,SumE), (true,PlusE) ] |
|
457 |
||
458 |
val unsafe_brls = |
|
459 |
[ (false,PlusI_inl), (false,PlusI_inr), (false,SumI), |
|
460 |
(true,subst_prodE) ] |
|
461 |
||
462 |
(*0 subgoals vs 1 or more*) |
|
463 |
val (safe0_brls, safep_brls) = |
|
464 |
List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls |
|
465 |
||
466 |
fun safestep_tac thms i = |
|
467 |
form_tac ORELSE |
|
468 |
resolve_tac thms i ORELSE |
|
469 |
biresolve_tac safe0_brls i ORELSE mp_tac i ORELSE |
|
470 |
DETERM (biresolve_tac safep_brls i) |
|
471 |
||
472 |
fun safe_tac thms i = DEPTH_SOLVE_1 (safestep_tac thms i) |
|
473 |
||
474 |
fun step_tac thms = safestep_tac thms ORELSE' biresolve_tac unsafe_brls |
|
475 |
||
476 |
(*Fails unless it solves the goal!*) |
|
477 |
fun pc_tac thms = DEPTH_SOLVE_1 o (step_tac thms) |
|
478 |
||
479 |
end |
|
480 |
*} |
|
481 |
||
482 |
use "rew.ML" |
|
483 |
||
484 |
||
485 |
subsection {* The elimination rules for fst/snd *} |
|
486 |
||
487 |
lemma SumE_fst: "p : Sum(A,B) ==> fst(p) : A" |
|
488 |
apply (unfold basic_defs) |
|
489 |
apply (erule SumE) |
|
490 |
apply assumption |
|
491 |
done |
|
492 |
||
493 |
(*The first premise must be p:Sum(A,B) !!*) |
|
494 |
lemma SumE_snd: |
|
495 |
assumes major: "p: Sum(A,B)" |
|
496 |
and "A type" |
|
497 |
and "!!x. x:A ==> B(x) type" |
|
498 |
shows "snd(p) : B(fst(p))" |
|
499 |
apply (unfold basic_defs) |
|
500 |
apply (rule major [THEN SumE]) |
|
501 |
apply (rule SumC [THEN subst_eqtyparg, THEN replace_type]) |
|
502 |
apply (tactic {* typechk_tac (thms "prems") *}) |
|
503 |
done |
|
504 |
||
505 |
end |