author | nipkow |
Thu, 11 Jul 2002 10:48:30 +0200 | |
changeset 13347 | 867f876589e7 |
parent 12110 | f8b4b11cd79d |
child 14565 | c6dc17aab88a |
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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Constructive Type Theory |
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*) |
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CTT = Pure + |
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types |
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i |
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t |
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o |
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arities |
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i,t,o :: logic |
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consts |
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(*Types*) |
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F,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,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,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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(*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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parents:
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changeset
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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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"@PROD" :: "[idt,t,t]=>t" ("(3PROD _:_./ _)" 10) |
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"@SUM" :: "[idt,t,t]=>t" ("(3SUM _:_./ _)" 10) |
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"+" :: "[t,t]=>t" (infixr 40) |
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(*Invisible infixes!*) |
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"@-->" :: "[t,t]=>t" ("(_ -->/ _)" [31,30] 30) |
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"@*" :: "[t,t]=>t" ("(_ */ _)" [51,50] 50) |
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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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translations |
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"PROD x:A. B" => "Prod(A, %x. B)" |
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"A --> B" => "Prod(A, _K(B))" |
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"SUM x:A. B" => "Sum(A, %x. B)" |
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"A * B" => "Sum(A, _K(B))" |
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x-symbol support for Pi, Sigma, -->, : (membership)
paulson
parents:
3837
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changeset
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syntax (xsymbols) |
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x-symbol support for Pi, Sigma, -->, : (membership)
paulson
parents:
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changeset
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"@-->" :: "[t,t]=>t" ("(_ \\<longrightarrow>/ _)" [31,30] 30) |
e6e7205e9e91
x-symbol support for Pi, Sigma, -->, : (membership)
paulson
parents:
3837
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changeset
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"@*" :: "[t,t]=>t" ("(_ \\<times>/ _)" [51,50] 50) |
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eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
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Elem :: "[i, t]=>prop" ("(_ /\\<in> _)" [10,10] 5) |
f8b4b11cd79d
eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
10467
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Eqelem :: "[i,i,t]=>prop" ("(2_ =/ _ \\<in>/ _)" [10,10,10] 5) |
f8b4b11cd79d
eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
10467
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changeset
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"@SUM" :: "[idt,t,t] => t" ("(3\\<Sigma> _\\<in>_./ _)" 10) |
f8b4b11cd79d
eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
10467
diff
changeset
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"@PROD" :: "[idt,t,t] => t" ("(3\\<Pi> _\\<in>_./ _)" 10) |
f8b4b11cd79d
eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
10467
diff
changeset
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"lam " :: "[idts, i] => i" ("(3\\<lambda>\\<lambda>_./ _)" 10) |
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paulson
parents:
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rules |
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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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end |
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ML |
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val print_translation = |
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[("Prod", dependent_tr' ("@PROD", "@-->")), |
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("Sum", dependent_tr' ("@SUM", "@*"))]; |
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