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(* Title: CTT/ex/equal
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1991 University of Cambridge
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Equality reasoning by rewriting.
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*)
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val prems =
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goal CTT.thy "p : Sum(A,B) ==> split(p,pair) = p : Sum(A,B)";
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by (rtac EqE 1);
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by (resolve_tac elim_rls 1 THEN resolve_tac prems 1);
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by (rew_tac prems);
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1294
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qed "split_eq";
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0
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val prems =
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goal CTT.thy
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"[| A type; B type; p : A+B |] ==> when(p,inl,inr) = p : A + B";
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by (rtac EqE 1);
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by (resolve_tac elim_rls 1 THEN resolve_tac prems 1);
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by (rew_tac prems);
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1294
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qed "when_eq";
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(*in the "rec" formulation of addition, 0+n=n *)
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val prems =
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goal CTT.thy "p:N ==> rec(p,0, %y z.succ(y)) = p : N";
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by (rtac EqE 1);
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by (resolve_tac elim_rls 1 THEN resolve_tac prems 1);
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by (rew_tac prems);
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result();
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(*the harder version, n+0=n: recursive, uses induction hypothesis*)
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val prems =
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goal CTT.thy "p:N ==> rec(p,0, %y z.succ(z)) = p : N";
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by (rtac EqE 1);
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by (resolve_tac elim_rls 1 THEN resolve_tac prems 1);
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by (hyp_rew_tac prems);
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result();
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(*Associativity of addition*)
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val prems =
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goal CTT.thy
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"[| a:N; b:N; c:N |] ==> rec(rec(a, b, %x y.succ(y)), c, %x y.succ(y)) = \
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\ rec(a, rec(b, c, %x y.succ(y)), %x y.succ(y)) : N";
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by (NE_tac "a" 1);
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by (hyp_rew_tac prems);
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result();
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(*Martin-Lof (1984) page 62: pairing is surjective*)
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val prems =
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goal CTT.thy
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"p : Sum(A,B) ==> <split(p,%x y.x), split(p,%x y.y)> = p : Sum(A,B)";
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by (rtac EqE 1);
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by (resolve_tac elim_rls 1 THEN resolve_tac prems 1);
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by (DEPTH_SOLVE_1 (rew_tac prems)); (*!!!!!!!*)
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result();
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val prems =
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goal CTT.thy "[| a : A; b : B |] ==> \
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\ (lam u. split(u, %v w.<w,v>)) ` <a,b> = <b,a> : SUM x:B.A";
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by (rew_tac prems);
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result();
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(*a contrived, complicated simplication, requires sum-elimination also*)
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val prems =
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goal CTT.thy
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"(lam f. lam x. f`(f`x)) ` (lam u. split(u, %v w.<w,v>)) = \
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\ lam x. x : PROD x:(SUM y:N.N). (SUM y:N.N)";
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by (resolve_tac reduction_rls 1);
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by (resolve_tac intrL_rls 3);
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by (rtac EqE 4);
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by (rtac SumE 4 THEN assume_tac 4);
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(*order of unifiers is essential here*)
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by (rew_tac prems);
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result();
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writeln"Reached end of file.";
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(*28 August 1988: loaded this file in 34 seconds*)
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(*2 September 1988: loaded this file in 48 seconds*)
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