| author | wenzelm |
| Sun, 01 Dec 2013 17:09:35 +0100 | |
| changeset 54664 | 9dd9d0f023be |
| parent 35762 | af3ff2ba4c54 |
| child 58889 | 5b7a9633cfa8 |
| permissions | -rw-r--r-- |
| 19761 | 1 |
(* Title: CTT/ex/Equality.thy |
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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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*) |
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header "Equality reasoning by rewriting" |
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theory Equality |
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imports CTT |
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begin |
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lemma split_eq: "p : Sum(A,B) ==> split(p,pair) = p : Sum(A,B)" |
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apply (rule EqE) |
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apply (rule elim_rls, assumption) |
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apply (tactic "rew_tac []") |
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done |
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lemma when_eq: "[| A type; B type; p : A+B |] ==> when(p,inl,inr) = p : A + B" |
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apply (rule EqE) |
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apply (rule elim_rls, assumption) |
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apply (tactic "rew_tac []") |
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done |
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(*in the "rec" formulation of addition, 0+n=n *) |
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lemma "p:N ==> rec(p,0, %y z. succ(y)) = p : N" |
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apply (rule EqE) |
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apply (rule elim_rls, assumption) |
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apply (tactic "rew_tac []") |
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done |
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(*the harder version, n+0=n: recursive, uses induction hypothesis*) |
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lemma "p:N ==> rec(p,0, %y z. succ(z)) = p : N" |
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apply (rule EqE) |
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apply (rule elim_rls, assumption) |
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apply (tactic "hyp_rew_tac []") |
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done |
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(*Associativity of addition*) |
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lemma "[| a:N; b:N; c:N |] |
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==> 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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27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
19761
diff
changeset
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apply (tactic {* NE_tac @{context} "a" 1 *})
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apply (tactic "hyp_rew_tac []") |
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done |
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(*Martin-Lof (1984) page 62: pairing is surjective*) |
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lemma "p : Sum(A,B) ==> <split(p,%x y. x), split(p,%x y. y)> = p : Sum(A,B)" |
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apply (rule EqE) |
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apply (rule elim_rls, assumption) |
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apply (tactic {* DEPTH_SOLVE_1 (rew_tac []) *}) (*!!!!!!!*)
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done |
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lemma "[| 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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apply (tactic "rew_tac []") |
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done |
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(*a contrived, complicated simplication, requires sum-elimination also*) |
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lemma "(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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apply (rule reduction_rls) |
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apply (rule_tac [3] intrL_rls) |
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apply (rule_tac [4] EqE) |
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apply (rule_tac [4] SumE, tactic "assume_tac 4") |
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(*order of unifiers is essential here*) |
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apply (tactic "rew_tac []") |
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done |
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end |