author | wenzelm |
Sat, 01 Jul 2000 19:55:22 +0200 | |
changeset 9230 | 17ae63f82ad8 |
parent 9020 | 1056cbbaeb29 |
child 9345 | 2f5f6824f888 |
permissions | -rw-r--r-- |
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(* Title: HOL/prod |
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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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||
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Ordered Pairs, the Cartesian product type, the unit type |
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*) |
8 |
||
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(*This counts as a non-emptiness result for admitting 'a * 'b as a type*) |
|
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Goalw [Prod_def] "Pair_Rep a b : Prod"; |
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by (EVERY1 [rtac CollectI, rtac exI, rtac exI, rtac refl]); |
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qed "ProdI"; |
|
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||
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val [major] = goalw Prod.thy [Pair_Rep_def] |
|
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"Pair_Rep a b = Pair_Rep a' b' ==> a=a' & b=b'"; |
|
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by (EVERY1 [rtac (major RS fun_cong RS fun_cong RS subst), |
|
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rtac conjI, rtac refl, rtac refl]); |
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qed "Pair_Rep_inject"; |
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||
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Goal "inj_on Abs_Prod Prod"; |
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by (rtac inj_on_inverseI 1); |
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by (etac Abs_Prod_inverse 1); |
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qed "inj_on_Abs_Prod"; |
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|
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val prems = Goalw [Pair_def] |
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"[| (a, b) = (a',b'); [| a=a'; b=b' |] ==> R |] ==> R"; |
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by (rtac (inj_on_Abs_Prod RS inj_onD RS Pair_Rep_inject RS conjE) 1); |
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by (REPEAT (ares_tac (prems@[ProdI]) 1)); |
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qed "Pair_inject"; |
|
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||
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Goal "((a,b) = (a',b')) = (a=a' & b=b')"; |
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by (blast_tac (claset() addSEs [Pair_inject]) 1); |
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qed "Pair_eq"; |
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AddIffs [Pair_eq]; |
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|
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Goalw [fst_def] "fst (a,b) = a"; |
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by (Blast_tac 1); |
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qed "fst_conv"; |
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Goalw [snd_def] "snd (a,b) = b"; |
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by (Blast_tac 1); |
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qed "snd_conv"; |
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Addsimps [fst_conv, snd_conv]; |
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|
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Goal "fst (x, y) = a ==> x = a"; |
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by (Asm_full_simp_tac 1); |
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qed "fst_eqD"; |
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Goal "snd (x, y) = a ==> y = a"; |
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by (Asm_full_simp_tac 1); |
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qed "snd_eqD"; |
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|
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Goalw [Pair_def] "? x y. p = (x,y)"; |
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by (rtac (rewrite_rule [Prod_def] Rep_Prod RS CollectE) 1); |
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by (EVERY1[etac exE, etac exE, rtac exI, rtac exI, |
|
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rtac (Rep_Prod_inverse RS sym RS trans), etac arg_cong]); |
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qed "PairE_lemma"; |
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val [prem] = Goal "[| !!x y. p = (x,y) ==> Q |] ==> Q"; |
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by (rtac (PairE_lemma RS exE) 1); |
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by (REPEAT (eresolve_tac [prem,exE] 1)); |
|
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qed "PairE"; |
|
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fun pair_tac s = EVERY' [res_inst_tac [("p",s)] PairE, hyp_subst_tac, |
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K prune_params_tac]; |
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|
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(* Do not add as rewrite rule: invalidates some proofs in IMP *) |
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Goal "p = (fst(p),snd(p))"; |
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by (pair_tac "p" 1); |
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by (Asm_simp_tac 1); |
|
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qed "surjective_pairing"; |
|
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||
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Goal "? x y. z = (x, y)"; |
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by (rtac exI 1); |
|
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by (rtac exI 1); |
|
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by (rtac surjective_pairing 1); |
|
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qed "surj_pair"; |
|
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Addsimps [surj_pair]; |
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||
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||
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bind_thm ("split_paired_all", |
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SplitPairedAll.rule (standard (surjective_pairing RS eq_reflection))); |
|
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(* |
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Addsimps [split_paired_all] does not work with simplifier |
|
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because it also affects premises in congrence rules, |
|
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where is can lead to premises of the form !!a b. ... = ?P(a,b) |
|
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which cannot be solved by reflexivity. |
|
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*) |
|
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||
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(* replace parameters of product type by individual component parameters *) |
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local |
|
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fun is_pair (_,Type("*",_)) = true |
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| is_pair _ = false; |
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fun exists_paired_all prem = exists is_pair (Logic.strip_params prem); |
93 |
val split_tac = full_simp_tac (HOL_basic_ss addsimps [split_paired_all]); |
|
1301 | 94 |
in |
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val split_all_tac = SUBGOAL (fn (prem,i) => |
96 |
if exists_paired_all prem then split_tac i else no_tac); |
|
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end; |
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||
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claset_ref() := claset() addSWrapper ("split_all_tac", |
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fn tac2 => split_all_tac ORELSE' tac2); |
|
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Goal "(!x. P x) = (!a b. P(a,b))"; |
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by (Fast_tac 1); |
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qed "split_paired_All"; |
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Addsimps [split_paired_All]; |
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(* AddIffs is not a good idea because it makes Blast_tac loop *) |
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107 |
|
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bind_thm ("prod_induct", |
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allI RS (allI RS (split_paired_All RS iffD2)) RS spec); |
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|
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Goal "(? x. P x) = (? a b. P(a,b))"; |
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by (Fast_tac 1); |
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qed "split_paired_Ex"; |
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Addsimps [split_paired_Ex]; |
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|
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Goalw [split_def] "split c (a,b) = c a b"; |
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by (Simp_tac 1); |
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qed "split"; |
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Addsimps [split]; |
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|
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(*Subsumes the old split_Pair when f is the identity function*) |
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Goal "split (%x y. f(x,y)) = f"; |
|
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by (rtac ext 1); |
|
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by (pair_tac "x" 1); |
|
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by (Simp_tac 1); |
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qed "split_Pair_apply"; |
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(*Can't be added to simpset: loops!*) |
|
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Goal "(SOME x. P x) = (SOME (a,b). P(a,b))"; |
|
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by (simp_tac (simpset() addsimps [split_Pair_apply]) 1); |
|
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qed "split_paired_Eps"; |
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Goal "!!s t. (s=t) = (fst(s)=fst(t) & snd(s)=snd(t))"; |
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by (split_all_tac 1); |
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by (Asm_simp_tac 1); |
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qed "Pair_fst_snd_eq"; |
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(*Prevents simplification of c: much faster*) |
|
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val [prem] = goal Prod.thy |
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"p=q ==> split c p = split c q"; |
|
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by (rtac (prem RS arg_cong) 1); |
|
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qed "split_weak_cong"; |
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|
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Goal "(%(x,y). f(x,y)) = f"; |
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by (rtac ext 1); |
|
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by (split_all_tac 1); |
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by (rtac split 1); |
|
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qed "split_eta"; |
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val prems = Goal "(!!x y. f x y = g(x,y)) ==> (%(x,y). f x y) = g"; |
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by (asm_simp_tac (simpset() addsimps prems@[split_eta]) 1); |
|
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qed "cond_split_eta"; |
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|
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(*simplification procedure for cond_split_eta. |
|
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using split_eta a rewrite rule is not general enough, and using |
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cond_split_eta directly would render some existing proofs very inefficient. |
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similarly for split_beta. *) |
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local |
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fun Pair_pat k 0 (Bound m) = (m = k) |
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| Pair_pat k i (Const ("Pair", _) $ Bound m $ t) = i > 0 andalso |
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m = k+i andalso Pair_pat k (i-1) t |
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| Pair_pat _ _ _ = false; |
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fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t |
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| no_args k i (t $ u) = no_args k i t andalso no_args k i u |
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165 |
| no_args k i (Bound m) = m < k orelse m > k+i |
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| no_args _ _ _ = true; |
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fun split_pat tp i (Abs (_,_,t)) = if tp 0 i t then Some (i,t) else None |
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168 |
| split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t |
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| split_pat tp i _ = None; |
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fun metaeq sg lhs rhs = mk_meta_eq (prove_goalw_cterm [] |
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(cterm_of sg (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))) |
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172 |
(K [simp_tac (HOL_basic_ss addsimps [cond_split_eta]) 1])); |
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173 |
fun simproc name patstr = Simplifier.mk_simproc name |
8100 | 174 |
[Thm.read_cterm (sign_of Prod.thy) (patstr, HOLogic.termT)]; |
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|
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val beta_patstr = "split f z"; |
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val eta_patstr = "split f"; |
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fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t |
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| beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse |
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(beta_term_pat k i t andalso beta_term_pat k i u) |
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| beta_term_pat k i t = no_args k i t; |
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182 |
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg |
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183 |
| eta_term_pat _ _ _ = false; |
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184 |
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) |
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185 |
| subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg |
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186 |
else (subst arg k i t $ subst arg k i u) |
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187 |
| subst arg k i t = t; |
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188 |
fun beta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t) $ arg) = |
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189 |
(case split_pat beta_term_pat 1 t of |
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190 |
Some (i,f) => Some (metaeq sg s (subst arg 0 i f)) |
5294 | 191 |
| None => None) |
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192 |
| beta_proc _ _ _ = None; |
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193 |
fun eta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t)) = |
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194 |
(case split_pat eta_term_pat 1 t of |
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195 |
Some (_,ft) => Some (metaeq sg s (let val (f $ arg) = ft in f end)) |
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196 |
| None => None) |
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197 |
| eta_proc _ _ _ = None; |
5294 | 198 |
in |
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199 |
val split_beta_proc = simproc "split_beta" beta_patstr beta_proc; |
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200 |
val split_eta_proc = simproc "split_eta" eta_patstr eta_proc; |
5294 | 201 |
end; |
202 |
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|
203 |
Addsimprocs [split_beta_proc,split_eta_proc]; |
5294 | 204 |
|
7031 | 205 |
Goal "(%(x,y). P x y) z = P (fst z) (snd z)"; |
206 |
by (stac surjective_pairing 1 THEN rtac split 1); |
|
207 |
qed "split_beta"; |
|
4134 | 208 |
|
923 | 209 |
(*For use with split_tac and the simplifier*) |
5069 | 210 |
Goal "R (split c p) = (! x y. p = (x,y) --> R (c x y))"; |
923 | 211 |
by (stac surjective_pairing 1); |
212 |
by (stac split 1); |
|
2935 | 213 |
by (Blast_tac 1); |
4830 | 214 |
qed "split_split"; |
923 | 215 |
|
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|
216 |
(* could be done after split_tac has been speeded up significantly: |
4830 | 217 |
simpset_ref() := simpset() addsplits [split_split]; |
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|
218 |
precompute the constants involved and don't do anything unless |
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|
219 |
the current goal contains one of those constants |
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|
220 |
*) |
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|
221 |
|
5069 | 222 |
Goal "R (split c p) = (~(? x y. p = (x,y) & (~R (c x y))))"; |
4830 | 223 |
by (stac split_split 1); |
4435 | 224 |
by (Simp_tac 1); |
225 |
qed "expand_split_asm"; |
|
226 |
||
923 | 227 |
(** split used as a logical connective or set former **) |
228 |
||
2935 | 229 |
(*These rules are for use with blast_tac. |
923 | 230 |
Could instead call simp_tac/asm_full_simp_tac using split as rewrite.*) |
231 |
||
5069 | 232 |
Goal "!!p. [| !!a b. p=(a,b) ==> c a b |] ==> split c p"; |
1552 | 233 |
by (split_all_tac 1); |
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diff
changeset
|
234 |
by (Asm_simp_tac 1); |
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diff
changeset
|
235 |
qed "splitI2"; |
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|
236 |
|
7958 | 237 |
Goal "!!p. [| !!a b. (a,b)=p ==> c a b x |] ==> split c p x"; |
238 |
by (split_all_tac 1); |
|
239 |
by (Asm_simp_tac 1); |
|
240 |
qed "splitI2'"; |
|
241 |
||
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changeset
|
242 |
Goal "c a b ==> split c (a,b)"; |
1264 | 243 |
by (Asm_simp_tac 1); |
923 | 244 |
qed "splitI"; |
245 |
||
5316 | 246 |
val prems = Goalw [split_def] |
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diff
changeset
|
247 |
"[| split c p; !!x y. [| p = (x,y); c x y |] ==> Q |] ==> Q"; |
923 | 248 |
by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1)); |
249 |
qed "splitE"; |
|
250 |
||
8157 | 251 |
val prems = Goalw [split_def] |
252 |
"[| split c p z; !!x y. [| p = (x,y); c x y z |] ==> Q |] ==> Q"; |
|
253 |
by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1)); |
|
254 |
qed "splitE'"; |
|
255 |
||
7031 | 256 |
val major::prems = goal Prod.thy |
257 |
"[| Q (split P z); !!x y. [|z = (x, y); Q (P x y)|] ==> R \ |
|
258 |
\ |] ==> R"; |
|
259 |
by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1)); |
|
260 |
by (rtac (split_beta RS subst) 1 THEN rtac major 1); |
|
261 |
qed "splitE2"; |
|
4134 | 262 |
|
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|
263 |
Goal "split R (a,b) ==> R a b"; |
923 | 264 |
by (etac (split RS iffD1) 1); |
265 |
qed "splitD"; |
|
266 |
||
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changeset
|
267 |
Goal "z: c a b ==> z: split c (a,b)"; |
1264 | 268 |
by (Asm_simp_tac 1); |
923 | 269 |
qed "mem_splitI"; |
270 |
||
5069 | 271 |
Goal "!!p. [| !!a b. p=(a,b) ==> z: c a b |] ==> z: split c p"; |
1552 | 272 |
by (split_all_tac 1); |
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|
273 |
by (Asm_simp_tac 1); |
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changeset
|
274 |
qed "mem_splitI2"; |
d0266c81a85e
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diff
changeset
|
275 |
|
5316 | 276 |
val prems = Goalw [split_def] |
972
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923
diff
changeset
|
277 |
"[| z: split c p; !!x y. [| p = (x,y); z: c x y |] ==> Q |] ==> Q"; |
923 | 278 |
by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1)); |
279 |
qed "mem_splitE"; |
|
280 |
||
7958 | 281 |
AddSIs [splitI, splitI2, splitI2', mem_splitI, mem_splitI2]; |
8157 | 282 |
AddSEs [splitE, splitE', mem_splitE]; |
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changeset
|
283 |
|
9020
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changeset
|
284 |
Goal "(%u. ? x y. u = (x, y) & P (x, y)) = P"; |
1056cbbaeb29
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diff
changeset
|
285 |
by (rtac ext 1); |
1056cbbaeb29
added split_eta_SetCompr2 (also to simpset), generalized SetCompr_Sigma_eq
oheimb
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diff
changeset
|
286 |
by (Fast_tac 1); |
1056cbbaeb29
added split_eta_SetCompr2 (also to simpset), generalized SetCompr_Sigma_eq
oheimb
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diff
changeset
|
287 |
qed "split_eta_SetCompr"; |
1056cbbaeb29
added split_eta_SetCompr2 (also to simpset), generalized SetCompr_Sigma_eq
oheimb
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diff
changeset
|
288 |
Addsimps [split_eta_SetCompr]; |
1056cbbaeb29
added split_eta_SetCompr2 (also to simpset), generalized SetCompr_Sigma_eq
oheimb
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changeset
|
289 |
|
1056cbbaeb29
added split_eta_SetCompr2 (also to simpset), generalized SetCompr_Sigma_eq
oheimb
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diff
changeset
|
290 |
Goal "(%u. ? x y. u = (x, y) & P x y) = split P"; |
1056cbbaeb29
added split_eta_SetCompr2 (also to simpset), generalized SetCompr_Sigma_eq
oheimb
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diff
changeset
|
291 |
br ext 1; |
1056cbbaeb29
added split_eta_SetCompr2 (also to simpset), generalized SetCompr_Sigma_eq
oheimb
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8703
diff
changeset
|
292 |
by (Fast_tac 1); |
1056cbbaeb29
added split_eta_SetCompr2 (also to simpset), generalized SetCompr_Sigma_eq
oheimb
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diff
changeset
|
293 |
qed "split_eta_SetCompr2"; |
1056cbbaeb29
added split_eta_SetCompr2 (also to simpset), generalized SetCompr_Sigma_eq
oheimb
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8703
diff
changeset
|
294 |
Addsimps [split_eta_SetCompr2]; |
1056cbbaeb29
added split_eta_SetCompr2 (also to simpset), generalized SetCompr_Sigma_eq
oheimb
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diff
changeset
|
295 |
|
4534 | 296 |
(* allows simplifications of nested splits in case of independent predicates *) |
5069 | 297 |
Goal "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"; |
4534 | 298 |
by (rtac ext 1); |
299 |
by (Blast_tac 1); |
|
300 |
qed "split_part"; |
|
301 |
Addsimps [split_part]; |
|
302 |
||
5069 | 303 |
Goal "(@(x',y'). x = x' & y = y') = (x,y)"; |
4534 | 304 |
by (Blast_tac 1); |
305 |
qed "Eps_split_eq"; |
|
306 |
Addsimps [Eps_split_eq]; |
|
307 |
(* |
|
308 |
the following would be slightly more general, |
|
309 |
but cannot be used as rewrite rule: |
|
310 |
### Cannot add premise as rewrite rule because it contains (type) unknowns: |
|
311 |
### ?y = .x |
|
5143
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paulson
parents:
5132
diff
changeset
|
312 |
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"; |
4534 | 313 |
by (rtac select_equality 1); |
314 |
by ( Simp_tac 1); |
|
315 |
by (split_all_tac 1); |
|
316 |
by (Asm_full_simp_tac 1); |
|
317 |
qed "Eps_split_eq"; |
|
318 |
*) |
|
319 |
||
923 | 320 |
(*** prod_fun -- action of the product functor upon functions ***) |
321 |
||
5069 | 322 |
Goalw [prod_fun_def] "prod_fun f g (a,b) = (f(a),g(b))"; |
923 | 323 |
by (rtac split 1); |
324 |
qed "prod_fun"; |
|
4521 | 325 |
Addsimps [prod_fun]; |
923 | 326 |
|
5278 | 327 |
Goal "prod_fun (f1 o f2) (g1 o g2) = ((prod_fun f1 g1) o (prod_fun f2 g2))"; |
923 | 328 |
by (rtac ext 1); |
4828 | 329 |
by (pair_tac "x" 1); |
4521 | 330 |
by (Asm_simp_tac 1); |
923 | 331 |
qed "prod_fun_compose"; |
332 |
||
5069 | 333 |
Goal "prod_fun (%x. x) (%y. y) = (%z. z)"; |
923 | 334 |
by (rtac ext 1); |
4828 | 335 |
by (pair_tac "z" 1); |
4521 | 336 |
by (Asm_simp_tac 1); |
923 | 337 |
qed "prod_fun_ident"; |
4521 | 338 |
Addsimps [prod_fun_ident]; |
923 | 339 |
|
5316 | 340 |
Goal "(a,b):r ==> (f(a),g(b)) : (prod_fun f g)``r"; |
923 | 341 |
by (rtac image_eqI 1); |
342 |
by (rtac (prod_fun RS sym) 1); |
|
5316 | 343 |
by (assume_tac 1); |
923 | 344 |
qed "prod_fun_imageI"; |
345 |
||
5316 | 346 |
val major::prems = Goal |
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
923
diff
changeset
|
347 |
"[| c: (prod_fun f g)``r; !!x y. [| c=(f(x),g(y)); (x,y):r |] ==> P \ |
923 | 348 |
\ |] ==> P"; |
349 |
by (rtac (major RS imageE) 1); |
|
350 |
by (res_inst_tac [("p","x")] PairE 1); |
|
351 |
by (resolve_tac prems 1); |
|
2935 | 352 |
by (Blast_tac 2); |
4089 | 353 |
by (blast_tac (claset() addIs [prod_fun]) 1); |
923 | 354 |
qed "prod_fun_imageE"; |
355 |
||
5788 | 356 |
AddIs [prod_fun_imageI]; |
357 |
AddSEs [prod_fun_imageE]; |
|
358 |
||
4521 | 359 |
|
923 | 360 |
(*** Disjoint union of a family of sets - Sigma ***) |
361 |
||
7031 | 362 |
Goalw [Sigma_def] "[| a:A; b:B(a) |] ==> (a,b) : Sigma A B"; |
363 |
by (REPEAT (ares_tac [singletonI,UN_I] 1)); |
|
364 |
qed "SigmaI"; |
|
923 | 365 |
|
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Reorganization of how classical rules are installed
paulson
parents:
2637
diff
changeset
|
366 |
AddSIs [SigmaI]; |
cdb908486a96
Reorganization of how classical rules are installed
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diff
changeset
|
367 |
|
923 | 368 |
(*The general elimination rule*) |
7031 | 369 |
val major::prems = Goalw [Sigma_def] |
923 | 370 |
"[| c: Sigma A B; \ |
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
923
diff
changeset
|
371 |
\ !!x y.[| x:A; y:B(x); c=(x,y) |] ==> P \ |
7031 | 372 |
\ |] ==> P"; |
373 |
by (cut_facts_tac [major] 1); |
|
374 |
by (REPEAT (eresolve_tac [UN_E, singletonE] 1 ORELSE ares_tac prems 1)) ; |
|
375 |
qed "SigmaE"; |
|
923 | 376 |
|
972
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changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
923
diff
changeset
|
377 |
(** Elimination of (a,b):A*B -- introduces no eigenvariables **) |
7007 | 378 |
|
379 |
Goal "(a,b) : Sigma A B ==> a : A"; |
|
380 |
by (etac SigmaE 1); |
|
381 |
by (REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ; |
|
382 |
qed "SigmaD1"; |
|
923 | 383 |
|
7007 | 384 |
Goal "(a,b) : Sigma A B ==> b : B(a)"; |
385 |
by (etac SigmaE 1); |
|
386 |
by (REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ; |
|
387 |
qed "SigmaD2"; |
|
923 | 388 |
|
7007 | 389 |
val [major,minor]= goal Prod.thy |
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
923
diff
changeset
|
390 |
"[| (a,b) : Sigma A B; \ |
923 | 391 |
\ [| a:A; b:B(a) |] ==> P \ |
7007 | 392 |
\ |] ==> P"; |
393 |
by (rtac minor 1); |
|
394 |
by (rtac (major RS SigmaD1) 1); |
|
395 |
by (rtac (major RS SigmaD2) 1) ; |
|
396 |
qed "SigmaE2"; |
|
923 | 397 |
|
2856
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Reorganization of how classical rules are installed
paulson
parents:
2637
diff
changeset
|
398 |
AddSEs [SigmaE2, SigmaE]; |
cdb908486a96
Reorganization of how classical rules are installed
paulson
parents:
2637
diff
changeset
|
399 |
|
5316 | 400 |
val prems = Goal |
1642 | 401 |
"[| A<=C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"; |
1515 | 402 |
by (cut_facts_tac prems 1); |
4089 | 403 |
by (blast_tac (claset() addIs (prems RL [subsetD])) 1); |
1515 | 404 |
qed "Sigma_mono"; |
405 |
||
7007 | 406 |
Goal "Sigma {} B = {}"; |
407 |
by (Blast_tac 1) ; |
|
408 |
qed "Sigma_empty1"; |
|
1618 | 409 |
|
8703 | 410 |
Goal "A <*> {} = {}"; |
7007 | 411 |
by (Blast_tac 1) ; |
412 |
qed "Sigma_empty2"; |
|
1618 | 413 |
|
8703 | 414 |
Addsimps [Sigma_empty1,Sigma_empty2]; |
415 |
||
416 |
Goal "UNIV <*> UNIV = UNIV"; |
|
417 |
by Auto_tac; |
|
418 |
qed "UNIV_Times_UNIV"; |
|
419 |
Addsimps [UNIV_Times_UNIV]; |
|
420 |
||
421 |
Goal "- (UNIV <*> A) = UNIV <*> (-A)"; |
|
422 |
by Auto_tac; |
|
423 |
qed "Compl_Times_UNIV1"; |
|
424 |
||
425 |
Goal "- (A <*> UNIV) = (-A) <*> UNIV"; |
|
426 |
by Auto_tac; |
|
427 |
qed "Compl_Times_UNIV2"; |
|
428 |
||
429 |
Addsimps [Compl_Times_UNIV1, Compl_Times_UNIV2]; |
|
1618 | 430 |
|
5069 | 431 |
Goal "((a,b): Sigma A B) = (a:A & b:B(a))"; |
2935 | 432 |
by (Blast_tac 1); |
1618 | 433 |
qed "mem_Sigma_iff"; |
3568
36ff1ab12021
Prod.ML: Added split_paired_EX and lots of comments about failed attempts to
nipkow
parents:
3429
diff
changeset
|
434 |
AddIffs [mem_Sigma_iff]; |
1618 | 435 |
|
8703 | 436 |
Goal "x:C ==> (A <*> C <= B <*> C) = (A <= B)"; |
6016 | 437 |
by (Blast_tac 1); |
438 |
qed "Times_subset_cancel2"; |
|
439 |
||
8703 | 440 |
Goal "x:C ==> (A <*> C = B <*> C) = (A = B)"; |
6016 | 441 |
by (blast_tac (claset() addEs [equalityE]) 1); |
442 |
qed "Times_eq_cancel2"; |
|
443 |
||
9020
1056cbbaeb29
added split_eta_SetCompr2 (also to simpset), generalized SetCompr_Sigma_eq
oheimb
parents:
8703
diff
changeset
|
444 |
Goal "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"; |
8261 | 445 |
by (Fast_tac 1); |
446 |
qed "SetCompr_Sigma_eq"; |
|
5810 | 447 |
|
448 |
(*** Complex rules for Sigma ***) |
|
449 |
||
8703 | 450 |
Goal "{(a,b). P a & Q b} = Collect P <*> Collect Q"; |
7031 | 451 |
by (Blast_tac 1); |
452 |
qed "Collect_split"; |
|
453 |
||
4534 | 454 |
Addsimps [Collect_split]; |
1515 | 455 |
|
2856
cdb908486a96
Reorganization of how classical rules are installed
paulson
parents:
2637
diff
changeset
|
456 |
(*Suggested by Pierre Chartier*) |
8703 | 457 |
Goal "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"; |
2935 | 458 |
by (Blast_tac 1); |
6830
f8aed3706af7
renamed UNION_... to UN_... (to fit the convention)
paulson
parents:
6394
diff
changeset
|
459 |
qed "UN_Times_distrib"; |
2856
cdb908486a96
Reorganization of how classical rules are installed
paulson
parents:
2637
diff
changeset
|
460 |
|
6016 | 461 |
Goal "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"; |
5810 | 462 |
by (Fast_tac 1); |
6016 | 463 |
qed "split_paired_Ball_Sigma"; |
464 |
Addsimps [split_paired_Ball_Sigma]; |
|
5810 | 465 |
|
6016 | 466 |
Goal "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"; |
5810 | 467 |
by (Fast_tac 1); |
6016 | 468 |
qed "split_paired_Bex_Sigma"; |
469 |
Addsimps [split_paired_Bex_Sigma]; |
|
5810 | 470 |
|
471 |
Goal "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"; |
|
472 |
by (Blast_tac 1); |
|
473 |
qed "Sigma_Un_distrib1"; |
|
474 |
||
475 |
Goal "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"; |
|
476 |
by (Blast_tac 1); |
|
477 |
qed "Sigma_Un_distrib2"; |
|
478 |
||
479 |
Goal "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"; |
|
480 |
by (Blast_tac 1); |
|
481 |
qed "Sigma_Int_distrib1"; |
|
482 |
||
483 |
Goal "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"; |
|
484 |
by (Blast_tac 1); |
|
485 |
qed "Sigma_Int_distrib2"; |
|
486 |
||
487 |
Goal "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"; |
|
488 |
by (Blast_tac 1); |
|
489 |
qed "Sigma_Diff_distrib1"; |
|
490 |
||
491 |
Goal "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"; |
|
492 |
by (Blast_tac 1); |
|
493 |
qed "Sigma_Diff_distrib2"; |
|
494 |
||
6016 | 495 |
Goal "Sigma (Union X) B = (UN A:X. Sigma A B)"; |
496 |
by (Blast_tac 1); |
|
497 |
qed "Sigma_Union"; |
|
498 |
||
8255 | 499 |
(*Non-dependent versions are needed to avoid the need for higher-order |
500 |
matching, especially when the rules are re-oriented*) |
|
8703 | 501 |
Goal "(A Un B) <*> C = (A <*> C) Un (B <*> C)"; |
8255 | 502 |
by (Blast_tac 1); |
503 |
qed "Times_Un_distrib1"; |
|
504 |
||
8703 | 505 |
Goal "(A Int B) <*> C = (A <*> C) Int (B <*> C)"; |
8255 | 506 |
by (Blast_tac 1); |
507 |
qed "Times_Int_distrib1"; |
|
508 |
||
8703 | 509 |
Goal "(A - B) <*> C = (A <*> C) - (B <*> C)"; |
8255 | 510 |
by (Blast_tac 1); |
511 |
qed "Times_Diff_distrib1"; |
|
5810 | 512 |
|
923 | 513 |
(** Exhaustion rule for unit -- a degenerate form of induction **) |
514 |
||
5069 | 515 |
Goalw [Unity_def] |
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
923
diff
changeset
|
516 |
"u = ()"; |
2886 | 517 |
by (stac (rewrite_rule [unit_def] Rep_unit RS singletonD RS sym) 1); |
2880 | 518 |
by (rtac (Rep_unit_inverse RS sym) 1); |
923 | 519 |
qed "unit_eq"; |
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1746
diff
changeset
|
520 |
|
5088
e4aa78d1312f
New rewrite unit_abs_eta_conv to compensate for unit_eq_proc
paulson
parents:
5083
diff
changeset
|
521 |
(*simplification procedure for unit_eq. |
e4aa78d1312f
New rewrite unit_abs_eta_conv to compensate for unit_eq_proc
paulson
parents:
5083
diff
changeset
|
522 |
Cannot use this rule directly -- it loops!*) |
5083 | 523 |
local |
6394 | 524 |
val unit_pat = Thm.cterm_of (Theory.sign_of thy) (Free ("x", HOLogic.unitT)); |
5553 | 525 |
val unit_meta_eq = standard (mk_meta_eq unit_eq); |
5083 | 526 |
fun proc _ _ t = |
527 |
if HOLogic.is_unit t then None |
|
528 |
else Some unit_meta_eq; |
|
529 |
in |
|
530 |
val unit_eq_proc = Simplifier.mk_simproc "unit_eq" [unit_pat] proc; |
|
531 |
end; |
|
532 |
||
533 |
Addsimprocs [unit_eq_proc]; |
|
534 |
||
535 |
||
5761 | 536 |
Goal "P () ==> P x"; |
537 |
by (Simp_tac 1); |
|
538 |
qed "unit_induct"; |
|
539 |
||
540 |
||
5088
e4aa78d1312f
New rewrite unit_abs_eta_conv to compensate for unit_eq_proc
paulson
parents:
5083
diff
changeset
|
541 |
(*This rewrite counters the effect of unit_eq_proc on (%u::unit. f u), |
e4aa78d1312f
New rewrite unit_abs_eta_conv to compensate for unit_eq_proc
paulson
parents:
5083
diff
changeset
|
542 |
replacing it by f rather than by %u.f(). *) |
e4aa78d1312f
New rewrite unit_abs_eta_conv to compensate for unit_eq_proc
paulson
parents:
5083
diff
changeset
|
543 |
Goal "(%u::unit. f()) = f"; |
e4aa78d1312f
New rewrite unit_abs_eta_conv to compensate for unit_eq_proc
paulson
parents:
5083
diff
changeset
|
544 |
by (rtac ext 1); |
e4aa78d1312f
New rewrite unit_abs_eta_conv to compensate for unit_eq_proc
paulson
parents:
5083
diff
changeset
|
545 |
by (Simp_tac 1); |
e4aa78d1312f
New rewrite unit_abs_eta_conv to compensate for unit_eq_proc
paulson
parents:
5083
diff
changeset
|
546 |
qed "unit_abs_eta_conv"; |
e4aa78d1312f
New rewrite unit_abs_eta_conv to compensate for unit_eq_proc
paulson
parents:
5083
diff
changeset
|
547 |
Addsimps [unit_abs_eta_conv]; |
e4aa78d1312f
New rewrite unit_abs_eta_conv to compensate for unit_eq_proc
paulson
parents:
5083
diff
changeset
|
548 |
|
e4aa78d1312f
New rewrite unit_abs_eta_conv to compensate for unit_eq_proc
paulson
parents:
5083
diff
changeset
|
549 |
|
5096
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
550 |
(*Attempts to remove occurrences of split, and pair-valued parameters*) |
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
551 |
val remove_split = rewrite_rule [split RS eq_reflection] o |
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
552 |
rule_by_tactic (TRYALL split_all_tac); |
1746
f0c6aabc6c02
Moved split_rule et al from ind_syntax.ML to Prod.ML.
nipkow
parents:
1727
diff
changeset
|
553 |
|
5096
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
554 |
local |
1746
f0c6aabc6c02
Moved split_rule et al from ind_syntax.ML to Prod.ML.
nipkow
parents:
1727
diff
changeset
|
555 |
|
f0c6aabc6c02
Moved split_rule et al from ind_syntax.ML to Prod.ML.
nipkow
parents:
1727
diff
changeset
|
556 |
(*In ap_split S T u, term u expects separate arguments for the factors of S, |
f0c6aabc6c02
Moved split_rule et al from ind_syntax.ML to Prod.ML.
nipkow
parents:
1727
diff
changeset
|
557 |
with result type T. The call creates a new term expecting one argument |
f0c6aabc6c02
Moved split_rule et al from ind_syntax.ML to Prod.ML.
nipkow
parents:
1727
diff
changeset
|
558 |
of type S.*) |
5096
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
559 |
fun ap_split (Type ("*", [T1, T2])) T3 u = |
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
560 |
HOLogic.split_const (T1, T2, T3) $ |
1746
f0c6aabc6c02
Moved split_rule et al from ind_syntax.ML to Prod.ML.
nipkow
parents:
1727
diff
changeset
|
561 |
Abs("v", T1, |
2031 | 562 |
ap_split T2 T3 |
5096
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
563 |
((ap_split T1 (HOLogic.prodT_factors T2 ---> T3) (incr_boundvars 1 u)) $ |
2031 | 564 |
Bound 0)) |
1746
f0c6aabc6c02
Moved split_rule et al from ind_syntax.ML to Prod.ML.
nipkow
parents:
1727
diff
changeset
|
565 |
| ap_split T T3 u = u; |
f0c6aabc6c02
Moved split_rule et al from ind_syntax.ML to Prod.ML.
nipkow
parents:
1727
diff
changeset
|
566 |
|
5096
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
567 |
(*Curries any Var of function type in the rule*) |
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
568 |
fun split_rule_var' (t as Var (v, Type ("fun", [T1, T2])), rl) = |
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
569 |
let val T' = HOLogic.prodT_factors T1 ---> T2 |
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
570 |
val newt = ap_split T1 T2 (Var (v, T')) |
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
571 |
val cterm = Thm.cterm_of (#sign (rep_thm rl)) |
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
572 |
in |
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
573 |
instantiate ([], [(cterm t, cterm newt)]) rl |
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
574 |
end |
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
575 |
| split_rule_var' (t, rl) = rl; |
1746
f0c6aabc6c02
Moved split_rule et al from ind_syntax.ML to Prod.ML.
nipkow
parents:
1727
diff
changeset
|
576 |
|
5096
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
577 |
in |
1746
f0c6aabc6c02
Moved split_rule et al from ind_syntax.ML to Prod.ML.
nipkow
parents:
1727
diff
changeset
|
578 |
|
5096
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
579 |
val split_rule_var = standard o remove_split o split_rule_var'; |
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
580 |
|
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
581 |
(*Curries ALL function variables occurring in a rule's conclusion*) |
84b00be693b4
Moved most of the Prod_Syntax - stuff to HOLogic.
berghofe
parents:
5088
diff
changeset
|
582 |
fun split_rule rl = remove_split (foldr split_rule_var' (term_vars (concl_of rl), rl)) |
1746
f0c6aabc6c02
Moved split_rule et al from ind_syntax.ML to Prod.ML.
nipkow
parents:
1727
diff
changeset
|
583 |
|> standard; |
f0c6aabc6c02
Moved split_rule et al from ind_syntax.ML to Prod.ML.
nipkow
parents:
1727
diff
changeset
|
584 |
|
f0c6aabc6c02
Moved split_rule et al from ind_syntax.ML to Prod.ML.
nipkow
parents:
1727
diff
changeset
|
585 |
end; |
5810 | 586 |
|
587 |