author | paulson |
Thu, 21 Mar 1996 13:02:26 +0100 | |
changeset 1601 | 0ef6ea27ab15 |
parent 1461 | 6bcb44e4d6e5 |
child 1779 | 1155c06fa956 |
permissions | -rw-r--r-- |
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(* Title: HOLCF/sprod2.ML |
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ID: $Id$ |
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Author: Franz Regensburger |
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Copyright 1993 Technische Universitaet Muenchen |
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|
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Lemmas for sprod2.thy |
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*) |
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open Sprod2; |
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(* ------------------------------------------------------------------------ *) |
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(* access to less_sprod in class po *) |
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(* ------------------------------------------------------------------------ *) |
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qed_goal "less_sprod3a" Sprod2.thy |
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"p1=Ispair UU UU ==> p1 << p2" |
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(fn prems => |
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[ |
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(cut_facts_tac prems 1), |
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(rtac (inst_sprod_po RS ssubst) 1), |
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(etac less_sprod1a 1) |
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]); |
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qed_goal "less_sprod3b" Sprod2.thy |
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"p1~=Ispair UU UU ==>\ |
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\ (p1<<p2) = (Isfst(p1)<<Isfst(p2) & Issnd(p1)<<Issnd(p2))" |
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(fn prems => |
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[ |
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(cut_facts_tac prems 1), |
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(rtac (inst_sprod_po RS ssubst) 1), |
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(etac less_sprod1b 1) |
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]); |
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qed_goal "less_sprod4b" Sprod2.thy |
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"p << Ispair UU UU ==> p = Ispair UU UU" |
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(fn prems => |
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[ |
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(cut_facts_tac prems 1), |
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(rtac less_sprod2b 1), |
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(etac (inst_sprod_po RS subst) 1) |
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]); |
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val less_sprod4a = (less_sprod4b RS defined_Ispair_rev); |
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(* Ispair ?a ?b << Ispair UU UU ==> ?a = UU | ?b = UU *) |
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qed_goal "less_sprod4c" Sprod2.thy |
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"[|Ispair xa ya << Ispair x y; xa~=UU; ya~=UU; x~=UU; y~=UU|] ==>\ |
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\ xa<<x & ya << y" |
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(fn prems => |
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[ |
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(cut_facts_tac prems 1), |
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(rtac less_sprod2c 1), |
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(etac (inst_sprod_po RS subst) 1), |
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(REPEAT (atac 1)) |
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]); |
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(* ------------------------------------------------------------------------ *) |
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(* type sprod is pointed *) |
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(* ------------------------------------------------------------------------ *) |
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qed_goal "minimal_sprod" Sprod2.thy "Ispair UU UU << p" |
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(fn prems => |
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[ |
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(rtac less_sprod3a 1), |
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(rtac refl 1) |
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]); |
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(* ------------------------------------------------------------------------ *) |
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(* Ispair is monotone in both arguments *) |
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(* ------------------------------------------------------------------------ *) |
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qed_goalw "monofun_Ispair1" Sprod2.thy [monofun] "monofun(Ispair)" |
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(fn prems => |
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[ |
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(strip_tac 1), |
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(rtac (less_fun RS iffD2) 1), |
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(strip_tac 1), |
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(res_inst_tac [("Q", |
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" Ispair y xa = Ispair UU UU")] (excluded_middle RS disjE) 1), |
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(res_inst_tac [("Q", |
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" Ispair x xa = Ispair UU UU")] (excluded_middle RS disjE) 1), |
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(rtac (less_sprod3b RS iffD2) 1), |
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(atac 1), |
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(rtac conjI 1), |
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(rtac (Isfst RS ssubst) 1), |
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(etac (strict_Ispair_rev RS conjunct1) 1), |
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(etac (strict_Ispair_rev RS conjunct2) 1), |
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(rtac (Isfst RS ssubst) 1), |
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(etac (strict_Ispair_rev RS conjunct1) 1), |
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(etac (strict_Ispair_rev RS conjunct2) 1), |
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(atac 1), |
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(rtac (Issnd RS ssubst) 1), |
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(etac (strict_Ispair_rev RS conjunct1) 1), |
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(etac (strict_Ispair_rev RS conjunct2) 1), |
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(rtac (Issnd RS ssubst) 1), |
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(etac (strict_Ispair_rev RS conjunct1) 1), |
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(etac (strict_Ispair_rev RS conjunct2) 1), |
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(rtac refl_less 1), |
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(etac less_sprod3a 1), |
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(res_inst_tac [("Q", |
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" Ispair x xa = Ispair UU UU")] (excluded_middle RS disjE) 1), |
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(etac less_sprod3a 2), |
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(res_inst_tac [("P","Ispair y xa = Ispair UU UU")] notE 1), |
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(atac 2), |
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(rtac defined_Ispair 1), |
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(etac notUU_I 1), |
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(etac (strict_Ispair_rev RS conjunct1) 1), |
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(etac (strict_Ispair_rev RS conjunct2) 1) |
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]); |
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qed_goalw "monofun_Ispair2" Sprod2.thy [monofun] "monofun(Ispair(x))" |
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(fn prems => |
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[ |
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(strip_tac 1), |
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(res_inst_tac [("Q", |
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" Ispair x y = Ispair UU UU")] (excluded_middle RS disjE) 1), |
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(res_inst_tac [("Q", |
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" Ispair x xa = Ispair UU UU")] (excluded_middle RS disjE) 1), |
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(rtac (less_sprod3b RS iffD2) 1), |
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(atac 1), |
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(rtac conjI 1), |
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(rtac (Isfst RS ssubst) 1), |
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(etac (strict_Ispair_rev RS conjunct1) 1), |
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(etac (strict_Ispair_rev RS conjunct2) 1), |
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(rtac (Isfst RS ssubst) 1), |
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(etac (strict_Ispair_rev RS conjunct1) 1), |
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(etac (strict_Ispair_rev RS conjunct2) 1), |
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(rtac refl_less 1), |
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(rtac (Issnd RS ssubst) 1), |
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(etac (strict_Ispair_rev RS conjunct1) 1), |
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(etac (strict_Ispair_rev RS conjunct2) 1), |
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(rtac (Issnd RS ssubst) 1), |
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(etac (strict_Ispair_rev RS conjunct1) 1), |
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(etac (strict_Ispair_rev RS conjunct2) 1), |
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(atac 1), |
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(etac less_sprod3a 1), |
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(res_inst_tac [("Q", |
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" Ispair x xa = Ispair UU UU")] (excluded_middle RS disjE) 1), |
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(etac less_sprod3a 2), |
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(res_inst_tac [("P","Ispair x y = Ispair UU UU")] notE 1), |
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(atac 2), |
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(rtac defined_Ispair 1), |
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(etac (strict_Ispair_rev RS conjunct1) 1), |
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(etac notUU_I 1), |
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(etac (strict_Ispair_rev RS conjunct2) 1) |
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]); |
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qed_goal " monofun_Ispair" Sprod2.thy |
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"[|x1<<x2; y1<<y2|] ==> Ispair x1 y1 << Ispair x2 y2" |
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(fn prems => |
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[ |
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(cut_facts_tac prems 1), |
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(rtac trans_less 1), |
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(rtac (monofun_Ispair1 RS monofunE RS spec RS spec RS mp RS |
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(less_fun RS iffD1 RS spec)) 1), |
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(rtac (monofun_Ispair2 RS monofunE RS spec RS spec RS mp) 2), |
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(atac 1), |
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(atac 1) |
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]); |
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(* ------------------------------------------------------------------------ *) |
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(* Isfst and Issnd are monotone *) |
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(* ------------------------------------------------------------------------ *) |
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qed_goalw " monofun_Isfst" Sprod2.thy [monofun] "monofun(Isfst)" |
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(fn prems => |
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[ |
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(strip_tac 1), |
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(res_inst_tac [("p","x")] IsprodE 1), |
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(hyp_subst_tac 1), |
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(rtac trans_less 1), |
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(rtac minimal 2), |
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(rtac (strict_Isfst1 RS ssubst) 1), |
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(rtac refl_less 1), |
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(hyp_subst_tac 1), |
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(res_inst_tac [("p","y")] IsprodE 1), |
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(hyp_subst_tac 1), |
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(res_inst_tac [("t","Isfst(Ispair xa ya)")] subst 1), |
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(rtac refl_less 2), |
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(etac (less_sprod4b RS sym RS arg_cong) 1), |
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(hyp_subst_tac 1), |
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(rtac (Isfst RS ssubst) 1), |
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(atac 1), |
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(atac 1), |
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(rtac (Isfst RS ssubst) 1), |
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(atac 1), |
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(atac 1), |
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(etac (less_sprod4c RS conjunct1) 1), |
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(REPEAT (atac 1)) |
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]); |
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qed_goalw "monofun_Issnd" Sprod2.thy [monofun] "monofun(Issnd)" |
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(fn prems => |
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[ |
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(strip_tac 1), |
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(res_inst_tac [("p","x")] IsprodE 1), |
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(hyp_subst_tac 1), |
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(rtac trans_less 1), |
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(rtac minimal 2), |
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(rtac (strict_Issnd1 RS ssubst) 1), |
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(rtac refl_less 1), |
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(hyp_subst_tac 1), |
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(res_inst_tac [("p","y")] IsprodE 1), |
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(hyp_subst_tac 1), |
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(res_inst_tac [("t","Issnd(Ispair xa ya)")] subst 1), |
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(rtac refl_less 2), |
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(etac (less_sprod4b RS sym RS arg_cong) 1), |
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(hyp_subst_tac 1), |
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(rtac (Issnd RS ssubst) 1), |
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(atac 1), |
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(atac 1), |
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(rtac (Issnd RS ssubst) 1), |
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(atac 1), |
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(atac 1), |
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(etac (less_sprod4c RS conjunct2) 1), |
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(REPEAT (atac 1)) |
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221 |
]); |
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(* ------------------------------------------------------------------------ *) |
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(* the type 'a ** 'b is a cpo *) |
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(* ------------------------------------------------------------------------ *) |
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qed_goal "lub_sprod" Sprod2.thy |
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"[|is_chain(S)|] ==> range(S) <<| \ |
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\ Ispair (lub(range(%i.Isfst(S i)))) (lub(range(%i.Issnd(S i))))" |
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(fn prems => |
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[ |
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(cut_facts_tac prems 1), |
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(rtac is_lubI 1), |
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(rtac conjI 1), |
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(rtac ub_rangeI 1), |
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(rtac allI 1), |
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(res_inst_tac [("t","S(i)")] (surjective_pairing_Sprod RS ssubst) 1), |
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(rtac monofun_Ispair 1), |
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(rtac is_ub_thelub 1), |
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(etac (monofun_Isfst RS ch2ch_monofun) 1), |
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(rtac is_ub_thelub 1), |
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(etac (monofun_Issnd RS ch2ch_monofun) 1), |
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(strip_tac 1), |
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(res_inst_tac [("t","u")] (surjective_pairing_Sprod RS ssubst) 1), |
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(rtac monofun_Ispair 1), |
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(rtac is_lub_thelub 1), |
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(etac (monofun_Isfst RS ch2ch_monofun) 1), |
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(etac (monofun_Isfst RS ub2ub_monofun) 1), |
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(rtac is_lub_thelub 1), |
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(etac (monofun_Issnd RS ch2ch_monofun) 1), |
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(etac (monofun_Issnd RS ub2ub_monofun) 1) |
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]); |
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val thelub_sprod = (lub_sprod RS thelubI); |
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qed_goal "cpo_sprod" Sprod2.thy |
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"is_chain(S::nat=>'a**'b)==>? x.range(S)<<| x" |
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(fn prems => |
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[ |
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(cut_facts_tac prems 1), |
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(rtac exI 1), |
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(etac lub_sprod 1) |
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]); |
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