9791
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(* Title: HOL/BCV/Semilat.ML
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ID: $Id$
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Author: Tobias Nipkow
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Copyright 2000 TUM
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*)
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Goalw [order_def] "order r ==> x <=_r x";
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by(Asm_simp_tac 1);
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qed "order_refl";
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Addsimps [order_refl];
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AddIs [order_refl];
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Goalw [order_def] "[| order r; x <=_r y; y <=_r x |] ==> x = y";
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by(Asm_simp_tac 1);
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qed "order_antisym";
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Goalw [order_def] "[| order r; x <=_r y; y <=_r z |] ==> x <=_r z";
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by(Blast_tac 1);
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qed "order_trans";
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Goalw [order_def,lesssub_def] "order r ==> ~ x <_r x";
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by(Blast_tac 1);
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qed "order_less_irrefl";
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AddIs [order_less_irrefl];
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Addsimps [order_less_irrefl];
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Goalw [order_def,lesssub_def] "[| order r; x <_r y; y <_r z |] ==> x <_r z";
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by(Blast_tac 1);
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qed "order_less_trans";
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Goalw [top_def] "top r T ==> x <=_r T";
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by(Asm_simp_tac 1);
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qed "topD";
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Addsimps [topD];
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AddIs [topD];
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Goal "[| order r; top r T |] ==> (T <=_r x) = (x = T)";
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by(blast_tac (claset() addIs [order_antisym]) 1);
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qed "top_le_conv";
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Addsimps [top_le_conv];
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Goalw [semilat_def,split RS eq_reflection]
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"semilat(A,r,f) == order r & closed A f & \
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\ (!x:A. !y:A. x <=_r x +_f y) & \
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\ (!x:A. !y:A. y <=_r x +_f y) & \
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\ (!x:A. !y:A. !z:A. x <=_r z & y <=_r z --> x +_f y <=_r z)";
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br (refl RS eq_reflection) 1;
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qed "semilat_Def";
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Goalw [semilat_Def] "semilat(A,r,f) ==> order r";
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by(Asm_simp_tac 1);
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qed "semilatDorderI";
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Addsimps [semilatDorderI];
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AddIs [semilatDorderI];
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Goalw [semilat_Def] "semilat(A,r,f) ==> closed A f";
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by(Asm_simp_tac 1);
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qed "semilatDclosedI";
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Addsimps [semilatDclosedI];
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AddIs [semilatDclosedI];
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Goalw [semilat_Def] "[| semilat(A,r,f); x:A; y:A |] ==> x <=_r x +_f y";
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by(Asm_simp_tac 1);
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qed "semilat_ub1";
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Goalw [semilat_Def] "[| semilat(A,r,f); x:A; y:A |] ==> y <=_r x +_f y";
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by(Asm_simp_tac 1);
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qed "semilat_ub2";
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Goalw [semilat_Def]
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"[| x <=_r z; y <=_r z; semilat(A,r,f); x:A; y:A; z:A |] ==> x +_f y <=_r z";
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by(Asm_simp_tac 1);
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qed "semilat_lub";
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Addsimps [semilat_ub1,semilat_ub2,semilat_lub];
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Goalw [semilat_Def]
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"[| x:A; y:A; z:A; semilat(A,r,f) |] ==> \
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\ (x +_f y <=_r z) = (x <=_r z & y <=_r z)";
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by (blast_tac (claset() addIs [semilat_ub1,semilat_ub2,semilat_lub,order_trans]) 1);
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qed "plus_le_conv";
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Addsimps [plus_le_conv];
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Goal "[| x:A; y:A; semilat(A,r,f) |] ==> (x <=_r y) = (x +_f y = y)";
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by (rtac iffI 1);
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by (REPEAT(ares_tac [semilatDorderI,order_antisym,semilat_lub,order_refl,semilat_ub2] 1));
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by (etac subst 1);
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by (Asm_simp_tac 1);
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qed "le_iff_plus_unchanged";
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Goal "[| x:A; y:A; semilat(A,r,f) |] ==> (x <=_r y) = (y +_f x = y)";
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by (rtac iffI 1);
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by (REPEAT(ares_tac [semilatDorderI,order_antisym,semilat_lub,order_refl,semilat_ub1] 1));
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by (etac subst 1);
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by (Asm_simp_tac 1);
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qed "le_iff_plus_unchanged2";
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(*** closed ***)
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Goalw [closed_def] "[| closed A f; x:A; y:A |] ==> x +_f y : A";
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by(Blast_tac 1);
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qed "closedD";
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Goalw [closed_def] "closed UNIV f";
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by(Simp_tac 1);
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qed "closed_UNIV";
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AddIffs [closed_UNIV];
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(*** lub ***)
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Goalw [is_lub_def]
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"is_lub r x y u ==> is_ub r x y u & (!z. is_ub r x y z --> (u,z):r)";
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ba 1;
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qed "is_lubD";
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Goalw [is_ub_def]
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"[| (x,u) : r; (y,u) : r |] ==> is_ub r x y u";
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by(Asm_simp_tac 1);
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qed "is_ubI";
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Goalw [is_ub_def]
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"is_ub r x y u ==> (x,u) : r & (y,u) : r";
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ba 1;
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qed "is_ubD";
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Goalw [is_lub_def,is_ub_def] "is_lub (r^*) x y y = ((x,y):r^*)";
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by (Blast_tac 1);
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qed "is_lub_bigger1";
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AddIffs [is_lub_bigger1];
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Goalw [is_lub_def,is_ub_def] "is_lub (r^*) x y x = ((y,x):r^*)";
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by (Blast_tac 1);
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qed "is_lub_bigger2";
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AddIffs [is_lub_bigger2];
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Goalw [is_lub_def,is_ub_def]
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"[| univalent r; is_lub (r^*) x y u; (x',x) : r |] ==> \
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\ EX v. is_lub (r^*) x' y v";
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by(case_tac "(y,x) : r^*" 1);
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by(case_tac "(y,x') : r^*" 1);
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by (Blast_tac 1);
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by(blast_tac (claset() addIs [r_into_rtrancl] addEs [converse_rtranclE]
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addDs [univalentD]) 1);
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br exI 1;
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br conjI 1;
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by(blast_tac (claset() addIs [rtrancl_into_rtrancl2]addDs [univalentD]) 1);
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by(blast_tac (claset() addIs [rtrancl_into_rtrancl,rtrancl_into_rtrancl2]
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addEs [converse_rtranclE] addDs [univalentD]) 1);
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qed "extend_lub";
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Goal "[| univalent r; (x,u) : r^* |] ==> (!y. (y,u) : r^* --> \
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\ (EX z. is_lub (r^*) x y z))";
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by(etac converse_rtrancl_induct 1);
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by(Clarify_tac 1);
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by(etac converse_rtrancl_induct 1);
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by(Blast_tac 1);
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by(blast_tac (claset() addIs [rtrancl_into_rtrancl2]) 1);
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by(blast_tac (claset() addIs [extend_lub]) 1);
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qed_spec_mp "univalent_has_lubs";
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Goalw [some_lub_def,is_lub_def]
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"[| acyclic r; is_lub (r^*) x y u |] ==> some_lub (r^*) x y = u";
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br someI2 1;
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ba 1;
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by(blast_tac (claset() addIs [antisymD]
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addSDs [acyclic_impl_antisym_rtrancl]) 1);
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qed "some_lub_conv";
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Goal
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"[| univalent r; acyclic r; (x,u):r^*; (y,u):r^* |] ==> \
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\ is_lub (r^*) x y (some_lub (r^*) x y)";
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by(fast_tac
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(claset() addDs [univalent_has_lubs]
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addss (simpset() addsimps [some_lub_conv])) 1);
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qed "is_lub_some_lub";
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