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(* Title: HOL/BCV/SemiLattice.ML
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ID: $Id$
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Author: Tobias Nipkow
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Copyright 1999 TUM
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*)
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Goalw [semilat_def] "[| x:A; y:A; semilat A |] ==> x <= x+y";
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by (Blast_tac 1);
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qed "semilat_ub1";
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Goalw [semilat_def] "[| x:A; y:A; semilat A |] ==> y <= x+y";
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by (Blast_tac 1);
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qed "semilat_ub2";
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Goalw [semilat_def] "[| x:A; y:A; z:A; x <= z; y <= z; semilat A |] ==> x+y <= z";
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by (Blast_tac 1);
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qed "semilat_lub";
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Goalw [semilat_def] "[| x:A; y:A; semilat A |] ==> x+y : A";
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by (Blast_tac 1);
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qed "semilat_plus";
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Addsimps [semilat_ub1,semilat_ub2,semilat_lub,semilat_plus];
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Goal "[| x:A; y:A; semilat A |] ==> (x <= y) = (x+y = y)";
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by (rtac iffI 1);
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by (REPEAT(ares_tac [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 |] ==> (x <= y) = (y = x+y)";
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by (rtac iffI 1);
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by (REPEAT(ares_tac [order_antisym,semilat_lub,order_refl,semilat_ub2] 1));
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by (etac ssubst 1);
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by (Asm_simp_tac 1);
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qed "le_iff_plus_unchanged2";
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Goal "[| x:A; y:A; z:A; semilat A; x <= y | x <= z |] ==> x <= y+z";
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by (blast_tac (claset() addIs [order_trans,semilat_ub1,semilat_ub2]) 1);
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qed "plus_mono";
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Goal "[| x:A; semilat A |] ==> x+x = x";
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by (REPEAT(ares_tac [order_refl RSN (4,le_iff_plus_unchanged RS iffD1)] 1));
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qed "semilat_idemp";
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Addsimps [semilat_idemp];
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Goal "[| x:A; y:A; z:A; semilat A |] ==> (x+y)+z = x+(y+z)";
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by (blast_tac (HOL_cs addSIs [order_refl,order_antisym,semilat_lub,semilat_plus]
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addIs [plus_mono]) 1);
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qed "semilat_assoc";
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Goal "[| x:A; y:A; semilat A |] ==> x+(x+y) = x+y";
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by (asm_simp_tac (simpset() addsimps [semilat_assoc RS sym]) 1);
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qed "semilat_assoc_idemp";
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Addsimps [semilat_assoc_idemp];
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Goal "[| x:A; y:A; z:A; semilat A |] ==> (x+y <= z) = (x <= z & y <= 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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(** option **)
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Goal "semilat A ==> semilat (option A)";
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by (simp_tac (simpset() addsimps [semilat_def,le_option,plus_option]
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addsplits [option.split]) 1);
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by (blast_tac (claset() addIs [semilat_ub1,semilat_ub2,semilat_lub,semilat_plus]) 1);
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qed "semilat_option";
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Addsimps [semilat_option];
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AddSIs [semilat_option];
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(** list **)
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Goalw [plus_list] "[] + [] = []";
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by (Simp_tac 1);
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qed "plus_Nil_Nil";
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Addsimps [plus_Nil_Nil];
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Goalw [plus_list] "(x#xs) + (y#ys) = (x+y)#(xs+ys)";
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by (Simp_tac 1);
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qed "plus_Cons_Cons";
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Addsimps [plus_Cons_Cons];
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Goal
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"!xs ys i. length xs = n--> length ys = n--> i<n--> (xs+ys)!i = xs!i + ys!i";
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by (induct_tac "n" 1);
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by (Simp_tac 1);
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by (Clarify_tac 1);
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by (cases_tac "xs" 1);
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by (Asm_full_simp_tac 1);
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by (cases_tac "ys" 1);
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by (Asm_full_simp_tac 1);
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by (asm_full_simp_tac (simpset() addsimps [nth_Cons] addsplits [nat.split]) 1);
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qed_spec_mp "nth_plus";
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Addsimps [nth_plus];
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Goalw [le_list]
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"semilat A ==> !xs:listsn n A. !ys:listsn n A. xs <= xs+ys";
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by (asm_simp_tac (simpset() addsimps [le_max_iff_disj]) 1);
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by (Clarify_tac 1);
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by (blast_tac (HOL_cs addIs [nth_in,nth_plus,listsnE_length,listsnE_set,semilat_ub1]) 1);
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qed_spec_mp "plus_list_ub1";
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Goalw [le_list]
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"semilat A ==> !xs:listsn n A. !ys:listsn n A. ys <= xs+ys";
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by (asm_simp_tac (simpset() addsimps [le_max_iff_disj]) 1);
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by (Clarify_tac 1);
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by (blast_tac (HOL_cs addIs [nth_in,nth_plus,listsnE_length,listsnE_set,semilat_ub2]) 1);
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qed_spec_mp "plus_list_ub2";
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Goalw [le_list]
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"semilat A ==> !xs:listsn n A. !ys:listsn n A. !zs:listsn n A. \
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\ xs <= zs & ys <= zs --> xs+ys <= zs";
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by (asm_simp_tac (simpset() addsimps [le_max_iff_disj] addcongs [conj_cong]) 1);
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by (Clarify_tac 1);
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by (blast_tac (HOL_cs addIs [nth_in,nth_plus,listsnE_length,listsnE_set,semilat_lub]) 1);
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qed_spec_mp "plus_list_lub";
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Goalw [listsn_def]
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"semilat A ==> !xs:listsn n A. !ys:listsn n A. xs+ys : listsn n A";
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by (induct_tac "n" 1);
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by (Simp_tac 1);
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by (Asm_full_simp_tac 1);
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by (Clarify_tac 1);
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by (cases_tac "xs" 1);
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by (Asm_full_simp_tac 1);
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by (cases_tac "ys" 1);
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by (Asm_full_simp_tac 1);
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by (Asm_full_simp_tac 1);
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by (blast_tac (claset() addIs [semilat_plus]) 1);
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qed_spec_mp "plus_list_closed";
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Goal "semilat A ==> semilat (listsn n A)";
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by (simp_tac (HOL_basic_ss addsimps [semilat_def]) 1);
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by (asm_simp_tac (HOL_ss addsimps [plus_list_ub1,plus_list_ub2,plus_list_lub,plus_list_closed]) 1);
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qed "semilat_listsn";
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Addsimps [semilat_listsn];
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AddSIs [semilat_listsn];
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Goalw [le_list]
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"!i xs. xs : listsn n A --> x:A --> semilat A --> i<n --> xs <= xs[i := x + xs!i]";
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by (Simp_tac 1);
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by (induct_tac "n" 1);
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by (Simp_tac 1);
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by (asm_simp_tac (simpset() addsimps [in_listsn_Suc_iff]) 1);
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by (Clarify_tac 1);
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by (asm_simp_tac (simpset() addsimps [nth_Cons] addsplits [nat.split]) 1);
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qed_spec_mp "list_update_incr";
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(* product *)
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Goalw [semilat_def,plus_prod] "[| semilat A; semilat B |] ==> semilat (A Times B)";
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by (Asm_simp_tac 1);
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qed "semilat_Times";
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