7998
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(*
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Ring homomorphism
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$Id$
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Author: Clemens Ballarin, started 15 April 1997
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*)
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(* Ring homomorphism *)
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Goalw [homo_def]
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"!! f. [| !! a b. f (a + b) = f a + f b; !! a b. f (a * b) = f a * f b; \
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13735
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\ f 1 = 1 |] ==> homo f";
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7998
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by Auto_tac;
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qed "homoI";
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Goalw [homo_def] "!! f. homo f ==> f (a + b) = f a + f b";
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by (Fast_tac 1);
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qed "homo_add";
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Goalw [homo_def] "!! f. homo f ==> f (a * b) = f a * f b";
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by (Fast_tac 1);
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qed "homo_mult";
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13735
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Goalw [homo_def] "!! f. homo f ==> f 1 = 1";
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7998
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by (Fast_tac 1);
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qed "homo_one";
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11093
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Goal "!! f::('a::ring=>'b::ring). homo f ==> f 0 = 0";
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by (res_inst_tac [("a", "f 0")] a_lcancel 1);
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7998
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by (asm_simp_tac (simpset() addsimps [homo_add RS sym]) 1);
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qed "homo_zero";
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Goal
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"!! f::('a::ring=>'b::ring). homo f ==> f (-a) = - f a";
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by (res_inst_tac [("a", "f a")] a_lcancel 1);
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by (ftac homo_zero 1);
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by (asm_simp_tac (simpset() addsimps [homo_add RS sym]) 1);
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qed "homo_uminus";
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Goal "!! f::('a::ring=>'b::ring). homo f ==> f (a ^ n) = f a ^ n";
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8707
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by (induct_tac "n" 1);
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7998
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by (dtac homo_one 1);
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by (Asm_simp_tac 1);
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by (dres_inst_tac [("a", "a^n"), ("b", "a")] homo_mult 1);
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by (Asm_full_simp_tac 1);
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7998
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qed "homo_power";
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Goal
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11093
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"!! f::('a::ring=>'b::ring). \
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\ homo f ==> f (setsum g {..n::nat}) = setsum (f o g) {..n}";
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8707
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by (induct_tac "n" 1);
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7998
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by (Asm_simp_tac 1);
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by (Simp_tac 1);
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11093
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by (dres_inst_tac [("a", "g (Suc n)"), ("b", "setsum g {..n}")] homo_add 1);
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7998
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by (Asm_simp_tac 1);
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qed "homo_SUM";
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Addsimps [homo_add, homo_mult, homo_one, homo_zero,
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homo_uminus, homo_power, homo_SUM];
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Goal "homo (%x. x)";
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by (fast_tac (claset() addSIs [homoI]) 1);
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qed "id_homo";
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Addsimps [id_homo];
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