7998
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(*
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Universal property and evaluation homomorphism of univariate polynomials
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$Id$
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Author: Clemens Ballarin, started 16 April 1997
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*)
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(* Ring homomorphisms and polynomials *)
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Goal "homo (const::'a::ring=>'a up)";
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by (auto_tac (claset() addSIs [homoI], simpset()));
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qed "const_homo";
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Delsimps [const_add, const_mult, const_one, const_zero];
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Addsimps [const_homo];
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Goal
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"!!f::'a::ring up=>'b::ring. homo f ==> f (a *s p) = f (const a) * f p";
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by (asm_simp_tac (simpset() addsimps [const_mult_is_smult RS sym]) 1);
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qed "homo_smult";
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Addsimps [homo_smult];
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(* Evaluation homomorphism *)
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Goal
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"!! phi::('a::ring=>'b::ring). homo phi ==> homo (EVAL2 phi a)";
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by (rtac homoI 1);
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by (rewtac EVAL2_def);
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(* + commutes *)
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(* degree estimations:
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bound of all sums can be extended to max (deg aa) (deg b) *)
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by (res_inst_tac [("m", "deg (aa + b)"), ("n", "max (deg aa) (deg b)")]
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SUM_shrink 1);
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by (rtac deg_add 1);
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by (asm_simp_tac (simpset() delsimps [coeff_add] addsimps [deg_aboveD]) 1);
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by (res_inst_tac [("m", "deg aa"), ("n", "max (deg aa) (deg b)")]
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SUM_shrink 1);
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by (rtac le_maxI1 1);
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by (asm_simp_tac (simpset() addsimps [deg_aboveD]) 1);
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by (res_inst_tac [("m", "deg b"), ("n", "max (deg aa) (deg b)")]
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SUM_shrink 1);
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by (rtac le_maxI2 1);
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by (asm_simp_tac (simpset() addsimps [deg_aboveD]) 1);
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(* actual homom property + *)
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by (asm_simp_tac (simpset() addsimps [l_distr, SUM_add]) 1);
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(* * commutes *)
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by (res_inst_tac [("m", "deg (aa * b)"), ("n", "deg aa + deg b")]
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SUM_shrink 1);
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by (rtac deg_mult_ring 1);
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by (asm_simp_tac (simpset() delsimps [coeff_mult] addsimps [deg_aboveD]) 1);
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by (rtac trans 1);
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by (rtac CauchySum 2);
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by (asm_simp_tac (simpset() addsimps [boundI, deg_aboveD]) 2);
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by (asm_simp_tac (simpset() addsimps [boundI, deg_aboveD]) 2);
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(* getting a^i and a^(k-i) together is difficult, so we do is manually *)
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by (res_inst_tac [("s",
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11093
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" setsum \
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\ (%k. setsum \
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7998
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\ (%i. phi (coeff i aa) * (phi (coeff (k - i) b) * \
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11093
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\ (a ^ i * a ^ (k - i)))) {..k}) \
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\ {..deg aa + deg b}")] trans 1);
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7998
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by (asm_simp_tac (simpset()
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addsimps [power_mult, leD RS add_diff_inverse, SUM_ldistr]) 1);
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by (simp_tac (simpset() addsimps m_ac) 1);
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by (simp_tac (simpset() addsimps m_ac) 1);
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(* <1> commutes *)
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by (Asm_simp_tac 1);
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qed "EVAL2_homo";
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Goalw [EVAL2_def]
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"!! phi::'a::ring=>'b::ring. EVAL2 phi a (const b) = phi b";
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by (Simp_tac 1);
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qed "EVAL2_const";
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Goalw [EVAL2_def]
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"!! phi::'a::domain=>'b::ring. homo phi ==> EVAL2 phi a (monom 1) = a";
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(* Must be able to distinguish 0 from <1>, hence 'a::domain *)
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by (Asm_simp_tac 1);
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qed "EVAL2_monom1";
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Goalw [EVAL2_def]
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"!! phi::'a::domain=>'b::ring. homo phi ==> EVAL2 phi a (monom n) = a ^ n";
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7998
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by (Simp_tac 1);
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8707
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by (case_tac "n" 1);
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by Auto_tac;
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qed "EVAL2_monom";
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Goal
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"!! phi::'a::ring=>'b::ring. \
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\ homo phi ==> EVAL2 phi a (b *s p) = phi b * EVAL2 phi a p";
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by (asm_simp_tac
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(simpset() addsimps [const_mult_is_smult RS sym, EVAL2_homo, EVAL2_const]) 1);
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qed "EVAL2_smult";
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Goalw [EVAL_def] "!! a::'a::ring. homo (EVAL a)";
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by (asm_simp_tac (simpset() addsimps [EVAL2_homo, id_homo]) 1);
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qed "EVAL_homo";
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Goalw [EVAL_def] "!! a::'a::ring. EVAL a (const b) = b";
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by (asm_simp_tac (simpset() addsimps [EVAL2_const]) 1);
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qed "EVAL_const";
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11093
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Goalw [EVAL_def] "!! a::'a::domain. EVAL a (monom n) = a ^ n";
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by (asm_simp_tac (simpset() addsimps [EVAL2_monom]) 1);
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qed "EVAL_monom";
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Goalw [EVAL_def] "!! a::'a::ring. EVAL a (b *s p) = b * EVAL a p";
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by (asm_simp_tac (simpset() addsimps [EVAL2_smult]) 1);
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qed "EVAL_smult";
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(* Examples *)
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Goal
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"EVAL (x::'a::domain) (a*X^2 + b*X^1 + c*X^0) = a * x ^ 2 + b * x ^ 1 + c";
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7998
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by (asm_simp_tac (simpset() delsimps [power_Suc]
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addsimps [EVAL_homo, EVAL_monom, EVAL_smult]) 1);
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result();
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Goal
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"EVAL (y::'a::domain) \
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7998
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\ (EVAL (const x) (monom 1 + const (a*X^2 + b*X^1 + c*X^0))) = \
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\ x ^ 1 + (a * y ^ 2 + b * y ^ 1 + c)";
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by (asm_simp_tac (simpset() delsimps [power_Suc]
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addsimps [EVAL_homo, EVAL_monom, EVAL_smult, EVAL_const]) 1);
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result();
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