13957
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(* Title: ComplexArith0.ML
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Author: Jacques D. Fleuriot
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Copyright: 2001 University of Edinburgh
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Description: Assorted facts that need binary literals
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Also, common factor cancellation (see e.g. HyperArith0)
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*)
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(** Division and inverse **)
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Goal "0/x = (0::complex)";
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by (simp_tac (simpset() addsimps [complex_divide_def]) 1);
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qed "complex_0_divide";
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Addsimps [complex_0_divide];
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Goalw [complex_divide_def] "x/(0::complex) = 0";
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by (stac COMPLEX_INVERSE_ZERO 1);
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by (Simp_tac 1);
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qed "COMPLEX_DIVIDE_ZERO";
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Goal "inverse (x::complex) = 1/x";
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by (simp_tac (simpset() addsimps [complex_divide_def]) 1);
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qed "complex_inverse_eq_divide";
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Goal "(inverse(x::complex) = 0) = (x = 0)";
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by (auto_tac (claset(),
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simpset() addsimps [COMPLEX_INVERSE_ZERO]));
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qed "complex_inverse_zero_iff";
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Addsimps [complex_inverse_zero_iff];
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Goal "(x/y = 0) = (x=0 | y=(0::complex))";
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by (auto_tac (claset(), simpset() addsimps [complex_divide_def]));
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qed "complex_divide_eq_0_iff";
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Addsimps [complex_divide_eq_0_iff];
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Goal "h ~= (0::complex) ==> h/h = 1";
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by (asm_simp_tac
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(simpset() addsimps [complex_divide_def]) 1);
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qed "complex_divide_self_eq";
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Addsimps [complex_divide_self_eq];
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bind_thm ("complex_mult_minus_right", complex_minus_mult_eq2 RS sym);
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Goal "!!k::complex. (k*m = k*n) = (k = 0 | m=n)";
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by (case_tac "k=0" 1);
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by (auto_tac (claset(), simpset() addsimps [complex_mult_left_cancel]));
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qed "complex_mult_eq_cancel1";
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Goal "!!k::complex. (m*k = n*k) = (k = 0 | m=n)";
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by (case_tac "k=0" 1);
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by (auto_tac (claset(), simpset() addsimps [complex_mult_right_cancel]));
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qed "complex_mult_eq_cancel2";
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Goal "!!k::complex. k~=0 ==> (k*m) / (k*n) = (m/n)";
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by (asm_simp_tac
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(simpset() addsimps [complex_divide_def, complex_inverse_distrib]) 1);
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by (subgoal_tac "k * m * (inverse k * inverse n) = \
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\ (k * inverse k) * (m * inverse n)" 1);
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by (Asm_full_simp_tac 1);
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by (asm_full_simp_tac (HOL_ss addsimps complex_mult_ac) 1);
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qed "complex_mult_div_cancel1";
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(*For ExtractCommonTerm*)
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Goal "(k*m) / (k*n) = (if k = (0::complex) then 0 else m/n)";
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by (simp_tac (simpset() addsimps [complex_mult_div_cancel1]) 1);
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qed "complex_mult_div_cancel_disj";
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local
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open Complex_Numeral_Simprocs
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in
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val rel_complex_number_of = [eq_complex_number_of];
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structure CancelNumeralFactorCommon =
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struct
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val mk_coeff = mk_coeff
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val dest_coeff = dest_coeff 1
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val trans_tac = Real_Numeral_Simprocs.trans_tac
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val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps complex_minus_from_mult_simps @ mult_1s))
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THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@complex_mult_minus_simps))
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THEN ALLGOALS (simp_tac (HOL_ss addsimps complex_mult_ac))
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val numeral_simp_tac = ALLGOALS (simp_tac (HOL_ss addsimps rel_complex_number_of@bin_simps))
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val simplify_meta_eq = simplify_meta_eq
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end
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structure DivCancelNumeralFactor = CancelNumeralFactorFun
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(open CancelNumeralFactorCommon
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val prove_conv = Bin_Simprocs.prove_conv
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val mk_bal = HOLogic.mk_binop "HOL.divide"
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val dest_bal = HOLogic.dest_bin "HOL.divide" complexT
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val cancel = complex_mult_div_cancel1 RS trans
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val neg_exchanges = false
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)
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structure EqCancelNumeralFactor = CancelNumeralFactorFun
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(open CancelNumeralFactorCommon
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val prove_conv = Bin_Simprocs.prove_conv
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val mk_bal = HOLogic.mk_eq
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val dest_bal = HOLogic.dest_bin "op =" complexT
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val cancel = complex_mult_eq_cancel1 RS trans
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val neg_exchanges = false
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)
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val complex_cancel_numeral_factors_relations =
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map prep_simproc
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[("complexeq_cancel_numeral_factor",
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["(l::complex) * m = n", "(l::complex) = m * n"],
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EqCancelNumeralFactor.proc)];
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val complex_cancel_numeral_factors_divide = prep_simproc
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("complexdiv_cancel_numeral_factor",
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["((l::complex) * m) / n", "(l::complex) / (m * n)",
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"((number_of v)::complex) / (number_of w)"],
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DivCancelNumeralFactor.proc);
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val complex_cancel_numeral_factors =
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complex_cancel_numeral_factors_relations @
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[complex_cancel_numeral_factors_divide];
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end;
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Addsimprocs complex_cancel_numeral_factors;
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(*examples:
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print_depth 22;
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set timing;
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set trace_simp;
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fun test s = (Goal s; by (Simp_tac 1));
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test "9*x = 12 * (y::complex)";
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test "(9*x) / (12 * (y::complex)) = z";
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test "-99*x = 132 * (y::complex)";
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test "999*x = -396 * (y::complex)";
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test "(999*x) / (-396 * (y::complex)) = z";
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test "-99*x = -81 * (y::complex)";
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test "(-99*x) / (-81 * (y::complex)) = z";
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test "-2 * x = -1 * (y::complex)";
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test "-2 * x = -(y::complex)";
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test "(-2 * x) / (-1 * (y::complex)) = z";
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*)
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(** Declarations for ExtractCommonTerm **)
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local
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open Complex_Numeral_Simprocs
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in
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structure CancelFactorCommon =
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struct
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val mk_sum = long_mk_prod
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val dest_sum = dest_prod
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val mk_coeff = mk_coeff
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val dest_coeff = dest_coeff
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val find_first = find_first []
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val trans_tac = Real_Numeral_Simprocs.trans_tac
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val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps mult_1s@complex_mult_ac))
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end;
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structure EqCancelFactor = ExtractCommonTermFun
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(open CancelFactorCommon
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val prove_conv = Bin_Simprocs.prove_conv
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val mk_bal = HOLogic.mk_eq
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val dest_bal = HOLogic.dest_bin "op =" complexT
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val simplify_meta_eq = cancel_simplify_meta_eq complex_mult_eq_cancel1
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);
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structure DivideCancelFactor = ExtractCommonTermFun
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(open CancelFactorCommon
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val prove_conv = Bin_Simprocs.prove_conv
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val mk_bal = HOLogic.mk_binop "HOL.divide"
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val dest_bal = HOLogic.dest_bin "HOL.divide" complexT
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val simplify_meta_eq = cancel_simplify_meta_eq complex_mult_div_cancel_disj
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);
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val complex_cancel_factor =
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map prep_simproc
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[("complex_eq_cancel_factor", ["(l::complex) * m = n", "(l::complex) = m * n"],
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EqCancelFactor.proc),
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("complex_divide_cancel_factor", ["((l::complex) * m) / n", "(l::complex) / (m * n)"],
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DivideCancelFactor.proc)];
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end;
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Addsimprocs complex_cancel_factor;
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(*examples:
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print_depth 22;
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set timing;
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set trace_simp;
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fun test s = (Goal s; by (Asm_simp_tac 1));
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test "x*k = k*(y::complex)";
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test "k = k*(y::complex)";
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test "a*(b*c) = (b::complex)";
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test "a*(b*c) = d*(b::complex)*(x*a)";
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test "(x*k) / (k*(y::complex)) = (uu::complex)";
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test "(k) / (k*(y::complex)) = (uu::complex)";
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test "(a*(b*c)) / ((b::complex)) = (uu::complex)";
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test "(a*(b*c)) / (d*(b::complex)*(x*a)) = (uu::complex)";
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(*FIXME: what do we do about this?*)
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test "a*(b*c)/(y*z) = d*(b::complex)*(x*a)/z";
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*)
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Goal "z~=0 ==> ((x::complex) = y/z) = (x*z = y)";
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by (subgoal_tac "(x*z = y) = (x*z = (y/z)*z)" 1);
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by (asm_simp_tac (simpset() addsimps [complex_divide_def, complex_mult_assoc]) 2);
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by (etac ssubst 1);
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by (stac complex_mult_eq_cancel2 1);
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by (Asm_simp_tac 1);
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qed "complex_eq_divide_eq";
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Addsimps [inst "z" "number_of ?w" complex_eq_divide_eq];
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Goal "z~=0 ==> (y/z = (x::complex)) = (y = x*z)";
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by (subgoal_tac "(y = x*z) = ((y/z)*z = x*z)" 1);
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by (asm_simp_tac (simpset() addsimps [complex_divide_def, complex_mult_assoc]) 2);
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by (etac ssubst 1);
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by (stac complex_mult_eq_cancel2 1);
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by (Asm_simp_tac 1);
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qed "complex_divide_eq_eq";
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Addsimps [inst "z" "number_of ?w" complex_divide_eq_eq];
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Goal "(m/k = n/k) = (k = 0 | m = (n::complex))";
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by (case_tac "k=0" 1);
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by (asm_simp_tac (simpset() addsimps [COMPLEX_DIVIDE_ZERO]) 1);
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by (asm_simp_tac (simpset() addsimps [complex_divide_eq_eq, complex_eq_divide_eq,
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complex_mult_eq_cancel2]) 1);
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qed "complex_divide_eq_cancel2";
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Goal "(k/m = k/n) = (k = 0 | m = (n::complex))";
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by (case_tac "m=0 | n = 0" 1);
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by (auto_tac (claset(),
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simpset() addsimps [COMPLEX_DIVIDE_ZERO, complex_divide_eq_eq,
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complex_eq_divide_eq, complex_mult_eq_cancel1]));
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qed "complex_divide_eq_cancel1";
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(** Division by 1, -1 **)
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Goal "(x::complex)/1 = x";
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by (simp_tac (simpset() addsimps [complex_divide_def]) 1);
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qed "complex_divide_1";
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Addsimps [complex_divide_1];
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Goal "x/-1 = -(x::complex)";
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by (Simp_tac 1);
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qed "complex_divide_minus1";
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Addsimps [complex_divide_minus1];
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Goal "-1/(x::complex) = - (1/x)";
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by (simp_tac (simpset() addsimps [complex_divide_def, complex_minus_inverse]) 1);
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qed "complex_minus1_divide";
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Addsimps [complex_minus1_divide];
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Goal "(x = - y) = (y = - (x::complex))";
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by Auto_tac;
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qed "complex_equation_minus";
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Goal "(- x = y) = (- (y::complex) = x)";
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by Auto_tac;
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qed "complex_minus_equation";
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Goal "(x + - a = (0::complex)) = (x=a)";
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by (simp_tac (simpset() addsimps [complex_diff_eq_eq,symmetric complex_diff_def]) 1);
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qed "complex_add_minus_iff";
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Addsimps [complex_add_minus_iff];
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Goal "(-b = -a) = (b = (a::complex))";
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by Auto_tac;
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qed "complex_minus_eq_cancel";
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Addsimps [complex_minus_eq_cancel];
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(*Distributive laws for literals*)
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Addsimps (map (inst "w" "number_of ?v")
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[complex_add_mult_distrib, complex_add_mult_distrib2,
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complex_diff_mult_distrib, complex_diff_mult_distrib2]);
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Addsimps [inst "x" "number_of ?v" complex_equation_minus];
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Addsimps [inst "y" "number_of ?v" complex_minus_equation];
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Goal "(x+y = (0::complex)) = (y = -x)";
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by Auto_tac;
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by (dtac (sym RS (complex_diff_eq_eq RS iffD2)) 1);
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by Auto_tac;
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qed "complex_add_eq_0_iff";
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AddIffs [complex_add_eq_0_iff];
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Goalw [complex_diff_def]"-(x-y) = y - (x::complex)";
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by (auto_tac (claset(),simpset() addsimps [complex_add_commute]));
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qed "complex_minus_diff_eq";
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Addsimps [complex_minus_diff_eq];
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Addsimps [inst "x" "number_of ?w" complex_inverse_eq_divide];
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