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theory ComputeNumeraltheory ComputeNumeral imports ComputeHOL ComputeFloat begin (* normalization of bit strings *) lemmas bitnorm = normalize_bin_simps (* neg for bit strings *) lemma neg1: "neg Int.Pls = False" by (simp add: Int.Pls_def) lemma neg2: "neg Int.Min = True" apply (subst Int.Min_def) by auto lemma neg3: "neg (Int.Bit0 x) = neg x" apply (simp add: neg_def) apply (subst Bit0_def) by auto lemma neg4: "neg (Int.Bit1 x) = neg x" apply (simp add: neg_def) apply (subst Bit1_def) by auto lemmas bitneg = neg1 neg2 neg3 neg4 (* iszero for bit strings *) lemma iszero1: "iszero Int.Pls = True" by (simp add: Int.Pls_def iszero_def) lemma iszero2: "iszero Int.Min = False" apply (subst Int.Min_def) apply (subst iszero_def) by simp lemma iszero3: "iszero (Int.Bit0 x) = iszero x" apply (subst Int.Bit0_def) apply (subst iszero_def)+ by auto lemma iszero4: "iszero (Int.Bit1 x) = False" apply (subst Int.Bit1_def) apply (subst iszero_def)+ apply simp by arith lemmas bitiszero = iszero1 iszero2 iszero3 iszero4 (* lezero for bit strings *) constdefs "lezero x == (x ≤ 0)" lemma lezero1: "lezero Int.Pls = True" unfolding Int.Pls_def lezero_def by auto lemma lezero2: "lezero Int.Min = True" unfolding Int.Min_def lezero_def by auto lemma lezero3: "lezero (Int.Bit0 x) = lezero x" unfolding Int.Bit0_def lezero_def by auto lemma lezero4: "lezero (Int.Bit1 x) = neg x" unfolding Int.Bit1_def lezero_def neg_def by auto lemmas bitlezero = lezero1 lezero2 lezero3 lezero4 (* equality for bit strings *) lemmas biteq = eq_bin_simps (* x < y for bit strings *) lemmas bitless = less_bin_simps (* x ≤ y for bit strings *) lemmas bitle = le_bin_simps (* succ for bit strings *) lemmas bitsucc = succ_bin_simps (* pred for bit strings *) lemmas bitpred = pred_bin_simps (* unary minus for bit strings *) lemmas bituminus = minus_bin_simps (* addition for bit strings *) lemmas bitadd = add_bin_simps (* multiplication for bit strings *) lemma mult_Pls_right: "x * Int.Pls = Int.Pls" by (simp add: Pls_def) lemma mult_Min_right: "x * Int.Min = - x" by (subst mult_commute, simp add: mult_Min) lemma multb0x: "(Int.Bit0 x) * y = Int.Bit0 (x * y)" by (rule mult_Bit0) lemma multxb0: "x * (Int.Bit0 y) = Int.Bit0 (x * y)" unfolding Bit0_def by simp lemma multb1: "(Int.Bit1 x) * (Int.Bit1 y) = Int.Bit1 (Int.Bit0 (x * y) + x + y)" unfolding Bit0_def Bit1_def by (simp add: algebra_simps) lemmas bitmul = mult_Pls mult_Min mult_Pls_right mult_Min_right multb0x multxb0 multb1 lemmas bitarith = bitnorm bitiszero bitneg bitlezero biteq bitless bitle bitsucc bitpred bituminus bitadd bitmul constdefs "nat_norm_number_of (x::nat) == x" lemma nat_norm_number_of: "nat_norm_number_of (number_of w) = (if lezero w then 0 else number_of w)" apply (simp add: nat_norm_number_of_def) unfolding lezero_def iszero_def neg_def apply (simp add: numeral_simps) done (* Normalization of nat literals *) lemma natnorm0: "(0::nat) = number_of (Int.Pls)" by auto lemma natnorm1: "(1 :: nat) = number_of (Int.Bit1 Int.Pls)" by auto lemmas natnorm = natnorm0 natnorm1 nat_norm_number_of (* Suc *) lemma natsuc: "Suc (number_of x) = (if neg x then 1 else number_of (Int.succ x))" by (auto simp add: number_of_is_id) (* Addition for nat *) lemma natadd: "number_of x + ((number_of y)::nat) = (if neg x then (number_of y) else (if neg y then number_of x else (number_of (x + y))))" unfolding nat_number_of_def number_of_is_id neg_def by auto (* Subtraction for nat *) lemma natsub: "(number_of x) - ((number_of y)::nat) = (if neg x then 0 else (if neg y then number_of x else (nat_norm_number_of (number_of (x + (- y))))))" unfolding nat_norm_number_of by (auto simp add: number_of_is_id neg_def lezero_def iszero_def Let_def nat_number_of_def) (* Multiplication for nat *) lemma natmul: "(number_of x) * ((number_of y)::nat) = (if neg x then 0 else (if neg y then 0 else number_of (x * y)))" unfolding nat_number_of_def number_of_is_id neg_def by (simp add: nat_mult_distrib) lemma nateq: "(((number_of x)::nat) = (number_of y)) = ((lezero x ∧ lezero y) ∨ (x = y))" by (auto simp add: iszero_def lezero_def neg_def number_of_is_id) lemma natless: "(((number_of x)::nat) < (number_of y)) = ((x < y) ∧ (¬ (lezero y)))" by (simp add: lezero_def numeral_simps not_le) lemma natle: "(((number_of x)::nat) ≤ (number_of y)) = (y < x --> lezero x)" by (auto simp add: number_of_is_id lezero_def nat_number_of_def) fun natfac :: "nat => nat" where "natfac n = (if n = 0 then 1 else n * (natfac (n - 1)))" lemmas compute_natarith = bitarith natnorm natsuc natadd natsub natmul nateq natless natle natfac.simps lemma number_eq: "(((number_of x)::'a::{number_ring, ordered_idom}) = (number_of y)) = (x = y)" unfolding number_of_eq apply simp done lemma number_le: "(((number_of x)::'a::{number_ring, ordered_idom}) ≤ (number_of y)) = (x ≤ y)" unfolding number_of_eq apply simp done lemma number_less: "(((number_of x)::'a::{number_ring, ordered_idom}) < (number_of y)) = (x < y)" unfolding number_of_eq apply simp done lemma number_diff: "((number_of x)::'a::{number_ring, ordered_idom}) - number_of y = number_of (x + (- y))" apply (subst diff_number_of_eq) apply simp done lemmas number_norm = number_of_Pls[symmetric] numeral_1_eq_1[symmetric] lemmas compute_numberarith = number_of_minus[symmetric] number_of_add[symmetric] number_diff number_of_mult[symmetric] number_norm number_eq number_le number_less lemma compute_real_of_nat_number_of: "real ((number_of v)::nat) = (if neg v then 0 else number_of v)" by (simp only: real_of_nat_number_of number_of_is_id) lemma compute_nat_of_int_number_of: "nat ((number_of v)::int) = (number_of v)" by simp lemmas compute_num_conversions = compute_real_of_nat_number_of compute_nat_of_int_number_of real_number_of lemmas zpowerarith = zpower_number_of_even zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring] zpower_Pls zpower_Min (* div, mod *) lemma adjust: "adjust b (q, r) = (if 0 ≤ r - b then (2 * q + 1, r - b) else (2 * q, r))" by (auto simp only: adjust_def) lemma negateSnd: "negateSnd (q, r) = (q, -r)" by (simp add: negateSnd_def) lemma divmod: "IntDiv.divmod a b = (if 0≤a then if 0≤b then posDivAlg a b else if a=0 then (0, 0) else negateSnd (negDivAlg (-a) (-b)) else if 0<b then negDivAlg a b else negateSnd (posDivAlg (-a) (-b)))" by (auto simp only: IntDiv.divmod_def) lemmas compute_div_mod = div_def mod_def divmod adjust negateSnd posDivAlg.simps negDivAlg.simps (* collecting all the theorems *) lemma even_Pls: "even (Int.Pls) = True" apply (unfold Pls_def even_def) by simp lemma even_Min: "even (Int.Min) = False" apply (unfold Min_def even_def) by simp lemma even_B0: "even (Int.Bit0 x) = True" apply (unfold Bit0_def) by simp lemma even_B1: "even (Int.Bit1 x) = False" apply (unfold Bit1_def) by simp lemma even_number_of: "even ((number_of w)::int) = even w" by (simp only: number_of_is_id) lemmas compute_even = even_Pls even_Min even_B0 even_B1 even_number_of lemmas compute_numeral = compute_if compute_let compute_pair compute_bool compute_natarith compute_numberarith max_def min_def compute_num_conversions zpowerarith compute_div_mod compute_even end