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theory NatSimprocs(* Title: HOL/NatSimprocs.thy ID: $Id: NatSimprocs.thy,v 1.22 2005/09/17 16:25:11 wenzelm Exp $ Copyright 2003 TU Muenchen *) header {*Simprocs for the Naturals*} theory NatSimprocs imports NatBin uses "int_factor_simprocs.ML" "nat_simprocs.ML" begin setup nat_simprocs_setup subsection{*For simplifying @{term "Suc m - K"} and @{term "K - Suc m"}*} text{*Where K above is a literal*} lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)" by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split) text {*Now just instantiating @{text n} to @{text "number_of v"} does the right simplification, but with some redundant inequality tests.*} lemma neg_number_of_bin_pred_iff_0: "neg (number_of (bin_pred v)::int) = (number_of v = (0::nat))" apply (subgoal_tac "neg (number_of (bin_pred v)) = (number_of v < Suc 0) ") apply (simp only: less_Suc_eq_le le_0_eq) apply (subst less_number_of_Suc, simp) done text{*No longer required as a simprule because of the @{text inverse_fold} simproc*} lemma Suc_diff_number_of: "neg (number_of (bin_minus v)::int) ==> Suc m - (number_of v) = m - (number_of (bin_pred v))" apply (subst Suc_diff_eq_diff_pred) apply (simp add: ); apply (simp del: nat_numeral_1_eq_1); apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric] neg_number_of_bin_pred_iff_0) done lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n" by (simp add: numerals split add: nat_diff_split) subsection{*For @{term nat_case} and @{term nat_rec}*} lemma nat_case_number_of [simp]: "nat_case a f (number_of v) = (let pv = number_of (bin_pred v) in if neg pv then a else f (nat pv))" by (simp split add: nat.split add: Let_def neg_number_of_bin_pred_iff_0) lemma nat_case_add_eq_if [simp]: "nat_case a f ((number_of v) + n) = (let pv = number_of (bin_pred v) in if neg pv then nat_case a f n else f (nat pv + n))" apply (subst add_eq_if) apply (simp split add: nat.split del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric] Let_def neg_imp_number_of_eq_0 neg_number_of_bin_pred_iff_0) done lemma nat_rec_number_of [simp]: "nat_rec a f (number_of v) = (let pv = number_of (bin_pred v) in if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))" apply (case_tac " (number_of v) ::nat") apply (simp_all (no_asm_simp) add: Let_def neg_number_of_bin_pred_iff_0) apply (simp split add: split_if_asm) done lemma nat_rec_add_eq_if [simp]: "nat_rec a f (number_of v + n) = (let pv = number_of (bin_pred v) in if neg pv then nat_rec a f n else f (nat pv + n) (nat_rec a f (nat pv + n)))" apply (subst add_eq_if) apply (simp split add: nat.split del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric] Let_def neg_imp_number_of_eq_0 neg_number_of_bin_pred_iff_0) done subsection{*Various Other Lemmas*} subsubsection{*Evens and Odds, for Mutilated Chess Board*} text{*Lemmas for specialist use, NOT as default simprules*} lemma nat_mult_2: "2 * z = (z+z::nat)" proof - have "2*z = (1 + 1)*z" by simp also have "... = z+z" by (simp add: left_distrib) finally show ?thesis . qed lemma nat_mult_2_right: "z * 2 = (z+z::nat)" by (subst mult_commute, rule nat_mult_2) text{*Case analysis on @{term "n<2"}*} lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0" by arith lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)" by arith lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)" by (simp add: nat_mult_2 [symmetric]) lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2" apply (subgoal_tac "m mod 2 < 2") apply (erule less_2_cases [THEN disjE]) apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1) done lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)" apply (subgoal_tac "m mod 2 < 2") apply (force simp del: mod_less_divisor, simp) done subsubsection{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*} lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)" by simp lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)" by simp text{*Can be used to eliminate long strings of Sucs, but not by default*} lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n" by simp text{*These lemmas collapse some needless occurrences of Suc: at least three Sucs, since two and fewer are rewritten back to Suc again! We already have some rules to simplify operands smaller than 3.*} lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)" by (simp add: Suc3_eq_add_3) lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)" by (simp add: Suc3_eq_add_3) lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n" by (simp add: Suc3_eq_add_3) lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n" by (simp add: Suc3_eq_add_3) lemmas Suc_div_eq_add3_div_number_of = Suc_div_eq_add3_div [of _ "number_of v", standard] declare Suc_div_eq_add3_div_number_of [simp] lemmas Suc_mod_eq_add3_mod_number_of = Suc_mod_eq_add3_mod [of _ "number_of v", standard] declare Suc_mod_eq_add3_mod_number_of [simp] subsection{*Special Simplification for Constants*} text{*These belong here, late in the development of HOL, to prevent their interfering with proofs of abstract properties of instances of the function @{term number_of}*} text{*These distributive laws move literals inside sums and differences.*} lemmas left_distrib_number_of = left_distrib [of _ _ "number_of v", standard] declare left_distrib_number_of [simp] lemmas right_distrib_number_of = right_distrib [of "number_of v", standard] declare right_distrib_number_of [simp] lemmas left_diff_distrib_number_of = left_diff_distrib [of _ _ "number_of v", standard] declare left_diff_distrib_number_of [simp] lemmas right_diff_distrib_number_of = right_diff_distrib [of "number_of v", standard] declare right_diff_distrib_number_of [simp] text{*These are actually for fields, like real: but where else to put them?*} lemmas zero_less_divide_iff_number_of = zero_less_divide_iff [of "number_of w", standard] declare zero_less_divide_iff_number_of [simp] lemmas divide_less_0_iff_number_of = divide_less_0_iff [of "number_of w", standard] declare divide_less_0_iff_number_of [simp] lemmas zero_le_divide_iff_number_of = zero_le_divide_iff [of "number_of w", standard] declare zero_le_divide_iff_number_of [simp] lemmas divide_le_0_iff_number_of = divide_le_0_iff [of "number_of w", standard] declare divide_le_0_iff_number_of [simp] (**** IF times_divide_eq_right and times_divide_eq_left are removed as simprules, then these special-case declarations may be useful. text{*These simprules move numerals into numerators and denominators.*} lemma times_recip_eq_right [simp]: "a * (1/c) = a / (c::'a::field)" by (simp add: times_divide_eq) lemma times_recip_eq_left [simp]: "(1/c) * a = a / (c::'a::field)" by (simp add: times_divide_eq) lemmas times_divide_eq_right_number_of = times_divide_eq_right [of "number_of w", standard] declare times_divide_eq_right_number_of [simp] lemmas times_divide_eq_right_number_of = times_divide_eq_right [of _ _ "number_of w", standard] declare times_divide_eq_right_number_of [simp] lemmas times_divide_eq_left_number_of = times_divide_eq_left [of _ "number_of w", standard] declare times_divide_eq_left_number_of [simp] lemmas times_divide_eq_left_number_of = times_divide_eq_left [of _ _ "number_of w", standard] declare times_divide_eq_left_number_of [simp] ****) text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}. It looks strange, but then other simprocs simplify the quotient.*} lemmas inverse_eq_divide_number_of = inverse_eq_divide [of "number_of w", standard] declare inverse_eq_divide_number_of [simp] text{*These laws simplify inequalities, moving unary minus from a term into the literal.*} lemmas less_minus_iff_number_of = less_minus_iff [of "number_of v", standard] declare less_minus_iff_number_of [simp] lemmas le_minus_iff_number_of = le_minus_iff [of "number_of v", standard] declare le_minus_iff_number_of [simp] lemmas equation_minus_iff_number_of = equation_minus_iff [of "number_of v", standard] declare equation_minus_iff_number_of [simp] lemmas minus_less_iff_number_of = minus_less_iff [of _ "number_of v", standard] declare minus_less_iff_number_of [simp] lemmas minus_le_iff_number_of = minus_le_iff [of _ "number_of v", standard] declare minus_le_iff_number_of [simp] lemmas minus_equation_iff_number_of = minus_equation_iff [of _ "number_of v", standard] declare minus_equation_iff_number_of [simp] text{*These simplify inequalities where one side is the constant 1.*} lemmas less_minus_iff_1 = less_minus_iff [of 1, simplified] declare less_minus_iff_1 [simp] lemmas le_minus_iff_1 = le_minus_iff [of 1, simplified] declare le_minus_iff_1 [simp] lemmas equation_minus_iff_1 = equation_minus_iff [of 1, simplified] declare equation_minus_iff_1 [simp] lemmas minus_less_iff_1 = minus_less_iff [of _ 1, simplified] declare minus_less_iff_1 [simp] lemmas minus_le_iff_1 = minus_le_iff [of _ 1, simplified] declare minus_le_iff_1 [simp] lemmas minus_equation_iff_1 = minus_equation_iff [of _ 1, simplified] declare minus_equation_iff_1 [simp] text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "≤"}) *} lemmas mult_less_cancel_left_number_of = mult_less_cancel_left [of "number_of v", standard] declare mult_less_cancel_left_number_of [simp] lemmas mult_less_cancel_right_number_of = mult_less_cancel_right [of _ "number_of v", standard] declare mult_less_cancel_right_number_of [simp] lemmas mult_le_cancel_left_number_of = mult_le_cancel_left [of "number_of v", standard] declare mult_le_cancel_left_number_of [simp] lemmas mult_le_cancel_right_number_of = mult_le_cancel_right [of _ "number_of v", standard] declare mult_le_cancel_right_number_of [simp] text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "≤"} and @{text "="}) *} lemmas le_divide_eq_number_of = le_divide_eq [of _ _ "number_of w", standard] declare le_divide_eq_number_of [simp] lemmas divide_le_eq_number_of = divide_le_eq [of _ "number_of w", standard] declare divide_le_eq_number_of [simp] lemmas less_divide_eq_number_of = less_divide_eq [of _ _ "number_of w", standard] declare less_divide_eq_number_of [simp] lemmas divide_less_eq_number_of = divide_less_eq [of _ "number_of w", standard] declare divide_less_eq_number_of [simp] lemmas eq_divide_eq_number_of = eq_divide_eq [of _ _ "number_of w", standard] declare eq_divide_eq_number_of [simp] lemmas divide_eq_eq_number_of = divide_eq_eq [of _ "number_of w", standard] declare divide_eq_eq_number_of [simp] subsection{*Optional Simplification Rules Involving Constants*} text{*Simplify quotients that are compared with a literal constant.*} lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard] lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard] lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard] lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard] lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard] lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard] text{*Not good as automatic simprules because they cause case splits.*} lemmas divide_const_simps = le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1 subsubsection{*Division By @{text "-1"}*} lemma divide_minus1 [simp]: "x/-1 = -(x::'a::{field,division_by_zero,number_ring})" by simp lemma minus1_divide [simp]: "-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)" by (simp add: divide_inverse inverse_minus_eq) lemma half_gt_zero_iff: "(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))" by auto lemmas half_gt_zero = half_gt_zero_iff [THEN iffD2, simp] (* The following lemma should appear in Divides.thy, but there the proof doesn't work. *) lemma nat_dvd_not_less: "[| 0 < m; m < n |] ==> ¬ n dvd (m::nat)" by (unfold dvd_def) auto ML {* val divide_minus1 = thm "divide_minus1"; val minus1_divide = thm "minus1_divide"; *} end
theorem int_mult_div_cancel1:
k ≠ 0 ==> k * m div (k * n) = m div n
theorem int_mult_div_cancel_disj:
k * m div (k * n) = (if k = 0 then 0 else m div n)
lemma Suc_diff_eq_diff_pred:
Numeral0 < n ==> Suc m - n = m - (n - Numeral1)
lemma neg_number_of_bin_pred_iff_0:
neg (number_of (bin_pred v)) = (number_of v = 0)
lemma Suc_diff_number_of:
neg (number_of (bin_minus v)) ==> Suc m - number_of v = m - number_of (bin_pred v)
lemma diff_Suc_eq_diff_pred:
m - Suc n = m - 1 - n
lemma nat_case_number_of:
nat_case a f (number_of v) = (let pv = number_of (bin_pred v) in if neg pv then a else f (nat pv))
lemma nat_case_add_eq_if:
nat_case a f (number_of v + n) = (let pv = number_of (bin_pred v) in if neg pv then nat_case a f n else f (nat pv + n))
lemma nat_rec_number_of:
nat_rec a f (number_of v) = (let pv = number_of (bin_pred v) in if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))
lemma nat_rec_add_eq_if:
nat_rec a f (number_of v + n) = (let pv = number_of (bin_pred v) in if neg pv then nat_rec a f n else f (nat pv + n) (nat_rec a f (nat pv + n)))
lemma nat_mult_2:
2 * z = z + z
lemma nat_mult_2_right:
z * 2 = z + z
lemma less_2_cases:
n < 2 ==> n = 0 ∨ n = Suc 0
lemma div2_Suc_Suc:
Suc (Suc m) div 2 = Suc (m div 2)
lemma add_self_div_2:
(m + m) div 2 = m
lemma mod2_Suc_Suc:
Suc (Suc m) mod 2 = m mod 2
lemma mod2_gr_0:
(0 < m mod 2) = (m mod 2 = 1)
lemma add_2_eq_Suc:
2 + n = Suc (Suc n)
lemma add_2_eq_Suc':
n + 2 = Suc (Suc n)
lemma Suc3_eq_add_3:
Suc (Suc (Suc n)) = 3 + n
lemma div_Suc_eq_div_add3:
m div Suc (Suc (Suc n)) = m div (3 + n)
lemma mod_Suc_eq_mod_add3:
m mod Suc (Suc (Suc n)) = m mod (3 + n)
lemma Suc_div_eq_add3_div:
Suc (Suc (Suc m)) div n = (3 + m) div n
lemma Suc_mod_eq_add3_mod:
Suc (Suc (Suc m)) mod n = (3 + m) mod n
lemmas Suc_div_eq_add3_div_number_of:
Suc (Suc (Suc m)) div number_of v = (3 + m) div number_of v
lemmas Suc_div_eq_add3_div_number_of:
Suc (Suc (Suc m)) div number_of v = (3 + m) div number_of v
lemmas Suc_mod_eq_add3_mod_number_of:
Suc (Suc (Suc m)) mod number_of v = (3 + m) mod number_of v
lemmas Suc_mod_eq_add3_mod_number_of:
Suc (Suc (Suc m)) mod number_of v = (3 + m) mod number_of v
lemmas left_distrib_number_of:
(a + b) * number_of v = a * number_of v + b * number_of v
lemmas left_distrib_number_of:
(a + b) * number_of v = a * number_of v + b * number_of v
lemmas right_distrib_number_of:
number_of v * (b + c) = number_of v * b + number_of v * c
lemmas right_distrib_number_of:
number_of v * (b + c) = number_of v * b + number_of v * c
lemmas left_diff_distrib_number_of:
(a - b) * number_of v = a * number_of v - b * number_of v
lemmas left_diff_distrib_number_of:
(a - b) * number_of v = a * number_of v - b * number_of v
lemmas right_diff_distrib_number_of:
number_of v * (b - c) = number_of v * b - number_of v * c
lemmas right_diff_distrib_number_of:
number_of v * (b - c) = number_of v * b - number_of v * c
lemmas zero_less_divide_iff_number_of:
((0::'b) < number_of w / b) = ((0::'b) < number_of w ∧ (0::'b) < b ∨ number_of w < (0::'b) ∧ b < (0::'b))
lemmas zero_less_divide_iff_number_of:
((0::'b) < number_of w / b) = ((0::'b) < number_of w ∧ (0::'b) < b ∨ number_of w < (0::'b) ∧ b < (0::'b))
lemmas divide_less_0_iff_number_of:
(number_of w / b < (0::'b)) = ((0::'b) < number_of w ∧ b < (0::'b) ∨ number_of w < (0::'b) ∧ (0::'b) < b)
lemmas divide_less_0_iff_number_of:
(number_of w / b < (0::'b)) = ((0::'b) < number_of w ∧ b < (0::'b) ∨ number_of w < (0::'b) ∧ (0::'b) < b)
lemmas zero_le_divide_iff_number_of:
((0::'b) ≤ number_of w / b) = ((0::'b) ≤ number_of w ∧ (0::'b) ≤ b ∨ number_of w ≤ (0::'b) ∧ b ≤ (0::'b))
lemmas zero_le_divide_iff_number_of:
((0::'b) ≤ number_of w / b) = ((0::'b) ≤ number_of w ∧ (0::'b) ≤ b ∨ number_of w ≤ (0::'b) ∧ b ≤ (0::'b))
lemmas divide_le_0_iff_number_of:
(number_of w / b ≤ (0::'b)) = ((0::'b) ≤ number_of w ∧ b ≤ (0::'b) ∨ number_of w ≤ (0::'b) ∧ (0::'b) ≤ b)
lemmas divide_le_0_iff_number_of:
(number_of w / b ≤ (0::'b)) = ((0::'b) ≤ number_of w ∧ b ≤ (0::'b) ∨ number_of w ≤ (0::'b) ∧ (0::'b) ≤ b)
lemmas inverse_eq_divide_number_of:
inverse (number_of w) = (1::'b) / number_of w
lemmas inverse_eq_divide_number_of:
inverse (number_of w) = (1::'b) / number_of w
lemmas less_minus_iff_number_of:
(number_of v < - b) = (b < - number_of v)
lemmas less_minus_iff_number_of:
(number_of v < - b) = (b < - number_of v)
lemmas le_minus_iff_number_of:
(number_of v ≤ - b) = (b ≤ - number_of v)
lemmas le_minus_iff_number_of:
(number_of v ≤ - b) = (b ≤ - number_of v)
lemmas equation_minus_iff_number_of:
(number_of v = - b) = (b = - number_of v)
lemmas equation_minus_iff_number_of:
(number_of v = - b) = (b = - number_of v)
lemmas minus_less_iff_number_of:
(- a < number_of v) = (- number_of v < a)
lemmas minus_less_iff_number_of:
(- a < number_of v) = (- number_of v < a)
lemmas minus_le_iff_number_of:
(- a ≤ number_of v) = (- number_of v ≤ a)
lemmas minus_le_iff_number_of:
(- a ≤ number_of v) = (- number_of v ≤ a)
lemmas minus_equation_iff_number_of:
(- a = number_of v) = (- number_of v = a)
lemmas minus_equation_iff_number_of:
(- a = number_of v) = (- number_of v = a)
lemmas less_minus_iff_1:
((1::'b) < - b) = (b < - (1::'b))
lemmas less_minus_iff_1:
((1::'b) < - b) = (b < - (1::'b))
lemmas le_minus_iff_1:
((1::'b) ≤ - b) = (b ≤ - (1::'b))
lemmas le_minus_iff_1:
((1::'b) ≤ - b) = (b ≤ - (1::'b))
lemmas equation_minus_iff_1:
(- b = (1::'b)) = (b = - (1::'b))
lemmas equation_minus_iff_1:
(- b = (1::'b)) = (b = - (1::'b))
lemmas minus_less_iff_1:
(- a < (1::'b)) = (- (1::'b) < a)
lemmas minus_less_iff_1:
(- a < (1::'b)) = (- (1::'b) < a)
lemmas minus_le_iff_1:
(- a ≤ (1::'b)) = (- (1::'b) ≤ a)
lemmas minus_le_iff_1:
(- a ≤ (1::'b)) = (- (1::'b) ≤ a)
lemmas minus_equation_iff_1:
(a = - (1::'b)) = (- (1::'b) = a)
lemmas minus_equation_iff_1:
(a = - (1::'b)) = (- (1::'b) = a)
lemmas mult_less_cancel_left_number_of:
(number_of v * a < number_of v * b) = (((0::'b) ≤ number_of v --> a < b) ∧ (number_of v ≤ (0::'b) --> b < a))
lemmas mult_less_cancel_left_number_of:
(number_of v * a < number_of v * b) = (((0::'b) ≤ number_of v --> a < b) ∧ (number_of v ≤ (0::'b) --> b < a))
lemmas mult_less_cancel_right_number_of:
(a * number_of v < b * number_of v) = (((0::'b) ≤ number_of v --> a < b) ∧ (number_of v ≤ (0::'b) --> b < a))
lemmas mult_less_cancel_right_number_of:
(a * number_of v < b * number_of v) = (((0::'b) ≤ number_of v --> a < b) ∧ (number_of v ≤ (0::'b) --> b < a))
lemmas mult_le_cancel_left_number_of:
(number_of v * a ≤ number_of v * b) = (((0::'b) < number_of v --> a ≤ b) ∧ (number_of v < (0::'b) --> b ≤ a))
lemmas mult_le_cancel_left_number_of:
(number_of v * a ≤ number_of v * b) = (((0::'b) < number_of v --> a ≤ b) ∧ (number_of v < (0::'b) --> b ≤ a))
lemmas mult_le_cancel_right_number_of:
(a * number_of v ≤ b * number_of v) = (((0::'b) < number_of v --> a ≤ b) ∧ (number_of v < (0::'b) --> b ≤ a))
lemmas mult_le_cancel_right_number_of:
(a * number_of v ≤ b * number_of v) = (((0::'b) < number_of v --> a ≤ b) ∧ (number_of v < (0::'b) --> b ≤ a))
lemmas le_divide_eq_number_of:
(a ≤ b / number_of w) = (if (0::'b) < number_of w then a * number_of w ≤ b else if number_of w < (0::'b) then b ≤ a * number_of w else a ≤ (0::'b))
lemmas le_divide_eq_number_of:
(a ≤ b / number_of w) = (if (0::'b) < number_of w then a * number_of w ≤ b else if number_of w < (0::'b) then b ≤ a * number_of w else a ≤ (0::'b))
lemmas divide_le_eq_number_of:
(b / number_of w ≤ a) = (if (0::'b) < number_of w then b ≤ a * number_of w else if number_of w < (0::'b) then a * number_of w ≤ b else (0::'b) ≤ a)
lemmas divide_le_eq_number_of:
(b / number_of w ≤ a) = (if (0::'b) < number_of w then b ≤ a * number_of w else if number_of w < (0::'b) then a * number_of w ≤ b else (0::'b) ≤ a)
lemmas less_divide_eq_number_of:
(a < b / number_of w) = (if (0::'b) < number_of w then a * number_of w < b else if number_of w < (0::'b) then b < a * number_of w else a < (0::'b))
lemmas less_divide_eq_number_of:
(a < b / number_of w) = (if (0::'b) < number_of w then a * number_of w < b else if number_of w < (0::'b) then b < a * number_of w else a < (0::'b))
lemmas divide_less_eq_number_of:
(b / number_of w < a) = (if (0::'b) < number_of w then b < a * number_of w else if number_of w < (0::'b) then a * number_of w < b else (0::'b) < a)
lemmas divide_less_eq_number_of:
(b / number_of w < a) = (if (0::'b) < number_of w then b < a * number_of w else if number_of w < (0::'b) then a * number_of w < b else (0::'b) < a)
lemmas eq_divide_eq_number_of:
(a = b / number_of w) = (if number_of w ≠ (0::'b) then a * number_of w = b else a = (0::'b))
lemmas eq_divide_eq_number_of:
(a = b / number_of w) = (if number_of w ≠ (0::'b) then a * number_of w = b else a = (0::'b))
lemmas divide_eq_eq_number_of:
(b / number_of w = a) = (if number_of w ≠ (0::'b) then b = a * number_of w else a = (0::'b))
lemmas divide_eq_eq_number_of:
(b / number_of w = a) = (if number_of w ≠ (0::'b) then b = a * number_of w else a = (0::'b))
lemmas le_divide_eq_number_of:
(number_of w ≤ b / c) = (if (0::'b) < c then number_of w * c ≤ b else if c < (0::'b) then b ≤ number_of w * c else number_of w ≤ (0::'b))
lemmas le_divide_eq_number_of:
(number_of w ≤ b / c) = (if (0::'b) < c then number_of w * c ≤ b else if c < (0::'b) then b ≤ number_of w * c else number_of w ≤ (0::'b))
lemmas divide_le_eq_number_of:
(b / c ≤ number_of w) = (if (0::'b) < c then b ≤ number_of w * c else if c < (0::'b) then number_of w * c ≤ b else (0::'b) ≤ number_of w)
lemmas divide_le_eq_number_of:
(b / c ≤ number_of w) = (if (0::'b) < c then b ≤ number_of w * c else if c < (0::'b) then number_of w * c ≤ b else (0::'b) ≤ number_of w)
lemmas less_divide_eq_number_of:
(number_of w < b / c) = (if (0::'b) < c then number_of w * c < b else if c < (0::'b) then b < number_of w * c else number_of w < (0::'b))
lemmas less_divide_eq_number_of:
(number_of w < b / c) = (if (0::'b) < c then number_of w * c < b else if c < (0::'b) then b < number_of w * c else number_of w < (0::'b))
lemmas divide_less_eq_number_of:
(b / c < number_of w) = (if (0::'b) < c then b < number_of w * c else if c < (0::'b) then number_of w * c < b else (0::'b) < number_of w)
lemmas divide_less_eq_number_of:
(b / c < number_of w) = (if (0::'b) < c then b < number_of w * c else if c < (0::'b) then number_of w * c < b else (0::'b) < number_of w)
lemmas eq_divide_eq_number_of:
(number_of w = b / c) = (if c ≠ (0::'b) then number_of w * c = b else number_of w = (0::'b))
lemmas eq_divide_eq_number_of:
(number_of w = b / c) = (if c ≠ (0::'b) then number_of w * c = b else number_of w = (0::'b))
lemmas divide_eq_eq_number_of:
(b / c = number_of w) = (if c ≠ (0::'b) then b = number_of w * c else number_of w = (0::'b))
lemmas divide_eq_eq_number_of:
(b / c = number_of w) = (if c ≠ (0::'b) then b = number_of w * c else number_of w = (0::'b))
lemmas divide_const_simps:
(number_of w ≤ b / c) = (if (0::'b) < c then number_of w * c ≤ b else if c < (0::'b) then b ≤ number_of w * c else number_of w ≤ (0::'b))
(b / c ≤ number_of w) = (if (0::'b) < c then b ≤ number_of w * c else if c < (0::'b) then number_of w * c ≤ b else (0::'b) ≤ number_of w)
(number_of w < b / c) = (if (0::'b) < c then number_of w * c < b else if c < (0::'b) then b < number_of w * c else number_of w < (0::'b))
(b / c < number_of w) = (if (0::'b) < c then b < number_of w * c else if c < (0::'b) then number_of w * c < b else (0::'b) < number_of w)
(number_of w = b / c) = (if c ≠ (0::'b) then number_of w * c = b else number_of w = (0::'b))
(b / c = number_of w) = (if c ≠ (0::'b) then b = number_of w * c else number_of w = (0::'b))
((1::'a) ≤ b / a) = ((0::'a) < a ∧ a ≤ b ∨ a < (0::'a) ∧ b ≤ a)
(b / a ≤ (1::'a)) = ((0::'a) < a ∧ b ≤ a ∨ a < (0::'a) ∧ a ≤ b ∨ a = (0::'a))
((1::'a) < b / a) = ((0::'a) < a ∧ a < b ∨ a < (0::'a) ∧ b < a)
(b / a < (1::'a)) = ((0::'a) < a ∧ b < a ∨ a < (0::'a) ∧ a < b ∨ a = (0::'a))
lemmas divide_const_simps:
(number_of w ≤ b / c) = (if (0::'b) < c then number_of w * c ≤ b else if c < (0::'b) then b ≤ number_of w * c else number_of w ≤ (0::'b))
(b / c ≤ number_of w) = (if (0::'b) < c then b ≤ number_of w * c else if c < (0::'b) then number_of w * c ≤ b else (0::'b) ≤ number_of w)
(number_of w < b / c) = (if (0::'b) < c then number_of w * c < b else if c < (0::'b) then b < number_of w * c else number_of w < (0::'b))
(b / c < number_of w) = (if (0::'b) < c then b < number_of w * c else if c < (0::'b) then number_of w * c < b else (0::'b) < number_of w)
(number_of w = b / c) = (if c ≠ (0::'b) then number_of w * c = b else number_of w = (0::'b))
(b / c = number_of w) = (if c ≠ (0::'b) then b = number_of w * c else number_of w = (0::'b))
((1::'a) ≤ b / a) = ((0::'a) < a ∧ a ≤ b ∨ a < (0::'a) ∧ b ≤ a)
(b / a ≤ (1::'a)) = ((0::'a) < a ∧ b ≤ a ∨ a < (0::'a) ∧ a ≤ b ∨ a = (0::'a))
((1::'a) < b / a) = ((0::'a) < a ∧ a < b ∨ a < (0::'a) ∧ b < a)
(b / a < (1::'a)) = ((0::'a) < a ∧ b < a ∨ a < (0::'a) ∧ a < b ∨ a = (0::'a))
lemma divide_minus1:
x / (-1::'a) = - x
lemma minus1_divide:
(-1::'a) / x = - ((1::'a) / x)
lemma half_gt_zero_iff:
((0::'a) < r / (2::'a)) = ((0::'a) < r)
lemmas half_gt_zero:
(0::'a1) < r1 ==> (0::'a1) < r1 / (2::'a1)
lemmas half_gt_zero:
(0::'a1) < r1 ==> (0::'a1) < r1 / (2::'a1)
lemma nat_dvd_not_less:
[| 0 < m; m < n |] ==> ¬ n dvd m