src/HOL/Nat_Numeral.thy
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(*  Title:      HOL/Nat_Numeral.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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header {* Binary numerals for the natural numbers *}
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theory Nat_Numeral
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imports Int
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begin
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subsection {* Numerals for natural numbers *}
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text {*
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  Arithmetic for naturals is reduced to that for the non-negative integers.
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*}
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instantiation nat :: number_semiring
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begin
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definition
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  nat_number_of_def [code_unfold, code del]: "number_of v = nat (number_of v)"
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instance proof
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  fix n show "number_of (int n) = (of_nat n :: nat)"
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    unfolding nat_number_of_def number_of_eq by simp
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qed
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end
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lemma [code_post]:
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  "nat (number_of v) = number_of v"
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  unfolding nat_number_of_def ..
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subsection {* Special case: squares and cubes *}
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lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
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  by (simp add: nat_number_of_def)
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lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
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  by (simp add: nat_number_of_def)
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context power
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begin
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abbreviation (xsymbols)
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  power2 :: "'a \<Rightarrow> 'a"  ("(_\<twosuperior>)" [1000] 999) where
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  "x\<twosuperior> \<equiv> x ^ 2"
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notation (latex output)
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  power2  ("(_\<twosuperior>)" [1000] 999)
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notation (HTML output)
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  power2  ("(_\<twosuperior>)" [1000] 999)
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end
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context monoid_mult
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begin
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lemma power2_eq_square: "a\<twosuperior> = a * a"
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  by (simp add: numeral_2_eq_2)
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lemma power3_eq_cube: "a ^ 3 = a * a * a"
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  by (simp add: numeral_3_eq_3 mult_assoc)
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lemma power_even_eq:
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  "a ^ (2*n) = (a ^ n) ^ 2"
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  by (subst mult_commute) (simp add: power_mult)
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lemma power_odd_eq:
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  "a ^ Suc (2*n) = a * (a ^ n) ^ 2"
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  by (simp add: power_even_eq)
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end
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context semiring_1
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begin
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lemma zero_power2 [simp]: "0\<twosuperior> = 0"
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  by (simp add: power2_eq_square)
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lemma one_power2 [simp]: "1\<twosuperior> = 1"
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  by (simp add: power2_eq_square)
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end
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context ring_1
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begin
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lemma power2_minus [simp]:
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  "(- a)\<twosuperior> = a\<twosuperior>"
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  by (simp add: power2_eq_square)
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text{*
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  We cannot prove general results about the numeral @{term "-1"},
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  so we have to use @{term "- 1"} instead.
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*}
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lemma power_minus1_even [simp]:
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  "(- 1) ^ (2*n) = 1"
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proof (induct n)
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  case 0 show ?case by simp
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next
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  case (Suc n) then show ?case by (simp add: power_add)
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qed
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lemma power_minus1_odd:
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  "(- 1) ^ Suc (2*n) = - 1"
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  by simp
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lemma power_minus_even [simp]:
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  "(-a) ^ (2*n) = a ^ (2*n)"
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  by (simp add: power_minus [of a]) 
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end
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context ring_1_no_zero_divisors
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begin
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lemma zero_eq_power2 [simp]:
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  "a\<twosuperior> = 0 \<longleftrightarrow> a = 0"
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  unfolding power2_eq_square by simp
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lemma power2_eq_1_iff:
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  "a\<twosuperior> = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
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  unfolding power2_eq_square by (rule square_eq_1_iff)
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end
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context idom
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begin
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lemma power2_eq_iff: "x\<twosuperior> = y\<twosuperior> \<longleftrightarrow> x = y \<or> x = - y"
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  unfolding power2_eq_square by (rule square_eq_iff)
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end
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context linordered_ring
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begin
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lemma sum_squares_ge_zero:
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  "0 \<le> x * x + y * y"
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  by (intro add_nonneg_nonneg zero_le_square)
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lemma not_sum_squares_lt_zero:
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  "\<not> x * x + y * y < 0"
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  by (simp add: not_less sum_squares_ge_zero)
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end
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context linordered_ring_strict
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begin
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lemma sum_squares_eq_zero_iff:
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  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
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  by (simp add: add_nonneg_eq_0_iff)
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lemma sum_squares_le_zero_iff:
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  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
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  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
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lemma sum_squares_gt_zero_iff:
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  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
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0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
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  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
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   168
end
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context linordered_semidom
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begin
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lemma power2_le_imp_le:
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  "x\<twosuperior> \<le> y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
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   175
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
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parents: 31002
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   176
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lemma power2_less_imp_less:
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  "x\<twosuperior> < y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
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   179
  by (rule power_less_imp_less_base)
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lemma power2_eq_imp_eq:
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  "x\<twosuperior> = y\<twosuperior> \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
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   183
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
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   185
end
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context linordered_idom
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begin
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   189
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lemma zero_le_power2 [simp]:
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   191
  "0 \<le> a\<twosuperior>"
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  by (simp add: power2_eq_square)
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   193
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lemma zero_less_power2 [simp]:
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  "0 < a\<twosuperior> \<longleftrightarrow> a \<noteq> 0"
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   196
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
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   197
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lemma power2_less_0 [simp]:
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  "\<not> a\<twosuperior> < 0"
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  by (force simp add: power2_eq_square mult_less_0_iff) 
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parents: 31002
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   201
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lemma abs_power2 [simp]:
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  "abs (a\<twosuperior>) = a\<twosuperior>"
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  by (simp add: power2_eq_square abs_mult abs_mult_self)
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parents: 31002
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   205
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   206
lemma power2_abs [simp]:
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  "(abs a)\<twosuperior> = a\<twosuperior>"
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  by (simp add: power2_eq_square abs_mult_self)
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parents: 31002
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   209
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   210
lemma odd_power_less_zero:
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  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
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   212
proof (induct n)
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  case 0
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  then show ?case by simp
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   215
next
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   216
  case (Suc n)
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   217
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
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    by (simp add: mult_ac power_add power2_eq_square)
79f0858d9d49 collected square lemmas in Nat_Numeral
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  thus ?case
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    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
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   221
qed
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   222
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   223
lemma odd_0_le_power_imp_0_le:
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  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
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   225
  using odd_power_less_zero [of a n]
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    by (force simp add: linorder_not_less [symmetric]) 
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   227
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   228
lemma zero_le_even_power'[simp]:
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   229
  "0 \<le> a ^ (2*n)"
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   230
proof (induct n)
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   231
  case 0
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   232
    show ?case by simp
31014
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   233
next
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   234
  case (Suc n)
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   235
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
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   236
      by (simp add: mult_ac power_add power2_eq_square)
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   237
    thus ?case
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   238
      by (simp add: Suc zero_le_mult_iff)
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qed
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   240
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   241
lemma sum_power2_ge_zero:
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  "0 \<le> x\<twosuperior> + y\<twosuperior>"
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parents: 31002
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   243
  unfolding power2_eq_square by (rule sum_squares_ge_zero)
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   244
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   245
lemma not_sum_power2_lt_zero:
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   246
  "\<not> x\<twosuperior> + y\<twosuperior> < 0"
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parents: 31002
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   247
  unfolding power2_eq_square by (rule not_sum_squares_lt_zero)
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   248
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   249
lemma sum_power2_eq_zero_iff:
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   250
  "x\<twosuperior> + y\<twosuperior> = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
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parents: 31002
diff changeset
   251
  unfolding power2_eq_square by (rule sum_squares_eq_zero_iff)
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   252
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   253
lemma sum_power2_le_zero_iff:
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   254
  "x\<twosuperior> + y\<twosuperior> \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
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parents: 31002
diff changeset
   255
  unfolding power2_eq_square by (rule sum_squares_le_zero_iff)
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haftmann
parents: 31002
diff changeset
   256
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   257
lemma sum_power2_gt_zero_iff:
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   258
  "0 < x\<twosuperior> + y\<twosuperior> \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
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parents: 31002
diff changeset
   259
  unfolding power2_eq_square by (rule sum_squares_gt_zero_iff)
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   260
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haftmann
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   261
end
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diff changeset
   262
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   263
lemma power2_sum:
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cc46a678faaf added number_semiring class, plus a few new lemmas;
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   264
  fixes x y :: "'a::number_semiring"
31014
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   265
  shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y"
43531
cc46a678faaf added number_semiring class, plus a few new lemmas;
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diff changeset
   266
  by (simp add: algebra_simps power2_eq_square semiring_mult_2_right)
31014
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   267
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   268
lemma power2_diff:
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   269
  fixes x y :: "'a::number_ring"
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parents: 31002
diff changeset
   270
  shows "(x - y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> - 2 * x * y"
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   271
  by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute)
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diff changeset
   272
23164
69e55066dbca moved Integ files to canonical place;
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parents:
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   273
29040
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   274
subsection {* Predicate for negative binary numbers *}
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   275
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
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   276
definition neg  :: "int \<Rightarrow> bool" where
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  "neg Z \<longleftrightarrow> Z < 0"
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   278
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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lemma not_neg_int [simp]: "~ neg (of_nat n)"
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   280
by (simp add: neg_def)
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   281
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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   282
lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
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   283
by (simp add: neg_def del: of_nat_Suc)
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   284
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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   285
lemmas neg_eq_less_0 = neg_def
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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diff changeset
   286
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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   287
lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
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   288
by (simp add: neg_def linorder_not_less)
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   289
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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   290
text{*To simplify inequalities when Numeral1 can get simplified to 1*}
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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   291
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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   292
lemma not_neg_0: "~ neg 0"
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   293
by (simp add: One_int_def neg_def)
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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   294
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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   295
lemma not_neg_1: "~ neg 1"
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   296
by (simp add: neg_def linorder_not_less)
29040
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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   297
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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   298
lemma neg_nat: "neg z ==> nat z = 0"
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diff changeset
   299
by (simp add: neg_def order_less_imp_le) 
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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diff changeset
   300
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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   301
lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
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diff changeset
   302
by (simp add: linorder_not_less neg_def)
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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diff changeset
   303
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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   304
text {*
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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diff changeset
   305
  If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
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   306
  @{term Numeral0} IS @{term "number_of Pls"}
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diff changeset
   307
*}
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diff changeset
   308
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
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   309
lemma not_neg_number_of_Pls: "~ neg (number_of Int.Pls)"
286c669d3a7a move all neg-related lemmas to NatBin; make type of neg specific to int
huffman
parents: 29039
diff changeset
   310
  by (simp add: neg_def)
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lemma neg_number_of_Min: "neg (number_of Int.Min)"
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  by (simp add: neg_def)
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lemma neg_number_of_Bit0:
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  "neg (number_of (Int.Bit0 w)) = neg (number_of w)"
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  by (simp add: neg_def)
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lemma neg_number_of_Bit1:
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  "neg (number_of (Int.Bit1 w)) = neg (number_of w)"
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  by (simp add: neg_def)
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lemmas neg_simps [simp] =
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  not_neg_0 not_neg_1
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  not_neg_number_of_Pls neg_number_of_Min
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  neg_number_of_Bit0 neg_number_of_Bit1
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subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
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declare nat_1 [simp]
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lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
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by (simp add: nat_number_of_def)
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31998
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lemma nat_numeral_0_eq_0 [simp, code_post]: "Numeral0 = (0::nat)"
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by (simp add: nat_number_of_def)
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lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
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by (simp add: nat_number_of_def)
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lemma Numeral1_eq1_nat:
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  "(1::nat) = Numeral1"
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  by simp
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lemma numeral_1_eq_Suc_0 [code_post]: "Numeral1 = Suc 0"
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by (simp only: nat_numeral_1_eq_1 One_nat_def)
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subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
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lemma int_nat_number_of [simp]:
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     "int (number_of v) =  
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         (if neg (number_of v :: int) then 0  
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          else (number_of v :: int))"
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   356
  unfolding nat_number_of_def number_of_is_id neg_def
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  by simp
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lemma nonneg_int_cases:
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   360
  fixes k :: int assumes "0 \<le> k" obtains n where "k = of_nat n"
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  using assms by (cases k, simp, simp)
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subsubsection{*Successor *}
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   364
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lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
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apply (rule sym)
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apply (simp add: nat_eq_iff int_Suc)
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   368
done
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   369
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lemma Suc_nat_number_of_add:
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   371
     "Suc (number_of v + n) =  
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   372
        (if neg (number_of v :: int) then 1+n else number_of (Int.succ v) + n)"
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   373
  unfolding nat_number_of_def number_of_is_id neg_def numeral_simps
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   374
  by (simp add: Suc_nat_eq_nat_zadd1 add_ac)
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   375
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lemma Suc_nat_number_of [simp]:
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     "Suc (number_of v) =  
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parents: 25571
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   378
        (if neg (number_of v :: int) then 1 else number_of (Int.succ v))"
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   379
apply (cut_tac n = 0 in Suc_nat_number_of_add)
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   380
apply (simp cong del: if_weak_cong)
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   381
done
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subsubsection{*Addition *}
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   385
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lemma add_nat_number_of [simp]:
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parents:
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   387
     "(number_of v :: nat) + number_of v' =  
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   388
         (if v < Int.Pls then number_of v'  
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   389
          else if v' < Int.Pls then number_of v  
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   390
          else number_of (v + v'))"
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parents: 29011
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   391
  unfolding nat_number_of_def number_of_is_id numeral_simps
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   392
  by (simp add: nat_add_distrib)
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   393
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   394
lemma nat_number_of_add_1 [simp]:
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   395
  "number_of v + (1::nat) =
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parents: 30079
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   396
    (if v < Int.Pls then 1 else number_of (Int.succ v))"
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   397
  unfolding nat_number_of_def number_of_is_id numeral_simps
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   398
  by (simp add: nat_add_distrib)
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parents: 30079
diff changeset
   399
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   400
lemma nat_1_add_number_of [simp]:
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parents: 30079
diff changeset
   401
  "(1::nat) + number_of v =
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parents: 30079
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   402
    (if v < Int.Pls then 1 else number_of (Int.succ v))"
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parents: 30079
diff changeset
   403
  unfolding nat_number_of_def number_of_is_id numeral_simps
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parents: 30079
diff changeset
   404
  by (simp add: nat_add_distrib)
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parents: 30079
diff changeset
   405
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parents: 30079
diff changeset
   406
lemma nat_1_add_1 [simp]: "1 + 1 = (2::nat)"
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parents: 43526
diff changeset
   407
  by (rule semiring_one_add_one_is_two)
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parents: 43526
diff changeset
   408
cc46a678faaf added number_semiring class, plus a few new lemmas;
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parents: 43526
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   409
text {* TODO: replace simp rules above with these generic ones: *}
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parents: 43526
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   410
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parents: 43526
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   411
lemma semiring_add_number_of:
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parents: 43526
diff changeset
   412
  "\<lbrakk>Int.Pls \<le> v; Int.Pls \<le> v'\<rbrakk> \<Longrightarrow>
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parents: 43526
diff changeset
   413
    (number_of v :: 'a::number_semiring) + number_of v' = number_of (v + v')"
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huffman
parents: 43526
diff changeset
   414
  unfolding Int.Pls_def
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huffman
parents: 43526
diff changeset
   415
  by (elim nonneg_int_cases,
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parents: 43526
diff changeset
   416
    simp only: number_of_int of_nat_add [symmetric])
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huffman
parents: 43526
diff changeset
   417
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parents: 43526
diff changeset
   418
lemma semiring_number_of_add_1:
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parents: 43526
diff changeset
   419
  "Int.Pls \<le> v \<Longrightarrow>
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parents: 43526
diff changeset
   420
    number_of v + (1::'a::number_semiring) = number_of (Int.succ v)"
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parents: 43526
diff changeset
   421
  unfolding Int.Pls_def Int.succ_def
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huffman
parents: 43526
diff changeset
   422
  by (elim nonneg_int_cases,
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huffman
parents: 43526
diff changeset
   423
    simp only: number_of_int add_commute [where b=1] of_nat_Suc [symmetric])
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parents: 43526
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   424
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parents: 43526
diff changeset
   425
lemma semiring_1_add_number_of:
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huffman
parents: 43526
diff changeset
   426
  "Int.Pls \<le> v \<Longrightarrow>
cc46a678faaf added number_semiring class, plus a few new lemmas;
huffman
parents: 43526
diff changeset
   427
    (1::'a::number_semiring) + number_of v = number_of (Int.succ v)"
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huffman
parents: 43526
diff changeset
   428
  unfolding Int.Pls_def Int.succ_def
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huffman
parents: 43526
diff changeset
   429
  by (elim nonneg_int_cases,
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huffman
parents: 43526
diff changeset
   430
    simp only: number_of_int add_commute [where b=1] of_nat_Suc [symmetric])
30081
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parents: 30079
diff changeset
   431
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   432
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   433
subsubsection{*Subtraction *}
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parents:
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   434
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parents:
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   435
lemma diff_nat_eq_if:
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wenzelm
parents:
diff changeset
   436
     "nat z - nat z' =  
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parents:
diff changeset
   437
        (if neg z' then nat z   
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wenzelm
parents:
diff changeset
   438
         else let d = z-z' in     
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wenzelm
parents:
diff changeset
   439
              if neg d then 0 else nat d)"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26342
diff changeset
   440
by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
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haftmann
parents: 26342
diff changeset
   441
23164
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parents:
diff changeset
   442
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
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   443
lemma diff_nat_number_of [simp]: 
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wenzelm
parents:
diff changeset
   444
     "(number_of v :: nat) - number_of v' =  
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huffman
parents: 29011
diff changeset
   445
        (if v' < Int.Pls then number_of v  
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   446
         else let d = number_of (v + uminus v') in     
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wenzelm
parents:
diff changeset
   447
              if neg d then 0 else nat d)"
29012
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huffman
parents: 29011
diff changeset
   448
  unfolding nat_number_of_def number_of_is_id numeral_simps neg_def
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huffman
parents: 29011
diff changeset
   449
  by auto
23164
69e55066dbca moved Integ files to canonical place;
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parents:
diff changeset
   450
30081
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parents: 30079
diff changeset
   451
lemma nat_number_of_diff_1 [simp]:
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parents: 30079
diff changeset
   452
  "number_of v - (1::nat) =
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huffman
parents: 30079
diff changeset
   453
    (if v \<le> Int.Pls then 0 else number_of (Int.pred v))"
46b9c8ae3897 add simp rules for numerals with 1::nat
huffman
parents: 30079
diff changeset
   454
  unfolding nat_number_of_def number_of_is_id numeral_simps
46b9c8ae3897 add simp rules for numerals with 1::nat
huffman
parents: 30079
diff changeset
   455
  by auto
46b9c8ae3897 add simp rules for numerals with 1::nat
huffman
parents: 30079
diff changeset
   456
23164
69e55066dbca moved Integ files to canonical place;
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parents:
diff changeset
   457
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parents:
diff changeset
   458
subsubsection{*Multiplication *}
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wenzelm
parents:
diff changeset
   459
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wenzelm
parents:
diff changeset
   460
lemma mult_nat_number_of [simp]:
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wenzelm
parents:
diff changeset
   461
     "(number_of v :: nat) * number_of v' =  
29012
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parents: 29011
diff changeset
   462
       (if v < Int.Pls then 0 else number_of (v * v'))"
9140227dc8c5 change lemmas to avoid using neg
huffman
parents: 29011
diff changeset
   463
  unfolding nat_number_of_def number_of_is_id numeral_simps
28984
060832a1f087 change arith_special simps to avoid using neg
huffman
parents: 28969
diff changeset
   464
  by (simp add: nat_mult_distrib)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   465
43531
cc46a678faaf added number_semiring class, plus a few new lemmas;
huffman
parents: 43526
diff changeset
   466
(* TODO: replace mult_nat_number_of with this next rule *)
cc46a678faaf added number_semiring class, plus a few new lemmas;
huffman
parents: 43526
diff changeset
   467
lemma semiring_mult_number_of:
cc46a678faaf added number_semiring class, plus a few new lemmas;
huffman
parents: 43526
diff changeset
   468
  "\<lbrakk>Int.Pls \<le> v; Int.Pls \<le> v'\<rbrakk> \<Longrightarrow>
cc46a678faaf added number_semiring class, plus a few new lemmas;
huffman
parents: 43526
diff changeset
   469
    (number_of v :: 'a::number_semiring) * number_of v' = number_of (v * v')"
cc46a678faaf added number_semiring class, plus a few new lemmas;
huffman
parents: 43526
diff changeset
   470
  unfolding Int.Pls_def
cc46a678faaf added number_semiring class, plus a few new lemmas;
huffman
parents: 43526
diff changeset
   471
  by (elim nonneg_int_cases,
cc46a678faaf added number_semiring class, plus a few new lemmas;
huffman
parents: 43526
diff changeset
   472
    simp only: number_of_int of_nat_mult [symmetric])
cc46a678faaf added number_semiring class, plus a few new lemmas;
huffman
parents: 43526
diff changeset
   473
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   474
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   475
subsection{*Comparisons*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   476
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   477
subsubsection{*Equals (=) *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   478
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   479
lemma eq_nat_number_of [simp]:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   480
     "((number_of v :: nat) = number_of v') =  
28969
4ed63cdda799 change more lemmas to avoid using iszero
huffman
parents: 28968
diff changeset
   481
      (if neg (number_of v :: int) then (number_of v' :: int) \<le> 0
4ed63cdda799 change more lemmas to avoid using iszero
huffman
parents: 28968
diff changeset
   482
       else if neg (number_of v' :: int) then (number_of v :: int) = 0
4ed63cdda799 change more lemmas to avoid using iszero
huffman
parents: 28968
diff changeset
   483
       else v = v')"
4ed63cdda799 change more lemmas to avoid using iszero
huffman
parents: 28968
diff changeset
   484
  unfolding nat_number_of_def number_of_is_id neg_def
4ed63cdda799 change more lemmas to avoid using iszero
huffman
parents: 28968
diff changeset
   485
  by auto
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   486
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   487
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   488
subsubsection{*Less-than (<) *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   489
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   490
lemma less_nat_number_of [simp]:
29011
a47003001699 simplify less_nat_number_of
huffman
parents: 29010
diff changeset
   491
  "(number_of v :: nat) < number_of v' \<longleftrightarrow>
a47003001699 simplify less_nat_number_of
huffman
parents: 29010
diff changeset
   492
    (if v < v' then Int.Pls < v' else False)"
a47003001699 simplify less_nat_number_of
huffman
parents: 29010
diff changeset
   493
  unfolding nat_number_of_def number_of_is_id numeral_simps
28961
9f33ab8e15db simplify proof of less_nat_number_of
huffman
parents: 28562
diff changeset
   494
  by auto
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   495
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   496
29010
5cd646abf6bc add lemma le_nat_number_of
huffman
parents: 28984
diff changeset
   497
subsubsection{*Less-than-or-equal *}
5cd646abf6bc add lemma le_nat_number_of
huffman
parents: 28984
diff changeset
   498
5cd646abf6bc add lemma le_nat_number_of
huffman
parents: 28984
diff changeset
   499
lemma le_nat_number_of [simp]:
5cd646abf6bc add lemma le_nat_number_of
huffman
parents: 28984
diff changeset
   500
  "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow>
5cd646abf6bc add lemma le_nat_number_of
huffman
parents: 28984
diff changeset
   501
    (if v \<le> v' then True else v \<le> Int.Pls)"
5cd646abf6bc add lemma le_nat_number_of
huffman
parents: 28984
diff changeset
   502
  unfolding nat_number_of_def number_of_is_id numeral_simps
5cd646abf6bc add lemma le_nat_number_of
huffman
parents: 28984
diff changeset
   503
  by auto
5cd646abf6bc add lemma le_nat_number_of
huffman
parents: 28984
diff changeset
   504
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   505
(*Maps #n to n for n = 0, 1, 2*)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   506
lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   507
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   508
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   509
subsection{*Powers with Numeric Exponents*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   510
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   511
text{*Squares of literal numerals will be evaluated.*}
31014
79f0858d9d49 collected square lemmas in Nat_Numeral
haftmann
parents: 31002
diff changeset
   512
lemmas power2_eq_square_number_of [simp] =
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   513
    power2_eq_square [of "number_of w", standard]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   514
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   515
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   516
text{*Simprules for comparisons where common factors can be cancelled.*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   517
lemmas zero_compare_simps =
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   518
    add_strict_increasing add_strict_increasing2 add_increasing
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   519
    zero_le_mult_iff zero_le_divide_iff 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   520
    zero_less_mult_iff zero_less_divide_iff 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   521
    mult_le_0_iff divide_le_0_iff 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   522
    mult_less_0_iff divide_less_0_iff 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   523
    zero_le_power2 power2_less_0
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   524
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   525
subsubsection{*Nat *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   526
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   527
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35047
diff changeset
   528
by simp
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   529
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   530
(*Expresses a natural number constant as the Suc of another one.
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   531
  NOT suitable for rewriting because n recurs in the condition.*)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   532
lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   533
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   534
subsubsection{*Arith *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   535
31790
05c92381363c corrected and unified thm names
nipkow
parents: 31182
diff changeset
   536
lemma Suc_eq_plus1: "Suc n = n + 1"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35047
diff changeset
   537
  unfolding One_nat_def by simp
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   538
31790
05c92381363c corrected and unified thm names
nipkow
parents: 31182
diff changeset
   539
lemma Suc_eq_plus1_left: "Suc n = 1 + n"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35047
diff changeset
   540
  unfolding One_nat_def by simp
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   541
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   542
(* These two can be useful when m = number_of... *)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   543
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   544
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 29958
diff changeset
   545
  unfolding One_nat_def by (cases m) simp_all
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   546
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   547
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 29958
diff changeset
   548
  unfolding One_nat_def by (cases m) simp_all
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   549
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   550
lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 29958
diff changeset
   551
  unfolding One_nat_def by (cases m) simp_all
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   552
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   553
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   554
subsection{*Comparisons involving (0::nat) *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   555
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   556
text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   557
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   558
lemma eq_number_of_0 [simp]:
29012
9140227dc8c5 change lemmas to avoid using neg
huffman
parents: 29011
diff changeset
   559
  "number_of v = (0::nat) \<longleftrightarrow> v \<le> Int.Pls"
9140227dc8c5 change lemmas to avoid using neg
huffman
parents: 29011
diff changeset
   560
  unfolding nat_number_of_def number_of_is_id numeral_simps
9140227dc8c5 change lemmas to avoid using neg
huffman
parents: 29011
diff changeset
   561
  by auto
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   562
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   563
lemma eq_0_number_of [simp]:
29012
9140227dc8c5 change lemmas to avoid using neg
huffman
parents: 29011
diff changeset
   564
  "(0::nat) = number_of v \<longleftrightarrow> v \<le> Int.Pls"
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   565
by (rule trans [OF eq_sym_conv eq_number_of_0])
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   566
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   567
lemma less_0_number_of [simp]:
29012
9140227dc8c5 change lemmas to avoid using neg
huffman
parents: 29011
diff changeset
   568
   "(0::nat) < number_of v \<longleftrightarrow> Int.Pls < v"
9140227dc8c5 change lemmas to avoid using neg
huffman
parents: 29011
diff changeset
   569
  unfolding nat_number_of_def number_of_is_id numeral_simps
9140227dc8c5 change lemmas to avoid using neg
huffman
parents: 29011
diff changeset
   570
  by simp
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   571
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   572
lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
28969
4ed63cdda799 change more lemmas to avoid using iszero
huffman
parents: 28968
diff changeset
   573
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   574
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   575
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   576
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   577
subsection{*Comparisons involving  @{term Suc} *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   578
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   579
lemma eq_number_of_Suc [simp]:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   580
     "(number_of v = Suc n) =  
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25571
diff changeset
   581
        (let pv = number_of (Int.pred v) in  
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   582
         if neg pv then False else nat pv = n)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   583
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   584
                  number_of_pred nat_number_of_def 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   585
            split add: split_if)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   586
apply (rule_tac x = "number_of v" in spec)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   587
apply (auto simp add: nat_eq_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   588
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   589
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   590
lemma Suc_eq_number_of [simp]:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   591
     "(Suc n = number_of v) =  
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25571
diff changeset
   592
        (let pv = number_of (Int.pred v) in  
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   593
         if neg pv then False else nat pv = n)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   594
by (rule trans [OF eq_sym_conv eq_number_of_Suc])
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   595
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   596
lemma less_number_of_Suc [simp]:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   597
     "(number_of v < Suc n) =  
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25571
diff changeset
   598
        (let pv = number_of (Int.pred v) in  
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   599
         if neg pv then True else nat pv < n)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   600
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   601
                  number_of_pred nat_number_of_def  
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   602
            split add: split_if)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   603
apply (rule_tac x = "number_of v" in spec)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   604
apply (auto simp add: nat_less_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   605
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   606
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   607
lemma less_Suc_number_of [simp]:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   608
     "(Suc n < number_of v) =  
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25571
diff changeset
   609
        (let pv = number_of (Int.pred v) in  
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   610
         if neg pv then False else n < nat pv)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   611
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   612
                  number_of_pred nat_number_of_def
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   613
            split add: split_if)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   614
apply (rule_tac x = "number_of v" in spec)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   615
apply (auto simp add: zless_nat_eq_int_zless)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   616
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   617
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   618
lemma le_number_of_Suc [simp]:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   619
     "(number_of v <= Suc n) =  
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25571
diff changeset
   620
        (let pv = number_of (Int.pred v) in  
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   621
         if neg pv then True else nat pv <= n)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35047
diff changeset
   622
by (simp add: Let_def linorder_not_less [symmetric])
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   623
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   624
lemma le_Suc_number_of [simp]:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   625
     "(Suc n <= number_of v) =  
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25571
diff changeset
   626
        (let pv = number_of (Int.pred v) in  
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   627
         if neg pv then False else n <= nat pv)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35047
diff changeset
   628
by (simp add: Let_def linorder_not_less [symmetric])
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   629
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   630
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25571
diff changeset
   631
lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Int.Min"
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   632
by auto
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   633
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   634
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   635
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   636
subsection{*Max and Min Combined with @{term Suc} *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   637
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   638
lemma max_number_of_Suc [simp]:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   639
     "max (Suc n) (number_of v) =  
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25571
diff changeset
   640
        (let pv = number_of (Int.pred v) in  
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   641
         if neg pv then Suc n else Suc(max n (nat pv)))"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   642
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   643
            split add: split_if nat.split)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   644
apply (rule_tac x = "number_of v" in spec) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   645
apply auto
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   646
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   647
 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   648
lemma max_Suc_number_of [simp]:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   649
     "max (number_of v) (Suc n) =  
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25571
diff changeset
   650
        (let pv = number_of (Int.pred v) in  
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   651
         if neg pv then Suc n else Suc(max (nat pv) n))"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   652
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   653
            split add: split_if nat.split)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   654
apply (rule_tac x = "number_of v" in spec) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   655
apply auto
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   656
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   657
 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   658
lemma min_number_of_Suc [simp]:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   659
     "min (Suc n) (number_of v) =  
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25571
diff changeset
   660
        (let pv = number_of (Int.pred v) in  
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   661
         if neg pv then 0 else Suc(min n (nat pv)))"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   662
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   663
            split add: split_if nat.split)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   664
apply (rule_tac x = "number_of v" in spec) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   665
apply auto
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   666
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   667
 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   668
lemma min_Suc_number_of [simp]:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   669
     "min (number_of v) (Suc n) =  
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25571
diff changeset
   670
        (let pv = number_of (Int.pred v) in  
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   671
         if neg pv then 0 else Suc(min (nat pv) n))"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   672
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   673
            split add: split_if nat.split)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   674
apply (rule_tac x = "number_of v" in spec) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   675
apply auto
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   676
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   677
 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   678
subsection{*Literal arithmetic involving powers*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   679
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   680
lemma power_nat_number_of:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   681
     "(number_of v :: nat) ^ n =  
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   682
       (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   683
by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   684
         split add: split_if cong: imp_cong)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   685
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   686
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   687
lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   688
declare power_nat_number_of_number_of [simp]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   689
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   690
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   691
23294
9302a50a5bc9 generalize zpower_number_of_{even,odd} lemmas
huffman
parents: 23277
diff changeset
   692
text{*For arbitrary rings*}
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   693
23294
9302a50a5bc9 generalize zpower_number_of_{even,odd} lemmas
huffman
parents: 23277
diff changeset
   694
lemma power_number_of_even:
43526
2b92a6943915 generalize lemmas power_number_of_even and power_number_of_odd
huffman
parents: 40690
diff changeset
   695
  fixes z :: "'a::monoid_mult"
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25965
diff changeset
   696
  shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)"
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   697
by (cases "w \<ge> 0") (auto simp add: Let_def Bit0_def nat_number_of_def number_of_is_id
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   698
  nat_add_distrib power_add simp del: nat_number_of)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   699
23294
9302a50a5bc9 generalize zpower_number_of_{even,odd} lemmas
huffman
parents: 23277
diff changeset
   700
lemma power_number_of_odd:
43526
2b92a6943915 generalize lemmas power_number_of_even and power_number_of_odd
huffman
parents: 40690
diff changeset
   701
  fixes z :: "'a::monoid_mult"
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25965
diff changeset
   702
  shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   703
     then (let w = z ^ (number_of w) in z * w * w) else 1)"
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35631
diff changeset
   704
unfolding Let_def Bit1_def nat_number_of_def number_of_is_id
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35631
diff changeset
   705
apply (cases "0 <= w")
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35631
diff changeset
   706
apply (simp only: mult_assoc nat_add_distrib power_add, simp)
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   707
apply (simp add: not_le mult_2 [symmetric] add_assoc)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   708
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   709
23294
9302a50a5bc9 generalize zpower_number_of_{even,odd} lemmas
huffman
parents: 23277
diff changeset
   710
lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
9302a50a5bc9 generalize zpower_number_of_{even,odd} lemmas
huffman
parents: 23277
diff changeset
   711
lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   712
23294
9302a50a5bc9 generalize zpower_number_of_{even,odd} lemmas
huffman
parents: 23277
diff changeset
   713
lemmas power_number_of_even_number_of [simp] =
9302a50a5bc9 generalize zpower_number_of_{even,odd} lemmas
huffman
parents: 23277
diff changeset
   714
    power_number_of_even [of "number_of v", standard]
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   715
23294
9302a50a5bc9 generalize zpower_number_of_{even,odd} lemmas
huffman
parents: 23277
diff changeset
   716
lemmas power_number_of_odd_number_of [simp] =
9302a50a5bc9 generalize zpower_number_of_{even,odd} lemmas
huffman
parents: 23277
diff changeset
   717
    power_number_of_odd [of "number_of v", standard]
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   718
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   719
lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35047
diff changeset
   720
  by (simp add: nat_number_of_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   721
40690
3f472e57446a added "no_atp" for fact that confuses the SMT normalizer and that doesn't help ATPs anyway
blanchet
parents: 40077
diff changeset
   722
lemma nat_number_of_Min [no_atp]: "number_of Int.Min = (0::nat)"
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   723
  apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   724
  done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   725
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25965
diff changeset
   726
lemma nat_number_of_Bit0:
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25965
diff changeset
   727
    "number_of (Int.Bit0 w) = (let n::nat = number_of w in n + n)"
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   728
by (cases "w \<ge> 0") (auto simp add: Let_def Bit0_def nat_number_of_def number_of_is_id
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   729
  nat_add_distrib simp del: nat_number_of)
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25965
diff changeset
   730
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25965
diff changeset
   731
lemma nat_number_of_Bit1:
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25965
diff changeset
   732
  "number_of (Int.Bit1 w) =
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   733
    (if neg (number_of w :: int) then 0
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   734
     else let n = number_of w in Suc (n + n))"
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35631
diff changeset
   735
unfolding Let_def Bit1_def nat_number_of_def number_of_is_id neg_def
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35631
diff changeset
   736
apply (cases "w < 0")
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   737
apply (simp add: mult_2 [symmetric] add_assoc)
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35631
diff changeset
   738
apply (simp only: nat_add_distrib, simp)
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   739
done
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   740
40077
c8a9eaaa2f59 nat_number -> eval_nat_numeral
nipkow
parents: 36964
diff changeset
   741
lemmas eval_nat_numeral =
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35047
diff changeset
   742
  nat_number_of_Bit0 nat_number_of_Bit1
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35047
diff changeset
   743
36699
816da1023508 moved nat_arith ot Nat_Numeral: clarified normalizer setup
haftmann
parents: 35815
diff changeset
   744
lemmas nat_arith =
816da1023508 moved nat_arith ot Nat_Numeral: clarified normalizer setup
haftmann
parents: 35815
diff changeset
   745
  add_nat_number_of
816da1023508 moved nat_arith ot Nat_Numeral: clarified normalizer setup
haftmann
parents: 35815
diff changeset
   746
  diff_nat_number_of
816da1023508 moved nat_arith ot Nat_Numeral: clarified normalizer setup
haftmann
parents: 35815
diff changeset
   747
  mult_nat_number_of
816da1023508 moved nat_arith ot Nat_Numeral: clarified normalizer setup
haftmann
parents: 35815
diff changeset
   748
  eq_nat_number_of
816da1023508 moved nat_arith ot Nat_Numeral: clarified normalizer setup
haftmann
parents: 35815
diff changeset
   749
  less_nat_number_of
816da1023508 moved nat_arith ot Nat_Numeral: clarified normalizer setup
haftmann
parents: 35815
diff changeset
   750
36716
b09f3ad3208f moved generic lemmas to appropriate places
haftmann
parents: 36699
diff changeset
   751
lemmas semiring_norm =
b09f3ad3208f moved generic lemmas to appropriate places
haftmann
parents: 36699
diff changeset
   752
  Let_def arith_simps nat_arith rel_simps neg_simps if_False
b09f3ad3208f moved generic lemmas to appropriate places
haftmann
parents: 36699
diff changeset
   753
  if_True add_0 add_Suc add_number_of_left mult_number_of_left
b09f3ad3208f moved generic lemmas to appropriate places
haftmann
parents: 36699
diff changeset
   754
  numeral_1_eq_1 [symmetric] Suc_eq_plus1
b09f3ad3208f moved generic lemmas to appropriate places
haftmann
parents: 36699
diff changeset
   755
  numeral_0_eq_0 [symmetric] numerals [symmetric]
36841
02df88789641 include iszero_simps in semiring_norm just once (they are already included in rel_simps)
huffman
parents: 36823
diff changeset
   756
  not_iszero_Numeral1
36716
b09f3ad3208f moved generic lemmas to appropriate places
haftmann
parents: 36699
diff changeset
   757
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   758
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   759
  by (fact Let_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   760
31014
79f0858d9d49 collected square lemmas in Nat_Numeral
haftmann
parents: 31002
diff changeset
   761
lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring})"
79f0858d9d49 collected square lemmas in Nat_Numeral
haftmann
parents: 31002
diff changeset
   762
  by (simp only: number_of_Min power_minus1_even)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   763
31014
79f0858d9d49 collected square lemmas in Nat_Numeral
haftmann
parents: 31002
diff changeset
   764
lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring})"
79f0858d9d49 collected square lemmas in Nat_Numeral
haftmann
parents: 31002
diff changeset
   765
  by (simp only: number_of_Min power_minus1_odd)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   766
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   767
lemma nat_number_of_add_left:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   768
     "number_of v + (number_of v' + (k::nat)) =  
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   769
         (if neg (number_of v :: int) then number_of v' + k  
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   770
          else if neg (number_of v' :: int) then number_of v + k  
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   771
          else number_of (v + v') + k)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   772
by (auto simp add: neg_def)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   773
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   774
lemma nat_number_of_mult_left:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   775
     "number_of v * (number_of v' * (k::nat)) =  
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   776
         (if v < Int.Pls then 0
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   777
          else number_of (v * v') * k)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   778
by (auto simp add: not_less Pls_def nat_number_of_def number_of_is_id
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   779
  nat_mult_distrib simp del: nat_number_of)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   780
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   781
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   782
subsection{*Literal arithmetic and @{term of_nat}*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   783
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   784
lemma of_nat_double:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   785
     "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   786
by (simp only: mult_2 nat_add_distrib of_nat_add) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   787
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   788
lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   789
by (simp only: nat_number_of_def)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   790
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   791
lemma of_nat_number_of_lemma:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   792
     "of_nat (number_of v :: nat) =  
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   793
         (if 0 \<le> (number_of v :: int) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   794
          then (number_of v :: 'a :: number_ring)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   795
          else 0)"
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   796
by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   797
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   798
lemma of_nat_number_of_eq [simp]:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   799
     "of_nat (number_of v :: nat) =  
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   800
         (if neg (number_of v :: int) then 0  
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   801
          else (number_of v :: 'a :: number_ring))"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   802
by (simp only: of_nat_number_of_lemma neg_def, simp) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   803
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   804
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   805
subsubsection{*For simplifying @{term "Suc m - K"} and  @{term "K - Suc m"}*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   806
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   807
text{*Where K above is a literal*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   808
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   809
lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35047
diff changeset
   810
by (simp split: nat_diff_split)
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   811
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   812
text {*Now just instantiating @{text n} to @{text "number_of v"} does
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   813
  the right simplification, but with some redundant inequality
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   814
  tests.*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   815
lemma neg_number_of_pred_iff_0:
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   816
  "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   817
apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   818
apply (simp only: less_Suc_eq_le le_0_eq)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   819
apply (subst less_number_of_Suc, simp)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   820
done
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   821
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   822
text{*No longer required as a simprule because of the @{text inverse_fold}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   823
   simproc*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   824
lemma Suc_diff_number_of:
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   825
     "Int.Pls < v ==>
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   826
      Suc m - (number_of v) = m - (number_of (Int.pred v))"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   827
apply (subst Suc_diff_eq_diff_pred)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   828
apply simp
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   829
apply (simp del: nat_numeral_1_eq_1)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   830
apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   831
                        neg_number_of_pred_iff_0)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   832
done
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   833
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   834
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35047
diff changeset
   835
by (simp split: nat_diff_split)
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   836
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   837
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   838
subsubsection{*For @{term nat_case} and @{term nat_rec}*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   839
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   840
lemma nat_case_number_of [simp]:
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   841
     "nat_case a f (number_of v) =
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   842
        (let pv = number_of (Int.pred v) in
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   843
         if neg pv then a else f (nat pv))"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   844
by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   845
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   846
lemma nat_case_add_eq_if [simp]:
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   847
     "nat_case a f ((number_of v) + n) =
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   848
       (let pv = number_of (Int.pred v) in
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   849
         if neg pv then nat_case a f n else f (nat pv + n))"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   850
apply (subst add_eq_if)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   851
apply (simp split add: nat.split
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   852
            del: nat_numeral_1_eq_1
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   853
            add: nat_numeral_1_eq_1 [symmetric]
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   854
                 numeral_1_eq_Suc_0 [symmetric]
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   855
                 neg_number_of_pred_iff_0)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   856
done
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   857
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   858
lemma nat_rec_number_of [simp]:
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   859
     "nat_rec a f (number_of v) =
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   860
        (let pv = number_of (Int.pred v) in
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   861
         if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   862
apply (case_tac " (number_of v) ::nat")
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   863
apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   864
apply (simp split add: split_if_asm)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   865
done
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   866
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   867
lemma nat_rec_add_eq_if [simp]:
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   868
     "nat_rec a f (number_of v + n) =
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   869
        (let pv = number_of (Int.pred v) in
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   870
         if neg pv then nat_rec a f n
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   871
                   else f (nat pv + n) (nat_rec a f (nat pv + n)))"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   872
apply (subst add_eq_if)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   873
apply (simp split add: nat.split
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   874
            del: nat_numeral_1_eq_1
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   875
            add: nat_numeral_1_eq_1 [symmetric]
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   876
                 numeral_1_eq_Suc_0 [symmetric]
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   877
                 neg_number_of_pred_iff_0)
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   878
done
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   879
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   880
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   881
subsubsection{*Various Other Lemmas*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   882
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31068
diff changeset
   883
lemma card_UNIV_bool[simp]: "card (UNIV :: bool set) = 2"
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31068
diff changeset
   884
by(simp add: UNIV_bool)
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31068
diff changeset
   885
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   886
text {*Evens and Odds, for Mutilated Chess Board*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   887
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   888
text{*Lemmas for specialist use, NOT as default simprules*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   889
lemma nat_mult_2: "2 * z = (z+z::nat)"
43531
cc46a678faaf added number_semiring class, plus a few new lemmas;
huffman
parents: 43526
diff changeset
   890
by (rule semiring_mult_2)
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   891
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   892
lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
43531
cc46a678faaf added number_semiring class, plus a few new lemmas;
huffman
parents: 43526
diff changeset
   893
by (rule semiring_mult_2_right)
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   894
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   895
text{*Case analysis on @{term "n<2"}*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   896
lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 32069
diff changeset
   897
by (auto simp add: nat_1_add_1 [symmetric])
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   898
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   899
text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   900
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   901
lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   902
by simp
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   903
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   904
lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   905
by simp
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   906
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   907
text{*Can be used to eliminate long strings of Sucs, but not by default*}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   908
lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   909
by simp
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30497
diff changeset
   910
31096
e546e15089ef newline at end of file
huffman
parents: 31080
diff changeset
   911
end