src/HOL/Integ/IntArith.thy
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(*  Title:      HOL/Integ/IntArith.thy
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    ID:         $Id$
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    Authors:    Larry Paulson and Tobias Nipkow
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*)
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header {* Integer arithmetic *}
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theory IntArith
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imports Numeral
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uses ("int_arith1.ML")
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begin
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text{*Duplicate: can't understand why it's necessary*}
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declare numeral_0_eq_0 [simp]
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subsection{*Instantiating Binary Arithmetic for the Integers*}
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instance int :: number_ring
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  int_number_of_def: "number_of w \<equiv> of_int w"
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  by intro_classes (simp only: int_number_of_def)
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subsection{*Inequality Reasoning for the Arithmetic Simproc*}
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lemma add_numeral_0: "Numeral0 + a = (a::'a::number_ring)"
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by simp 
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lemma add_numeral_0_right: "a + Numeral0 = (a::'a::number_ring)"
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by simp
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lemma mult_numeral_1: "Numeral1 * a = (a::'a::number_ring)"
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by simp 
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lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::number_ring)"
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by simp
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text{*Theorem lists for the cancellation simprocs. The use of binary numerals
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for 0 and 1 reduces the number of special cases.*}
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lemmas add_0s = add_numeral_0 add_numeral_0_right
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lemmas mult_1s = mult_numeral_1 mult_numeral_1_right 
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                 mult_minus1 mult_minus1_right
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subsection{*Special Arithmetic Rules for Abstract 0 and 1*}
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text{*Arithmetic computations are defined for binary literals, which leaves 0
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and 1 as special cases. Addition already has rules for 0, but not 1.
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Multiplication and unary minus already have rules for both 0 and 1.*}
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lemma binop_eq: "[|f x y = g x y; x = x'; y = y'|] ==> f x' y' = g x' y'"
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by simp
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lemmas add_number_of_eq = number_of_add [symmetric]
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text{*Allow 1 on either or both sides*}
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lemma one_add_one_is_two: "1 + 1 = (2::'a::number_ring)"
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by (simp del: numeral_1_eq_1 add: numeral_1_eq_1 [symmetric] add_number_of_eq)
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lemmas add_special =
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    one_add_one_is_two
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    binop_eq [of "op +", OF add_number_of_eq numeral_1_eq_1 refl, standard]
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    binop_eq [of "op +", OF add_number_of_eq refl numeral_1_eq_1, standard]
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text{*Allow 1 on either or both sides (1-1 already simplifies to 0)*}
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lemmas diff_special =
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    binop_eq [of "op -", OF diff_number_of_eq numeral_1_eq_1 refl, standard]
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    binop_eq [of "op -", OF diff_number_of_eq refl numeral_1_eq_1, standard]
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text{*Allow 0 or 1 on either side with a binary numeral on the other*}
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lemmas eq_special =
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    binop_eq [of "op =", OF eq_number_of_eq numeral_0_eq_0 refl, standard]
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    binop_eq [of "op =", OF eq_number_of_eq numeral_1_eq_1 refl, standard]
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    binop_eq [of "op =", OF eq_number_of_eq refl numeral_0_eq_0, standard]
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    binop_eq [of "op =", OF eq_number_of_eq refl numeral_1_eq_1, standard]
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text{*Allow 0 or 1 on either side with a binary numeral on the other*}
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lemmas less_special =
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  binop_eq [of "op <", OF less_number_of_eq_neg numeral_0_eq_0 refl, standard]
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  binop_eq [of "op <", OF less_number_of_eq_neg numeral_1_eq_1 refl, standard]
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  binop_eq [of "op <", OF less_number_of_eq_neg refl numeral_0_eq_0, standard]
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  binop_eq [of "op <", OF less_number_of_eq_neg refl numeral_1_eq_1, standard]
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text{*Allow 0 or 1 on either side with a binary numeral on the other*}
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lemmas le_special =
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    binop_eq [of "op \<le>", OF le_number_of_eq numeral_0_eq_0 refl, standard]
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    binop_eq [of "op \<le>", OF le_number_of_eq numeral_1_eq_1 refl, standard]
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    binop_eq [of "op \<le>", OF le_number_of_eq refl numeral_0_eq_0, standard]
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    binop_eq [of "op \<le>", OF le_number_of_eq refl numeral_1_eq_1, standard]
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lemmas arith_special = 
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       add_special diff_special eq_special less_special le_special
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use "int_arith1.ML"
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setup int_arith_setup
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subsection{*Lemmas About Small Numerals*}
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lemma of_int_m1 [simp]: "of_int -1 = (-1 :: 'a :: number_ring)"
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proof -
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  have "(of_int -1 :: 'a) = of_int (- 1)" by simp
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  also have "... = - of_int 1" by (simp only: of_int_minus)
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  also have "... = -1" by simp
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  finally show ?thesis .
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qed
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lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{ordered_idom,number_ring})"
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by (simp add: abs_if)
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lemma abs_power_minus_one [simp]:
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     "abs(-1 ^ n) = (1::'a::{ordered_idom,number_ring,recpower})"
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by (simp add: power_abs)
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lemma of_int_number_of_eq:
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     "of_int (number_of v) = (number_of v :: 'a :: number_ring)"
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by (simp add: number_of_eq) 
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text{*Lemmas for specialist use, NOT as default simprules*}
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lemma mult_2: "2 * z = (z+z::'a::number_ring)"
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proof -
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  have "2*z = (1 + 1)*z" by simp
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  also have "... = z+z" by (simp add: left_distrib)
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  finally show ?thesis .
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qed
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lemma mult_2_right: "z * 2 = (z+z::'a::number_ring)"
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by (subst mult_commute, rule mult_2)
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subsection{*More Inequality Reasoning*}
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lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
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by arith
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lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
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by arith
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lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
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by arith
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lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
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by arith
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lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
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by arith
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subsection{*The Functions @{term nat} and @{term int}*}
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text{*Simplify the terms @{term "int 0"}, @{term "int(Suc 0)"} and
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  @{term "w + - z"}*}
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declare Zero_int_def [symmetric, simp]
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declare One_int_def [symmetric, simp]
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text{*cooper.ML refers to this theorem*}
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lemmas diff_int_def_symmetric = diff_int_def [symmetric, simp]
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lemma nat_0: "nat 0 = 0"
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by (simp add: nat_eq_iff)
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lemma nat_1: "nat 1 = Suc 0"
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by (subst nat_eq_iff, simp)
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lemma nat_2: "nat 2 = Suc (Suc 0)"
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by (subst nat_eq_iff, simp)
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lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
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apply (insert zless_nat_conj [of 1 z])
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apply (auto simp add: nat_1)
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done
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text{*This simplifies expressions of the form @{term "int n = z"} where
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      z is an integer literal.*}
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lemmas int_eq_iff_number_of = int_eq_iff [of _ "number_of v", standard]
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declare int_eq_iff_number_of [simp]
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lemma split_nat [arith_split]:
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  "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
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  (is "?P = (?L & ?R)")
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proof (cases "i < 0")
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  case True thus ?thesis by simp
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next
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  case False
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  have "?P = ?L"
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  proof
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    assume ?P thus ?L using False by clarsimp
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  next
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    assume ?L thus ?P using False by simp
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  qed
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  with False show ?thesis by simp
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qed
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(*Analogous to zadd_int*)
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lemma zdiff_int: "n \<le> m ==> int m - int n = int (m-n)" 
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by (induct m n rule: diff_induct, simp_all)
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lemma nat_mult_distrib: "(0::int) \<le> z ==> nat (z*z') = nat z * nat z'"
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apply (case_tac "0 \<le> z'")
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apply (rule inj_int [THEN injD])
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apply (simp add: int_mult zero_le_mult_iff)
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apply (simp add: mult_le_0_iff)
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done
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lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
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apply (rule trans)
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apply (rule_tac [2] nat_mult_distrib, auto)
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done
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lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
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apply (case_tac "z=0 | w=0")
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apply (auto simp add: abs_if nat_mult_distrib [symmetric] 
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                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
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done
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subsection "Induction principles for int"
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                     (* `set:int': dummy construction *)
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theorem int_ge_induct[case_names base step,induct set:int]:
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  assumes ge: "k \<le> (i::int)" and
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        base: "P(k)" and
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        step: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
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  shows "P i"
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proof -
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  { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
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    proof (induct n)
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      case 0
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      hence "i = k" by arith
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      thus "P i" using base by simp
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    next
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      case (Suc n)
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      hence "n = nat((i - 1) - k)" by arith
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      moreover
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      have ki1: "k \<le> i - 1" using Suc.prems by arith
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      ultimately
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      have "P(i - 1)" by(rule Suc.hyps)
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      from step[OF ki1 this] show ?case by simp
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    qed
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   246
  }
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  with ge show ?thesis by fast
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qed
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                     (* `set:int': dummy construction *)
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theorem int_gr_induct[case_names base step,induct set:int]:
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  assumes gr: "k < (i::int)" and
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        base: "P(k+1)" and
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        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
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  shows "P i"
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apply(rule int_ge_induct[of "k + 1"])
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  using gr apply arith
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 apply(rule base)
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apply (rule step, simp+)
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   260
done
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   261
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theorem int_le_induct[consumes 1,case_names base step]:
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  assumes le: "i \<le> (k::int)" and
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        base: "P(k)" and
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        step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
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   266
  shows "P i"
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   267
proof -
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  { fix n have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
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    proof (induct n)
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      case 0
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      hence "i = k" by arith
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      thus "P i" using base by simp
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    next
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      case (Suc n)
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      hence "n = nat(k - (i+1))" by arith
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   276
      moreover
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   277
      have ki1: "i + 1 \<le> k" using Suc.prems by arith
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   278
      ultimately
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   279
      have "P(i+1)" by(rule Suc.hyps)
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   280
      from step[OF ki1 this] show ?case by simp
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    qed
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   282
  }
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   283
  with le show ?thesis by fast
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   284
qed
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   285
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   286
theorem int_less_induct [consumes 1,case_names base step]:
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   287
  assumes less: "(i::int) < k" and
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   288
        base: "P(k - 1)" and
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   289
        step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
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   290
  shows "P i"
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   291
apply(rule int_le_induct[of _ "k - 1"])
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   292
  using less apply arith
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   293
 apply(rule base)
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   294
apply (rule step, simp+)
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   295
done
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   296
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   297
subsection{*Intermediate value theorems*}
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   298
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   299
lemma int_val_lemma:
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   300
     "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->  
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   301
      f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
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8ed6989228bb Simplification of the development of Integers
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   302
apply (induct_tac "n", simp)
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   303
apply (intro strip)
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   304
apply (erule impE, simp)
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   305
apply (erule_tac x = n in allE, simp)
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   306
apply (case_tac "k = f (n+1) ")
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   307
 apply force
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   308
apply (erule impE)
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6145dd7538d7 replaced monomorphic abs definitions by abs_if
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diff changeset
   309
 apply (simp add: abs_if split add: split_if_asm)
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   310
apply (blast intro: le_SucI)
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   311
done
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   312
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   313
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
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lemma nat_intermed_int_val:
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     "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;  
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         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
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apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k 
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       in int_val_lemma)
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apply simp
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apply (erule exE)
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apply (rule_tac x = "i+m" in exI, arith)
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done
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subsection{*Products and 1, by T. M. Rasmussen*}
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lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
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by arith
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lemma abs_zmult_eq_1: "(\<bar>m * n\<bar> = 1) ==> \<bar>m\<bar> = (1::int)"
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apply (case_tac "\<bar>n\<bar>=1") 
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apply (simp add: abs_mult) 
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apply (rule ccontr) 
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apply (auto simp add: linorder_neq_iff abs_mult) 
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apply (subgoal_tac "2 \<le> \<bar>m\<bar> & 2 \<le> \<bar>n\<bar>")
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 prefer 2 apply arith 
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apply (subgoal_tac "2*2 \<le> \<bar>m\<bar> * \<bar>n\<bar>", simp) 
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apply (rule mult_mono, auto) 
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done
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lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
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by (insert abs_zmult_eq_1 [of m n], arith)
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lemma pos_zmult_eq_1_iff: "0 < (m::int) ==> (m * n = 1) = (m = 1 & n = 1)"
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apply (auto dest: pos_zmult_eq_1_iff_lemma) 
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apply (simp add: mult_commute [of m]) 
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apply (frule pos_zmult_eq_1_iff_lemma, auto) 
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done
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lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
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apply (rule iffI) 
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 apply (frule pos_zmult_eq_1_iff_lemma)
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 apply (simp add: mult_commute [of m]) 
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 apply (frule pos_zmult_eq_1_iff_lemma, auto) 
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done
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subsection {* code generator setup *}
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code_modulename SML
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  Numeral Integer
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lemma Numeral_Pls_refl [code func]:
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  "Numeral.Pls = Numeral.Pls" ..
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lemma Numeral_Min_refl [code func]:
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  "Numeral.Min = Numeral.Min" ..
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lemma zero_int_refl [code func]:
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  "(0\<Colon>int) = 0" ..
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lemma one_int_refl [code func]:
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  "(1\<Colon>int) = 1" ..
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lemma number_of_int_refl [code func]:
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  "(number_of \<Colon> int \<Rightarrow> int) = number_of" ..
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lemma number_of_is_id:
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  "number_of (k::int) = k"
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  unfolding int_number_of_def by simp
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lemma zero_is_num_zero [code inline, symmetric, normal post]:
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  "(0::int) = number_of Numeral.Pls" 
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  by simp
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lemma one_is_num_one [code inline, symmetric, normal post]:
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  "(1::int) = number_of  (Numeral.Pls BIT bit.B1)" 
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  by simp 
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   390
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lemmas int_code_rewrites =
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  arith_simps(5-27)
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  arith_extra_simps(1-5) [where 'a = int]
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declare int_code_rewrites [code func]
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code_type bit
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  (SML "bool")
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  (Haskell "Bool")
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code_const "Numeral.bit.B0" and "Numeral.bit.B1"
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  (SML "false" and "true")
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  (Haskell "False" and "True")
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   403
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code_const "number_of \<Colon> int \<Rightarrow> int"
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  and "Numeral.Pls" and "Numeral.Min" and "Numeral.Bit"
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  and "Numeral.succ" and "Numeral.pred"
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  (SML "_"
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     and "0/ :/ IntInf.int"
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     and "~1/ :/ IntInf.int"
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     and "(_; _; raise Fail \"BIT\")"
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     and "IntInf.+/ (_,/ 1)"
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     and "IntInf.-/ (_,/ 1))")
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  (Haskell "_"
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     and "0"
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     and "negate/ 1"
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     and "error/ \"BIT\""
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     and "(+)/ 1"
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     and "(-)/ _/ 1")
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setup {*
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  CodegenPackage.add_appconst ("Numeral.Bit", CodegenPackage.appgen_numeral (try HOLogic.dest_numeral))
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*}
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subsection {* legacy ML bindings *}
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   426
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ML
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{*
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val zle_diff1_eq = thm "zle_diff1_eq";
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val zle_add1_eq_le = thm "zle_add1_eq_le";
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   431
val nonneg_eq_int = thm "nonneg_eq_int";
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   432
val abs_minus_one = thm "abs_minus_one";
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   433
val of_int_number_of_eq = thm"of_int_number_of_eq";
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val nat_eq_iff = thm "nat_eq_iff";
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   435
val nat_eq_iff2 = thm "nat_eq_iff2";
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   436
val nat_less_iff = thm "nat_less_iff";
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   437
val int_eq_iff = thm "int_eq_iff";
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   438
val nat_0 = thm "nat_0";
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   439
val nat_1 = thm "nat_1";
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val nat_2 = thm "nat_2";
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val nat_less_eq_zless = thm "nat_less_eq_zless";
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val nat_le_eq_zle = thm "nat_le_eq_zle";
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   443
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val nat_intermed_int_val = thm "nat_intermed_int_val";
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val pos_zmult_eq_1_iff = thm "pos_zmult_eq_1_iff";
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   446
val zmult_eq_1_iff = thm "zmult_eq_1_iff";
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val nat_add_distrib = thm "nat_add_distrib";
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val nat_diff_distrib = thm "nat_diff_distrib";
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val nat_mult_distrib = thm "nat_mult_distrib";
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val nat_mult_distrib_neg = thm "nat_mult_distrib_neg";
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val nat_abs_mult_distrib = thm "nat_abs_mult_distrib";
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*}
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   453
7707
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   454
end