--- a/src/HOL/NatArith.thy Wed Nov 08 11:23:09 2006 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,246 +0,0 @@
-(* Title: HOL/NatArith.thy
- ID: $Id$
- Author: Tobias Nipkow and Markus Wenzel
-*)
-
-header {*Further Arithmetic Facts Concerning the Natural Numbers*}
-
-theory NatArith
-imports Nat
-uses "arith_data.ML"
-begin
-
-setup arith_setup
-
-text{*The following proofs may rely on the arithmetic proof procedures.*}
-
-lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
- by (auto simp: le_eq_less_or_eq dest: less_imp_Suc_add)
-
-lemma pred_nat_trancl_eq_le: "((m, n) : pred_nat^*) = (m \<le> n)"
-by (simp add: less_eq reflcl_trancl [symmetric]
- del: reflcl_trancl, arith)
-
-lemma nat_diff_split:
- "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
- -- {* elimination of @{text -} on @{text nat} *}
- by (cases "a<b" rule: case_split)
- (auto simp add: diff_is_0_eq [THEN iffD2])
-
-lemma nat_diff_split_asm:
- "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
- -- {* elimination of @{text -} on @{text nat} in assumptions *}
- by (simp split: nat_diff_split)
-
-lemmas [arith_split] = nat_diff_split split_min split_max
-
-
-
-lemma le_square: "m \<le> m*(m::nat)"
-by (induct_tac "m", auto)
-
-lemma le_cube: "(m::nat) \<le> m*(m*m)"
-by (induct_tac "m", auto)
-
-
-text{*Subtraction laws, mostly by Clemens Ballarin*}
-
-lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
-by arith
-
-lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
-by arith
-
-lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
-by arith
-
-lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
-by arith
-
-lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
-by arith
-
-lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
-by arith
-
-(*Replaces the previous diff_less and le_diff_less, which had the stronger
- second premise n\<le>m*)
-lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
-by arith
-
-
-(** Simplification of relational expressions involving subtraction **)
-
-lemma diff_diff_eq: "[| k \<le> m; k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
-by (simp split add: nat_diff_split)
-
-lemma eq_diff_iff: "[| k \<le> m; k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
-by (auto split add: nat_diff_split)
-
-lemma less_diff_iff: "[| k \<le> m; k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
-by (auto split add: nat_diff_split)
-
-lemma le_diff_iff: "[| k \<le> m; k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
-by (auto split add: nat_diff_split)
-
-
-text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
-
-(* Monotonicity of subtraction in first argument *)
-lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
-by (simp split add: nat_diff_split)
-
-lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
-by (simp split add: nat_diff_split)
-
-lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
-by (simp split add: nat_diff_split)
-
-lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==> m=n"
-by (simp split add: nat_diff_split)
-
-text{*Lemmas for ex/Factorization*}
-
-lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
-by (case_tac "m", auto)
-
-lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
-by (case_tac "m", auto)
-
-lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
-by (case_tac "m", auto)
-
-
-text{*Rewriting to pull differences out*}
-
-lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
-by arith
-
-lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
-by arith
-
-lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
-by arith
-
-(*The others are
- i - j - k = i - (j + k),
- k \<le> j ==> j - k + i = j + i - k,
- k \<le> j ==> i + (j - k) = i + j - k *)
-lemmas add_diff_assoc = diff_add_assoc [symmetric]
-lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
-declare diff_diff_left [simp] add_diff_assoc [simp] add_diff_assoc2[simp]
-
-text{*At present we prove no analogue of @{text not_less_Least} or @{text
-Least_Suc}, since there appears to be no need.*}
-
-ML
-{*
-val pred_nat_trancl_eq_le = thm "pred_nat_trancl_eq_le";
-val nat_diff_split = thm "nat_diff_split";
-val nat_diff_split_asm = thm "nat_diff_split_asm";
-val le_square = thm "le_square";
-val le_cube = thm "le_cube";
-val diff_less_mono = thm "diff_less_mono";
-val less_diff_conv = thm "less_diff_conv";
-val le_diff_conv = thm "le_diff_conv";
-val le_diff_conv2 = thm "le_diff_conv2";
-val diff_diff_cancel = thm "diff_diff_cancel";
-val le_add_diff = thm "le_add_diff";
-val diff_less = thm "diff_less";
-val diff_diff_eq = thm "diff_diff_eq";
-val eq_diff_iff = thm "eq_diff_iff";
-val less_diff_iff = thm "less_diff_iff";
-val le_diff_iff = thm "le_diff_iff";
-val diff_le_mono = thm "diff_le_mono";
-val diff_le_mono2 = thm "diff_le_mono2";
-val diff_less_mono2 = thm "diff_less_mono2";
-val diffs0_imp_equal = thm "diffs0_imp_equal";
-val one_less_mult = thm "one_less_mult";
-val n_less_m_mult_n = thm "n_less_m_mult_n";
-val n_less_n_mult_m = thm "n_less_n_mult_m";
-val diff_diff_right = thm "diff_diff_right";
-val diff_Suc_diff_eq1 = thm "diff_Suc_diff_eq1";
-val diff_Suc_diff_eq2 = thm "diff_Suc_diff_eq2";
-*}
-
-subsection{*Embedding of the Naturals into any @{text
-semiring_1_cancel}: @{term of_nat}*}
-
-consts of_nat :: "nat => 'a::semiring_1_cancel"
-
-primrec
- of_nat_0: "of_nat 0 = 0"
- of_nat_Suc: "of_nat (Suc m) = of_nat m + 1"
-
-lemma of_nat_1 [simp]: "of_nat 1 = 1"
-by simp
-
-lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n"
-apply (induct m)
-apply (simp_all add: add_ac)
-done
-
-lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n"
-apply (induct m)
-apply (simp_all add: add_ac left_distrib)
-done
-
-lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semidom)"
-apply (induct m, simp_all)
-apply (erule order_trans)
-apply (rule less_add_one [THEN order_less_imp_le])
-done
-
-lemma less_imp_of_nat_less:
- "m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)"
-apply (induct m n rule: diff_induct, simp_all)
-apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force)
-done
-
-lemma of_nat_less_imp_less:
- "of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n"
-apply (induct m n rule: diff_induct, simp_all)
-apply (insert zero_le_imp_of_nat)
-apply (force simp add: linorder_not_less [symmetric])
-done
-
-lemma of_nat_less_iff [simp]:
- "(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)"
-by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
-
-text{*Special cases where either operand is zero*}
-lemmas of_nat_0_less_iff = of_nat_less_iff [of 0, simplified]
-lemmas of_nat_less_0_iff = of_nat_less_iff [of _ 0, simplified]
-declare of_nat_0_less_iff [simp]
-declare of_nat_less_0_iff [simp]
-
-lemma of_nat_le_iff [simp]:
- "(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)"
-by (simp add: linorder_not_less [symmetric])
-
-text{*Special cases where either operand is zero*}
-lemmas of_nat_0_le_iff = of_nat_le_iff [of 0, simplified]
-lemmas of_nat_le_0_iff = of_nat_le_iff [of _ 0, simplified]
-declare of_nat_0_le_iff [simp]
-declare of_nat_le_0_iff [simp]
-
-text{*The ordering on the @{text semiring_1_cancel} is necessary
-to exclude the possibility of a finite field, which indeed wraps back to
-zero.*}
-lemma of_nat_eq_iff [simp]:
- "(of_nat m = (of_nat n::'a::ordered_semidom)) = (m = n)"
-by (simp add: order_eq_iff)
-
-text{*Special cases where either operand is zero*}
-lemmas of_nat_0_eq_iff = of_nat_eq_iff [of 0, simplified]
-lemmas of_nat_eq_0_iff = of_nat_eq_iff [of _ 0, simplified]
-declare of_nat_0_eq_iff [simp]
-declare of_nat_eq_0_iff [simp]
-
-lemma of_nat_diff [simp]:
- "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::ring_1)"
-by (simp del: of_nat_add
- add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)
-
-
-end