author | paulson |
Fri, 29 Oct 2004 15:16:02 +0200 | |
changeset 15270 | 8b3f707a78a7 |
parent 15251 | bb6f072c8d10 |
child 15281 | bd4611956c7b |
permissions | -rw-r--r-- |
923 | 1 |
(* Title: HOL/Nat.thy |
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ID: $Id$ |
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Author: Tobias Nipkow and Lawrence C Paulson |
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Type "nat" is a linear order, and a datatype; arithmetic operators + - |
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and * (for div, mod and dvd, see theory Divides). |
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*) |
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header {* Natural numbers *} |
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theory Nat |
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imports Wellfounded_Recursion Ring_and_Field |
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begin |
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subsection {* Type @{text ind} *} |
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typedecl ind |
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consts |
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Zero_Rep :: ind |
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Suc_Rep :: "ind => ind" |
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axioms |
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-- {* the axiom of infinity in 2 parts *} |
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inj_Suc_Rep: "inj Suc_Rep" |
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Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep" |
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subsection {* Type nat *} |
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text {* Type definition *} |
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consts |
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Nat :: "ind set" |
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inductive Nat |
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intros |
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Zero_RepI: "Zero_Rep : Nat" |
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Suc_RepI: "i : Nat ==> Suc_Rep i : Nat" |
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global |
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typedef (open Nat) |
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nat = Nat by (rule exI, rule Nat.Zero_RepI) |
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instance nat :: "{ord, zero, one}" .. |
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text {* Abstract constants and syntax *} |
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consts |
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Suc :: "nat => nat" |
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pred_nat :: "(nat * nat) set" |
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local |
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defs |
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Zero_nat_def: "0 == Abs_Nat Zero_Rep" |
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Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))" |
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One_nat_def [simp]: "1 == Suc 0" |
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-- {* nat operations *} |
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pred_nat_def: "pred_nat == {(m, n). n = Suc m}" |
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less_def: "m < n == (m, n) : trancl pred_nat" |
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le_def: "m \<le> (n::nat) == ~ (n < m)" |
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text {* Induction *} |
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theorem nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n" |
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apply (unfold Zero_nat_def Suc_def) |
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apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *} |
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apply (erule Rep_Nat [THEN Nat.induct]) |
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apply (rules elim: Abs_Nat_inverse [THEN subst]) |
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done |
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text {* Isomorphisms: @{text Abs_Nat} and @{text Rep_Nat} *} |
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lemma inj_Rep_Nat: "inj Rep_Nat" |
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apply (rule inj_on_inverseI) |
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apply (rule Rep_Nat_inverse) |
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done |
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lemma inj_on_Abs_Nat: "inj_on Abs_Nat Nat" |
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apply (rule inj_on_inverseI) |
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apply (erule Abs_Nat_inverse) |
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done |
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text {* Distinctness of constructors *} |
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lemma Suc_not_Zero [iff]: "Suc m \<noteq> 0" |
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apply (unfold Zero_nat_def Suc_def) |
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apply (rule inj_on_Abs_Nat [THEN inj_on_contraD]) |
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apply (rule Suc_Rep_not_Zero_Rep) |
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apply (rule Rep_Nat Nat.Suc_RepI Nat.Zero_RepI)+ |
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done |
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lemma Zero_not_Suc [iff]: "0 \<noteq> Suc m" |
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by (rule not_sym, rule Suc_not_Zero not_sym) |
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lemma Suc_neq_Zero: "Suc m = 0 ==> R" |
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by (rule notE, rule Suc_not_Zero) |
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lemma Zero_neq_Suc: "0 = Suc m ==> R" |
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by (rule Suc_neq_Zero, erule sym) |
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text {* Injectiveness of @{term Suc} *} |
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lemma inj_Suc: "inj Suc" |
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apply (unfold Suc_def) |
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apply (rule inj_onI) |
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apply (drule inj_on_Abs_Nat [THEN inj_onD]) |
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apply (rule Rep_Nat Nat.Suc_RepI)+ |
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apply (drule inj_Suc_Rep [THEN injD]) |
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apply (erule inj_Rep_Nat [THEN injD]) |
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done |
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lemma Suc_inject: "Suc x = Suc y ==> x = y" |
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by (rule inj_Suc [THEN injD]) |
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lemma Suc_Suc_eq [iff]: "(Suc m = Suc n) = (m = n)" |
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apply (rule iffI) |
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apply (erule Suc_inject) |
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apply (erule arg_cong) |
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done |
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lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False" |
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by auto |
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text {* @{typ nat} is a datatype *} |
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rep_datatype nat |
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distinct Suc_not_Zero Zero_not_Suc |
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inject Suc_Suc_eq |
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induction nat_induct |
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lemma n_not_Suc_n: "n \<noteq> Suc n" |
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by (induct n) simp_all |
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lemma Suc_n_not_n: "Suc t \<noteq> t" |
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by (rule not_sym, rule n_not_Suc_n) |
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text {* A special form of induction for reasoning |
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about @{term "m < n"} and @{term "m - n"} *} |
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theorem diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==> |
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(!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n" |
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apply (rule_tac x = m in spec) |
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apply (induct n) |
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prefer 2 |
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apply (rule allI) |
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apply (induct_tac x, rules+) |
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done |
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subsection {* Basic properties of "less than" *} |
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lemma wf_pred_nat: "wf pred_nat" |
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apply (unfold wf_def pred_nat_def, clarify) |
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apply (induct_tac x, blast+) |
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done |
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lemma wf_less: "wf {(x, y::nat). x < y}" |
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apply (unfold less_def) |
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apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_subset], blast) |
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done |
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lemma less_eq: "((m, n) : pred_nat^+) = (m < n)" |
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apply (unfold less_def) |
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apply (rule refl) |
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done |
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subsubsection {* Introduction properties *} |
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lemma less_trans: "i < j ==> j < k ==> i < (k::nat)" |
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apply (unfold less_def) |
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apply (rule trans_trancl [THEN transD], assumption+) |
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done |
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lemma lessI [iff]: "n < Suc n" |
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apply (unfold less_def pred_nat_def) |
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apply (simp add: r_into_trancl) |
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done |
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lemma less_SucI: "i < j ==> i < Suc j" |
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apply (rule less_trans, assumption) |
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apply (rule lessI) |
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done |
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lemma zero_less_Suc [iff]: "0 < Suc n" |
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apply (induct n) |
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apply (rule lessI) |
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apply (erule less_trans) |
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apply (rule lessI) |
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done |
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subsubsection {* Elimination properties *} |
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lemma less_not_sym: "n < m ==> ~ m < (n::nat)" |
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apply (unfold less_def) |
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apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym]) |
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done |
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lemma less_asym: |
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assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P |
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apply (rule contrapos_np) |
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apply (rule less_not_sym) |
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apply (rule h1) |
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apply (erule h2) |
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done |
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lemma less_not_refl: "~ n < (n::nat)" |
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apply (unfold less_def) |
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apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_not_refl]) |
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done |
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lemma less_irrefl [elim!]: "(n::nat) < n ==> R" |
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by (rule notE, rule less_not_refl) |
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lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)" by blast |
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lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t" |
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by (rule not_sym, rule less_not_refl2) |
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lemma lessE: |
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assumes major: "i < k" |
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and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P" |
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shows P |
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apply (rule major [unfolded less_def pred_nat_def, THEN tranclE], simp_all) |
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apply (erule p1) |
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apply (rule p2) |
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apply (simp add: less_def pred_nat_def, assumption) |
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done |
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lemma not_less0 [iff]: "~ n < (0::nat)" |
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by (blast elim: lessE) |
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lemma less_zeroE: "(n::nat) < 0 ==> R" |
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by (rule notE, rule not_less0) |
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lemma less_SucE: assumes major: "m < Suc n" |
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and less: "m < n ==> P" and eq: "m = n ==> P" shows P |
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apply (rule major [THEN lessE]) |
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apply (rule eq, blast) |
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apply (rule less, blast) |
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done |
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lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)" |
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by (blast elim!: less_SucE intro: less_trans) |
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lemma less_one [iff]: "(n < (1::nat)) = (n = 0)" |
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by (simp add: less_Suc_eq) |
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lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)" |
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by (simp add: less_Suc_eq) |
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lemma Suc_mono: "m < n ==> Suc m < Suc n" |
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by (induct n) (fast elim: less_trans lessE)+ |
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text {* "Less than" is a linear ordering *} |
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lemma less_linear: "m < n | m = n | n < (m::nat)" |
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apply (induct m) |
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apply (induct n) |
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apply (rule refl [THEN disjI1, THEN disjI2]) |
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apply (rule zero_less_Suc [THEN disjI1]) |
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apply (blast intro: Suc_mono less_SucI elim: lessE) |
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done |
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text {* "Less than" is antisymmetric, sort of *} |
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lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n" |
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apply(simp only:less_Suc_eq) |
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apply blast |
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done |
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lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)" |
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using less_linear by blast |
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lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m" |
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and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m" |
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shows "P n m" |
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apply (rule less_linear [THEN disjE]) |
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apply (erule_tac [2] disjE) |
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apply (erule lessCase) |
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apply (erule sym [THEN eqCase]) |
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apply (erule major) |
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done |
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subsubsection {* Inductive (?) properties *} |
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lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n" |
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apply (simp add: nat_neq_iff) |
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apply (blast elim!: less_irrefl less_SucE elim: less_asym) |
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done |
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lemma Suc_lessD: "Suc m < n ==> m < n" |
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apply (induct n) |
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apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+ |
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done |
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lemma Suc_lessE: assumes major: "Suc i < k" |
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and minor: "!!j. i < j ==> k = Suc j ==> P" shows P |
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apply (rule major [THEN lessE]) |
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apply (erule lessI [THEN minor]) |
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apply (erule Suc_lessD [THEN minor], assumption) |
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done |
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lemma Suc_less_SucD: "Suc m < Suc n ==> m < n" |
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by (blast elim: lessE dest: Suc_lessD) |
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lemma Suc_less_eq [iff]: "(Suc m < Suc n) = (m < n)" |
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apply (rule iffI) |
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apply (erule Suc_less_SucD) |
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apply (erule Suc_mono) |
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done |
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lemma less_trans_Suc: |
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assumes le: "i < j" shows "j < k ==> Suc i < k" |
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apply (induct k, simp_all) |
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apply (insert le) |
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apply (simp add: less_Suc_eq) |
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apply (blast dest: Suc_lessD) |
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done |
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text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *} |
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lemma not_less_eq: "(~ m < n) = (n < Suc m)" |
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by (rule_tac m = m and n = n in diff_induct, simp_all) |
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text {* Complete induction, aka course-of-values induction *} |
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lemma nat_less_induct: |
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assumes prem: "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n" |
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apply (rule_tac a=n in wf_induct) |
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apply (rule wf_pred_nat [THEN wf_trancl]) |
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apply (rule prem) |
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apply (unfold less_def, assumption) |
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done |
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lemmas less_induct = nat_less_induct [rule_format, case_names less] |
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subsection {* Properties of "less than or equal" *} |
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text {* Was @{text le_eq_less_Suc}, but this orientation is more useful *} |
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lemma less_Suc_eq_le: "(m < Suc n) = (m \<le> n)" |
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by (unfold le_def, rule not_less_eq [symmetric]) |
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lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n" |
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by (rule less_Suc_eq_le [THEN iffD2]) |
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lemma le0 [iff]: "(0::nat) \<le> n" |
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by (unfold le_def, rule not_less0) |
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lemma Suc_n_not_le_n: "~ Suc n \<le> n" |
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by (simp add: le_def) |
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lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)" |
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by (induct i) (simp_all add: le_def) |
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lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)" |
13449 | 361 |
by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq) |
362 |
||
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|
363 |
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R" |
13449 | 364 |
by (drule le_Suc_eq [THEN iffD1], rules+) |
365 |
||
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|
366 |
lemma leI: "~ n < m ==> m \<le> (n::nat)" by (simp add: le_def) |
13449 | 367 |
|
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|
368 |
lemma leD: "m \<le> n ==> ~ n < (m::nat)" |
13449 | 369 |
by (simp add: le_def) |
370 |
||
371 |
lemmas leE = leD [elim_format] |
|
372 |
||
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|
373 |
lemma not_less_iff_le: "(~ n < m) = (m \<le> (n::nat))" |
13449 | 374 |
by (blast intro: leI elim: leE) |
375 |
||
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|
376 |
lemma not_leE: "~ m \<le> n ==> n<(m::nat)" |
13449 | 377 |
by (simp add: le_def) |
378 |
||
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|
379 |
lemma not_le_iff_less: "(~ n \<le> m) = (m < (n::nat))" |
13449 | 380 |
by (simp add: le_def) |
381 |
||
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|
382 |
lemma Suc_leI: "m < n ==> Suc(m) \<le> n" |
13449 | 383 |
apply (simp add: le_def less_Suc_eq) |
384 |
apply (blast elim!: less_irrefl less_asym) |
|
385 |
done -- {* formerly called lessD *} |
|
386 |
||
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|
387 |
lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n" |
13449 | 388 |
by (simp add: le_def less_Suc_eq) |
389 |
||
390 |
text {* Stronger version of @{text Suc_leD} *} |
|
14267
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|
391 |
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n" |
13449 | 392 |
apply (simp add: le_def less_Suc_eq) |
393 |
using less_linear |
|
394 |
apply blast |
|
395 |
done |
|
396 |
||
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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|
397 |
lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)" |
13449 | 398 |
by (blast intro: Suc_leI Suc_le_lessD) |
399 |
||
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|
400 |
lemma le_SucI: "m \<le> n ==> m \<le> Suc n" |
13449 | 401 |
by (unfold le_def) (blast dest: Suc_lessD) |
402 |
||
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|
403 |
lemma less_imp_le: "m < n ==> m \<le> (n::nat)" |
13449 | 404 |
by (unfold le_def) (blast elim: less_asym) |
405 |
||
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|
406 |
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *} |
13449 | 407 |
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq |
408 |
||
409 |
||
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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|
410 |
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *} |
13449 | 411 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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|
412 |
lemma le_imp_less_or_eq: "m \<le> n ==> m < n | m = (n::nat)" |
13449 | 413 |
apply (unfold le_def) |
414 |
using less_linear |
|
415 |
apply (blast elim: less_irrefl less_asym) |
|
416 |
done |
|
417 |
||
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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|
418 |
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)" |
13449 | 419 |
apply (unfold le_def) |
420 |
using less_linear |
|
421 |
apply (blast elim!: less_irrefl elim: less_asym) |
|
422 |
done |
|
423 |
||
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changeset
|
424 |
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)" |
13449 | 425 |
by (rules intro: less_or_eq_imp_le le_imp_less_or_eq) |
426 |
||
427 |
text {* Useful with @{text Blast}. *} |
|
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|
428 |
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n" |
13449 | 429 |
by (rule less_or_eq_imp_le, rule disjI2) |
430 |
||
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|
431 |
lemma le_refl: "n \<le> (n::nat)" |
13449 | 432 |
by (simp add: le_eq_less_or_eq) |
433 |
||
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|
434 |
lemma le_less_trans: "[| i \<le> j; j < k |] ==> i < (k::nat)" |
13449 | 435 |
by (blast dest!: le_imp_less_or_eq intro: less_trans) |
436 |
||
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|
437 |
lemma less_le_trans: "[| i < j; j \<le> k |] ==> i < (k::nat)" |
13449 | 438 |
by (blast dest!: le_imp_less_or_eq intro: less_trans) |
439 |
||
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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|
440 |
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)" |
13449 | 441 |
by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans) |
442 |
||
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|
443 |
lemma le_anti_sym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)" |
13449 | 444 |
by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym) |
445 |
||
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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changeset
|
446 |
lemma Suc_le_mono [iff]: "(Suc n \<le> Suc m) = (n \<le> m)" |
13449 | 447 |
by (simp add: le_simps) |
448 |
||
449 |
text {* Axiom @{text order_less_le} of class @{text order}: *} |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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|
450 |
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)" |
13449 | 451 |
by (simp add: le_def nat_neq_iff) (blast elim!: less_asym) |
452 |
||
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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changeset
|
453 |
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n" |
13449 | 454 |
by (rule iffD2, rule nat_less_le, rule conjI) |
455 |
||
456 |
text {* Axiom @{text linorder_linear} of class @{text linorder}: *} |
|
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|
457 |
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m" |
13449 | 458 |
apply (simp add: le_eq_less_or_eq) |
459 |
using less_linear |
|
460 |
apply blast |
|
461 |
done |
|
462 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
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diff
changeset
|
463 |
text {* Type {@typ nat} is a wellfounded linear order *} |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
464 |
|
14691 | 465 |
instance nat :: "{order, linorder, wellorder}" |
466 |
by intro_classes |
|
467 |
(assumption | |
|
468 |
rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+ |
|
14341
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Ring_and_Field now requires axiom add_left_imp_eq for semirings.
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parents:
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diff
changeset
|
469 |
|
13449 | 470 |
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)" |
471 |
by (blast elim!: less_SucE) |
|
472 |
||
473 |
text {* |
|
474 |
Rewrite @{term "n < Suc m"} to @{term "n = m"} |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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|
475 |
if @{term "~ n < m"} or @{term "m \<le> n"} hold. |
13449 | 476 |
Not suitable as default simprules because they often lead to looping |
477 |
*} |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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changeset
|
478 |
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)" |
13449 | 479 |
by (rule not_less_less_Suc_eq, rule leD) |
480 |
||
481 |
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq |
|
482 |
||
483 |
||
484 |
text {* |
|
485 |
Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}. |
|
486 |
No longer added as simprules (they loop) |
|
487 |
but via @{text reorient_simproc} in Bin |
|
488 |
*} |
|
489 |
||
490 |
text {* Polymorphic, not just for @{typ nat} *} |
|
491 |
lemma zero_reorient: "(0 = x) = (x = 0)" |
|
492 |
by auto |
|
493 |
||
494 |
lemma one_reorient: "(1 = x) = (x = 1)" |
|
495 |
by auto |
|
496 |
||
497 |
subsection {* Arithmetic operators *} |
|
1660 | 498 |
|
12338
de0f4a63baa5
renamed class "term" to "type" (actually "HOL.type");
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parents:
11451
diff
changeset
|
499 |
axclass power < type |
10435 | 500 |
|
3370
5c5fdce3a4e4
Overloading of "^" requires new type class "power", with types "nat" and
paulson
parents:
2608
diff
changeset
|
501 |
consts |
13449 | 502 |
power :: "('a::power) => nat => 'a" (infixr "^" 80) |
3370
5c5fdce3a4e4
Overloading of "^" requires new type class "power", with types "nat" and
paulson
parents:
2608
diff
changeset
|
503 |
|
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
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7702
diff
changeset
|
504 |
|
13449 | 505 |
text {* arithmetic operators @{text "+ -"} and @{text "*"} *} |
506 |
||
14691 | 507 |
instance nat :: "{plus, minus, times, power}" .. |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
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diff
changeset
|
508 |
|
13449 | 509 |
text {* size of a datatype value; overloaded *} |
510 |
consts size :: "'a => nat" |
|
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset
|
511 |
|
13449 | 512 |
primrec |
513 |
add_0: "0 + n = n" |
|
514 |
add_Suc: "Suc m + n = Suc (m + n)" |
|
515 |
||
516 |
primrec |
|
517 |
diff_0: "m - 0 = m" |
|
518 |
diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)" |
|
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset
|
519 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset
|
520 |
primrec |
13449 | 521 |
mult_0: "0 * n = 0" |
522 |
mult_Suc: "Suc m * n = n + (m * n)" |
|
523 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
524 |
text {* These two rules ease the use of primitive recursion. |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
525 |
NOTE USE OF @{text "=="} *} |
13449 | 526 |
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c" |
527 |
by simp |
|
528 |
||
529 |
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)" |
|
530 |
by simp |
|
531 |
||
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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diff
changeset
|
532 |
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m" |
13449 | 533 |
by (case_tac n) simp_all |
534 |
||
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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changeset
|
535 |
lemma gr_implies_not0: "!!n::nat. m<n ==> n \<noteq> 0" |
13449 | 536 |
by (case_tac n) simp_all |
537 |
||
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
538 |
lemma neq0_conv [iff]: "!!n::nat. (n \<noteq> 0) = (0 < n)" |
13449 | 539 |
by (case_tac n) simp_all |
540 |
||
541 |
text {* This theorem is useful with @{text blast} *} |
|
542 |
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n" |
|
543 |
by (rule iffD1, rule neq0_conv, rules) |
|
544 |
||
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
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diff
changeset
|
545 |
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)" |
13449 | 546 |
by (fast intro: not0_implies_Suc) |
547 |
||
548 |
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)" |
|
549 |
apply (rule iffI) |
|
14208 | 550 |
apply (rule ccontr, simp_all) |
13449 | 551 |
done |
552 |
||
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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changeset
|
553 |
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)" |
13449 | 554 |
by (induct m') simp_all |
555 |
||
556 |
text {* Useful in certain inductive arguments *} |
|
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
557 |
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))" |
13449 | 558 |
by (case_tac m) simp_all |
559 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
560 |
lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n" |
13449 | 561 |
apply (rule nat_less_induct) |
562 |
apply (case_tac n) |
|
563 |
apply (case_tac [2] nat) |
|
564 |
apply (blast intro: less_trans)+ |
|
565 |
done |
|
566 |
||
567 |
subsection {* @{text LEAST} theorems for type @{typ nat} by specialization *} |
|
568 |
||
569 |
lemmas LeastI = wellorder_LeastI |
|
570 |
lemmas Least_le = wellorder_Least_le |
|
571 |
lemmas not_less_Least = wellorder_not_less_Least |
|
572 |
||
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
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diff
changeset
|
573 |
lemma Least_Suc: |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
574 |
"[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))" |
14208 | 575 |
apply (case_tac "n", auto) |
13449 | 576 |
apply (frule LeastI) |
577 |
apply (drule_tac P = "%x. P (Suc x) " in LeastI) |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
578 |
apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))") |
13449 | 579 |
apply (erule_tac [2] Least_le) |
14208 | 580 |
apply (case_tac "LEAST x. P x", auto) |
13449 | 581 |
apply (drule_tac P = "%x. P (Suc x) " in Least_le) |
582 |
apply (blast intro: order_antisym) |
|
583 |
done |
|
584 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
585 |
lemma Least_Suc2: |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
586 |
"[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
587 |
by (erule (1) Least_Suc [THEN ssubst], simp) |
13449 | 588 |
|
589 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
14208
diff
changeset
|
590 |
|
13449 | 591 |
subsection {* @{term min} and @{term max} *} |
592 |
||
593 |
lemma min_0L [simp]: "min 0 n = (0::nat)" |
|
594 |
by (rule min_leastL) simp |
|
595 |
||
596 |
lemma min_0R [simp]: "min n 0 = (0::nat)" |
|
597 |
by (rule min_leastR) simp |
|
598 |
||
599 |
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)" |
|
600 |
by (simp add: min_of_mono) |
|
601 |
||
602 |
lemma max_0L [simp]: "max 0 n = (n::nat)" |
|
603 |
by (rule max_leastL) simp |
|
604 |
||
605 |
lemma max_0R [simp]: "max n 0 = (n::nat)" |
|
606 |
by (rule max_leastR) simp |
|
607 |
||
608 |
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)" |
|
609 |
by (simp add: max_of_mono) |
|
610 |
||
611 |
||
612 |
subsection {* Basic rewrite rules for the arithmetic operators *} |
|
613 |
||
614 |
text {* Difference *} |
|
615 |
||
14193
30e41f63712e
Improved efficiency of code generated for + and -
berghofe
parents:
14131
diff
changeset
|
616 |
lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)" |
15251 | 617 |
by (induct n) simp_all |
13449 | 618 |
|
14193
30e41f63712e
Improved efficiency of code generated for + and -
berghofe
parents:
14131
diff
changeset
|
619 |
lemma diff_Suc_Suc [simp, code]: "Suc(m) - Suc(n) = m - n" |
15251 | 620 |
by (induct n) simp_all |
13449 | 621 |
|
622 |
||
623 |
text {* |
|
624 |
Could be (and is, below) generalized in various ways |
|
625 |
However, none of the generalizations are currently in the simpset, |
|
626 |
and I dread to think what happens if I put them in |
|
627 |
*} |
|
628 |
lemma Suc_pred [simp]: "0 < n ==> Suc (n - Suc 0) = n" |
|
629 |
by (simp split add: nat.split) |
|
630 |
||
14193
30e41f63712e
Improved efficiency of code generated for + and -
berghofe
parents:
14131
diff
changeset
|
631 |
declare diff_Suc [simp del, code del] |
13449 | 632 |
|
633 |
||
634 |
subsection {* Addition *} |
|
635 |
||
636 |
lemma add_0_right [simp]: "m + 0 = (m::nat)" |
|
637 |
by (induct m) simp_all |
|
638 |
||
639 |
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)" |
|
640 |
by (induct m) simp_all |
|
641 |
||
14193
30e41f63712e
Improved efficiency of code generated for + and -
berghofe
parents:
14131
diff
changeset
|
642 |
lemma [code]: "Suc m + n = m + Suc n" by simp |
30e41f63712e
Improved efficiency of code generated for + and -
berghofe
parents:
14131
diff
changeset
|
643 |
|
13449 | 644 |
|
645 |
text {* Associative law for addition *} |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
646 |
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)" |
13449 | 647 |
by (induct m) simp_all |
648 |
||
649 |
text {* Commutative law for addition *} |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
650 |
lemma nat_add_commute: "m + n = n + (m::nat)" |
13449 | 651 |
by (induct m) simp_all |
652 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
653 |
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)" |
13449 | 654 |
apply (rule mk_left_commute [of "op +"]) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
655 |
apply (rule nat_add_assoc) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
656 |
apply (rule nat_add_commute) |
13449 | 657 |
done |
658 |
||
14331 | 659 |
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))" |
13449 | 660 |
by (induct k) simp_all |
661 |
||
14331 | 662 |
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))" |
13449 | 663 |
by (induct k) simp_all |
664 |
||
14331 | 665 |
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))" |
13449 | 666 |
by (induct k) simp_all |
667 |
||
14331 | 668 |
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))" |
13449 | 669 |
by (induct k) simp_all |
670 |
||
671 |
text {* Reasoning about @{text "m + 0 = 0"}, etc. *} |
|
672 |
||
673 |
lemma add_is_0 [iff]: "!!m::nat. (m + n = 0) = (m = 0 & n = 0)" |
|
674 |
by (case_tac m) simp_all |
|
675 |
||
676 |
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)" |
|
677 |
by (case_tac m) simp_all |
|
678 |
||
679 |
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)" |
|
680 |
by (rule trans, rule eq_commute, rule add_is_1) |
|
681 |
||
682 |
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m | 0 < n)" |
|
683 |
by (simp del: neq0_conv add: neq0_conv [symmetric]) |
|
684 |
||
685 |
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0" |
|
686 |
apply (drule add_0_right [THEN ssubst]) |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
687 |
apply (simp add: nat_add_assoc del: add_0_right) |
13449 | 688 |
done |
689 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
690 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
691 |
subsection {* Multiplication *} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
692 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
693 |
text {* right annihilation in product *} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
694 |
lemma mult_0_right [simp]: "(m::nat) * 0 = 0" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
695 |
by (induct m) simp_all |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
696 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
697 |
text {* right successor law for multiplication *} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
698 |
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
699 |
by (induct m) (simp_all add: nat_add_left_commute) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
700 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
701 |
text {* Commutative law for multiplication *} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
702 |
lemma nat_mult_commute: "m * n = n * (m::nat)" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
703 |
by (induct m) simp_all |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
704 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
705 |
text {* addition distributes over multiplication *} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
706 |
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
707 |
by (induct m) (simp_all add: nat_add_assoc nat_add_left_commute) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
708 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
709 |
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
710 |
by (induct m) (simp_all add: nat_add_assoc) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
711 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
712 |
text {* Associative law for multiplication *} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
713 |
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
714 |
by (induct m) (simp_all add: add_mult_distrib) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
715 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
716 |
|
14740 | 717 |
text{*The naturals form a @{text comm_semiring_1_cancel}*} |
14738 | 718 |
instance nat :: comm_semiring_1_cancel |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
719 |
proof |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
720 |
fix i j k :: nat |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
721 |
show "(i + j) + k = i + (j + k)" by (rule nat_add_assoc) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
722 |
show "i + j = j + i" by (rule nat_add_commute) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
723 |
show "0 + i = i" by simp |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
724 |
show "(i * j) * k = i * (j * k)" by (rule nat_mult_assoc) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
725 |
show "i * j = j * i" by (rule nat_mult_commute) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
726 |
show "1 * i = i" by simp |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
727 |
show "(i + j) * k = i * k + j * k" by (simp add: add_mult_distrib) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
728 |
show "0 \<noteq> (1::nat)" by simp |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
729 |
assume "k+i = k+j" thus "i=j" by simp |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
730 |
qed |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
731 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
732 |
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)" |
15251 | 733 |
apply (induct m) |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
734 |
apply (induct_tac [2] n, simp_all) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
735 |
done |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
736 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
737 |
subsection {* Monotonicity of Addition *} |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
738 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
739 |
text {* strict, in 1st argument *} |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
740 |
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)" |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
741 |
by (induct k) simp_all |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
742 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
743 |
text {* strict, in both arguments *} |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
744 |
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)" |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
745 |
apply (rule add_less_mono1 [THEN less_trans], assumption+) |
15251 | 746 |
apply (induct j, simp_all) |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
747 |
done |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
748 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
749 |
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *} |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
750 |
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))" |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
751 |
apply (induct n) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
752 |
apply (simp_all add: order_le_less) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
753 |
apply (blast elim!: less_SucE |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
754 |
intro!: add_0_right [symmetric] add_Suc_right [symmetric]) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
755 |
done |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
756 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
757 |
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *} |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
758 |
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j" |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
759 |
apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
760 |
apply (induct_tac x) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
761 |
apply (simp_all add: add_less_mono) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
762 |
done |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
763 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
764 |
|
14740 | 765 |
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*} |
14738 | 766 |
instance nat :: ordered_semidom |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
767 |
proof |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
768 |
fix i j k :: nat |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
769 |
show "0 < (1::nat)" by simp |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
770 |
show "i \<le> j ==> k + i \<le> k + j" by simp |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
771 |
show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
772 |
qed |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
773 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
774 |
lemma nat_mult_1: "(1::nat) * n = n" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
775 |
by simp |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
776 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
777 |
lemma nat_mult_1_right: "n * (1::nat) = n" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
778 |
by simp |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
779 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
780 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
781 |
subsection {* Additional theorems about "less than" *} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
782 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
783 |
text {* A [clumsy] way of lifting @{text "<"} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
784 |
monotonicity to @{text "\<le>"} monotonicity *} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
785 |
lemma less_mono_imp_le_mono: |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
786 |
assumes lt_mono: "!!i j::nat. i < j ==> f i < f j" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
787 |
and le: "i \<le> j" shows "f i \<le> ((f j)::nat)" using le |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
788 |
apply (simp add: order_le_less) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
789 |
apply (blast intro!: lt_mono) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
790 |
done |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
791 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
792 |
text {* non-strict, in 1st argument *} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
793 |
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
794 |
by (rule add_right_mono) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
795 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
796 |
text {* non-strict, in both arguments *} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
797 |
lemma add_le_mono: "[| i \<le> j; k \<le> l |] ==> i + k \<le> j + (l::nat)" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
798 |
by (rule add_mono) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
799 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
800 |
lemma le_add2: "n \<le> ((m + n)::nat)" |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
801 |
by (insert add_right_mono [of 0 m n], simp) |
13449 | 802 |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
803 |
lemma le_add1: "n \<le> ((n + m)::nat)" |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
804 |
by (simp add: add_commute, rule le_add2) |
13449 | 805 |
|
806 |
lemma less_add_Suc1: "i < Suc (i + m)" |
|
807 |
by (rule le_less_trans, rule le_add1, rule lessI) |
|
808 |
||
809 |
lemma less_add_Suc2: "i < Suc (m + i)" |
|
810 |
by (rule le_less_trans, rule le_add2, rule lessI) |
|
811 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
812 |
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))" |
13449 | 813 |
by (rules intro!: less_add_Suc1 less_imp_Suc_add) |
814 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
815 |
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m" |
13449 | 816 |
by (rule le_trans, assumption, rule le_add1) |
817 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
818 |
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j" |
13449 | 819 |
by (rule le_trans, assumption, rule le_add2) |
820 |
||
821 |
lemma trans_less_add1: "(i::nat) < j ==> i < j + m" |
|
822 |
by (rule less_le_trans, assumption, rule le_add1) |
|
823 |
||
824 |
lemma trans_less_add2: "(i::nat) < j ==> i < m + j" |
|
825 |
by (rule less_le_trans, assumption, rule le_add2) |
|
826 |
||
827 |
lemma add_lessD1: "i + j < (k::nat) ==> i < k" |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
828 |
apply (rule le_less_trans [of _ "i+j"]) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
829 |
apply (simp_all add: le_add1) |
13449 | 830 |
done |
831 |
||
832 |
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))" |
|
833 |
apply (rule notI) |
|
834 |
apply (erule add_lessD1 [THEN less_irrefl]) |
|
835 |
done |
|
836 |
||
837 |
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))" |
|
838 |
by (simp add: add_commute not_add_less1) |
|
839 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
840 |
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)" |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
841 |
apply (rule order_trans [of _ "m+k"]) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
842 |
apply (simp_all add: le_add1) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
843 |
done |
13449 | 844 |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
845 |
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)" |
13449 | 846 |
apply (simp add: add_commute) |
847 |
apply (erule add_leD1) |
|
848 |
done |
|
849 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
850 |
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R" |
13449 | 851 |
by (blast dest: add_leD1 add_leD2) |
852 |
||
853 |
text {* needs @{text "!!k"} for @{text add_ac} to work *} |
|
854 |
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n" |
|
855 |
by (force simp del: add_Suc_right |
|
856 |
simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac) |
|
857 |
||
858 |
||
859 |
||
860 |
subsection {* Difference *} |
|
861 |
||
862 |
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0" |
|
863 |
by (induct m) simp_all |
|
864 |
||
865 |
text {* Addition is the inverse of subtraction: |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
866 |
if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *} |
13449 | 867 |
lemma add_diff_inverse: "~ m < n ==> n + (m - n) = (m::nat)" |
868 |
by (induct m n rule: diff_induct) simp_all |
|
869 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
870 |
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)" |
13449 | 871 |
by (simp add: add_diff_inverse not_less_iff_le) |
872 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
873 |
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)" |
13449 | 874 |
by (simp add: le_add_diff_inverse add_commute) |
875 |
||
876 |
||
877 |
subsection {* More results about difference *} |
|
878 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
879 |
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)" |
13449 | 880 |
by (induct m n rule: diff_induct) simp_all |
881 |
||
882 |
lemma diff_less_Suc: "m - n < Suc m" |
|
883 |
apply (induct m n rule: diff_induct) |
|
884 |
apply (erule_tac [3] less_SucE) |
|
885 |
apply (simp_all add: less_Suc_eq) |
|
886 |
done |
|
887 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
888 |
lemma diff_le_self [simp]: "m - n \<le> (m::nat)" |
13449 | 889 |
by (induct m n rule: diff_induct) (simp_all add: le_SucI) |
890 |
||
891 |
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k" |
|
892 |
by (rule le_less_trans, rule diff_le_self) |
|
893 |
||
894 |
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)" |
|
895 |
by (induct i j rule: diff_induct) simp_all |
|
896 |
||
897 |
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k" |
|
898 |
by (simp add: diff_diff_left) |
|
899 |
||
900 |
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n" |
|
14208 | 901 |
apply (case_tac "n", safe) |
13449 | 902 |
apply (simp add: le_simps) |
903 |
done |
|
904 |
||
905 |
text {* This and the next few suggested by Florian Kammueller *} |
|
906 |
lemma diff_commute: "(i::nat) - j - k = i - k - j" |
|
907 |
by (simp add: diff_diff_left add_commute) |
|
908 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
909 |
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)" |
13449 | 910 |
by (induct j k rule: diff_induct) simp_all |
911 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
912 |
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i" |
13449 | 913 |
by (simp add: add_commute diff_add_assoc) |
914 |
||
915 |
lemma diff_add_inverse: "(n + m) - n = (m::nat)" |
|
916 |
by (induct n) simp_all |
|
917 |
||
918 |
lemma diff_add_inverse2: "(m + n) - n = (m::nat)" |
|
919 |
by (simp add: diff_add_assoc) |
|
920 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
921 |
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)" |
13449 | 922 |
apply safe |
923 |
apply (simp_all add: diff_add_inverse2) |
|
924 |
done |
|
925 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
926 |
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)" |
13449 | 927 |
by (induct m n rule: diff_induct) simp_all |
928 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
929 |
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0" |
13449 | 930 |
by (rule iffD2, rule diff_is_0_eq) |
931 |
||
932 |
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)" |
|
933 |
by (induct m n rule: diff_induct) simp_all |
|
934 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
935 |
lemma less_imp_add_positive: "i < j ==> \<exists>k::nat. 0 < k & i + k = j" |
13449 | 936 |
apply (rule_tac x = "j - i" in exI) |
937 |
apply (simp (no_asm_simp) add: add_diff_inverse less_not_sym) |
|
938 |
done |
|
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset
|
939 |
|
13449 | 940 |
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)" |
941 |
apply (induct k i rule: diff_induct) |
|
942 |
apply (simp_all (no_asm)) |
|
943 |
apply rules |
|
944 |
done |
|
945 |
||
946 |
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0" |
|
947 |
apply (rule diff_self_eq_0 [THEN subst]) |
|
14208 | 948 |
apply (rule zero_induct_lemma, rules+) |
13449 | 949 |
done |
950 |
||
951 |
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)" |
|
952 |
by (induct k) simp_all |
|
953 |
||
954 |
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)" |
|
955 |
by (simp add: diff_cancel add_commute) |
|
956 |
||
957 |
lemma diff_add_0: "n - (n + m) = (0::nat)" |
|
958 |
by (induct n) simp_all |
|
959 |
||
960 |
||
961 |
text {* Difference distributes over multiplication *} |
|
962 |
||
963 |
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)" |
|
964 |
by (induct m n rule: diff_induct) (simp_all add: diff_cancel) |
|
965 |
||
966 |
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)" |
|
967 |
by (simp add: diff_mult_distrib mult_commute [of k]) |
|
968 |
-- {* NOT added as rewrites, since sometimes they are used from right-to-left *} |
|
969 |
||
970 |
lemmas nat_distrib = |
|
971 |
add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2 |
|
972 |
||
973 |
||
974 |
subsection {* Monotonicity of Multiplication *} |
|
975 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
976 |
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k" |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
977 |
by (simp add: mult_right_mono) |
13449 | 978 |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
979 |
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j" |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
980 |
by (simp add: mult_left_mono) |
13449 | 981 |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
982 |
text {* @{text "\<le>"} monotonicity, BOTH arguments *} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
983 |
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l" |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
984 |
by (simp add: mult_mono) |
13449 | 985 |
|
986 |
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k" |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
987 |
by (simp add: mult_strict_right_mono) |
13449 | 988 |
|
14266 | 989 |
text{*Differs from the standard @{text zero_less_mult_iff} in that |
990 |
there are no negative numbers.*} |
|
991 |
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)" |
|
13449 | 992 |
apply (induct m) |
14208 | 993 |
apply (case_tac [2] n, simp_all) |
13449 | 994 |
done |
995 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
996 |
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)" |
13449 | 997 |
apply (induct m) |
14208 | 998 |
apply (case_tac [2] n, simp_all) |
13449 | 999 |
done |
1000 |
||
1001 |
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)" |
|
15251 | 1002 |
apply (induct m, simp) |
1003 |
apply (induct n, simp, fastsimp) |
|
13449 | 1004 |
done |
1005 |
||
1006 |
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)" |
|
1007 |
apply (rule trans) |
|
14208 | 1008 |
apply (rule_tac [2] mult_eq_1_iff, fastsimp) |
13449 | 1009 |
done |
1010 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
1011 |
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)" |
13449 | 1012 |
apply (safe intro!: mult_less_mono1) |
14208 | 1013 |
apply (case_tac k, auto) |
13449 | 1014 |
apply (simp del: le_0_eq add: linorder_not_le [symmetric]) |
1015 |
apply (blast intro: mult_le_mono1) |
|
1016 |
done |
|
1017 |
||
1018 |
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)" |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
1019 |
by (simp add: mult_commute [of k]) |
13449 | 1020 |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
1021 |
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)" |
14208 | 1022 |
by (simp add: linorder_not_less [symmetric], auto) |
13449 | 1023 |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
1024 |
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)" |
14208 | 1025 |
by (simp add: linorder_not_less [symmetric], auto) |
13449 | 1026 |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
1027 |
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))" |
14208 | 1028 |
apply (cut_tac less_linear, safe, auto) |
13449 | 1029 |
apply (drule mult_less_mono1, assumption, simp)+ |
1030 |
done |
|
1031 |
||
1032 |
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))" |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
1033 |
by (simp add: mult_commute [of k]) |
13449 | 1034 |
|
1035 |
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)" |
|
1036 |
by (subst mult_less_cancel1) simp |
|
1037 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
1038 |
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)" |
13449 | 1039 |
by (subst mult_le_cancel1) simp |
1040 |
||
1041 |
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)" |
|
1042 |
by (subst mult_cancel1) simp |
|
1043 |
||
1044 |
text {* Lemma for @{text gcd} *} |
|
1045 |
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0" |
|
1046 |
apply (drule sym) |
|
1047 |
apply (rule disjCI) |
|
1048 |
apply (rule nat_less_cases, erule_tac [2] _) |
|
1049 |
apply (fastsimp elim!: less_SucE) |
|
1050 |
apply (fastsimp dest: mult_less_mono2) |
|
1051 |
done |
|
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset
|
1052 |
|
923 | 1053 |
end |