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theory Nat(* Title: HOL/Nat.thy ID: $Id: Nat.thy,v 1.138 2008/04/25 13:30:34 krauss Exp $ Author: Tobias Nipkow and Lawrence C Paulson and Markus Wenzel Type "nat" is a linear order, and a datatype; arithmetic operators + - and * (for div, mod and dvd, see theory Divides). *) header {* Natural numbers *} theory Nat imports Inductive Ring_and_Field uses "~~/src/Tools/rat.ML" "~~/src/Provers/Arith/cancel_sums.ML" ("arith_data.ML") "~~/src/Provers/Arith/fast_lin_arith.ML" ("Tools/lin_arith.ML") begin subsection {* Type @{text ind} *} typedecl ind axiomatization Zero_Rep :: ind and Suc_Rep :: "ind => ind" where -- {* the axiom of infinity in 2 parts *} inj_Suc_Rep: "inj Suc_Rep" and Suc_Rep_not_Zero_Rep: "Suc_Rep x ≠ Zero_Rep" subsection {* Type nat *} text {* Type definition *} inductive Nat :: "ind => bool" where Zero_RepI: "Nat Zero_Rep" | Suc_RepI: "Nat i ==> Nat (Suc_Rep i)" global typedef (open Nat) nat = "Collect Nat" by (rule exI, rule CollectI, rule Nat.Zero_RepI) constdefs Suc :: "nat => nat" Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))" local instantiation nat :: zero begin definition Zero_nat_def [code func del]: "0 = Abs_Nat Zero_Rep" instance .. end lemma nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n" apply (unfold Zero_nat_def Suc_def) apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *} apply (erule Rep_Nat [THEN CollectD, THEN Nat.induct]) apply (iprover elim: Abs_Nat_inverse [OF CollectI, THEN subst]) done lemma Suc_not_Zero [iff]: "Suc m ≠ 0" by (simp add: Zero_nat_def Suc_def Abs_Nat_inject Rep_Nat [THEN CollectD] Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep) lemma Zero_not_Suc [iff]: "0 ≠ Suc m" by (rule not_sym, rule Suc_not_Zero not_sym) lemma inj_Suc[simp]: "inj_on Suc N" by (simp add: Suc_def inj_on_def Abs_Nat_inject Rep_Nat [THEN CollectD] Suc_RepI inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject) lemma Suc_Suc_eq [iff]: "Suc m = Suc n <-> m = n" by (rule inj_Suc [THEN inj_eq]) rep_datatype nat distinct Suc_not_Zero Zero_not_Suc inject Suc_Suc_eq induction nat_induct declare nat.induct [case_names 0 Suc, induct type: nat] declare nat.exhaust [case_names 0 Suc, cases type: nat] lemmas nat_rec_0 = nat.recs(1) and nat_rec_Suc = nat.recs(2) lemmas nat_case_0 = nat.cases(1) and nat_case_Suc = nat.cases(2) text {* Injectiveness and distinctness lemmas *} lemma Suc_neq_Zero: "Suc m = 0 ==> R" by (rule notE, rule Suc_not_Zero) lemma Zero_neq_Suc: "0 = Suc m ==> R" by (rule Suc_neq_Zero, erule sym) lemma Suc_inject: "Suc x = Suc y ==> x = y" by (rule inj_Suc [THEN injD]) lemma n_not_Suc_n: "n ≠ Suc n" by (induct n) simp_all lemma Suc_n_not_n: "Suc n ≠ n" by (rule not_sym, rule n_not_Suc_n) text {* A special form of induction for reasoning about @{term "m < n"} and @{term "m - n"} *} lemma diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==> (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n" apply (rule_tac x = m in spec) apply (induct n) prefer 2 apply (rule allI) apply (induct_tac x, iprover+) done subsection {* Arithmetic operators *} instantiation nat :: "{minus, comm_monoid_add}" begin primrec plus_nat where add_0: "0 + n = (n::nat)" | add_Suc: "Suc m + n = Suc (m + n)" lemma add_0_right [simp]: "m + 0 = (m::nat)" by (induct m) simp_all lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)" by (induct m) simp_all lemma add_Suc_shift [code]: "Suc m + n = m + Suc n" by simp primrec minus_nat where diff_0: "m - 0 = (m::nat)" | diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)" declare diff_Suc [simp del, code del] lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)" by (induct n) (simp_all add: diff_Suc) lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n" by (induct n) (simp_all add: diff_Suc) instance proof fix n m q :: nat show "(n + m) + q = n + (m + q)" by (induct n) simp_all show "n + m = m + n" by (induct n) simp_all show "0 + n = n" by simp qed end instantiation nat :: comm_semiring_1_cancel begin definition One_nat_def [simp]: "1 = Suc 0" primrec times_nat where mult_0: "0 * n = (0::nat)" | mult_Suc: "Suc m * n = n + (m * n)" lemma mult_0_right [simp]: "(m::nat) * 0 = 0" by (induct m) simp_all lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)" by (induct m) (simp_all add: add_left_commute) lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)" by (induct m) (simp_all add: add_assoc) instance proof fix n m q :: nat show "0 ≠ (1::nat)" by simp show "1 * n = n" by simp show "n * m = m * n" by (induct n) simp_all show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib) show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib) assume "n + m = n + q" thus "m = q" by (induct n) simp_all qed end subsubsection {* Addition *} lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)" by (rule add_assoc) lemma nat_add_commute: "m + n = n + (m::nat)" by (rule add_commute) lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)" by (rule add_left_commute) lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))" by (rule add_left_cancel) lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))" by (rule add_right_cancel) text {* Reasoning about @{text "m + 0 = 0"}, etc. *} lemma add_is_0 [iff]: fixes m n :: nat shows "(m + n = 0) = (m = 0 & n = 0)" by (cases m) simp_all lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)" by (cases m) simp_all lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)" by (rule trans, rule eq_commute, rule add_is_1) lemma add_eq_self_zero: fixes m n :: nat shows "m + n = m ==> n = 0" by (induct m) simp_all lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N" apply (induct k) apply simp apply(drule comp_inj_on[OF _ inj_Suc]) apply (simp add:o_def) done subsubsection {* Difference *} lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0" by (induct m) simp_all lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)" by (induct i j rule: diff_induct) simp_all lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k" by (simp add: diff_diff_left) lemma diff_commute: "(i::nat) - j - k = i - k - j" by (simp add: diff_diff_left add_commute) lemma diff_add_inverse: "(n + m) - n = (m::nat)" by (induct n) simp_all lemma diff_add_inverse2: "(m + n) - n = (m::nat)" by (simp add: diff_add_inverse add_commute [of m n]) lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)" by (induct k) simp_all lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)" by (simp add: diff_cancel add_commute) lemma diff_add_0: "n - (n + m) = (0::nat)" by (induct n) simp_all text {* Difference distributes over multiplication *} lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)" by (induct m n rule: diff_induct) (simp_all add: diff_cancel) lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)" by (simp add: diff_mult_distrib mult_commute [of k]) -- {* NOT added as rewrites, since sometimes they are used from right-to-left *} subsubsection {* Multiplication *} lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)" by (rule mult_assoc) lemma nat_mult_commute: "m * n = n * (m::nat)" by (rule mult_commute) lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)" by (rule right_distrib) lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)" by (induct m) auto lemmas nat_distrib = add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2 lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)" apply (induct m) apply simp apply (induct n) apply auto done lemma one_eq_mult_iff [simp,noatp]: "(Suc 0 = m * n) = (m = 1 & n = 1)" apply (rule trans) apply (rule_tac [2] mult_eq_1_iff, fastsimp) done lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))" proof - have "k ≠ 0 ==> k * m = k * n ==> m = n" proof (induct n arbitrary: m) case 0 then show "m = 0" by simp next case (Suc n) then show "m = Suc n" by (cases m) (simp_all add: eq_commute [of "0"]) qed then show ?thesis by auto qed lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))" by (simp add: mult_commute) lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)" by (subst mult_cancel1) simp subsection {* Orders on @{typ nat} *} subsubsection {* Operation definition *} instantiation nat :: linorder begin primrec less_eq_nat where "(0::nat) ≤ n <-> True" | "Suc m ≤ n <-> (case n of 0 => False | Suc n => m ≤ n)" declare less_eq_nat.simps [simp del, code del] lemma [code]: "(0::nat) ≤ n <-> True" by (simp add: less_eq_nat.simps) lemma le0 [iff]: "0 ≤ (n::nat)" by (simp add: less_eq_nat.simps) definition less_nat where less_eq_Suc_le [code func del]: "n < m <-> Suc n ≤ m" lemma Suc_le_mono [iff]: "Suc n ≤ Suc m <-> n ≤ m" by (simp add: less_eq_nat.simps(2)) lemma Suc_le_eq [code]: "Suc m ≤ n <-> m < n" unfolding less_eq_Suc_le .. lemma le_0_eq [iff]: "(n::nat) ≤ 0 <-> n = 0" by (induct n) (simp_all add: less_eq_nat.simps(2)) lemma not_less0 [iff]: "¬ n < (0::nat)" by (simp add: less_eq_Suc_le) lemma less_nat_zero_code [code]: "n < (0::nat) <-> False" by simp lemma Suc_less_eq [iff]: "Suc m < Suc n <-> m < n" by (simp add: less_eq_Suc_le) lemma less_Suc_eq_le [code]: "m < Suc n <-> m ≤ n" by (simp add: less_eq_Suc_le) lemma le_SucI: "m ≤ n ==> m ≤ Suc n" by (induct m arbitrary: n) (simp_all add: less_eq_nat.simps(2) split: nat.splits) lemma Suc_leD: "Suc m ≤ n ==> m ≤ n" by (cases n) (auto intro: le_SucI) lemma less_SucI: "m < n ==> m < Suc n" by (simp add: less_eq_Suc_le) (erule Suc_leD) lemma Suc_lessD: "Suc m < n ==> m < n" by (simp add: less_eq_Suc_le) (erule Suc_leD) instance proof fix n m :: nat have less_imp_le: "n < m ==> n ≤ m" unfolding less_eq_Suc_le by (erule Suc_leD) have irrefl: "¬ m < m" by (induct m) auto have strict: "n ≤ m ==> n ≠ m ==> n < m" proof (induct n arbitrary: m) case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le) next case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le) qed show "n < m <-> n ≤ m ∧ n ≠ m" by (auto simp add: irrefl intro: less_imp_le strict) next fix n :: nat show "n ≤ n" by (induct n) simp_all next fix n m :: nat assume "n ≤ m" and "m ≤ n" then show "n = m" by (induct n arbitrary: m) (simp_all add: less_eq_nat.simps(2) split: nat.splits) next fix n m q :: nat assume "n ≤ m" and "m ≤ q" then show "n ≤ q" proof (induct n arbitrary: m q) case 0 show ?case by simp next case (Suc n) then show ?case by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify, simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify, simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits) qed next fix n m :: nat show "n ≤ m ∨ m ≤ n" by (induct n arbitrary: m) (simp_all add: less_eq_nat.simps(2) split: nat.splits) qed end subsubsection {* Introduction properties *} lemma lessI [iff]: "n < Suc n" by (simp add: less_Suc_eq_le) lemma zero_less_Suc [iff]: "0 < Suc n" by (simp add: less_Suc_eq_le) subsubsection {* Elimination properties *} lemma less_not_refl: "~ n < (n::nat)" by (rule order_less_irrefl) lemma less_not_refl2: "n < m ==> m ≠ (n::nat)" by (rule not_sym) (rule less_imp_neq) lemma less_not_refl3: "(s::nat) < t ==> s ≠ t" by (rule less_imp_neq) lemma less_irrefl_nat: "(n::nat) < n ==> R" by (rule notE, rule less_not_refl) lemma less_zeroE: "(n::nat) < 0 ==> R" by (rule notE) (rule not_less0) lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)" unfolding less_Suc_eq_le le_less .. lemma less_one [iff, noatp]: "(n < (1::nat)) = (n = 0)" by (simp add: less_Suc_eq) lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)" by (simp add: less_Suc_eq) lemma Suc_mono: "m < n ==> Suc m < Suc n" by simp text {* "Less than" is antisymmetric, sort of *} lemma less_antisym: "[| ¬ n < m; n < Suc m |] ==> m = n" unfolding not_less less_Suc_eq_le by (rule antisym) lemma nat_neq_iff: "((m::nat) ≠ n) = (m < n | n < m)" by (rule linorder_neq_iff) lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m" and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m" shows "P n m" apply (rule less_linear [THEN disjE]) apply (erule_tac [2] disjE) apply (erule lessCase) apply (erule sym [THEN eqCase]) apply (erule major) done subsubsection {* Inductive (?) properties *} lemma Suc_lessI: "m < n ==> Suc m ≠ n ==> Suc m < n" unfolding less_eq_Suc_le [of m] le_less by simp lemma lessE: assumes major: "i < k" and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P" shows P proof - from major have "∃j. i ≤ j ∧ k = Suc j" unfolding less_eq_Suc_le by (induct k) simp_all then have "(∃j. i < j ∧ k = Suc j) ∨ k = Suc i" by (clarsimp simp add: less_le) with p1 p2 show P by auto qed lemma less_SucE: assumes major: "m < Suc n" and less: "m < n ==> P" and eq: "m = n ==> P" shows P apply (rule major [THEN lessE]) apply (rule eq, blast) apply (rule less, blast) done lemma Suc_lessE: assumes major: "Suc i < k" and minor: "!!j. i < j ==> k = Suc j ==> P" shows P apply (rule major [THEN lessE]) apply (erule lessI [THEN minor]) apply (erule Suc_lessD [THEN minor], assumption) done lemma Suc_less_SucD: "Suc m < Suc n ==> m < n" by simp lemma less_trans_Suc: assumes le: "i < j" shows "j < k ==> Suc i < k" apply (induct k, simp_all) apply (insert le) apply (simp add: less_Suc_eq) apply (blast dest: Suc_lessD) done text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *} lemma not_less_eq: "¬ m < n <-> n < Suc m" unfolding not_less less_Suc_eq_le .. lemma not_less_eq_eq: "¬ m ≤ n <-> Suc n ≤ m" unfolding not_le Suc_le_eq .. text {* Properties of "less than or equal" *} lemma le_imp_less_Suc: "m ≤ n ==> m < Suc n" unfolding less_Suc_eq_le . lemma Suc_n_not_le_n: "~ Suc n ≤ n" unfolding not_le less_Suc_eq_le .. lemma le_Suc_eq: "(m ≤ Suc n) = (m ≤ n | m = Suc n)" by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq) lemma le_SucE: "m ≤ Suc n ==> (m ≤ n ==> R) ==> (m = Suc n ==> R) ==> R" by (drule le_Suc_eq [THEN iffD1], iprover+) lemma Suc_leI: "m < n ==> Suc(m) ≤ n" unfolding Suc_le_eq . text {* Stronger version of @{text Suc_leD} *} lemma Suc_le_lessD: "Suc m ≤ n ==> m < n" unfolding Suc_le_eq . lemma less_imp_le_nat: "m < n ==> m ≤ (n::nat)" unfolding less_eq_Suc_le by (rule Suc_leD) text {* For instance, @{text "(Suc m < Suc n) = (Suc m ≤ n) = (m < n)"} *} lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq text {* Equivalence of @{term "m ≤ n"} and @{term "m < n | m = n"} *} lemma less_or_eq_imp_le: "m < n | m = n ==> m ≤ (n::nat)" unfolding le_less . lemma le_eq_less_or_eq: "(m ≤ (n::nat)) = (m < n | m=n)" by (rule le_less) text {* Useful with @{text blast}. *} lemma eq_imp_le: "(m::nat) = n ==> m ≤ n" by auto lemma le_refl: "n ≤ (n::nat)" by simp lemma le_trans: "[| i ≤ j; j ≤ k |] ==> i ≤ (k::nat)" by (rule order_trans) lemma le_anti_sym: "[| m ≤ n; n ≤ m |] ==> m = (n::nat)" by (rule antisym) lemma nat_less_le: "((m::nat) < n) = (m ≤ n & m ≠ n)" by (rule less_le) lemma le_neq_implies_less: "(m::nat) ≤ n ==> m ≠ n ==> m < n" unfolding less_le .. lemma nat_le_linear: "(m::nat) ≤ n | n ≤ m" by (rule linear) lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat] lemma le_less_Suc_eq: "m ≤ n ==> (n < Suc m) = (n = m)" unfolding less_Suc_eq_le by auto lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)" unfolding not_less by (rule le_less_Suc_eq) lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq text {* These two rules ease the use of primitive recursion. NOTE USE OF @{text "=="} *} lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c" by simp lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)" by simp lemma not0_implies_Suc: "n ≠ 0 ==> ∃m. n = Suc m" by (cases n) simp_all lemma gr0_implies_Suc: "n > 0 ==> ∃m. n = Suc m" by (cases n) simp_all lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n ≠ 0" by (cases n) simp_all lemma neq0_conv[iff]: fixes n :: nat shows "(n ≠ 0) = (0 < n)" by (cases n) simp_all text {* This theorem is useful with @{text blast} *} lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n" by (rule neq0_conv[THEN iffD1], iprover) lemma gr0_conv_Suc: "(0 < n) = (∃m. n = Suc m)" by (fast intro: not0_implies_Suc) lemma not_gr0 [iff,noatp]: "!!n::nat. (~ (0 < n)) = (n = 0)" using neq0_conv by blast lemma Suc_le_D: "(Suc n ≤ m') ==> (? m. m' = Suc m)" by (induct m') simp_all text {* Useful in certain inductive arguments *} lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (∃j. m = Suc j & j < n))" by (cases m) simp_all subsubsection {* @{term min} and @{term max} *} lemma mono_Suc: "mono Suc" by (rule monoI) simp lemma min_0L [simp]: "min 0 n = (0::nat)" by (rule min_leastL) simp lemma min_0R [simp]: "min n 0 = (0::nat)" by (rule min_leastR) simp lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)" by (simp add: mono_Suc min_of_mono) lemma min_Suc1: "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))" by (simp split: nat.split) lemma min_Suc2: "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))" by (simp split: nat.split) lemma max_0L [simp]: "max 0 n = (n::nat)" by (rule max_leastL) simp lemma max_0R [simp]: "max n 0 = (n::nat)" by (rule max_leastR) simp lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)" by (simp add: mono_Suc max_of_mono) lemma max_Suc1: "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))" by (simp split: nat.split) lemma max_Suc2: "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))" by (simp split: nat.split) subsubsection {* Monotonicity of Addition *} lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n" by (simp add: diff_Suc split: nat.split) lemma nat_add_left_cancel_le [simp]: "(k + m ≤ k + n) = (m≤(n::nat))" by (induct k) simp_all lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))" by (induct k) simp_all lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)" by(auto dest:gr0_implies_Suc) text {* strict, in 1st argument *} lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)" by (induct k) simp_all text {* strict, in both arguments *} lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)" apply (rule add_less_mono1 [THEN less_trans], assumption+) apply (induct j, simp_all) done text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *} lemma less_imp_Suc_add: "m < n ==> (∃k. n = Suc (m + k))" apply (induct n) apply (simp_all add: order_le_less) apply (blast elim!: less_SucE intro!: add_0_right [symmetric] add_Suc_right [symmetric]) done text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *} lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j" apply(auto simp: gr0_conv_Suc) apply (induct_tac m) apply (simp_all add: add_less_mono) done text{*The naturals form an ordered @{text comm_semiring_1_cancel}*} instance nat :: ordered_semidom proof fix i j k :: nat show "0 < (1::nat)" by simp show "i ≤ j ==> k + i ≤ k + j" by simp show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2) qed lemma nat_mult_1: "(1::nat) * n = n" by simp lemma nat_mult_1_right: "n * (1::nat) = n" by simp subsubsection {* Additional theorems about @{term "op ≤"} *} text {* Complete induction, aka course-of-values induction *} lemma less_induct [case_names less]: fixes P :: "nat => bool" assumes step: "!!x. (!!y. y < x ==> P y) ==> P x" shows "P a" proof - have "!!z. z≤a ==> P z" proof (induct a) case (0 z) have "P 0" by (rule step) auto thus ?case using 0 by auto next case (Suc x z) then have "z ≤ x ∨ z = Suc x" by (simp add: le_Suc_eq) thus ?case proof assume "z ≤ x" thus "P z" by (rule Suc(1)) next assume z: "z = Suc x" show "P z" by (rule step) (rule Suc(1), simp add: z le_simps) qed qed thus ?thesis by auto qed lemma nat_less_induct: assumes "!!n. ∀m::nat. m < n --> P m ==> P n" shows "P n" using assms less_induct by blast lemma measure_induct_rule [case_names less]: fixes f :: "'a => nat" assumes step: "!!x. (!!y. f y < f x ==> P y) ==> P x" shows "P a" by (induct m≡"f a" arbitrary: a rule: less_induct) (auto intro: step) text {* old style induction rules: *} lemma measure_induct: fixes f :: "'a => nat" shows "(!!x. ∀y. f y < f x --> P y ==> P x) ==> P a" by (rule measure_induct_rule [of f P a]) iprover lemma full_nat_induct: assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)" shows "P n" by (rule less_induct) (auto intro: step simp:le_simps) text{*An induction rule for estabilishing binary relations*} lemma less_Suc_induct: assumes less: "i < j" and step: "!!i. P i (Suc i)" and trans: "!!i j k. P i j ==> P j k ==> P i k" shows "P i j" proof - from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add) have "P i (Suc (i + k))" proof (induct k) case 0 show ?case by (simp add: step) next case (Suc k) thus ?case by (auto intro: assms) qed thus "P i j" by (simp add: j) qed lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n" apply (rule nat_less_induct) apply (case_tac n) apply (case_tac [2] nat) apply (blast intro: less_trans)+ done text {* The method of infinite descent, frequently used in number theory. Provided by Roelof Oosterhuis. $P(n)$ is true for all $n\in\mathbb{N}$ if \begin{itemize} \item case ``0'': given $n=0$ prove $P(n)$, \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists a smaller integer $m$ such that $\neg P(m)$. \end{itemize} *} text{* A compact version without explicit base case: *} lemma infinite_descent: "[| !!n::nat. ¬ P n ==> ∃m<n. ¬ P m |] ==> P n" by (induct n rule: less_induct, auto) lemma infinite_descent0[case_names 0 smaller]: "[| P 0; !!n. n>0 ==> ¬ P n ==> (∃m::nat. m < n ∧ ¬P m) |] ==> P n" by (rule infinite_descent) (case_tac "n>0", auto) text {* Infinite descent using a mapping to $\mathbb{N}$: $P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and \begin{itemize} \item case ``0'': given $V(x)=0$ prove $P(x)$, \item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$. \end{itemize} NB: the proof also shows how to use the previous lemma. *} corollary infinite_descent0_measure [case_names 0 smaller]: assumes A0: "!!x. V x = (0::nat) ==> P x" and A1: "!!x. V x > 0 ==> ¬P x ==> (∃y. V y < V x ∧ ¬P y)" shows "P x" proof - obtain n where "n = V x" by auto moreover have "!!x. V x = n ==> P x" proof (induct n rule: infinite_descent0) case 0 -- "i.e. $V(x) = 0$" with A0 show "P x" by auto next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$" case (smaller n) then obtain x where vxn: "V x = n " and "V x > 0 ∧ ¬ P x" by auto with A1 obtain y where "V y < V x ∧ ¬ P y" by auto with vxn obtain m where "m = V y ∧ m<n ∧ ¬ P y" by auto then show ?case by auto qed ultimately show "P x" by auto qed text{* Again, without explicit base case: *} lemma infinite_descent_measure: assumes "!!x. ¬ P x ==> ∃y. (V::'a=>nat) y < V x ∧ ¬ P y" shows "P x" proof - from assms obtain n where "n = V x" by auto moreover have "!!x. V x = n ==> P x" proof (induct n rule: infinite_descent, auto) fix x assume "¬ P x" with assms show "∃m < V x. ∃y. V y = m ∧ ¬ P y" by auto qed ultimately show "P x" by auto qed text {* A [clumsy] way of lifting @{text "<"} monotonicity to @{text "≤"} monotonicity *} lemma less_mono_imp_le_mono: "[| !!i j::nat. i < j ==> f i < f j; i ≤ j |] ==> f i ≤ ((f j)::nat)" by (simp add: order_le_less) (blast) text {* non-strict, in 1st argument *} lemma add_le_mono1: "i ≤ j ==> i + k ≤ j + (k::nat)" by (rule add_right_mono) text {* non-strict, in both arguments *} lemma add_le_mono: "[| i ≤ j; k ≤ l |] ==> i + k ≤ j + (l::nat)" by (rule add_mono) lemma le_add2: "n ≤ ((m + n)::nat)" by (insert add_right_mono [of 0 m n], simp) lemma le_add1: "n ≤ ((n + m)::nat)" by (simp add: add_commute, rule le_add2) lemma less_add_Suc1: "i < Suc (i + m)" by (rule le_less_trans, rule le_add1, rule lessI) lemma less_add_Suc2: "i < Suc (m + i)" by (rule le_less_trans, rule le_add2, rule lessI) lemma less_iff_Suc_add: "(m < n) = (∃k. n = Suc (m + k))" by (iprover intro!: less_add_Suc1 less_imp_Suc_add) lemma trans_le_add1: "(i::nat) ≤ j ==> i ≤ j + m" by (rule le_trans, assumption, rule le_add1) lemma trans_le_add2: "(i::nat) ≤ j ==> i ≤ m + j" by (rule le_trans, assumption, rule le_add2) lemma trans_less_add1: "(i::nat) < j ==> i < j + m" by (rule less_le_trans, assumption, rule le_add1) lemma trans_less_add2: "(i::nat) < j ==> i < m + j" by (rule less_le_trans, assumption, rule le_add2) lemma add_lessD1: "i + j < (k::nat) ==> i < k" apply (rule le_less_trans [of _ "i+j"]) apply (simp_all add: le_add1) done lemma not_add_less1 [iff]: "~ (i + j < (i::nat))" apply (rule notI) apply (drule add_lessD1) apply (erule less_irrefl [THEN notE]) done lemma not_add_less2 [iff]: "~ (j + i < (i::nat))" by (simp add: add_commute) lemma add_leD1: "m + k ≤ n ==> m ≤ (n::nat)" apply (rule order_trans [of _ "m+k"]) apply (simp_all add: le_add1) done lemma add_leD2: "m + k ≤ n ==> k ≤ (n::nat)" apply (simp add: add_commute) apply (erule add_leD1) done lemma add_leE: "(m::nat) + k ≤ n ==> (m ≤ n ==> k ≤ n ==> R) ==> R" by (blast dest: add_leD1 add_leD2) text {* needs @{text "!!k"} for @{text add_ac} to work *} lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n" by (force simp del: add_Suc_right simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac) subsubsection {* More results about difference *} text {* Addition is the inverse of subtraction: if @{term "n ≤ m"} then @{term "n + (m - n) = m"}. *} lemma add_diff_inverse: "~ m < n ==> n + (m - n) = (m::nat)" by (induct m n rule: diff_induct) simp_all lemma le_add_diff_inverse [simp]: "n ≤ m ==> n + (m - n) = (m::nat)" by (simp add: add_diff_inverse linorder_not_less) lemma le_add_diff_inverse2 [simp]: "n ≤ m ==> (m - n) + n = (m::nat)" by (simp add: add_commute) lemma Suc_diff_le: "n ≤ m ==> Suc m - n = Suc (m - n)" by (induct m n rule: diff_induct) simp_all lemma diff_less_Suc: "m - n < Suc m" apply (induct m n rule: diff_induct) apply (erule_tac [3] less_SucE) apply (simp_all add: less_Suc_eq) done lemma diff_le_self [simp]: "m - n ≤ (m::nat)" by (induct m n rule: diff_induct) (simp_all add: le_SucI) lemma le_iff_add: "(m::nat) ≤ n = (∃k. n = m + k)" by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n]) lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k" by (rule le_less_trans, rule diff_le_self) lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n" by (cases n) (auto simp add: le_simps) lemma diff_add_assoc: "k ≤ (j::nat) ==> (i + j) - k = i + (j - k)" by (induct j k rule: diff_induct) simp_all lemma diff_add_assoc2: "k ≤ (j::nat) ==> (j + i) - k = (j - k) + i" by (simp add: add_commute diff_add_assoc) lemma le_imp_diff_is_add: "i ≤ (j::nat) ==> (j - i = k) = (j = k + i)" by (auto simp add: diff_add_inverse2) lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m ≤ n)" by (induct m n rule: diff_induct) simp_all lemma diff_is_0_eq' [simp]: "m ≤ n ==> (m::nat) - n = 0" by (rule iffD2, rule diff_is_0_eq) lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)" by (induct m n rule: diff_induct) simp_all lemma less_imp_add_positive: assumes "i < j" shows "∃k::nat. 0 < k & i + k = j" proof from assms show "0 < j - i & i + (j - i) = j" by (simp add: order_less_imp_le) qed text {* a nice rewrite for bounded subtraction *} lemma nat_minus_add_max: fixes n m :: nat shows "n - m + m = max n m" by (simp add: max_def not_le order_less_imp_le) 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") (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero) 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 (auto split: nat_diff_split) subsubsection {* Monotonicity of Multiplication *} lemma mult_le_mono1: "i ≤ (j::nat) ==> i * k ≤ j * k" by (simp add: mult_right_mono) lemma mult_le_mono2: "i ≤ (j::nat) ==> k * i ≤ k * j" by (simp add: mult_left_mono) text {* @{text "≤"} monotonicity, BOTH arguments *} lemma mult_le_mono: "i ≤ (j::nat) ==> k ≤ l ==> i * k ≤ j * l" by (simp add: mult_mono) lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k" by (simp add: mult_strict_right_mono) text{*Differs from the standard @{text zero_less_mult_iff} in that there are no negative numbers.*} lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)" apply (induct m) apply simp apply (case_tac n) apply simp_all done lemma one_le_mult_iff [simp]: "(Suc 0 ≤ m * n) = (1 ≤ m & 1 ≤ n)" apply (induct m) apply simp apply (case_tac n) apply simp_all done lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)" apply (safe intro!: mult_less_mono1) apply (case_tac k, auto) apply (simp del: le_0_eq add: linorder_not_le [symmetric]) apply (blast intro: mult_le_mono1) done lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)" by (simp add: mult_commute [of k]) lemma mult_le_cancel1 [simp]: "(k * (m::nat) ≤ k * n) = (0 < k --> m ≤ n)" by (simp add: linorder_not_less [symmetric], auto) lemma mult_le_cancel2 [simp]: "((m::nat) * k ≤ n * k) = (0 < k --> m ≤ n)" by (simp add: linorder_not_less [symmetric], auto) lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)" by (subst mult_less_cancel1) simp lemma Suc_mult_le_cancel1: "(Suc k * m ≤ Suc k * n) = (m ≤ n)" by (subst mult_le_cancel1) simp lemma le_square: "m ≤ m * (m::nat)" by (cases m) (auto intro: le_add1) lemma le_cube: "(m::nat) ≤ m * (m * m)" by (cases m) (auto intro: le_add1) text {* Lemma for @{text gcd} *} lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0" apply (drule sym) apply (rule disjCI) apply (rule nat_less_cases, erule_tac [2] _) apply (drule_tac [2] mult_less_mono2) apply (auto) done text {* the lattice order on @{typ nat} *} instantiation nat :: distrib_lattice begin definition "(inf :: nat => nat => nat) = min" definition "(sup :: nat => nat => nat) = max" instance by intro_classes (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def intro: order_less_imp_le antisym elim!: order_trans order_less_trans) end subsection {* Embedding of the Naturals into any @{text semiring_1}: @{term of_nat} *} context semiring_1 begin primrec of_nat :: "nat => 'a" where of_nat_0: "of_nat 0 = 0" | of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m" 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" by (induct m) (simp_all add: add_ac) lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n" by (induct m) (simp_all add: add_ac left_distrib) definition of_nat_aux :: "nat => 'a => 'a" where [code func del]: "of_nat_aux n i = of_nat n + i" lemma of_nat_aux_code [code]: "of_nat_aux 0 i = i" "of_nat_aux (Suc n) i = of_nat_aux n (i + 1)" -- {* tail recursive *} by (simp_all add: of_nat_aux_def add_ac) lemma of_nat_code [code]: "of_nat n = of_nat_aux n 0" by (simp add: of_nat_aux_def) end text{*Class for unital semirings with characteristic zero. Includes non-ordered rings like the complex numbers.*} class semiring_char_0 = semiring_1 + assumes of_nat_eq_iff [simp]: "of_nat m = of_nat n <-> m = n" begin text{*Special cases where either operand is zero*} lemma of_nat_0_eq_iff [simp, noatp]: "0 = of_nat n <-> 0 = n" by (rule of_nat_eq_iff [of 0, simplified]) lemma of_nat_eq_0_iff [simp, noatp]: "of_nat m = 0 <-> m = 0" by (rule of_nat_eq_iff [of _ 0, simplified]) lemma inj_of_nat: "inj of_nat" by (simp add: inj_on_def) end context ordered_semidom begin lemma zero_le_imp_of_nat: "0 ≤ of_nat m" apply (induct m, simp_all) apply (erule order_trans) apply (rule ord_le_eq_trans [OF _ add_commute]) apply (rule less_add_one [THEN less_imp_le]) done lemma less_imp_of_nat_less: "m < n ==> of_nat m < of_nat n" apply (induct m n rule: diff_induct, simp_all) apply (insert add_less_le_mono [OF zero_less_one zero_le_imp_of_nat], force) done lemma of_nat_less_imp_less: "of_nat m < of_nat n ==> m < n" apply (induct m n rule: diff_induct, simp_all) apply (insert zero_le_imp_of_nat) apply (force simp add: not_less [symmetric]) done lemma of_nat_less_iff [simp]: "of_nat m < of_nat n <-> m < n" by (blast intro: of_nat_less_imp_less less_imp_of_nat_less) lemma of_nat_le_iff [simp]: "of_nat m ≤ of_nat n <-> m ≤ n" by (simp add: not_less [symmetric] linorder_not_less [symmetric]) text{*Every @{text ordered_semidom} has characteristic zero.*} subclass semiring_char_0 by unfold_locales (simp add: eq_iff order_eq_iff) text{*Special cases where either operand is zero*} lemma of_nat_0_le_iff [simp]: "0 ≤ of_nat n" by (rule of_nat_le_iff [of 0, simplified]) lemma of_nat_le_0_iff [simp, noatp]: "of_nat m ≤ 0 <-> m = 0" by (rule of_nat_le_iff [of _ 0, simplified]) lemma of_nat_0_less_iff [simp]: "0 < of_nat n <-> 0 < n" by (rule of_nat_less_iff [of 0, simplified]) lemma of_nat_less_0_iff [simp]: "¬ of_nat m < 0" by (rule of_nat_less_iff [of _ 0, simplified]) end context ring_1 begin lemma of_nat_diff: "n ≤ m ==> of_nat (m - n) = of_nat m - of_nat n" by (simp add: compare_rls of_nat_add [symmetric]) end context ordered_idom begin lemma abs_of_nat [simp]: "¦of_nat n¦ = of_nat n" unfolding abs_if by auto end lemma of_nat_id [simp]: "of_nat n = n" by (induct n) auto lemma of_nat_eq_id [simp]: "of_nat = id" by (auto simp add: expand_fun_eq) subsection {* The Set of Natural Numbers *} context semiring_1 begin definition Nats :: "'a set" where "Nats = range of_nat" notation (xsymbols) Nats ("\<nat>") lemma of_nat_in_Nats [simp]: "of_nat n ∈ \<nat>" by (simp add: Nats_def) lemma Nats_0 [simp]: "0 ∈ \<nat>" apply (simp add: Nats_def) apply (rule range_eqI) apply (rule of_nat_0 [symmetric]) done lemma Nats_1 [simp]: "1 ∈ \<nat>" apply (simp add: Nats_def) apply (rule range_eqI) apply (rule of_nat_1 [symmetric]) done lemma Nats_add [simp]: "a ∈ \<nat> ==> b ∈ \<nat> ==> a + b ∈ \<nat>" apply (auto simp add: Nats_def) apply (rule range_eqI) apply (rule of_nat_add [symmetric]) done lemma Nats_mult [simp]: "a ∈ \<nat> ==> b ∈ \<nat> ==> a * b ∈ \<nat>" apply (auto simp add: Nats_def) apply (rule range_eqI) apply (rule of_nat_mult [symmetric]) done end subsection {* Further Arithmetic Facts Concerning the Natural Numbers *} lemma subst_equals: assumes 1: "t = s" and 2: "u = t" shows "u = s" using 2 1 by (rule trans) use "arith_data.ML" declaration {* K ArithData.setup *} use "Tools/lin_arith.ML" declaration {* K LinArith.setup *} lemmas [arith_split] = nat_diff_split split_min split_max text{*Subtraction laws, mostly by Clemens Ballarin*} lemma diff_less_mono: "[| a < (b::nat); c ≤ 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 ≤ (i::nat)) = (j ≤ i+k)" by arith lemma le_diff_conv2: "k ≤ j ==> (i ≤ j-k) = (i+k ≤ (j::nat))" by arith lemma diff_diff_cancel [simp]: "i ≤ (n::nat) ==> n - (n - i) = i" by arith lemma le_add_diff: "k ≤ (n::nat) ==> m ≤ n + m - k" by arith (*Replaces the previous diff_less and le_diff_less, which had the stronger second premise n≤m*) lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m" by arith text {* Simplification of relational expressions involving subtraction *} lemma diff_diff_eq: "[| k ≤ m; k ≤ (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)" by (simp split add: nat_diff_split) lemma eq_diff_iff: "[| k ≤ m; k ≤ (n::nat) |] ==> (m-k = n-k) = (m=n)" by (auto split add: nat_diff_split) lemma less_diff_iff: "[| k ≤ m; k ≤ (n::nat) |] ==> (m-k < n-k) = (m<n)" by (auto split add: nat_diff_split) lemma le_diff_iff: "[| k ≤ m; k ≤ (n::nat) |] ==> (m-k ≤ n-k) = (m≤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 ≤ (n::nat) ==> (m-l) ≤ (n-l)" by (simp split add: nat_diff_split) lemma diff_le_mono2: "m ≤ (n::nat) ==> (l-n) ≤ (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) lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i" unfolding min_def by auto lemma inj_on_diff_nat: assumes k_le_n: "∀n ∈ N. k ≤ (n::nat)" shows "inj_on (λn. n - k) N" proof (rule inj_onI) fix x y assume a: "x ∈ N" "y ∈ N" "x - k = y - k" with k_le_n have "x - k + k = y - k + k" by auto with a k_le_n show "x = y" by auto qed text{*Rewriting to pull differences out*} lemma diff_diff_right [simp]: "k≤j --> i - (j - k) = i + (k::nat) - j" by arith lemma diff_Suc_diff_eq1 [simp]: "k ≤ j ==> m - Suc (j - k) = m + k - Suc j" by arith lemma diff_Suc_diff_eq2 [simp]: "k ≤ j ==> Suc (j - k) - m = Suc j - (k + m)" by arith text{*Lemmas for ex/Factorization*} lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n" by (cases m) auto lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n" by (cases m) auto lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m" by (cases m) auto text {* Specialized induction principles that work "backwards": *} lemma inc_induct[consumes 1, case_names base step]: assumes less: "i <= j" assumes base: "P j" assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i" shows "P i" using less proof (induct d=="j - i" arbitrary: i) case (0 i) hence "i = j" by simp with base show ?case by simp next case (Suc d i) hence "i < j" "P (Suc i)" by simp_all thus "P i" by (rule step) qed lemma strict_inc_induct[consumes 1, case_names base step]: assumes less: "i < j" assumes base: "!!i. j = Suc i ==> P i" assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i" shows "P i" using less proof (induct d=="j - i - 1" arbitrary: i) case (0 i) with `i < j` have "j = Suc i" by simp with base show ?case by simp next case (Suc d i) hence "i < j" "P (Suc i)" by simp_all thus "P i" by (rule step) qed lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)" using inc_induct[of "k - i" k P, simplified] by blast lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0" using inc_induct[of 0 k P] by blast lemma nat_not_singleton: "(∀x. x = (0::nat)) = False" by auto (*The others are i - j - k = i - (j + k), k ≤ j ==> j - k + i = j + i - k, k ≤ 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.*} subsection {* size of a datatype value *} class size = type + fixes size :: "'a => nat" -- {* see further theory @{text Wellfounded} *} end
lemma nat_induct:
[| P 0; !!n. P n ==> P (Suc n) |] ==> P n
lemma Suc_not_Zero:
Suc m ≠ 0
lemma Zero_not_Suc:
0 ≠ Suc m
lemma inj_Suc:
inj_on Suc N
lemma Suc_Suc_eq:
(Suc m = Suc n) = (m = n)
lemma nat_rec_0:
nat_rec f1.0 f2.0 0 = f1.0
and nat_rec_Suc:
nat_rec f1.0 f2.0 (Suc nat) = f2.0 nat (nat_rec f1.0 f2.0 nat)
lemma nat_case_0:
nat_case f1.0 f2.0 0 = f1.0
and nat_case_Suc:
nat_case f1.0 f2.0 (Suc nat) = f2.0 nat
lemma Suc_neq_Zero:
Suc m = 0 ==> R
lemma Zero_neq_Suc:
0 = Suc m ==> R
lemma Suc_inject:
Suc x = Suc y ==> x = y
lemma n_not_Suc_n:
n ≠ Suc n
lemma Suc_n_not_n:
Suc n ≠ n
lemma diff_induct:
[| !!x. P x 0; !!y. P 0 (Suc y); !!x y. P x y ==> P (Suc x) (Suc y) |] ==> P m n
lemma add_0_right:
m + 0 = m
lemma add_Suc_right:
m + Suc n = Suc (m + n)
lemma add_Suc_shift:
Suc m + n = m + Suc n
lemma diff_0_eq_0:
0 - n = 0
lemma diff_Suc_Suc:
Suc m - Suc n = m - n
lemma mult_0_right:
m * 0 = 0
lemma mult_Suc_right:
m * Suc n = m + m * n
lemma add_mult_distrib:
(m + n) * k = m * k + n * k
lemma nat_add_assoc:
m + n + k = m + (n + k)
lemma nat_add_commute:
m + n = n + m
lemma nat_add_left_commute:
x + (y + z) = y + (x + z)
lemma nat_add_left_cancel:
(k + m = k + n) = (m = n)
lemma nat_add_right_cancel:
(m + k = n + k) = (m = n)
lemma add_is_0:
(m + n = 0) = (m = 0 ∧ n = 0)
lemma add_is_1:
(m + n = Suc 0) = (m = Suc 0 ∧ n = 0 ∨ m = 0 ∧ n = Suc 0)
lemma one_is_add:
(Suc 0 = m + n) = (m = Suc 0 ∧ n = 0 ∨ m = 0 ∧ n = Suc 0)
lemma add_eq_self_zero:
m + n = m ==> n = 0
lemma inj_on_add_nat:
inj_on (λn. n + k) N
lemma diff_self_eq_0:
m - m = 0
lemma diff_diff_left:
i - j - k = i - (j + k)
lemma Suc_diff_diff:
Suc m - n - Suc k = m - n - k
lemma diff_commute:
i - j - k = i - k - j
lemma diff_add_inverse:
n + m - n = m
lemma diff_add_inverse2:
m + n - n = m
lemma diff_cancel:
k + m - (k + n) = m - n
lemma diff_cancel2:
m + k - (n + k) = m - n
lemma diff_add_0:
n - (n + m) = 0
lemma diff_mult_distrib:
(m - n) * k = m * k - n * k
lemma diff_mult_distrib2:
k * (m - n) = k * m - k * n
lemma nat_mult_assoc:
m * n * k = m * (n * k)
lemma nat_mult_commute:
m * n = n * m
lemma add_mult_distrib2:
k * (m + n) = k * m + k * n
lemma mult_is_0:
(m * n = 0) = (m = 0 ∨ n = 0)
lemma nat_distrib:
(m + n) * k = m * k + n * k
k * (m + n) = k * m + k * n
(m - n) * k = m * k - n * k
k * (m - n) = k * m - k * n
lemma mult_eq_1_iff:
(m * n = Suc 0) = (m = 1 ∧ n = 1)
lemma one_eq_mult_iff:
(Suc 0 = m * n) = (m = 1 ∧ n = 1)
lemma mult_cancel1:
(k * m = k * n) = (m = n ∨ k = 0)
lemma mult_cancel2:
(m * k = n * k) = (m = n ∨ k = 0)
lemma Suc_mult_cancel1:
(Suc k * m = Suc k * n) = (m = n)
lemma
(0 ≤ n) = True
lemma le0:
0 ≤ n
lemma Suc_le_mono:
(Suc n ≤ Suc m) = (n ≤ m)
lemma Suc_le_eq:
(Suc m ≤ n) = (m < n)
lemma le_0_eq:
(n ≤ 0) = (n = 0)
lemma not_less0:
¬ n < 0
lemma less_nat_zero_code:
(n < 0) = False
lemma Suc_less_eq:
(Suc m < Suc n) = (m < n)
lemma less_Suc_eq_le:
(m < Suc n) = (m ≤ n)
lemma le_SucI:
m ≤ n ==> m ≤ Suc n
lemma Suc_leD:
Suc m ≤ n ==> m ≤ n
lemma less_SucI:
m < n ==> m < Suc n
lemma Suc_lessD:
Suc m < n ==> m < n
lemma lessI:
n < Suc n
lemma zero_less_Suc:
0 < Suc n
lemma less_not_refl:
¬ n < n
lemma less_not_refl2:
n < m ==> m ≠ n
lemma less_not_refl3:
s < t ==> s ≠ t
lemma less_irrefl_nat:
n < n ==> R
lemma less_zeroE:
n < 0 ==> R
lemma less_Suc_eq:
(m < Suc n) = (m < n ∨ m = n)
lemma less_one:
(n < 1) = (n = 0)
lemma less_Suc0:
(n < Suc 0) = (n = 0)
lemma Suc_mono:
m < n ==> Suc m < Suc n
lemma less_antisym:
[| ¬ n < m; n < Suc m |] ==> m = n
lemma nat_neq_iff:
(m ≠ n) = (m < n ∨ n < m)
lemma nat_less_cases:
[| m < n ==> P n m; m = n ==> P n m; n < m ==> P n m |] ==> P n m
lemma Suc_lessI:
[| m < n; Suc m ≠ n |] ==> Suc m < n
lemma lessE:
[| i < k; k = Suc i ==> P; !!j. [| i < j; k = Suc j |] ==> P |] ==> P
lemma less_SucE:
[| m < Suc n; m < n ==> P; m = n ==> P |] ==> P
lemma Suc_lessE:
[| Suc i < k; !!j. [| i < j; k = Suc j |] ==> P |] ==> P
lemma Suc_less_SucD:
Suc m < Suc n ==> m < n
lemma less_trans_Suc:
[| i < j; j < k |] ==> Suc i < k
lemma not_less_eq:
(¬ m < n) = (n < Suc m)
lemma not_less_eq_eq:
(¬ m ≤ n) = (Suc n ≤ m)
lemma le_imp_less_Suc:
m ≤ n ==> m < Suc n
lemma Suc_n_not_le_n:
¬ Suc n ≤ n
lemma le_Suc_eq:
(m ≤ Suc n) = (m ≤ n ∨ m = Suc n)
lemma le_SucE:
[| m ≤ Suc n; m ≤ n ==> R; m = Suc n ==> R |] ==> R
lemma Suc_leI:
m < n ==> Suc m ≤ n
lemma Suc_le_lessD:
Suc m ≤ n ==> m < n
lemma less_imp_le_nat:
m < n ==> m ≤ n
lemma le_simps:
m < n ==> m ≤ n
(m < Suc n) = (m ≤ n)
(Suc m ≤ n) = (m < n)
lemma less_or_eq_imp_le:
m < n ∨ m = n ==> m ≤ n
lemma le_eq_less_or_eq:
(m ≤ n) = (m < n ∨ m = n)
lemma eq_imp_le:
m = n ==> m ≤ n
lemma le_refl:
n ≤ n
lemma le_trans:
[| i ≤ j; j ≤ k |] ==> i ≤ k
lemma le_anti_sym:
[| m ≤ n; n ≤ m |] ==> m = n
lemma nat_less_le:
(m < n) = (m ≤ n ∧ m ≠ n)
lemma le_neq_implies_less:
[| m ≤ n; m ≠ n |] ==> m < n
lemma nat_le_linear:
m ≤ n ∨ n ≤ m
lemma linorder_neqE_nat:
[| x ≠ y; x < y ==> R; y < x ==> R |] ==> R
lemma le_less_Suc_eq:
m ≤ n ==> (n < Suc m) = (n = m)
lemma not_less_less_Suc_eq:
¬ n < m ==> (n < Suc m) = (n = m)
lemma not_less_simps:
¬ n < m ==> (n < Suc m) = (n = m)
m ≤ n ==> (n < Suc m) = (n = m)
lemma def_nat_rec_0:
(!!n. f n == nat_rec c h n) ==> f 0 = c
lemma def_nat_rec_Suc:
(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)
lemma not0_implies_Suc:
n ≠ 0 ==> ∃m. n = Suc m
lemma gr0_implies_Suc:
0 < n ==> ∃m. n = Suc m
lemma gr_implies_not0:
m < n ==> n ≠ 0
lemma neq0_conv:
(n ≠ 0) = (0 < n)
lemma gr0I:
(n = 0 ==> False) ==> 0 < n
lemma gr0_conv_Suc:
(0 < n) = (∃m. n = Suc m)
lemma not_gr0:
(¬ 0 < n) = (n = 0)
lemma Suc_le_D:
Suc n ≤ m' ==> ∃m. m' = Suc m
lemma less_Suc_eq_0_disj:
(m < Suc n) = (m = 0 ∨ (∃j. m = Suc j ∧ j < n))
lemma mono_Suc:
mono Suc
lemma min_0L:
min 0 n = 0
lemma min_0R:
min n 0 = 0
lemma min_Suc_Suc:
min (Suc m) (Suc n) = Suc (min m n)
lemma min_Suc1:
min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc (min n m'))
lemma min_Suc2:
min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc (min m' n))
lemma max_0L:
max 0 n = n
lemma max_0R:
max n 0 = n
lemma max_Suc_Suc:
max (Suc m) (Suc n) = Suc (max m n)
lemma max_Suc1:
max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc (max n m'))
lemma max_Suc2:
max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc (max m' n))
lemma Suc_pred:
0 < n ==> Suc (n - Suc 0) = n
lemma nat_add_left_cancel_le:
(k + m ≤ k + n) = (m ≤ n)
lemma nat_add_left_cancel_less:
(k + m < k + n) = (m < n)
lemma add_gr_0:
(0 < m + n) = (0 < m ∨ 0 < n)
lemma add_less_mono1:
i < j ==> i + k < j + k
lemma add_less_mono:
[| i < j; k < l |] ==> i + k < j + l
lemma less_imp_Suc_add:
m < n ==> ∃k. n = Suc (m + k)
lemma mult_less_mono2:
[| i < j; 0 < k |] ==> k * i < k * j
lemma nat_mult_1:
1 * n = n
lemma nat_mult_1_right:
n * 1 = n
lemma less_induct:
(!!x. (!!y. y < x ==> P y) ==> P x) ==> P a
lemma nat_less_induct:
(!!n. ∀m<n. P m ==> P n) ==> P n
lemma measure_induct_rule:
(!!x. (!!y. f y < f x ==> P y) ==> P x) ==> P a
lemma measure_induct:
(!!x. ∀y. f y < f x --> P y ==> P x) ==> P a
lemma full_nat_induct:
(!!n. ∀m. Suc m ≤ n --> P m ==> P n) ==> P n
lemma less_Suc_induct:
[| i < j; !!i. P i (Suc i); !!i j k. [| P i j; P j k |] ==> P i k |] ==> P i j
lemma nat_induct2:
[| P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k)) |] ==> P n
lemma infinite_descent:
(!!n. ¬ P n ==> ∃m<n. ¬ P m) ==> P n
lemma infinite_descent0:
[| P 0; !!n. [| 0 < n; ¬ P n |] ==> ∃m<n. ¬ P m |] ==> P n
corollary infinite_descent0_measure:
[| !!x. V x = 0 ==> P x; !!x. [| 0 < V x; ¬ P x |] ==> ∃y. V y < V x ∧ ¬ P y |]
==> P x
lemma infinite_descent_measure:
(!!x. ¬ P x ==> ∃y. V y < V x ∧ ¬ P y) ==> P x
lemma less_mono_imp_le_mono:
[| !!i j. i < j ==> f i < f j; i ≤ j |] ==> f i ≤ f j
lemma add_le_mono1:
i ≤ j ==> i + k ≤ j + k
lemma add_le_mono:
[| i ≤ j; k ≤ l |] ==> i + k ≤ j + l
lemma le_add2:
n ≤ m + n
lemma le_add1:
n ≤ n + m
lemma less_add_Suc1:
i < Suc (i + m)
lemma less_add_Suc2:
i < Suc (m + i)
lemma less_iff_Suc_add:
(m < n) = (∃k. n = Suc (m + k))
lemma trans_le_add1:
i ≤ j ==> i ≤ j + m
lemma trans_le_add2:
i ≤ j ==> i ≤ m + j
lemma trans_less_add1:
i < j ==> i < j + m
lemma trans_less_add2:
i < j ==> i < m + j
lemma add_lessD1:
i + j < k ==> i < k
lemma not_add_less1:
¬ i + j < i
lemma not_add_less2:
¬ j + i < i
lemma add_leD1:
m + k ≤ n ==> m ≤ n
lemma add_leD2:
m + k ≤ n ==> k ≤ n
lemma add_leE:
[| m + k ≤ n; [| m ≤ n; k ≤ n |] ==> R |] ==> R
lemma less_add_eq_less:
[| k < l; m + l = k + n |] ==> m < n
lemma add_diff_inverse:
¬ m < n ==> n + (m - n) = m
lemma le_add_diff_inverse:
n ≤ m ==> n + (m - n) = m
lemma le_add_diff_inverse2:
n ≤ m ==> m - n + n = m
lemma Suc_diff_le:
n ≤ m ==> Suc m - n = Suc (m - n)
lemma diff_less_Suc:
m - n < Suc m
lemma diff_le_self:
m - n ≤ m
lemma le_iff_add:
(m ≤ n) = (∃k. n = m + k)
lemma less_imp_diff_less:
j < k ==> j - n < k
lemma diff_Suc_less:
0 < n ==> n - Suc i < n
lemma diff_add_assoc:
k ≤ j ==> i + j - k = i + (j - k)
lemma diff_add_assoc2:
k ≤ j ==> j + i - k = j - k + i
lemma le_imp_diff_is_add:
i ≤ j ==> (j - i = k) = (j = k + i)
lemma diff_is_0_eq:
(m - n = 0) = (m ≤ n)
lemma diff_is_0_eq':
m ≤ n ==> m - n = 0
lemma zero_less_diff:
(0 < n - m) = (m < n)
lemma less_imp_add_positive:
i < j ==> ∃k>0. i + k = j
lemma nat_minus_add_max:
n - m + m = max n m
lemma nat_diff_split:
P (a - b) = ((a < b --> P 0) ∧ (∀d. a = b + d --> P d))
lemma nat_diff_split_asm:
P (a - b) = (¬ (a < b ∧ ¬ P 0 ∨ (∃d. a = b + d ∧ ¬ P d)))
lemma mult_le_mono1:
i ≤ j ==> i * k ≤ j * k
lemma mult_le_mono2:
i ≤ j ==> k * i ≤ k * j
lemma mult_le_mono:
[| i ≤ j; k ≤ l |] ==> i * k ≤ j * l
lemma mult_less_mono1:
[| i < j; 0 < k |] ==> i * k < j * k
lemma nat_0_less_mult_iff:
(0 < m * n) = (0 < m ∧ 0 < n)
lemma one_le_mult_iff:
(Suc 0 ≤ m * n) = (1 ≤ m ∧ 1 ≤ n)
lemma mult_less_cancel2:
(m * k < n * k) = (0 < k ∧ m < n)
lemma mult_less_cancel1:
(k * m < k * n) = (0 < k ∧ m < n)
lemma mult_le_cancel1:
(k * m ≤ k * n) = (0 < k --> m ≤ n)
lemma mult_le_cancel2:
(m * k ≤ n * k) = (0 < k --> m ≤ n)
lemma Suc_mult_less_cancel1:
(Suc k * m < Suc k * n) = (m < n)
lemma Suc_mult_le_cancel1:
(Suc k * m ≤ Suc k * n) = (m ≤ n)
lemma le_square:
m ≤ m * m
lemma le_cube:
m ≤ m * (m * m)
lemma mult_eq_self_implies_10:
m = m * n ==> n = 1 ∨ m = 0
lemma of_nat_1:
of_nat 1 = (1::'a)
lemma of_nat_add:
of_nat (m + n) = of_nat m + of_nat n
lemma of_nat_mult:
of_nat (m * n) = of_nat m * of_nat n
lemma of_nat_aux_code:
of_nat_aux 0 i = i
of_nat_aux (Suc n) i = of_nat_aux n (i + (1::'a))
lemma of_nat_code:
of_nat n = of_nat_aux n (0::'a)
lemma of_nat_0_eq_iff:
((0::'a) = of_nat n) = (0 = n)
lemma of_nat_eq_0_iff:
(of_nat m = (0::'a)) = (m = 0)
lemma inj_of_nat:
inj of_nat
lemma zero_le_imp_of_nat:
(0::'a) ≤ of_nat m
lemma less_imp_of_nat_less:
m < n ==> of_nat m < of_nat n
lemma of_nat_less_imp_less:
of_nat m < of_nat n ==> m < n
lemma of_nat_less_iff:
(of_nat m < of_nat n) = (m < n)
lemma of_nat_le_iff:
(of_nat m ≤ of_nat n) = (m ≤ n)
lemma of_nat_0_le_iff:
(0::'a) ≤ of_nat n
lemma of_nat_le_0_iff:
(of_nat m ≤ (0::'a)) = (m = 0)
lemma of_nat_0_less_iff:
((0::'a) < of_nat n) = (0 < n)
lemma of_nat_less_0_iff:
¬ of_nat m < (0::'a)
lemma of_nat_diff:
n ≤ m ==> of_nat (m - n) = of_nat m - of_nat n
lemma abs_of_nat:
¦of_nat n¦ = of_nat n
lemma of_nat_id:
of_nat n = n
lemma of_nat_eq_id:
of_nat = id
lemma of_nat_in_Nats:
of_nat n ∈ Nats
lemma Nats_0:
(0::'a) ∈ Nats
lemma Nats_1:
(1::'a) ∈ Nats
lemma Nats_add:
[| a ∈ Nats; b ∈ Nats |] ==> a + b ∈ Nats
lemma Nats_mult:
[| a ∈ Nats; b ∈ Nats |] ==> a * b ∈ Nats
lemma subst_equals:
[| t = s; u = t |] ==> u = s
lemma
P (a - b) = ((a < b --> P 0) ∧ (∀d. a = b + d --> P d))
P (min i j) = ((i ≤ j --> P i) ∧ (¬ i ≤ j --> P j))
P (max i j) = ((i ≤ j --> P j) ∧ (¬ i ≤ j --> P i))
lemma diff_less_mono:
[| a < b; c ≤ a |] ==> a - c < b - c
lemma less_diff_conv:
(i < j - k) = (i + k < j)
lemma le_diff_conv:
(j - k ≤ i) = (j ≤ i + k)
lemma le_diff_conv2:
k ≤ j ==> (i ≤ j - k) = (i + k ≤ j)
lemma diff_diff_cancel:
i ≤ n ==> n - (n - i) = i
lemma le_add_diff:
k ≤ n ==> m ≤ n + m - k
lemma diff_less:
[| 0 < n; 0 < m |] ==> m - n < m
lemma diff_diff_eq:
[| k ≤ m; k ≤ n |] ==> m - k - (n - k) = m - n
lemma eq_diff_iff:
[| k ≤ m; k ≤ n |] ==> (m - k = n - k) = (m = n)
lemma less_diff_iff:
[| k ≤ m; k ≤ n |] ==> (m - k < n - k) = (m < n)
lemma le_diff_iff:
[| k ≤ m; k ≤ n |] ==> (m - k ≤ n - k) = (m ≤ n)
lemma diff_le_mono:
m ≤ n ==> m - l ≤ n - l
lemma diff_le_mono2:
m ≤ n ==> l - n ≤ l - m
lemma diff_less_mono2:
[| m < n; m < l |] ==> l - n < l - m
lemma diffs0_imp_equal:
[| m - n = 0; n - m = 0 |] ==> m = n
lemma min_diff:
min (m - i) (n - i) = min m n - i
lemma inj_on_diff_nat:
∀n∈N. k ≤ n ==> inj_on (λn. n - k) N
lemma diff_diff_right:
k ≤ j --> i - (j - k) = i + k - j
lemma diff_Suc_diff_eq1:
k ≤ j ==> m - Suc (j - k) = m + k - Suc j
lemma diff_Suc_diff_eq2:
k ≤ j ==> Suc (j - k) - m = Suc j - (k + m)
lemma one_less_mult:
[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m * n
lemma n_less_m_mult_n:
[| Suc 0 < n; Suc 0 < m |] ==> n < m * n
lemma n_less_n_mult_m:
[| Suc 0 < n; Suc 0 < m |] ==> n < n * m
lemma inc_induct:
[| i ≤ j; P j; !!i. [| i < j; P (Suc i) |] ==> P i |] ==> P i
lemma strict_inc_induct:
[| i < j; !!i. j = Suc i ==> P i; !!i. [| i < j; P (Suc i) |] ==> P i |] ==> P i
lemma zero_induct_lemma:
[| P k; !!n. P (Suc n) ==> P n |] ==> P (k - i)
lemma zero_induct:
[| P k; !!n. P (Suc n) ==> P n |] ==> P 0
lemma nat_not_singleton:
(∀x. x = 0) = False
lemma add_diff_assoc:
k ≤ j ==> i + (j - k) = i + j - k
lemma add_diff_assoc2:
k ≤ j ==> j - k + i = j + i - k