(* Title: IntDef.thy ID: $Id: IntDef.thy,v 1.60 2005/09/27 10:14:39 berghofe Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1996 University of Cambridge *) header{*The Integers as Equivalence Classes over Pairs of Natural Numbers*} theory IntDef imports Equiv_Relations NatArith begin constdefs intrel :: "((nat * nat) * (nat * nat)) set" --{*the equivalence relation underlying the integers*} "intrel == {((x,y),(u,v)) | x y u v. x+v = u+y}" typedef (Integ) int = "UNIV//intrel" by (auto simp add: quotient_def) instance int :: "{ord, zero, one, plus, times, minus}" .. constdefs int :: "nat => int" "int m == Abs_Integ(intrel `` {(m,0)})" defs (overloaded) Zero_int_def: "0 == int 0" One_int_def: "1 == int 1" minus_int_def: "- z == Abs_Integ (\<Union>(x,y) ∈ Rep_Integ z. intrel``{(y,x)})" add_int_def: "z + w == Abs_Integ (\<Union>(x,y) ∈ Rep_Integ z. \<Union>(u,v) ∈ Rep_Integ w. intrel``{(x+u, y+v)})" diff_int_def: "z - (w::int) == z + (-w)" mult_int_def: "z * w == Abs_Integ (\<Union>(x,y) ∈ Rep_Integ z. \<Union>(u,v) ∈ Rep_Integ w. intrel``{(x*u + y*v, x*v + y*u)})" le_int_def: "z ≤ (w::int) == ∃x y u v. x+v ≤ u+y & (x,y) ∈ Rep_Integ z & (u,v) ∈ Rep_Integ w" less_int_def: "(z < (w::int)) == (z ≤ w & z ≠ w)" subsection{*Construction of the Integers*} subsubsection{*Preliminary Lemmas about the Equivalence Relation*} lemma intrel_iff [simp]: "(((x,y),(u,v)) ∈ intrel) = (x+v = u+y)" by (simp add: intrel_def) lemma equiv_intrel: "equiv UNIV intrel" by (simp add: intrel_def equiv_def refl_def sym_def trans_def) text{*Reduces equality of equivalence classes to the @{term intrel} relation: @{term "(intrel `` {x} = intrel `` {y}) = ((x,y) ∈ intrel)"} *} lemmas equiv_intrel_iff = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I] declare equiv_intrel_iff [simp] text{*All equivalence classes belong to set of representatives*} lemma [simp]: "intrel``{(x,y)} ∈ Integ" by (auto simp add: Integ_def intrel_def quotient_def) text{*Reduces equality on abstractions to equality on representatives: @{term "[|x ∈ Integ; y ∈ Integ|] ==> (Abs_Integ x = Abs_Integ y) = (x=y)"} *} declare Abs_Integ_inject [simp] Abs_Integ_inverse [simp] text{*Case analysis on the representation of an integer as an equivalence class of pairs of naturals.*} lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]: "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P" apply (rule Abs_Integ_cases [of z]) apply (auto simp add: Integ_def quotient_def) done subsubsection{*@{term int}: Embedding the Naturals into the Integers*} lemma inj_int: "inj int" by (simp add: inj_on_def int_def) lemma int_int_eq [iff]: "(int m = int n) = (m = n)" by (fast elim!: inj_int [THEN injD]) subsubsection{*Integer Unary Negation*} lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})" proof - have "(λ(x,y). intrel``{(y,x)}) respects intrel" by (simp add: congruent_def) thus ?thesis by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel]) qed lemma zminus_zminus: "- (- z) = (z::int)" by (cases z, simp add: minus) lemma zminus_0: "- 0 = (0::int)" by (simp add: int_def Zero_int_def minus) subsection{*Integer Addition*} lemma add: "Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) = Abs_Integ (intrel``{(x+u, y+v)})" proof - have "(λz w. (λ(x,y). (λ(u,v). intrel `` {(x+u, y+v)}) w) z) respects2 intrel" by (simp add: congruent2_def) thus ?thesis by (simp add: add_int_def UN_UN_split_split_eq UN_equiv_class2 [OF equiv_intrel equiv_intrel]) qed lemma zminus_zadd_distrib: "- (z + w) = (- z) + (- w::int)" by (cases z, cases w, simp add: minus add) lemma zadd_commute: "(z::int) + w = w + z" by (cases z, cases w, simp add: add_ac add) lemma zadd_assoc: "((z1::int) + z2) + z3 = z1 + (z2 + z3)" by (cases z1, cases z2, cases z3, simp add: add add_assoc) (*For AC rewriting*) lemma zadd_left_commute: "x + (y + z) = y + ((x + z) ::int)" apply (rule mk_left_commute [of "op +"]) apply (rule zadd_assoc) apply (rule zadd_commute) done lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute lemmas zmult_ac = OrderedGroup.mult_ac lemma zadd_int: "(int m) + (int n) = int (m + n)" by (simp add: int_def add) lemma zadd_int_left: "(int m) + (int n + z) = int (m + n) + z" by (simp add: zadd_int zadd_assoc [symmetric]) lemma int_Suc: "int (Suc m) = 1 + (int m)" by (simp add: One_int_def zadd_int) (*also for the instance declaration int :: comm_monoid_add*) lemma zadd_0: "(0::int) + z = z" apply (simp add: Zero_int_def int_def) apply (cases z, simp add: add) done lemma zadd_0_right: "z + (0::int) = z" by (rule trans [OF zadd_commute zadd_0]) lemma zadd_zminus_inverse2: "(- z) + z = (0::int)" by (cases z, simp add: int_def Zero_int_def minus add) subsection{*Integer Multiplication*} text{*Congruence property for multiplication*} lemma mult_congruent2: "(%p1 p2. (%(x,y). (%(u,v). intrel``{(x*u + y*v, x*v + y*u)}) p2) p1) respects2 intrel" apply (rule equiv_intrel [THEN congruent2_commuteI]) apply (force simp add: mult_ac, clarify) apply (simp add: congruent_def mult_ac) apply (rename_tac u v w x y z) apply (subgoal_tac "u*y + x*y = w*y + v*y & u*z + x*z = w*z + v*z") apply (simp add: mult_ac) apply (simp add: add_mult_distrib [symmetric]) done lemma mult: "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) = Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})" by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2 UN_equiv_class2 [OF equiv_intrel equiv_intrel]) lemma zmult_zminus: "(- z) * w = - (z * (w::int))" by (cases z, cases w, simp add: minus mult add_ac) lemma zmult_commute: "(z::int) * w = w * z" by (cases z, cases w, simp add: mult add_ac mult_ac) lemma zmult_assoc: "((z1::int) * z2) * z3 = z1 * (z2 * z3)" by (cases z1, cases z2, cases z3, simp add: mult add_mult_distrib2 mult_ac) lemma zadd_zmult_distrib: "((z1::int) + z2) * w = (z1 * w) + (z2 * w)" by (cases z1, cases z2, cases w, simp add: add mult add_mult_distrib2 mult_ac) lemma zadd_zmult_distrib2: "(w::int) * (z1 + z2) = (w * z1) + (w * z2)" by (simp add: zmult_commute [of w] zadd_zmult_distrib) lemma zdiff_zmult_distrib: "((z1::int) - z2) * w = (z1 * w) - (z2 * w)" by (simp add: diff_int_def zadd_zmult_distrib zmult_zminus) lemma zdiff_zmult_distrib2: "(w::int) * (z1 - z2) = (w * z1) - (w * z2)" by (simp add: zmult_commute [of w] zdiff_zmult_distrib) lemmas int_distrib = zadd_zmult_distrib zadd_zmult_distrib2 zdiff_zmult_distrib zdiff_zmult_distrib2 lemma int_mult: "int (m * n) = (int m) * (int n)" by (simp add: int_def mult) text{*Compatibility binding*} lemmas zmult_int = int_mult [symmetric] lemma zmult_1: "(1::int) * z = z" by (cases z, simp add: One_int_def int_def mult) lemma zmult_1_right: "z * (1::int) = z" by (rule trans [OF zmult_commute zmult_1]) text{*The integers form a @{text comm_ring_1}*} instance int :: comm_ring_1 proof fix i j k :: int show "(i + j) + k = i + (j + k)" by (simp add: zadd_assoc) show "i + j = j + i" by (simp add: zadd_commute) show "0 + i = i" by (rule zadd_0) show "- i + i = 0" by (rule zadd_zminus_inverse2) show "i - j = i + (-j)" by (simp add: diff_int_def) show "(i * j) * k = i * (j * k)" by (rule zmult_assoc) show "i * j = j * i" by (rule zmult_commute) show "1 * i = i" by (rule zmult_1) show "(i + j) * k = i * k + j * k" by (simp add: int_distrib) show "0 ≠ (1::int)" by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq) qed subsection{*The @{text "≤"} Ordering*} lemma le: "(Abs_Integ(intrel``{(x,y)}) ≤ Abs_Integ(intrel``{(u,v)})) = (x+v ≤ u+y)" by (force simp add: le_int_def) lemma zle_refl: "w ≤ (w::int)" by (cases w, simp add: le) lemma zle_trans: "[| i ≤ j; j ≤ k |] ==> i ≤ (k::int)" by (cases i, cases j, cases k, simp add: le) lemma zle_anti_sym: "[| z ≤ w; w ≤ z |] ==> z = (w::int)" by (cases w, cases z, simp add: le) (* Axiom 'order_less_le' of class 'order': *) lemma zless_le: "((w::int) < z) = (w ≤ z & w ≠ z)" by (simp add: less_int_def) instance int :: order by intro_classes (assumption | rule zle_refl zle_trans zle_anti_sym zless_le)+ (* Axiom 'linorder_linear' of class 'linorder': *) lemma zle_linear: "(z::int) ≤ w | w ≤ z" by (cases z, cases w) (simp add: le linorder_linear) instance int :: linorder by intro_classes (rule zle_linear) lemmas zless_linear = linorder_less_linear [where 'a = int] lemmas linorder_neqE_int = linorder_neqE[where 'a = int] lemma int_eq_0_conv [simp]: "(int n = 0) = (n = 0)" by (simp add: Zero_int_def) lemma zless_int [simp]: "(int m < int n) = (m<n)" by (simp add: le add int_def linorder_not_le [symmetric]) lemma int_less_0_conv [simp]: "~ (int k < 0)" by (simp add: Zero_int_def) lemma zero_less_int_conv [simp]: "(0 < int n) = (0 < n)" by (simp add: Zero_int_def) lemma int_0_less_1: "0 < (1::int)" by (simp only: Zero_int_def One_int_def One_nat_def zless_int) lemma int_0_neq_1 [simp]: "0 ≠ (1::int)" by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq) lemma zle_int [simp]: "(int m ≤ int n) = (m≤n)" by (simp add: linorder_not_less [symmetric]) lemma zero_zle_int [simp]: "(0 ≤ int n)" by (simp add: Zero_int_def) lemma int_le_0_conv [simp]: "(int n ≤ 0) = (n = 0)" by (simp add: Zero_int_def) lemma int_0 [simp]: "int 0 = (0::int)" by (simp add: Zero_int_def) lemma int_1 [simp]: "int 1 = 1" by (simp add: One_int_def) lemma int_Suc0_eq_1: "int (Suc 0) = 1" by (simp add: One_int_def One_nat_def) subsection{*Monotonicity results*} lemma zadd_left_mono: "i ≤ j ==> k + i ≤ k + (j::int)" by (cases i, cases j, cases k, simp add: le add) lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)" apply (cases i, cases j, cases k) apply (simp add: linorder_not_le [where 'a = int, symmetric] linorder_not_le [where 'a = nat] le add) done lemma zadd_zless_mono: "[| w'<w; z'≤z |] ==> w' + z' < w + (z::int)" by (rule order_less_le_trans [OF zadd_strict_right_mono zadd_left_mono]) subsection{*Strict Monotonicity of Multiplication*} text{*strict, in 1st argument; proof is by induction on k>0*} lemma zmult_zless_mono2_lemma: "i<j ==> 0<k ==> int k * i < int k * j" apply (induct "k", simp) apply (simp add: int_Suc) apply (case_tac "k=0") apply (simp_all add: zadd_zmult_distrib int_Suc0_eq_1 order_le_less) apply (simp add: zadd_zless_mono int_Suc0_eq_1 order_le_less) done lemma zero_le_imp_eq_int: "0 ≤ k ==> ∃n. k = int n" apply (cases k) apply (auto simp add: le add int_def Zero_int_def) apply (rule_tac x="x-y" in exI, simp) done lemma zmult_zless_mono2: "[| i<j; (0::int) < k |] ==> k*i < k*j" apply (frule order_less_imp_le [THEN zero_le_imp_eq_int]) apply (auto simp add: zmult_zless_mono2_lemma) done defs (overloaded) zabs_def: "abs(i::int) == if i < 0 then -i else i" text{*The integers form an ordered @{text comm_ring_1}*} instance int :: ordered_idom proof fix i j k :: int show "i ≤ j ==> k + i ≤ k + j" by (rule zadd_left_mono) show "i < j ==> 0 < k ==> k * i < k * j" by (rule zmult_zless_mono2) show "¦i¦ = (if i < 0 then -i else i)" by (simp only: zabs_def) qed lemma zless_imp_add1_zle: "w<z ==> w + (1::int) ≤ z" apply (cases w, cases z) apply (simp add: linorder_not_le [symmetric] le int_def add One_int_def) done subsection{*Magnitide of an Integer, as a Natural Number: @{term nat}*} constdefs nat :: "int => nat" "nat z == contents (\<Union>(x,y) ∈ Rep_Integ z. { x-y })" lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y" proof - have "(λ(x,y). {x-y}) respects intrel" by (simp add: congruent_def, arith) thus ?thesis by (simp add: nat_def UN_equiv_class [OF equiv_intrel]) qed lemma nat_int [simp]: "nat(int n) = n" by (simp add: nat int_def) lemma nat_zero [simp]: "nat 0 = 0" by (simp only: Zero_int_def nat_int) lemma int_nat_eq [simp]: "int (nat z) = (if 0 ≤ z then z else 0)" by (cases z, simp add: nat le int_def Zero_int_def) corollary nat_0_le: "0 ≤ z ==> int (nat z) = z" by simp lemma nat_le_0 [simp]: "z ≤ 0 ==> nat z = 0" by (cases z, simp add: nat le int_def Zero_int_def) lemma nat_le_eq_zle: "0 < w | 0 ≤ z ==> (nat w ≤ nat z) = (w≤z)" apply (cases w, cases z) apply (simp add: nat le linorder_not_le [symmetric] int_def Zero_int_def, arith) done text{*An alternative condition is @{term "0 ≤ w"} *} corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)" by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) corollary nat_less_eq_zless: "0 ≤ w ==> (nat w < nat z) = (w<z)" by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)" apply (cases w, cases z) apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith) done lemma nonneg_eq_int: "[| 0 ≤ z; !!m. z = int m ==> P |] ==> P" by (blast dest: nat_0_le sym) lemma nat_eq_iff: "(nat w = m) = (if 0 ≤ w then w = int m else m=0)" by (cases w, simp add: nat le int_def Zero_int_def, arith) corollary nat_eq_iff2: "(m = nat w) = (if 0 ≤ w then w = int m else m=0)" by (simp only: eq_commute [of m] nat_eq_iff) lemma nat_less_iff: "0 ≤ w ==> (nat w < m) = (w < int m)" apply (cases w) apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith) done lemma int_eq_iff: "(int m = z) = (m = nat z & 0 ≤ z)" by (auto simp add: nat_eq_iff2) lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)" by (insert zless_nat_conj [of 0], auto) lemma nat_add_distrib: "[| (0::int) ≤ z; 0 ≤ z' |] ==> nat (z+z') = nat z + nat z'" by (cases z, cases z', simp add: nat add le int_def Zero_int_def) lemma nat_diff_distrib: "[| (0::int) ≤ z'; z' ≤ z |] ==> nat (z-z') = nat z - nat z'" by (cases z, cases z', simp add: nat add minus diff_minus le int_def Zero_int_def) lemma nat_zminus_int [simp]: "nat (- (int n)) = 0" by (simp add: int_def minus nat Zero_int_def) lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)" by (cases z, simp add: nat le int_def linorder_not_le [symmetric], arith) subsection{*Lemmas about the Function @{term int} and Orderings*} lemma negative_zless_0: "- (int (Suc n)) < 0" by (simp add: order_less_le) lemma negative_zless [iff]: "- (int (Suc n)) < int m" by (rule negative_zless_0 [THEN order_less_le_trans], simp) lemma negative_zle_0: "- int n ≤ 0" by (simp add: minus_le_iff) lemma negative_zle [iff]: "- int n ≤ int m" by (rule order_trans [OF negative_zle_0 zero_zle_int]) lemma not_zle_0_negative [simp]: "~ (0 ≤ - (int (Suc n)))" by (subst le_minus_iff, simp) lemma int_zle_neg: "(int n ≤ - int m) = (n = 0 & m = 0)" by (simp add: int_def le minus Zero_int_def) lemma not_int_zless_negative [simp]: "~ (int n < - int m)" by (simp add: linorder_not_less) lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)" by (force simp add: order_eq_iff [of "- int n"] int_zle_neg) lemma zle_iff_zadd: "(w ≤ z) = (∃n. z = w + int n)" proof (cases w, cases z, simp add: le add int_def) fix a b c d assume "w = Abs_Integ (intrel `` {(a,b)})" "z = Abs_Integ (intrel `` {(c,d)})" show "(a+d ≤ c+b) = (∃n. c+b = a+n+d)" proof assume "a + d ≤ c + b" thus "∃n. c + b = a + n + d" by (auto intro!: exI [where x="c+b - (a+d)"]) next assume "∃n. c + b = a + n + d" then obtain n where "c + b = a + n + d" .. thus "a + d ≤ c + b" by arith qed qed lemma abs_int_eq [simp]: "abs (int m) = int m" by (simp add: abs_if) text{*This version is proved for all ordered rings, not just integers! It is proved here because attribute @{text arith_split} is not available in theory @{text Ring_and_Field}. But is it really better than just rewriting with @{text abs_if}?*} lemma abs_split [arith_split]: "P(abs(a::'a::ordered_idom)) = ((0 ≤ a --> P a) & (a < 0 --> P(-a)))" by (force dest: order_less_le_trans simp add: abs_if linorder_not_less) subsection{*The Constants @{term neg} and @{term iszero}*} constdefs neg :: "'a::ordered_idom => bool" "neg(Z) == Z < 0" (*For simplifying equalities*) iszero :: "'a::comm_semiring_1_cancel => bool" "iszero z == z = (0)" lemma not_neg_int [simp]: "~ neg(int n)" by (simp add: neg_def) lemma neg_zminus_int [simp]: "neg(- (int (Suc n)))" by (simp add: neg_def neg_less_0_iff_less) lemmas neg_eq_less_0 = neg_def lemma not_neg_eq_ge_0: "(~neg x) = (0 ≤ x)" by (simp add: neg_def linorder_not_less) subsection{*To simplify inequalities when Numeral1 can get simplified to 1*} lemma not_neg_0: "~ neg 0" by (simp add: One_int_def neg_def) lemma not_neg_1: "~ neg 1" by (simp add: neg_def linorder_not_less zero_le_one) lemma iszero_0: "iszero 0" by (simp add: iszero_def) lemma not_iszero_1: "~ iszero 1" by (simp add: iszero_def eq_commute) lemma neg_nat: "neg z ==> nat z = 0" by (simp add: neg_def order_less_imp_le) lemma not_neg_nat: "~ neg z ==> int (nat z) = z" by (simp add: linorder_not_less neg_def) subsection{*The Set of Natural Numbers*} constdefs Nats :: "'a::comm_semiring_1_cancel set" "Nats == range of_nat" syntax (xsymbols) Nats :: "'a set" ("\<nat>") lemma of_nat_in_Nats [simp]: "of_nat n ∈ Nats" by (simp add: Nats_def) lemma Nats_0 [simp]: "0 ∈ Nats" apply (simp add: Nats_def) apply (rule range_eqI) apply (rule of_nat_0 [symmetric]) done lemma Nats_1 [simp]: "1 ∈ Nats" apply (simp add: Nats_def) apply (rule range_eqI) apply (rule of_nat_1 [symmetric]) done lemma Nats_add [simp]: "[|a ∈ Nats; b ∈ Nats|] ==> a+b ∈ Nats" apply (auto simp add: Nats_def) apply (rule range_eqI) apply (rule of_nat_add [symmetric]) done lemma Nats_mult [simp]: "[|a ∈ Nats; b ∈ Nats|] ==> a*b ∈ Nats" apply (auto simp add: Nats_def) apply (rule range_eqI) apply (rule of_nat_mult [symmetric]) done text{*Agreement with the specific embedding for the integers*} lemma int_eq_of_nat: "int = (of_nat :: nat => int)" proof fix n show "int n = of_nat n" by (induct n, simp_all add: int_Suc add_ac) qed lemma of_nat_eq_id [simp]: "of_nat = (id :: nat => nat)" proof fix n show "of_nat n = id n" by (induct n, simp_all) qed subsection{*Embedding of the Integers into any @{text comm_ring_1}: @{term of_int}*} constdefs of_int :: "int => 'a::comm_ring_1" "of_int z == contents (\<Union>(i,j) ∈ Rep_Integ z. { of_nat i - of_nat j })" lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j" proof - have "(λ(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel" by (simp add: congruent_def compare_rls of_nat_add [symmetric] del: of_nat_add) thus ?thesis by (simp add: of_int_def UN_equiv_class [OF equiv_intrel]) qed lemma of_int_0 [simp]: "of_int 0 = 0" by (simp add: of_int Zero_int_def int_def) lemma of_int_1 [simp]: "of_int 1 = 1" by (simp add: of_int One_int_def int_def) lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z" by (cases w, cases z, simp add: compare_rls of_int add) lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)" by (cases z, simp add: compare_rls of_int minus) lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z" by (simp add: diff_minus) lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z" apply (cases w, cases z) apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib mult add_ac) done lemma of_int_le_iff [simp]: "(of_int w ≤ (of_int z::'a::ordered_idom)) = (w ≤ z)" apply (cases w) apply (cases z) apply (simp add: compare_rls of_int le diff_int_def add minus of_nat_add [symmetric] del: of_nat_add) done text{*Special cases where either operand is zero*} lemmas of_int_0_le_iff = of_int_le_iff [of 0, simplified] lemmas of_int_le_0_iff = of_int_le_iff [of _ 0, simplified] declare of_int_0_le_iff [simp] declare of_int_le_0_iff [simp] lemma of_int_less_iff [simp]: "(of_int w < (of_int z::'a::ordered_idom)) = (w < z)" by (simp add: linorder_not_le [symmetric]) text{*Special cases where either operand is zero*} lemmas of_int_0_less_iff = of_int_less_iff [of 0, simplified] lemmas of_int_less_0_iff = of_int_less_iff [of _ 0, simplified] declare of_int_0_less_iff [simp] declare of_int_less_0_iff [simp] text{*The ordering on the @{text comm_ring_1} is necessary. See @{text of_nat_eq_iff} above.*} lemma of_int_eq_iff [simp]: "(of_int w = (of_int z::'a::ordered_idom)) = (w = z)" by (simp add: order_eq_iff) text{*Special cases where either operand is zero*} lemmas of_int_0_eq_iff = of_int_eq_iff [of 0, simplified] lemmas of_int_eq_0_iff = of_int_eq_iff [of _ 0, simplified] declare of_int_0_eq_iff [simp] declare of_int_eq_0_iff [simp] lemma of_int_eq_id [simp]: "of_int = (id :: int => int)" proof fix z show "of_int z = id z" by (cases z, simp add: of_int add minus int_eq_of_nat [symmetric] int_def diff_minus) qed subsection{*The Set of Integers*} constdefs Ints :: "'a::comm_ring_1 set" "Ints == range of_int" syntax (xsymbols) Ints :: "'a set" ("\<int>") lemma Ints_0 [simp]: "0 ∈ Ints" apply (simp add: Ints_def) apply (rule range_eqI) apply (rule of_int_0 [symmetric]) done lemma Ints_1 [simp]: "1 ∈ Ints" apply (simp add: Ints_def) apply (rule range_eqI) apply (rule of_int_1 [symmetric]) done lemma Ints_add [simp]: "[|a ∈ Ints; b ∈ Ints|] ==> a+b ∈ Ints" apply (auto simp add: Ints_def) apply (rule range_eqI) apply (rule of_int_add [symmetric]) done lemma Ints_minus [simp]: "a ∈ Ints ==> -a ∈ Ints" apply (auto simp add: Ints_def) apply (rule range_eqI) apply (rule of_int_minus [symmetric]) done lemma Ints_diff [simp]: "[|a ∈ Ints; b ∈ Ints|] ==> a-b ∈ Ints" apply (auto simp add: Ints_def) apply (rule range_eqI) apply (rule of_int_diff [symmetric]) done lemma Ints_mult [simp]: "[|a ∈ Ints; b ∈ Ints|] ==> a*b ∈ Ints" apply (auto simp add: Ints_def) apply (rule range_eqI) apply (rule of_int_mult [symmetric]) done text{*Collapse nested embeddings*} lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n" by (induct n, auto) lemma of_int_int_eq [simp]: "of_int (int n) = of_nat n" by (simp add: int_eq_of_nat) lemma Ints_cases [case_names of_int, cases set: Ints]: "q ∈ \<int> ==> (!!z. q = of_int z ==> C) ==> C" proof (simp add: Ints_def) assume "!!z. q = of_int z ==> C" assume "q ∈ range of_int" thus C .. qed lemma Ints_induct [case_names of_int, induct set: Ints]: "q ∈ \<int> ==> (!!z. P (of_int z)) ==> P q" by (rule Ints_cases) auto (* int (Suc n) = 1 + int n *) declare int_Suc [simp] subsection{*More Properties of @{term setsum} and @{term setprod}*} text{*By Jeremy Avigad*} lemma of_nat_setsum: "of_nat (setsum f A) = (∑x∈A. of_nat(f x))" apply (case_tac "finite A") apply (erule finite_induct, auto) done lemma of_int_setsum: "of_int (setsum f A) = (∑x∈A. of_int(f x))" apply (case_tac "finite A") apply (erule finite_induct, auto) done lemma int_setsum: "int (setsum f A) = (∑x∈A. int(f x))" by (simp add: int_eq_of_nat of_nat_setsum) lemma of_nat_setprod: "of_nat (setprod f A) = (∏x∈A. of_nat(f x))" apply (case_tac "finite A") apply (erule finite_induct, auto) done lemma of_int_setprod: "of_int (setprod f A) = (∏x∈A. of_int(f x))" apply (case_tac "finite A") apply (erule finite_induct, auto) done lemma int_setprod: "int (setprod f A) = (∏x∈A. int(f x))" by (simp add: int_eq_of_nat of_nat_setprod) lemma setprod_nonzero_nat: "finite A ==> (∀x ∈ A. f x ≠ (0::nat)) ==> setprod f A ≠ 0" by (rule setprod_nonzero, auto) lemma setprod_zero_eq_nat: "finite A ==> (setprod f A = (0::nat)) = (∃x ∈ A. f x = 0)" by (rule setprod_zero_eq, auto) lemma setprod_nonzero_int: "finite A ==> (∀x ∈ A. f x ≠ (0::int)) ==> setprod f A ≠ 0" by (rule setprod_nonzero, auto) lemma setprod_zero_eq_int: "finite A ==> (setprod f A = (0::int)) = (∃x ∈ A. f x = 0)" by (rule setprod_zero_eq, auto) text{*Now we replace the case analysis rule by a more conventional one: whether an integer is negative or not.*} lemma zless_iff_Suc_zadd: "(w < z) = (∃n. z = w + int(Suc n))" apply (cases z, cases w) apply (auto simp add: le add int_def linorder_not_le [symmetric]) apply (rename_tac a b c d) apply (rule_tac x="a+d - Suc(c+b)" in exI) apply arith done lemma negD: "x<0 ==> ∃n. x = - (int (Suc n))" apply (cases x) apply (auto simp add: le minus Zero_int_def int_def order_less_le) apply (rule_tac x="y - Suc x" in exI, arith) done theorem int_cases [cases type: int, case_names nonneg neg]: "[|!! n. z = int n ==> P; !! n. z = - (int (Suc n)) ==> P |] ==> P" apply (case_tac "z < 0", blast dest!: negD) apply (simp add: linorder_not_less) apply (blast dest: nat_0_le [THEN sym]) done theorem int_induct [induct type: int, case_names nonneg neg]: "[|!! n. P (int n); !!n. P (- (int (Suc n))) |] ==> P z" by (cases z) auto text{*Contributed by Brian Huffman*} theorem int_diff_cases [case_names diff]: assumes prem: "!!m n. z = int m - int n ==> P" shows "P" apply (rule_tac z=z in int_cases) apply (rule_tac m=n and n=0 in prem, simp) apply (rule_tac m=0 and n="Suc n" in prem, simp) done lemma of_nat_nat: "0 ≤ z ==> of_nat (nat z) = of_int z" apply (cases z) apply (simp_all add: not_zle_0_negative del: int_Suc) done subsection {* Configuration of the code generator *} (*FIXME: the IntInf.fromInt below hides a dependence on fixed-precision ints!*) types_code "int" ("int") attach (term_of) {* val term_of_int = HOLogic.mk_int o IntInf.fromInt; *} attach (test) {* fun gen_int i = one_of [~1, 1] * random_range 0 i; *} constdefs int_aux :: "int => nat => int" "int_aux i n == (i + int n)" nat_aux :: "nat => int => nat" "nat_aux n i == (n + nat i)" lemma [code]: "int_aux i 0 = i" "int_aux i (Suc n) = int_aux (i + 1) n" -- {* tail recursive *} "int n = int_aux 0 n" by (simp add: int_aux_def)+ lemma [code]: "nat_aux n i = (if i <= 0 then n else nat_aux (Suc n) (i - 1))" -- {* tail recursive *} by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le dest: zless_imp_add1_zle) lemma [code]: "nat i = nat_aux 0 i" by (simp add: nat_aux_def) consts_code "0" :: "int" ("0") "1" :: "int" ("1") "uminus" :: "int => int" ("~") "op +" :: "int => int => int" ("(_ +/ _)") "op *" :: "int => int => int" ("(_ */ _)") "op <" :: "int => int => bool" ("(_ </ _)") "op <=" :: "int => int => bool" ("(_ <=/ _)") "neg" ("(_ < 0)") ML {* fun number_of_codegen thy defs gr dep module b (Const ("Numeral.number_of", Type ("fun", [_, T as Type ("IntDef.int", [])])) $ bin) = (SOME (fst (Codegen.invoke_tycodegen thy defs dep module false (gr, T)), Pretty.str (IntInf.toString (HOLogic.dest_binum bin))) handle TERM _ => NONE) | number_of_codegen thy defs gr s thyname b (Const ("Numeral.number_of", Type ("fun", [_, Type ("nat", [])])) $ bin) = SOME (Codegen.invoke_codegen thy defs s thyname b (gr, Const ("IntDef.nat", HOLogic.intT --> HOLogic.natT) $ (Const ("Numeral.number_of", HOLogic.binT --> HOLogic.intT) $ bin))) | number_of_codegen _ _ _ _ _ _ _ = NONE; *} setup {* [Codegen.add_codegen "number_of_codegen" number_of_codegen] *} quickcheck_params [default_type = int] (*Legacy ML bindings, but no longer the structure Int.*) ML {* val zabs_def = thm "zabs_def" val int_0 = thm "int_0"; val int_1 = thm "int_1"; val int_Suc0_eq_1 = thm "int_Suc0_eq_1"; val neg_eq_less_0 = thm "neg_eq_less_0"; val not_neg_eq_ge_0 = thm "not_neg_eq_ge_0"; val not_neg_0 = thm "not_neg_0"; val not_neg_1 = thm "not_neg_1"; val iszero_0 = thm "iszero_0"; val not_iszero_1 = thm "not_iszero_1"; val int_0_less_1 = thm "int_0_less_1"; val int_0_neq_1 = thm "int_0_neq_1"; val negative_zless = thm "negative_zless"; val negative_zle = thm "negative_zle"; val not_zle_0_negative = thm "not_zle_0_negative"; val not_int_zless_negative = thm "not_int_zless_negative"; val negative_eq_positive = thm "negative_eq_positive"; val zle_iff_zadd = thm "zle_iff_zadd"; val abs_int_eq = thm "abs_int_eq"; val abs_split = thm"abs_split"; val nat_int = thm "nat_int"; val nat_zminus_int = thm "nat_zminus_int"; val nat_zero = thm "nat_zero"; val not_neg_nat = thm "not_neg_nat"; val neg_nat = thm "neg_nat"; val zless_nat_eq_int_zless = thm "zless_nat_eq_int_zless"; val nat_0_le = thm "nat_0_le"; val nat_le_0 = thm "nat_le_0"; val zless_nat_conj = thm "zless_nat_conj"; val int_cases = thm "int_cases"; val int_def = thm "int_def"; val Zero_int_def = thm "Zero_int_def"; val One_int_def = thm "One_int_def"; val diff_int_def = thm "diff_int_def"; val inj_int = thm "inj_int"; val zminus_zminus = thm "zminus_zminus"; val zminus_0 = thm "zminus_0"; val zminus_zadd_distrib = thm "zminus_zadd_distrib"; val zadd_commute = thm "zadd_commute"; val zadd_assoc = thm "zadd_assoc"; val zadd_left_commute = thm "zadd_left_commute"; val zadd_ac = thms "zadd_ac"; val zmult_ac = thms "zmult_ac"; val zadd_int = thm "zadd_int"; val zadd_int_left = thm "zadd_int_left"; val int_Suc = thm "int_Suc"; val zadd_0 = thm "zadd_0"; val zadd_0_right = thm "zadd_0_right"; val zmult_zminus = thm "zmult_zminus"; val zmult_commute = thm "zmult_commute"; val zmult_assoc = thm "zmult_assoc"; val zadd_zmult_distrib = thm "zadd_zmult_distrib"; val zadd_zmult_distrib2 = thm "zadd_zmult_distrib2"; val zdiff_zmult_distrib = thm "zdiff_zmult_distrib"; val zdiff_zmult_distrib2 = thm "zdiff_zmult_distrib2"; val int_distrib = thms "int_distrib"; val zmult_int = thm "zmult_int"; val zmult_1 = thm "zmult_1"; val zmult_1_right = thm "zmult_1_right"; val int_int_eq = thm "int_int_eq"; val int_eq_0_conv = thm "int_eq_0_conv"; val zless_int = thm "zless_int"; val int_less_0_conv = thm "int_less_0_conv"; val zero_less_int_conv = thm "zero_less_int_conv"; val zle_int = thm "zle_int"; val zero_zle_int = thm "zero_zle_int"; val int_le_0_conv = thm "int_le_0_conv"; val zle_refl = thm "zle_refl"; val zle_linear = thm "zle_linear"; val zle_trans = thm "zle_trans"; val zle_anti_sym = thm "zle_anti_sym"; val Ints_def = thm "Ints_def"; val Nats_def = thm "Nats_def"; val of_nat_0 = thm "of_nat_0"; val of_nat_Suc = thm "of_nat_Suc"; val of_nat_1 = thm "of_nat_1"; val of_nat_add = thm "of_nat_add"; val of_nat_mult = thm "of_nat_mult"; val zero_le_imp_of_nat = thm "zero_le_imp_of_nat"; val less_imp_of_nat_less = thm "less_imp_of_nat_less"; val of_nat_less_imp_less = thm "of_nat_less_imp_less"; val of_nat_less_iff = thm "of_nat_less_iff"; val of_nat_le_iff = thm "of_nat_le_iff"; val of_nat_eq_iff = thm "of_nat_eq_iff"; val Nats_0 = thm "Nats_0"; val Nats_1 = thm "Nats_1"; val Nats_add = thm "Nats_add"; val Nats_mult = thm "Nats_mult"; val int_eq_of_nat = thm"int_eq_of_nat"; val of_int = thm "of_int"; val of_int_0 = thm "of_int_0"; val of_int_1 = thm "of_int_1"; val of_int_add = thm "of_int_add"; val of_int_minus = thm "of_int_minus"; val of_int_diff = thm "of_int_diff"; val of_int_mult = thm "of_int_mult"; val of_int_le_iff = thm "of_int_le_iff"; val of_int_less_iff = thm "of_int_less_iff"; val of_int_eq_iff = thm "of_int_eq_iff"; val Ints_0 = thm "Ints_0"; val Ints_1 = thm "Ints_1"; val Ints_add = thm "Ints_add"; val Ints_minus = thm "Ints_minus"; val Ints_diff = thm "Ints_diff"; val Ints_mult = thm "Ints_mult"; val of_int_of_nat_eq = thm"of_int_of_nat_eq"; val Ints_cases = thm "Ints_cases"; val Ints_induct = thm "Ints_induct"; *} end
lemma intrel_iff:
(((x, y), u, v) ∈ intrel) = (x + v = u + y)
lemma equiv_intrel:
equiv UNIV intrel
lemmas equiv_intrel_iff:
(intrel `` {x} = intrel `` {y}) = ((x, y) ∈ intrel)
lemmas equiv_intrel_iff:
(intrel `` {x} = intrel `` {y}) = ((x, y) ∈ intrel)
lemma
intrel `` {(x, y)} ∈ Integ
lemma eq_Abs_Integ:
(!!x y. z = Abs_Integ (intrel `` {(x, y)}) ==> P) ==> P
lemma inj_int:
inj int
lemma int_int_eq:
(int m = int n) = (m = n)
lemma minus:
- Abs_Integ (intrel `` {(x, y)}) = Abs_Integ (intrel `` {(y, x)})
lemma zminus_zminus:
- (- z) = z
lemma zminus_0:
- 0 = 0
lemma add:
Abs_Integ (intrel `` {(x, y)}) + Abs_Integ (intrel `` {(u, v)}) = Abs_Integ (intrel `` {(x + u, y + v)})
lemma zminus_zadd_distrib:
- (z + w) = - z + - w
lemma zadd_commute:
z + w = w + z
lemma zadd_assoc:
z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)
lemma zadd_left_commute:
x + (y + z) = y + (x + z)
lemmas zadd_ac:
z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)
z + w = w + z
x + (y + z) = y + (x + z)
lemmas zadd_ac:
z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)
z + w = w + z
x + (y + z) = y + (x + z)
lemmas zmult_ac:
a * b * c = a * (b * c)
a * b = b * a
a * (b * c) = b * (a * c)
lemmas zmult_ac:
a * b * c = a * (b * c)
a * b = b * a
a * (b * c) = b * (a * c)
lemma zadd_int:
int m + int n = int (m + n)
lemma zadd_int_left:
int m + (int n + z) = int (m + n) + z
lemma int_Suc:
int (Suc m) = 1 + int m
lemma zadd_0:
0 + z = z
lemma zadd_0_right:
z + 0 = z
lemma zadd_zminus_inverse2:
- z + z = 0
lemma mult_congruent2:
congruent2 intrel intrel (%p1 p2. (%(x, y). (%(u, v). intrel `` {(x * u + y * v, x * v + y * u)}) p2) p1)
lemma mult:
Abs_Integ (intrel `` {(x, y)}) * Abs_Integ (intrel `` {(u, v)}) = Abs_Integ (intrel `` {(x * u + y * v, x * v + y * u)})
lemma zmult_zminus:
- z * w = - (z * w)
lemma zmult_commute:
z * w = w * z
lemma zmult_assoc:
z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)
lemma zadd_zmult_distrib:
(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w
lemma zadd_zmult_distrib2:
w * (z1.0 + z2.0) = w * z1.0 + w * z2.0
lemma zdiff_zmult_distrib:
(z1.0 - z2.0) * w = z1.0 * w - z2.0 * w
lemma zdiff_zmult_distrib2:
w * (z1.0 - z2.0) = w * z1.0 - w * z2.0
lemmas int_distrib:
(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w
w * (z1.0 + z2.0) = w * z1.0 + w * z2.0
(z1.0 - z2.0) * w = z1.0 * w - z2.0 * w
w * (z1.0 - z2.0) = w * z1.0 - w * z2.0
lemmas int_distrib:
(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w
w * (z1.0 + z2.0) = w * z1.0 + w * z2.0
(z1.0 - z2.0) * w = z1.0 * w - z2.0 * w
w * (z1.0 - z2.0) = w * z1.0 - w * z2.0
lemma int_mult:
int (m * n) = int m * int n
lemmas zmult_int:
int m1 * int n1 = int (m1 * n1)
lemmas zmult_int:
int m1 * int n1 = int (m1 * n1)
lemma zmult_1:
1 * z = z
lemma zmult_1_right:
z * 1 = z
lemma le:
(Abs_Integ (intrel `` {(x, y)}) ≤ Abs_Integ (intrel `` {(u, v)})) = (x + v ≤ u + y)
lemma zle_refl:
w ≤ w
lemma zle_trans:
[| i ≤ j; j ≤ k |] ==> i ≤ k
lemma zle_anti_sym:
[| z ≤ w; w ≤ z |] ==> z = w
lemma zless_le:
(w < z) = (w ≤ z ∧ w ≠ z)
lemma zle_linear:
z ≤ w ∨ w ≤ z
lemmas zless_linear:
x < y ∨ x = y ∨ y < x
lemmas zless_linear:
x < y ∨ x = y ∨ y < x
lemmas linorder_neqE_int:
[| x ≠ y; x < y ==> R; y < x ==> R |] ==> R
lemmas linorder_neqE_int:
[| x ≠ y; x < y ==> R; y < x ==> R |] ==> R
lemma int_eq_0_conv:
(int n = 0) = (n = 0)
lemma zless_int:
(int m < int n) = (m < n)
lemma int_less_0_conv:
¬ int k < 0
lemma zero_less_int_conv:
(0 < int n) = (0 < n)
lemma int_0_less_1:
0 < 1
lemma int_0_neq_1:
0 ≠ 1
lemma zle_int:
(int m ≤ int n) = (m ≤ n)
lemma zero_zle_int:
0 ≤ int n
lemma int_le_0_conv:
(int n ≤ 0) = (n = 0)
lemma int_0:
int 0 = 0
lemma int_1:
int 1 = 1
lemma int_Suc0_eq_1:
int (Suc 0) = 1
lemma zadd_left_mono:
i ≤ j ==> k + i ≤ k + j
lemma zadd_strict_right_mono:
i < j ==> i + k < j + k
lemma zadd_zless_mono:
[| w' < w; z' ≤ z |] ==> w' + z' < w + z
lemma zmult_zless_mono2_lemma:
[| i < j; 0 < k |] ==> int k * i < int k * j
lemma zero_le_imp_eq_int:
0 ≤ k ==> ∃n. k = int n
lemma zmult_zless_mono2:
[| i < j; 0 < k |] ==> k * i < k * j
lemma zless_imp_add1_zle:
w < z ==> w + 1 ≤ z
lemma nat:
nat (Abs_Integ (intrel `` {(x, y)})) = x - y
lemma nat_int:
nat (int n) = n
lemma nat_zero:
nat 0 = 0
lemma int_nat_eq:
int (nat z) = (if 0 ≤ z then z else 0)
corollary nat_0_le:
0 ≤ z ==> int (nat z) = z
lemma nat_le_0:
z ≤ 0 ==> nat z = 0
lemma nat_le_eq_zle:
0 < w ∨ 0 ≤ z ==> (nat w ≤ nat z) = (w ≤ z)
corollary nat_mono_iff:
0 < z ==> (nat w < nat z) = (w < z)
corollary nat_less_eq_zless:
0 ≤ w ==> (nat w < nat z) = (w < z)
lemma zless_nat_conj:
(nat w < nat z) = (0 < z ∧ w < z)
lemma nonneg_eq_int:
[| 0 ≤ z; !!m. z = int m ==> P |] ==> P
lemma nat_eq_iff:
(nat w = m) = (if 0 ≤ w then w = int m else m = 0)
corollary nat_eq_iff2:
(m = nat w) = (if 0 ≤ w then w = int m else m = 0)
lemma nat_less_iff:
0 ≤ w ==> (nat w < m) = (w < int m)
lemma int_eq_iff:
(int m = z) = (m = nat z ∧ 0 ≤ z)
lemma zero_less_nat_eq:
(0 < nat z) = (0 < z)
lemma nat_add_distrib:
[| 0 ≤ z; 0 ≤ z' |] ==> nat (z + z') = nat z + nat z'
lemma nat_diff_distrib:
[| 0 ≤ z'; z' ≤ z |] ==> nat (z - z') = nat z - nat z'
lemma nat_zminus_int:
nat (- int n) = 0
lemma zless_nat_eq_int_zless:
(m < nat z) = (int m < z)
lemma negative_zless_0:
- int (Suc n) < 0
lemma negative_zless:
- int (Suc n) < int m
lemma negative_zle_0:
- int n ≤ 0
lemma negative_zle:
- int n ≤ int m
lemma not_zle_0_negative:
¬ 0 ≤ - int (Suc n)
lemma int_zle_neg:
(int n ≤ - int m) = (n = 0 ∧ m = 0)
lemma not_int_zless_negative:
¬ int n < - int m
lemma negative_eq_positive:
(- int n = int m) = (n = 0 ∧ m = 0)
lemma zle_iff_zadd:
(w ≤ z) = (∃n. z = w + int n)
lemma abs_int_eq:
¦int m¦ = int m
lemma abs_split:
P ¦a¦ = (((0::'a) ≤ a --> P a) ∧ (a < (0::'a) --> P (- a)))
lemma not_neg_int:
¬ neg (int n)
lemma neg_zminus_int:
neg (- int (Suc n))
lemmas neg_eq_less_0:
neg Z == Z < (0::'a)
lemmas neg_eq_less_0:
neg Z == Z < (0::'a)
lemma not_neg_eq_ge_0:
(¬ neg x) = ((0::'a) ≤ x)
lemma not_neg_0:
¬ neg (0::'a)
lemma not_neg_1:
¬ neg (1::'a)
lemma iszero_0:
iszero (0::'a)
lemma not_iszero_1:
¬ iszero (1::'a)
lemma neg_nat:
neg z ==> nat z = 0
lemma not_neg_nat:
¬ neg z ==> int (nat z) = z
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 int_eq_of_nat:
int = of_nat
lemma of_nat_eq_id:
of_nat = id
lemma of_int:
of_int (Abs_Integ (intrel `` {(i, j)})) = of_nat i - of_nat j
lemma of_int_0:
of_int 0 = (0::'a)
lemma of_int_1:
of_int 1 = (1::'a)
lemma of_int_add:
of_int (w + z) = of_int w + of_int z
lemma of_int_minus:
of_int (- z) = - of_int z
lemma of_int_diff:
of_int (w - z) = of_int w - of_int z
lemma of_int_mult:
of_int (w * z) = of_int w * of_int z
lemma of_int_le_iff:
(of_int w ≤ of_int z) = (w ≤ z)
lemmas of_int_0_le_iff:
((0::'a) ≤ of_int z) = (0 ≤ z)
lemmas of_int_0_le_iff:
((0::'a) ≤ of_int z) = (0 ≤ z)
lemmas of_int_le_0_iff:
(of_int w ≤ (0::'a)) = (w ≤ 0)
lemmas of_int_le_0_iff:
(of_int w ≤ (0::'a)) = (w ≤ 0)
lemma of_int_less_iff:
(of_int w < of_int z) = (w < z)
lemmas of_int_0_less_iff:
((0::'a) < of_int z) = (0 < z)
lemmas of_int_0_less_iff:
((0::'a) < of_int z) = (0 < z)
lemmas of_int_less_0_iff:
(of_int w < (0::'a)) = (w < 0)
lemmas of_int_less_0_iff:
(of_int w < (0::'a)) = (w < 0)
lemma of_int_eq_iff:
(of_int w = of_int z) = (w = z)
lemmas of_int_0_eq_iff:
((0::'a) = of_int z) = (0 = z)
lemmas of_int_0_eq_iff:
((0::'a) = of_int z) = (0 = z)
lemmas of_int_eq_0_iff:
(of_int w = (0::'a)) = (w = 0)
lemmas of_int_eq_0_iff:
(of_int w = (0::'a)) = (w = 0)
lemma of_int_eq_id:
of_int = id
lemma Ints_0:
(0::'a) ∈ Ints
lemma Ints_1:
(1::'a) ∈ Ints
lemma Ints_add:
[| a ∈ Ints; b ∈ Ints |] ==> a + b ∈ Ints
lemma Ints_minus:
a ∈ Ints ==> - a ∈ Ints
lemma Ints_diff:
[| a ∈ Ints; b ∈ Ints |] ==> a - b ∈ Ints
lemma Ints_mult:
[| a ∈ Ints; b ∈ Ints |] ==> a * b ∈ Ints
lemma of_int_of_nat_eq:
of_int (of_nat n) = of_nat n
lemma of_int_int_eq:
of_int (int n) = of_nat n
lemma Ints_cases:
[| q ∈ Ints; !!z. q = of_int z ==> C |] ==> C
lemma Ints_induct:
[| q ∈ Ints; !!z. P (of_int z) |] ==> P q
lemma of_nat_setsum:
of_nat (setsum f A) = (∑x∈A. of_nat (f x))
lemma of_int_setsum:
of_int (setsum f A) = (∑x∈A. of_int (f x))
lemma int_setsum:
int (setsum f A) = (∑x∈A. int (f x))
lemma of_nat_setprod:
of_nat (setprod f A) = (∏x∈A. of_nat (f x))
lemma of_int_setprod:
of_int (setprod f A) = (∏x∈A. of_int (f x))
lemma int_setprod:
int (setprod f A) = (∏x∈A. int (f x))
lemma setprod_nonzero_nat:
[| finite A; ∀x∈A. f x ≠ 0 |] ==> setprod f A ≠ 0
lemma setprod_zero_eq_nat:
finite A ==> (setprod f A = 0) = (∃x∈A. f x = 0)
lemma setprod_nonzero_int:
[| finite A; ∀x∈A. f x ≠ 0 |] ==> setprod f A ≠ 0
lemma setprod_zero_eq_int:
finite A ==> (setprod f A = 0) = (∃x∈A. f x = 0)
lemma zless_iff_Suc_zadd:
(w < z) = (∃n. z = w + int (Suc n))
lemma negD:
x < 0 ==> ∃n. x = - int (Suc n)
theorem int_cases:
[| !!n. z = int n ==> P; !!n. z = - int (Suc n) ==> P |] ==> P
theorem int_induct:
[| !!n. P (int n); !!n. P (- int (Suc n)) |] ==> P z
theorem int_diff_cases:
(!!m n. z = int m - int n ==> P) ==> P
lemma of_nat_nat:
0 ≤ z ==> of_nat (nat z) = of_int z
lemma
int_aux i 0 = i
int_aux i (Suc n) = int_aux (i + 1) n
int n = int_aux 0 n
lemma
nat_aux n i = (if i ≤ 0 then n else nat_aux (Suc n) (i - 1))
lemma
nat i = nat_aux 0 i