(* Title: IntDef.thy ID: $Id: IntDef.thy,v 1.26 2007/11/08 19:08:00 wenzelm 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 Nat begin text {* the equivalence relation underlying the integers *} definition intrel :: "((nat × nat) × (nat × nat)) set" where "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 :: zero Zero_int_def: "0 ≡ Abs_Integ (intrel `` {(0, 0)})" .. instance int :: one One_int_def: "1 ≡ Abs_Integ (intrel `` {(1, 0)})" .. instance int :: plus add_int_def: "z + w ≡ Abs_Integ (\<Union>(x, y) ∈ Rep_Integ z. \<Union>(u, v) ∈ Rep_Integ w. intrel `` {(x + u, y + v)})" .. instance int :: minus minus_int_def: "- z ≡ Abs_Integ (\<Union>(x, y) ∈ Rep_Integ z. intrel `` {(y, x)})" diff_int_def: "z - w ≡ z + (-w)" .. instance int :: times 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)})" .. instance int :: ord le_int_def: "z ≤ w ≡ ∃x y u v. x+v ≤ u+y ∧ (x, y) ∈ Rep_Integ z ∧ (u, v) ∈ Rep_Integ w" less_int_def: "z < w ≡ z ≤ w ∧ z ≠ w" .. lemmas [code func del] = Zero_int_def One_int_def add_int_def minus_int_def mult_int_def le_int_def less_int_def subsection{*Construction of the Integers*} 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 [simp] = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I] 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: @{prop "[|x ∈ Integ; y ∈ Integ|] ==> (Abs_Integ x = Abs_Integ y) = (x=y)"} *} declare Abs_Integ_inject [simp,noatp] Abs_Integ_inverse [simp,noatp] 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 subsection{*Arithmetic Operations*} 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 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 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]) 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 (cases i, cases j, cases k) (simp add: add add_assoc) show "i + j = j + i" by (cases i, cases j) (simp add: add_ac add) show "0 + i = i" by (cases i) (simp add: Zero_int_def add) show "- i + i = 0" by (cases i) (simp add: Zero_int_def minus add) show "i - j = i + - j" by (simp add: diff_int_def) show "(i * j) * k = i * (j * k)" by (cases i, cases j, cases k) (simp add: mult ring_simps) show "i * j = j * i" by (cases i, cases j) (simp add: mult ring_simps) show "1 * i = i" by (cases i) (simp add: One_int_def mult) show "(i + j) * k = i * k + j * k" by (cases i, cases j, cases k) (simp add: add mult ring_simps) show "0 ≠ (1::int)" by (simp add: Zero_int_def One_int_def) qed lemma int_def: "of_nat m = Abs_Integ (intrel `` {(m, 0)})" by (induct m, simp_all add: Zero_int_def One_int_def add) 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 less: "(Abs_Integ(intrel``{(x,y)}) < Abs_Integ(intrel``{(u,v)})) = (x+v < u+y)" by (simp add: less_int_def le order_less_le) instance int :: linorder proof fix i j k :: int show "(i < j) = (i ≤ j ∧ i ≠ j)" by (simp add: less_int_def) show "i ≤ i" by (cases i) (simp add: le) show "i ≤ j ==> j ≤ k ==> i ≤ k" by (cases i, cases j, cases k) (simp add: le) show "i ≤ j ==> j ≤ i ==> i = j" by (cases i, cases j) (simp add: le) show "i ≤ j ∨ j ≤ i" by (cases i, cases j) (simp add: le linorder_linear) qed instance int :: pordered_cancel_ab_semigroup_add proof fix i j k :: int show "i ≤ j ==> k + i ≤ k + j" by (cases i, cases j, cases k) (simp add: le add) qed text{*Strict Monotonicity of Multiplication*} text{*strict, in 1st argument; proof is by induction on k>0*} lemma zmult_zless_mono2_lemma: "(i::int)<j ==> 0<k ==> of_nat k * i < of_nat k * j" apply (induct "k", simp) apply (simp add: left_distrib) apply (case_tac "k=0") apply (simp_all add: add_strict_mono) done lemma zero_le_imp_eq_int: "(0::int) ≤ k ==> ∃n. k = of_nat 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 zero_less_imp_eq_int: "(0::int) < k ==> ∃n>0. k = of_nat n" apply (cases k) apply (simp add: less 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 (drule zero_less_imp_eq_int) apply (auto simp add: zmult_zless_mono2_lemma) done instance int :: abs zabs_def: "¦i::int¦ ≡ if i < 0 then - i else i" .. instance int :: sgn zsgn_def: "sgn(i::int) ≡ (if i=0 then 0 else if 0<i then 1 else - 1)" .. instance int :: distrib_lattice "inf ≡ min" "sup ≡ max" by intro_classes (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1) text{*The integers form an ordered integral domain*} instance int :: ordered_idom proof fix i j k :: int 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) show "sgn(i::int) = (if i=0 then 0 else if 0<i then 1 else - 1)" by (simp only: zsgn_def) qed lemma zless_imp_add1_zle: "w<z ==> w + (1::int) ≤ z" apply (cases w, cases z) apply (simp add: less le add One_int_def) done subsection{*Magnitude of an Integer, as a Natural Number: @{term nat}*} definition nat :: "int => nat" where [code func del]: "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 (of_nat n) = n" by (simp add: nat int_def) lemma nat_zero [simp]: "nat 0 = 0" by (simp add: Zero_int_def nat) lemma int_nat_eq [simp]: "of_nat (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 ==> of_nat (nat z) = z" by simp lemma nat_le_0 [simp]: "z ≤ 0 ==> nat z = 0" by (cases z, simp add: nat le 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] 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 [simp]: "(nat w < nat z) = (0 < z & w < z)" apply (cases w, cases z) apply (simp add: nat le Zero_int_def linorder_not_le [symmetric], arith) done lemma nonneg_eq_int: fixes z :: int assumes "0 ≤ z" and "!!m. z = of_nat m ==> P" shows P using assms by (blast dest: nat_0_le sym) lemma nat_eq_iff: "(nat w = m) = (if 0 ≤ w then w = of_nat 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 = of_nat m else m=0)" by (simp only: eq_commute [of m] nat_eq_iff) lemma nat_less_iff: "0 ≤ w ==> (nat w < m) = (w < of_nat m)" apply (cases w) apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith) done lemma int_eq_iff: "(of_nat 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 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 Zero_int_def) lemma nat_zminus_int [simp]: "nat (- (of_nat n)) = 0" by (simp add: int_def minus nat Zero_int_def) lemma zless_nat_eq_int_zless: "(m < nat z) = (of_nat m < z)" by (cases z, simp add: nat less int_def, arith) subsection{*Lemmas about the Function @{term of_nat} and Orderings*} lemma negative_zless_0: "- (of_nat (Suc n)) < (0 :: int)" by (simp add: order_less_le del: of_nat_Suc) lemma negative_zless [iff]: "- (of_nat (Suc n)) < (of_nat m :: int)" by (rule negative_zless_0 [THEN order_less_le_trans], simp) lemma negative_zle_0: "- of_nat n ≤ (0 :: int)" by (simp add: minus_le_iff) lemma negative_zle [iff]: "- of_nat n ≤ (of_nat m :: int)" by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff]) lemma not_zle_0_negative [simp]: "~ (0 ≤ - (of_nat (Suc n) :: int))" by (subst le_minus_iff, simp del: of_nat_Suc) lemma int_zle_neg: "((of_nat n :: int) ≤ - of_nat m) = (n = 0 & m = 0)" by (simp add: int_def le minus Zero_int_def) lemma not_int_zless_negative [simp]: "~ ((of_nat n :: int) < - of_nat m)" by (simp add: linorder_not_less) lemma negative_eq_positive [simp]: "((- of_nat n :: int) = of_nat m) = (n = 0 & m = 0)" by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg) lemma zle_iff_zadd: "(w::int) ≤ z <-> (∃n. z = w + of_nat n)" proof - have "(w ≤ z) = (0 ≤ z - w)" by (simp only: le_diff_eq add_0_left) also have "… = (∃n. z - w = of_nat n)" by (auto elim: zero_le_imp_eq_int) also have "… = (∃n. z = w + of_nat n)" by (simp only: group_simps) finally show ?thesis . qed lemma zadd_int_left: "of_nat m + (of_nat n + z) = of_nat (m + n) + (z::int)" by simp lemma int_Suc0_eq_1: "of_nat (Suc 0) = (1::int)" by simp 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,noatp]: "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 {* Constants @{term neg} and @{term iszero} *} definition neg :: "'a::ordered_idom => bool" where "neg Z <-> Z < 0" definition (*for simplifying equalities*) iszero :: "'a::semiring_1 => bool" where "iszero z <-> z = 0" lemma not_neg_int [simp]: "~ neg (of_nat n)" by (simp add: neg_def) lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))" by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc) 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) text{*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 ==> of_nat (nat z) = z" by (simp add: linorder_not_less neg_def) subsection{*Embedding of the Integers into any @{text ring_1}: @{term of_int}*} context ring_1 begin term of_nat definition of_int :: "int => 'a" where "of_int z = contents (\<Union>(i, j) ∈ Rep_Integ z. { of_nat i - of_nat j })" lemmas [code func del] = of_int_def 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) lemma of_int_1 [simp]: "of_int 1 = 1" by (simp add: of_int One_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_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 of_nat_mult) done text{*Collapse nested embeddings*} lemma of_int_of_nat_eq [simp]: "of_int (Nat.of_nat n) = of_nat n" by (induct n, auto) end lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z" by (simp add: diff_minus) 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 [simp] = of_int_le_iff [of 0, simplified] lemmas of_int_le_0_iff [simp] = of_int_le_iff [of _ 0, simplified] 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 [simp] = of_int_less_iff [of 0, simplified] lemmas of_int_less_0_iff [simp] = of_int_less_iff [of _ 0, simplified] text{*Class for unital rings with characteristic zero. Includes non-ordered rings like the complex numbers.*} class ring_char_0 = ring_1 + semiring_char_0 begin lemma of_int_eq_iff [simp]: "of_int w = of_int z <-> w = z" apply (cases w, cases z, simp add: of_int) apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq) apply (simp only: of_nat_add [symmetric] of_nat_eq_iff) done text{*Special cases where either operand is zero*} lemmas of_int_0_eq_iff [simp] = of_int_eq_iff [of 0, simplified] lemmas of_int_eq_0_iff [simp] = of_int_eq_iff [of _ 0, simplified] end text{*Every @{text ordered_idom} has characteristic zero.*} instance ordered_idom ⊆ ring_char_0 .. lemma of_int_eq_id [simp]: "of_int = id" proof fix z show "of_int z = id z" by (cases z) (simp add: of_int add minus int_def diff_minus) qed context ring_1 begin lemma of_nat_nat: "0 ≤ z ==> of_nat (nat z) = of_int z" by (cases z rule: eq_Abs_Integ) (simp add: nat le of_int Zero_int_def of_nat_diff) end subsection{*The Set of Integers*} context ring_1 begin definition Ints :: "'a set" where "Ints = range of_int" end notation (xsymbols) Ints ("\<int>") context ring_1 begin lemma Ints_0 [simp]: "0 ∈ \<int>" apply (simp add: Ints_def) apply (rule range_eqI) apply (rule of_int_0 [symmetric]) done lemma Ints_1 [simp]: "1 ∈ \<int>" apply (simp add: Ints_def) apply (rule range_eqI) apply (rule of_int_1 [symmetric]) done lemma Ints_add [simp]: "a ∈ \<int> ==> b ∈ \<int> ==> a + b ∈ \<int>" apply (auto simp add: Ints_def) apply (rule range_eqI) apply (rule of_int_add [symmetric]) done lemma Ints_minus [simp]: "a ∈ \<int> ==> -a ∈ \<int>" apply (auto simp add: Ints_def) apply (rule range_eqI) apply (rule of_int_minus [symmetric]) done lemma Ints_mult [simp]: "a ∈ \<int> ==> b ∈ \<int> ==> a * b ∈ \<int>" apply (auto simp add: Ints_def) apply (rule range_eqI) apply (rule of_int_mult [symmetric]) done lemma Ints_cases [cases set: Ints]: assumes "q ∈ \<int>" obtains (of_int) z where "q = of_int z" unfolding Ints_def proof - from `q ∈ \<int>` have "q ∈ range of_int" unfolding Ints_def . then obtain z where "q = of_int z" .. then show thesis .. 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 end lemma Ints_diff [simp]: "a ∈ \<int> ==> b ∈ \<int> ==> a-b ∈ \<int>" apply (auto simp add: Ints_def) apply (rule range_eqI) apply (rule of_int_diff [symmetric]) done subsection {* @{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 (cases "finite A") apply (erule finite_induct, auto) done lemma of_int_setsum: "of_int (setsum f A) = (∑x∈A. of_int(f x))" apply (cases "finite A") apply (erule finite_induct, auto) done lemma of_nat_setprod: "of_nat (setprod f A) = (∏x∈A. of_nat(f x))" apply (cases "finite A") apply (erule finite_induct, auto simp add: of_nat_mult) done lemma of_int_setprod: "of_int (setprod f A) = (∏x∈A. of_int(f x))" apply (cases "finite A") apply (erule finite_induct, auto) done 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) lemmas int_setsum = of_nat_setsum [where 'a=int] lemmas int_setprod = of_nat_setprod [where 'a=int] subsection {* Further properties *} 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 :: int) < z <-> (∃n. z = w + of_nat (Suc n))" apply (cases z, cases w) apply (auto simp add: less add int_def) apply (rename_tac a b c d) apply (rule_tac x="a+d - Suc(c+b)" in exI) apply arith done lemma negD: "(x :: int) < 0 ==> ∃n. x = - (of_nat (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) = of_nat n ==> P; !! n. z = - (of_nat (Suc n)) ==> P |] ==> P" apply (cases "z < 0", blast dest!: negD) apply (simp add: linorder_not_less del: of_nat_Suc) apply (blast dest: nat_0_le [THEN sym]) done theorem int_induct [induct type: int, case_names nonneg neg]: "[|!! n. P (of_nat n :: int); !!n. P (- (of_nat (Suc n))) |] ==> P z" by (cases z rule: int_cases) auto text{*Contributed by Brian Huffman*} theorem int_diff_cases: obtains (diff) m n where "(z::int) = of_nat m - of_nat n" apply (cases z rule: eq_Abs_Integ) apply (rule_tac m=x and n=y in diff) apply (simp add: int_def diff_def minus add) done subsection {* Legacy theorems *} lemmas zminus_zminus = minus_minus [of "z::int", standard] lemmas zminus_0 = minus_zero [where 'a=int] lemmas zminus_zadd_distrib = minus_add_distrib [of "z::int" "w", standard] lemmas zadd_commute = add_commute [of "z::int" "w", standard] lemmas zadd_assoc = add_assoc [of "z1::int" "z2" "z3", standard] lemmas zadd_left_commute = add_left_commute [of "x::int" "y" "z", standard] lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute lemmas zmult_ac = OrderedGroup.mult_ac lemmas zadd_0 = OrderedGroup.add_0_left [of "z::int", standard] lemmas zadd_0_right = OrderedGroup.add_0_left [of "z::int", standard] lemmas zadd_zminus_inverse2 = left_minus [of "z::int", standard] lemmas zmult_zminus = mult_minus_left [of "z::int" "w", standard] lemmas zmult_commute = mult_commute [of "z::int" "w", standard] lemmas zmult_assoc = mult_assoc [of "z1::int" "z2" "z3", standard] lemmas zadd_zmult_distrib = left_distrib [of "z1::int" "z2" "w", standard] lemmas zadd_zmult_distrib2 = right_distrib [of "w::int" "z1" "z2", standard] lemmas zdiff_zmult_distrib = left_diff_distrib [of "z1::int" "z2" "w", standard] lemmas zdiff_zmult_distrib2 = right_diff_distrib [of "w::int" "z1" "z2", standard] lemmas int_distrib = zadd_zmult_distrib zadd_zmult_distrib2 zdiff_zmult_distrib zdiff_zmult_distrib2 lemmas zmult_1 = mult_1_left [of "z::int", standard] lemmas zmult_1_right = mult_1_right [of "z::int", standard] lemmas zle_refl = order_refl [of "w::int", standard] lemmas zle_trans = order_trans [where 'a=int and x="i" and y="j" and z="k", standard] lemmas zle_anti_sym = order_antisym [of "z::int" "w", standard] lemmas zle_linear = linorder_linear [of "z::int" "w", standard] lemmas zless_linear = linorder_less_linear [where 'a = int] lemmas zadd_left_mono = add_left_mono [of "i::int" "j" "k", standard] lemmas zadd_strict_right_mono = add_strict_right_mono [of "i::int" "j" "k", standard] lemmas zadd_zless_mono = add_less_le_mono [of "w'::int" "w" "z'" "z", standard] lemmas int_0_less_1 = zero_less_one [where 'a=int] lemmas int_0_neq_1 = zero_neq_one [where 'a=int] lemmas inj_int = inj_of_nat [where 'a=int] lemmas int_int_eq = of_nat_eq_iff [where 'a=int] lemmas zadd_int = of_nat_add [where 'a=int, symmetric] lemmas int_mult = of_nat_mult [where 'a=int] lemmas zmult_int = of_nat_mult [where 'a=int, symmetric] lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n", standard] lemmas zless_int = of_nat_less_iff [where 'a=int] lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k", standard] lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int] lemmas zle_int = of_nat_le_iff [where 'a=int] lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int] lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n", standard] lemmas int_0 = of_nat_0 [where 'a=int] lemmas int_1 = of_nat_1 [where 'a=int] lemmas int_Suc = of_nat_Suc [where 'a=int] lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m", standard] lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int] lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric] lemmas zless_le = less_int_def [THEN meta_eq_to_obj_eq] lemmas int_eq_of_nat = TrueI abbreviation int :: "nat => int" where "int ≡ of_nat" end
lemma
0 == Abs_Integ (intrel `` {(0, 0)})
1 == Abs_Integ (intrel `` {(1, 0)})
z + w ==
Abs_Integ
(UN (x, y):Rep_Integ z. UN (u, v):Rep_Integ w. intrel `` {(x + u, y + v)})
- z == Abs_Integ (UN (x, y):Rep_Integ z. intrel `` {(y, x)})
z * w ==
Abs_Integ
(UN (x, y):Rep_Integ z.
UN (u, v):Rep_Integ w. intrel `` {(x * u + y * v, x * v + y * u)})
z ≤ w == ∃x y u v. x + v ≤ u + y ∧ (x, y) ∈ Rep_Integ z ∧ (u, v) ∈ Rep_Integ w
z < w == z ≤ w ∧ z ≠ w
lemma intrel_iff:
(((x, y), u, v) ∈ intrel) = (x + v = u + y)
lemma equiv_intrel:
equiv UNIV intrel
lemma 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 minus:
- Abs_Integ (intrel `` {(x, y)}) = Abs_Integ (intrel `` {(y, x)})
lemma add:
Abs_Integ (intrel `` {(x, y)}) + Abs_Integ (intrel `` {(u, v)}) =
Abs_Integ (intrel `` {(x + u, y + v)})
lemma mult_congruent2:
(λp1 p2.
(λ(x, y). (λ(u, v). intrel `` {(x * u + y * v, x * v + y * u)}) p2)
p1) respects2
intrel
lemma mult:
Abs_Integ (intrel `` {(x, y)}) * Abs_Integ (intrel `` {(u, v)}) =
Abs_Integ (intrel `` {(x * u + y * v, x * v + y * u)})
lemma int_def:
of_nat m = Abs_Integ (intrel `` {(m, 0)})
lemma le:
(Abs_Integ (intrel `` {(x, y)}) ≤ Abs_Integ (intrel `` {(u, v)})) =
(x + v ≤ u + y)
lemma less:
(Abs_Integ (intrel `` {(x, y)}) < Abs_Integ (intrel `` {(u, v)})) =
(x + v < u + y)
lemma zmult_zless_mono2_lemma:
[| i < j; 0 < k |] ==> of_nat k * i < of_nat k * j
lemma zero_le_imp_eq_int:
0 ≤ k ==> ∃n. k = of_nat n
lemma zero_less_imp_eq_int:
0 < k ==> ∃n>0. k = of_nat 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 (of_nat n) = n
lemma nat_zero:
nat 0 = 0
lemma int_nat_eq:
of_nat (nat z) = (if 0 ≤ z then z else 0)
corollary nat_0_le:
0 ≤ z ==> of_nat (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 = of_nat m ==> P |] ==> P
lemma nat_eq_iff:
(nat w = m) = (if 0 ≤ w then w = of_nat m else m = 0)
corollary nat_eq_iff2:
(m = nat w) = (if 0 ≤ w then w = of_nat m else m = 0)
lemma nat_less_iff:
0 ≤ w ==> (nat w < m) = (w < of_nat m)
lemma int_eq_iff:
(of_nat 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 (- of_nat n) = 0
lemma zless_nat_eq_int_zless:
(m < nat z) = (of_nat m < z)
lemma negative_zless_0:
- of_nat (Suc n) < 0
lemma negative_zless:
- of_nat (Suc n) < of_nat m
lemma negative_zle_0:
- of_nat n ≤ 0
lemma negative_zle:
- of_nat n ≤ of_nat m
lemma not_zle_0_negative:
¬ 0 ≤ - of_nat (Suc n)
lemma int_zle_neg:
(of_nat n ≤ - of_nat m) = (n = 0 ∧ m = 0)
lemma not_int_zless_negative:
¬ of_nat n < - of_nat m
lemma negative_eq_positive:
(- of_nat n = of_nat m) = (n = 0 ∧ m = 0)
lemma zle_iff_zadd:
(w ≤ z) = (∃n. z = w + of_nat n)
lemma zadd_int_left:
of_nat m + (of_nat n + z) = of_nat (m + n) + z
lemma int_Suc0_eq_1:
of_nat (Suc 0) = 1
lemma abs_split:
P ¦a¦ = (((0::'a) ≤ a --> P a) ∧ (a < (0::'a) --> P (- a)))
lemma not_neg_int:
¬ neg (of_nat n)
lemma neg_zminus_int:
neg (- of_nat (Suc n))
lemma 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 ==> of_nat (nat z) = z
lemma
of_int z = contents (UN (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
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_mult:
of_int (w * z) = of_int w * of_int z
lemma of_int_of_nat_eq:
of_int (of_nat n) = of_nat n
lemma of_int_diff:
of_int (w - z) = of_int w - of_int z
lemma of_int_le_iff:
(of_int w ≤ of_int z) = (w ≤ z)
lemma of_int_0_le_iff:
((0::'a) ≤ of_int z) = (0 ≤ z)
lemma 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)
lemma of_int_0_less_iff:
((0::'a) < of_int z) = (0 < z)
lemma 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)
lemma of_int_0_eq_iff:
((0::'a) = of_int z) = (0 = z)
lemma of_int_eq_0_iff:
(of_int w = (0::'a)) = (w = 0)
lemma of_int_eq_id:
of_int = id
lemma of_nat_nat:
0 ≤ z ==> of_nat (nat z) = of_int z
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_mult:
[| a ∈ Ints; b ∈ Ints |] ==> a * b ∈ Ints
lemma Ints_cases:
[| q ∈ Ints; !!z. q = of_int z ==> thesis |] ==> thesis
lemma Ints_induct:
[| q ∈ Ints; !!z. P (of_int z) |] ==> P q
lemma Ints_diff:
[| a ∈ Ints; b ∈ Ints |] ==> a - b ∈ Ints
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 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 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 int_setsum:
of_nat (setsum f A) = (∑x∈A. of_nat (f x))
lemma int_setprod:
of_nat (setprod f A) = (∏x∈A. of_nat (f x))
lemma zless_iff_Suc_zadd:
(w < z) = (∃n. z = w + of_nat (Suc n))
lemma negD:
x < 0 ==> ∃n. x = - of_nat (Suc n)
theorem int_cases:
[| !!n. z = of_nat n ==> P; !!n. z = - of_nat (Suc n) ==> P |] ==> P
theorem int_induct:
[| !!n. P (of_nat n); !!n. P (- of_nat (Suc n)) |] ==> P z
theorem int_diff_cases:
(!!m n. z = of_nat m - of_nat n ==> thesis) ==> thesis
lemma zminus_zminus:
- (- z) = z
lemma zminus_0:
- 0 = 0
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)
lemma zadd_ac:
z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)
z + w = w + z
x + (y + z) = y + (x + z)
lemma zmult_ac:
a * b * c = a * (b * c)
a * b = b * a
a * (b * c) = b * (a * c)
lemma zadd_0:
0 + z = z
lemma zadd_0_right:
0 + z = z
lemma zadd_zminus_inverse2:
- z + z = 0
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
lemma 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 zmult_1:
1 * z = z
lemma zmult_1_right:
z * 1 = z
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 zle_linear:
z ≤ w ∨ w ≤ z
lemma zless_linear:
x < y ∨ x = y ∨ y < x
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 int_0_less_1:
0 < 1
lemma int_0_neq_1:
0 ≠ 1
lemma inj_int:
inj of_nat
lemma int_int_eq:
(of_nat m = of_nat n) = (m = n)
lemma zadd_int:
of_nat m + of_nat n = of_nat (m + n)
lemma int_mult:
of_nat (m * n) = of_nat m * of_nat n
lemma zmult_int:
of_nat m * of_nat n = of_nat (m * n)
lemma int_eq_0_conv:
(of_nat n = 0) = (n = 0)
lemma zless_int:
(of_nat m < of_nat n) = (m < n)
lemma int_less_0_conv:
¬ of_nat k < 0
lemma zero_less_int_conv:
(0 < of_nat n) = (0 < n)
lemma zle_int:
(of_nat m ≤ of_nat n) = (m ≤ n)
lemma zero_zle_int:
0 ≤ of_nat n
lemma int_le_0_conv:
(of_nat n ≤ 0) = (n = 0)
lemma int_0:
of_nat 0 = 0
lemma int_1:
of_nat 1 = 1
lemma int_Suc:
of_nat (Suc m) = 1 + of_nat m
lemma abs_int_eq:
¦of_nat m¦ = of_nat m
lemma of_int_int_eq:
of_int (of_nat n) = of_nat n
lemma zdiff_int:
n ≤ m ==> of_nat m - of_nat n = of_nat (m - n)
lemma zless_le:
(z1 < w1) = (z1 ≤ w1 ∧ z1 ≠ w1)
lemma int_eq_of_nat:
True