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Diffstat (limited to 'theories/Numbers/Integer/TreeMod/ZTreeMod.v')
| -rw-r--r-- | theories/Numbers/Integer/TreeMod/ZTreeMod.v | 216 |
1 files changed, 216 insertions, 0 deletions
diff --git a/theories/Numbers/Integer/TreeMod/ZTreeMod.v b/theories/Numbers/Integer/TreeMod/ZTreeMod.v new file mode 100644 index 0000000000..7479868e9d --- /dev/null +++ b/theories/Numbers/Integer/TreeMod/ZTreeMod.v @@ -0,0 +1,216 @@ +Require Export NZAxioms. +Require Import NMake. (* contains W0Type *) +Require Import ZnZ. +Require Import Basic_type. (* contais base *) +Require Import ZAux. + +Module NZBigIntsAxiomsMod (Import BoundedIntsMod : W0Type) <: NZAxiomsSig. + +Open Local Scope Z_scope. + +Definition NZ := w. + +Definition NZ_to_Z : NZ -> Z := znz_to_Z w_op. +Definition Z_to_NZ : Z -> NZ := znz_of_Z w_op. +Notation Local wB := (base w_op.(znz_digits)). + +Notation Local "[| x |]" := (w_op.(znz_to_Z) x) (at level 0, x at level 99). + +Definition NZE (n m : NZ) := [| n |] = [| m |]. +Definition NZ0 := w_op.(znz_0). +Definition NZsucc := w_op.(znz_succ). +Definition NZpred := w_op.(znz_pred). +Definition NZplus := w_op.(znz_add). +Definition NZminus := w_op.(znz_sub). +Definition NZtimes := w_op.(znz_mul). + +Theorem NZE_equiv : equiv NZ NZE. +Proof. +unfold equiv, reflexive, symmetric, transitive, NZE; repeat split; intros; auto. +now transitivity [| y |]. +Qed. + +Add Relation NZ NZE + reflexivity proved by (proj1 NZE_equiv) + symmetry proved by (proj2 (proj2 NZE_equiv)) + transitivity proved by (proj1 (proj2 NZE_equiv)) +as NZE_rel. + +Add Morphism NZsucc with signature NZE ==> NZE as NZsucc_wd. +Proof. +unfold NZE; intros n m H. do 2 rewrite w_spec.(spec_succ). now rewrite H. +Qed. + +Add Morphism NZpred with signature NZE ==> NZE as NZpred_wd. +Proof. +unfold NZE; intros n m H. do 2 rewrite w_spec.(spec_pred). now rewrite H. +Qed. + +Add Morphism NZplus with signature NZE ==> NZE ==> NZE as NZplus_wd. +Proof. +unfold NZE; intros n1 n2 H1 m1 m2 H2. do 2 rewrite w_spec.(spec_add). +now rewrite H1, H2. +Qed. + +Add Morphism NZminus with signature NZE ==> NZE ==> NZE as NZminus_wd. +Proof. +unfold NZE; intros n1 n2 H1 m1 m2 H2. do 2 rewrite w_spec.(spec_sub). +now rewrite H1, H2. +Qed. + +Add Morphism NZtimes with signature NZE ==> NZE ==> NZE as NZtimes_wd. +Proof. +unfold NZE; intros n1 n2 H1 m1 m2 H2. do 2 rewrite w_spec.(spec_mul). +now rewrite H1, H2. +Qed. + +Delimit Scope IntScope with Int. +Bind Scope IntScope with NZ. +Open Local Scope IntScope. +Notation "x == y" := (NZE x y) (at level 70) : IntScope. +Notation "x ~= y" := (~ NZE x y) (at level 70) : IntScope. +Notation "0" := NZ0 : IntScope. +Notation "'S'" := NZsucc : IntScope. +Notation "'P'" := NZpred : IntScope. +(*Notation "1" := (S 0) : IntScope.*) +Notation "x + y" := (NZplus x y) : IntScope. +Notation "x - y" := (NZminus x y) : IntScope. +Notation "x * y" := (NZtimes x y) : IntScope. + +Theorem gt_wB_1 : 1 < wB. +Proof. +unfold base. apply Zgt_pow_1; unfold Zlt; auto with zarith. +Qed. + +Theorem gt_wB_0 : 0 < wB. +Proof. +pose proof gt_wB_1; auto with zarith. +Qed. + +Lemma NZsucc_mod_wB : forall n : Z, (n + 1) mod wB = ((n mod wB) + 1) mod wB. +Proof. +intro n. +pattern 1 at 2. replace 1 with (1 mod wB). rewrite <- Zmod_plus; [ | apply gt_wB_0]. +reflexivity. +now rewrite Zmod_def_small; [ | split; [auto with zarith | apply gt_wB_1]]. +Qed. + +Lemma NZpred_mod_wB : forall n : Z, (n - 1) mod wB = ((n mod wB) - 1) mod wB. +Proof. +intro n. +pattern 1 at 2. replace 1 with (1 mod wB). rewrite <- Zmod_minus; [ | apply gt_wB_0]. +reflexivity. +now rewrite Zmod_def_small; [ | split; [auto with zarith | apply gt_wB_1]]. +Qed. + +Lemma NZ_to_Z_mod : forall n : NZ, [| n |] mod wB = [| n |]. +Proof. +intro n; rewrite Zmod_def_small. reflexivity. apply w_spec.(spec_to_Z). +Qed. + +Theorem NZpred_succ : forall n : NZ, P (S n) == n. +Proof. +intro n; unfold NZsucc, NZpred, NZE. rewrite w_spec.(spec_pred), w_spec.(spec_succ). +rewrite <- NZpred_mod_wB. +replace ([| n |] + 1 - 1)%Z with [| n |] by auto with zarith. apply NZ_to_Z_mod. +Qed. + +Lemma Z_to_NZ_0 : Z_to_NZ 0%Z == 0%Int. +Proof. +unfold NZE, NZ_to_Z, Z_to_NZ. rewrite znz_of_Z_correct. +symmetry; apply w_spec.(spec_0). +exact w_spec. split; [auto with zarith |apply gt_wB_0]. +Qed. + +Section Induction. + +Variable A : NZ -> Prop. +Hypothesis A_wd : predicate_wd NZE A. +Hypothesis A0 : A 0. +Hypothesis AS : forall n : NZ, A n <-> A (S n). (* Below, we use only -> direction *) + +Add Morphism A with signature NZE ==> iff as A_morph. +Proof A_wd. + +Let B (n : Z) := A (Z_to_NZ n). + +Lemma B0 : B 0. +Proof. +unfold B. now rewrite Z_to_NZ_0. +Qed. + +Lemma BS : forall n : Z, 0 <= n -> n < wB - 1 -> B n -> B (n + 1). +Proof. +intros n H1 H2 H3. +unfold B in *. apply -> AS in H3. +setoid_replace (Z_to_NZ (n + 1)) with (S (Z_to_NZ n)) using relation NZE. assumption. +unfold NZE. rewrite w_spec.(spec_succ). +unfold NZ_to_Z, Z_to_NZ. +do 2 (rewrite znz_of_Z_correct; [ | exact w_spec | auto with zarith]). +symmetry; apply Zmod_def_small; auto with zarith. +Qed. + +Lemma B_holds : forall n : Z, 0 <= n < wB -> B n. +Proof. +intros n [H1 H2]. +apply Zbounded_induction with wB. +apply B0. apply BS. assumption. assumption. +Qed. + +Theorem NZinduction : forall n : NZ, A n. +Proof. +intro n. setoid_replace n with (Z_to_NZ (NZ_to_Z n)) using relation NZE. +apply B_holds. apply w_spec.(spec_to_Z). +unfold NZE, NZ_to_Z, Z_to_NZ; rewrite znz_of_Z_correct. +reflexivity. +exact w_spec. +apply w_spec.(spec_to_Z). +Qed. + +End Induction. + +Theorem NZplus_0_l : forall n : NZ, 0 + n == n. +Proof. +intro n; unfold NZplus, NZ0, NZE. rewrite w_spec.(spec_add). rewrite w_spec.(spec_0). +rewrite Zplus_0_l. rewrite Zmod_def_small; [reflexivity | apply w_spec.(spec_to_Z)]. +Qed. + +Theorem NZplus_succ_l : forall n m : NZ, (S n) + m == S (n + m). +Proof. +intros n m; unfold NZplus, NZsucc, NZE. rewrite w_spec.(spec_add). +do 2 rewrite w_spec.(spec_succ). rewrite w_spec.(spec_add). +rewrite NZsucc_mod_wB. repeat rewrite Zplus_mod_idemp_l; try apply gt_wB_0. +rewrite <- (Zplus_assoc ([| n |] mod wB) 1 [| m |]). rewrite Zplus_mod_idemp_l; [ |apply gt_wB_0]. +rewrite (Zplus_comm 1 [| m |]); now rewrite Zplus_assoc. +Qed. + +Theorem NZminus_0_r : forall n : NZ, n - 0 == n. +Proof. +intro n; unfold NZminus, NZ0, NZE. rewrite w_spec.(spec_sub). +rewrite w_spec.(spec_0). rewrite Zminus_0_r. apply NZ_to_Z_mod. +Qed. + +Theorem NZminus_succ_r : forall n m : NZ, n - (S m) == P (n - m). +Proof. +intros n m; unfold NZminus, NZsucc, NZpred, NZE. +rewrite w_spec.(spec_pred). do 2 rewrite w_spec.(spec_sub). +rewrite w_spec.(spec_succ). rewrite Zminus_mod_idemp_r; [ | apply gt_wB_0]. +rewrite Zminus_mod_idemp_l; [ | apply gt_wB_0]. +now replace ([|n|] - ([|m|] + 1))%Z with ([|n|] - [|m|] - 1)%Z by auto with zarith. +Qed. + +Theorem NZtimes_0_r : forall n : NZ, n * 0 == 0. +Proof. +intro n; unfold NZtimes, NZ0, NZ, NZE. rewrite w_spec.(spec_mul). +rewrite w_spec.(spec_0). now rewrite Zmult_0_r. +Qed. + +Theorem NZtimes_succ_r : forall n m : NZ, n * (S m) == n * m + n. +Proof. +intros n m; unfold NZtimes, NZsucc, NZplus, NZE. rewrite w_spec.(spec_mul). +rewrite w_spec.(spec_add). rewrite w_spec.(spec_mul). rewrite w_spec.(spec_succ). +rewrite Zplus_mod_idemp_l; [ | apply gt_wB_0]. rewrite Zmult_mod_idemp_r; [ | apply gt_wB_0]. +rewrite Zmult_plus_distr_r. now rewrite Zmult_1_r. +Qed. + +End NZBigIntsAxiomsMod. |