diff options
Diffstat (limited to 'theories/Numbers/Natural/Abstract/NBase.v')
| -rw-r--r-- | theories/Numbers/Natural/Abstract/NBase.v | 154 |
1 files changed, 46 insertions, 108 deletions
diff --git a/theories/Numbers/Natural/Abstract/NBase.v b/theories/Numbers/Natural/Abstract/NBase.v index 02d82bacd..6bf12ee5e 100644 --- a/theories/Numbers/Natural/Abstract/NBase.v +++ b/theories/Numbers/Natural/Abstract/NBase.v @@ -12,131 +12,75 @@ Require Export Decidable. Require Export NAxioms. -Require Import NZMulOrder. (* The last property functor on NZ, which subsumes all others *) +Require Import NZProperties. -Module NBasePropFunct (Import NAxiomsMod : NAxiomsSig). +Module NBasePropFunct (Import N : NAxiomsSig). +(** First, we import all known facts about both natural numbers and integers. *) +Include NZPropFunct N. +Local Open Scope NumScope. -Open Local Scope NatScope. - -(* We call the last property functor on NZ, which includes all the previous -ones, to get all properties of NZ at once. This way we will include them -only one time. *) - -Module Export NZMulOrderMod := NZMulOrderPropFunct NZOrdAxiomsMod. - -(* Here we probably need to re-prove all axioms declared in NAxioms.v to -make sure that the definitions like N, S and add are unfolded in them, -since unfolding is done only inside a functor. In fact, we'll do it in the -files that prove the corresponding properties. In those files, we will also -rename properties proved in NZ files by removing NZ from their names. In -this way, one only has to consult, for example, NAdd.v to see all -available properties for add, i.e., one does not have to go to NAxioms.v -for axioms and NZAdd.v for theorems. *) - -Theorem succ_wd : forall n1 n2 : N, n1 == n2 -> S n1 == S n2. -Proof NZsucc_wd. - -Theorem pred_wd : forall n1 n2 : N, n1 == n2 -> P n1 == P n2. -Proof NZpred_wd. - -Theorem pred_succ : forall n : N, P (S n) == n. -Proof NZpred_succ. - -Theorem pred_0 : P 0 == 0. -Proof pred_0. - -Theorem Neq_refl : forall n : N, n == n. -Proof (@Equivalence_Reflexive _ _ NZeq_equiv). - -Theorem Neq_sym : forall n m : N, n == m -> m == n. -Proof (@Equivalence_Symmetric _ _ NZeq_equiv). - -Theorem Neq_trans : forall n m p : N, n == m -> m == p -> n == p. -Proof (@Equivalence_Transitive _ _ NZeq_equiv). - -Theorem neq_sym : forall n m : N, n ~= m -> m ~= n. -Proof NZneq_sym. - -Theorem succ_inj : forall n1 n2 : N, S n1 == S n2 -> n1 == n2. -Proof NZsucc_inj. - -Theorem succ_inj_wd : forall n1 n2 : N, S n1 == S n2 <-> n1 == n2. -Proof NZsucc_inj_wd. - -Theorem succ_inj_wd_neg : forall n m : N, S n ~= S m <-> n ~= m. -Proof NZsucc_inj_wd_neg. - -(* Decidability and stability of equality was proved only in NZOrder, but -since it does not mention order, we'll put it here *) - -Theorem eq_dec : forall n m : N, decidable (n == m). -Proof NZeq_dec. - -Theorem eq_dne : forall n m : N, ~ ~ n == m <-> n == m. -Proof NZeq_dne. - -(* Now we prove that the successor of a number is not zero by defining a +(** We prove that the successor of a number is not zero by defining a function (by recursion) that maps 0 to false and the successor to true *) -Definition if_zero (A : Set) (a b : A) (n : N) : A := +Definition if_zero (A : Type) (a b : A) (n : N.t) : A := recursion a (fun _ _ => b) n. -Instance if_zero_wd (A : Set) : Proper (eq ==> eq ==> Neq ==> eq) (if_zero A). +Implicit Arguments if_zero [A]. + +Instance if_zero_wd (A : Type) : + Proper (Logic.eq ==> Logic.eq ==> N.eq ==> Logic.eq) (@if_zero A). Proof. intros; unfold if_zero. repeat red; intros. apply recursion_wd; auto. repeat red; auto. Qed. -Theorem if_zero_0 : forall (A : Set) (a b : A), if_zero A a b 0 = a. +Theorem if_zero_0 : forall (A : Type) (a b : A), if_zero a b 0 = a. Proof. unfold if_zero; intros; now rewrite recursion_0. Qed. -Theorem if_zero_succ : forall (A : Set) (a b : A) (n : N), if_zero A a b (S n) = b. +Theorem if_zero_succ : + forall (A : Type) (a b : A) (n : N.t), if_zero a b (S n) = b. Proof. intros; unfold if_zero. -now rewrite (@recursion_succ A (@eq A)). +now rewrite recursion_succ. Qed. -Implicit Arguments if_zero [A]. - -Theorem neq_succ_0 : forall n : N, S n ~= 0. +Theorem neq_succ_0 : forall n, S n ~= 0. Proof. intros n H. -assert (true = false); [| discriminate]. -replace true with (if_zero false true (S n)) by apply if_zero_succ. -pattern false at 2; replace false with (if_zero false true 0) by apply if_zero_0. -now rewrite H. +generalize (Logic.eq_refl (if_zero false true 0)). +rewrite <- H at 1. rewrite if_zero_0, if_zero_succ; discriminate. Qed. -Theorem neq_0_succ : forall n : N, 0 ~= S n. +Theorem neq_0_succ : forall n, 0 ~= S n. Proof. intro n; apply neq_sym; apply neq_succ_0. Qed. -(* Next, we show that all numbers are nonnegative and recover regular induction -from the bidirectional induction on NZ *) +(** Next, we show that all numbers are nonnegative and recover regular + induction from the bidirectional induction on NZ *) -Theorem le_0_l : forall n : N, 0 <= n. +Theorem le_0_l : forall n, 0 <= n. Proof. -NZinduct n. -now apply NZeq_le_incl. +nzinduct n. +now apply eq_le_incl. intro n; split. -apply NZle_le_succ_r. -intro H; apply -> NZle_succ_r in H; destruct H as [H | H]. +apply le_le_succ_r. +intro H; apply -> le_succ_r in H; destruct H as [H | H]. assumption. symmetry in H; false_hyp H neq_succ_0. Qed. Theorem induction : - forall A : N -> Prop, Proper (Neq==>iff) A -> - A 0 -> (forall n : N, A n -> A (S n)) -> forall n : N, A n. + forall A : N.t -> Prop, Proper (N.eq==>iff) A -> + A 0 -> (forall n, A n -> A (S n)) -> forall n, A n. Proof. -intros A A_wd A0 AS n; apply NZright_induction with 0; try assumption. +intros A A_wd A0 AS n; apply right_induction with 0; try assumption. intros; auto; apply le_0_l. apply le_0_l. Qed. -(* The theorems NZinduction, NZcentral_induction and the tactic NZinduct +(** The theorems [bi_induction], [central_induction] and the tactic [nzinduct] refer to bidirectional induction, which is not useful on natural numbers. Therefore, we define a new induction tactic for natural numbers. We do not have to call "Declare Left Step" and "Declare Right Step" @@ -146,8 +90,8 @@ from NZ. *) Ltac induct n := induction_maker n ltac:(apply induction). Theorem case_analysis : - forall A : N -> Prop, Proper (Neq==>iff) A -> - A 0 -> (forall n : N, A (S n)) -> forall n : N, A n. + forall A : N.t -> Prop, Proper (N.eq==>iff) A -> + A 0 -> (forall n, A (S n)) -> forall n, A n. Proof. intros; apply induction; auto. Qed. @@ -173,7 +117,7 @@ now left. intro n; right; now exists n. Qed. -Theorem eq_pred_0 : forall n : N, P n == 0 <-> n == 0 \/ n == 1. +Theorem eq_pred_0 : forall n, P n == 0 <-> n == 0 \/ n == 1. Proof. cases n. rewrite pred_0. setoid_replace (0 == 1) with False using relation iff. tauto. @@ -184,14 +128,14 @@ setoid_replace (S n == 0) with False using relation iff by rewrite succ_inj_wd. tauto. Qed. -Theorem succ_pred : forall n : N, n ~= 0 -> S (P n) == n. +Theorem succ_pred : forall n, n ~= 0 -> S (P n) == n. Proof. cases n. intro H; exfalso; now apply H. intros; now rewrite pred_succ. Qed. -Theorem pred_inj : forall n m : N, n ~= 0 -> m ~= 0 -> P n == P m -> n == m. +Theorem pred_inj : forall n m, n ~= 0 -> m ~= 0 -> P n == P m -> n == m. Proof. intros n m; cases n. intros H; exfalso; now apply H. @@ -200,13 +144,13 @@ intros H; exfalso; now apply H. intros m H2 H3. do 2 rewrite pred_succ in H3. now rewrite H3. Qed. -(* The following induction principle is useful for reasoning about, e.g., +(** The following induction principle is useful for reasoning about, e.g., Fibonacci numbers *) Section PairInduction. -Variable A : N -> Prop. -Hypothesis A_wd : Proper (Neq==>iff) A. +Variable A : N.t -> Prop. +Hypothesis A_wd : Proper (N.eq==>iff) A. Theorem pair_induction : A 0 -> A 1 -> @@ -219,13 +163,12 @@ Qed. End PairInduction. -(*Ltac pair_induct n := induction_maker n ltac:(apply pair_induction).*) +(** The following is useful for reasoning about, e.g., Ackermann function *) -(* The following is useful for reasoning about, e.g., Ackermann function *) Section TwoDimensionalInduction. -Variable R : N -> N -> Prop. -Hypothesis R_wd : Proper (Neq==>Neq==>iff) R. +Variable R : N.t -> N.t -> Prop. +Hypothesis R_wd : Proper (N.eq==>N.eq==>iff) R. Theorem two_dim_induction : R 0 0 -> @@ -241,21 +184,16 @@ Qed. End TwoDimensionalInduction. -(*Ltac two_dim_induct n m := - try intros until n; - try intros until m; - pattern n, m; apply two_dim_induction; clear n m; - [solve_relation_wd | | | ].*) Section DoubleInduction. -Variable R : N -> N -> Prop. -Hypothesis R_wd : Proper (Neq==>Neq==>iff) R. +Variable R : N.t -> N.t -> Prop. +Hypothesis R_wd : Proper (N.eq==>N.eq==>iff) R. Theorem double_induction : - (forall m : N, R 0 m) -> - (forall n : N, R (S n) 0) -> - (forall n m : N, R n m -> R (S n) (S m)) -> forall n m : N, R n m. + (forall m, R 0 m) -> + (forall n, R (S n) 0) -> + (forall n m, R n m -> R (S n) (S m)) -> forall n m, R n m. Proof. intros H1 H2 H3; induct n; auto. intros n H; cases m; auto. |