Natural numbers nat built from O and S are defined in Datatypes.v
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This module defines the following operations on natural numbers :
This module states various lemmas and theorems about natural numbers, including Peano's axioms of arithmetic (in Coq, these are in fact provable) Case analysis on |
Require Logic.
Require LogicSyntax.
Require Datatypes.
Definition eq_S := (f_equal nat nat S).
Hint eq_S : v62 := Resolve (f_equal nat nat S).
Hint eq_nat_unary : core := Resolve (f_equal nat).
| The predecessor function |
Definition pred : nat->nat := [n:nat](Cases n of O => O | (S u) => u end).
Hint eq_pred : v62 := Resolve (f_equal nat nat pred).
Theorem pred_Sn : (m:nat) m=(pred (S m)).
Proof.
Auto.
Qed.
Theorem eq_add_S : (n,m:nat) (S n)=(S m) -> n=m.
Proof.
Intros n m H ; Change (pred (S n))=(pred (S m)); Auto.
Qed.
Hints Immediate eq_add_S : core v62.
| A consequence of the previous axioms |
Theorem not_eq_S : (n,m:nat) ~(n=m) -> ~((S n)=(S m)).
Proof.
Red; Auto.
Qed.
Hints Resolve not_eq_S : core v62.
Definition IsSucc : nat->Prop
:= [n:nat]Cases n of O => False | (S p) => True end.
Theorem O_S : (n:nat)~(O=(S n)).
Proof.
Red;Intros n H.
Change (IsSucc O).
Elim (sym_eq nat O (S n));[Exact I | Assumption].
Qed.
Hints Resolve O_S : core v62.
Theorem n_Sn : (n:nat) ~(n=(S n)).
Proof.
Induction n ; Auto.
Qed.
Hints Resolve n_Sn : core v62.
| Addition |
Fixpoint plus [n:nat] : nat -> nat :=
[m:nat]Cases n of
O => m
| (S p) => (S (plus p m)) end.
Hint eq_plus : v62 := Resolve (f_equal2 nat nat nat plus).
Hint eq_nat_binary : core := Resolve (f_equal2 nat nat).
Lemma plus_n_O : (n:nat) n=(plus n O).
Proof.
Induction n ; Simpl ; Auto.
Qed.
Hints Resolve plus_n_O : core v62.
Lemma plus_n_Sm : (n,m:nat) (S (plus n m))=(plus n (S m)).
Proof.
Intros m n; Elim m; Simpl; Auto.
Qed.
Hints Resolve plus_n_Sm : core v62.
| Multiplication |
Fixpoint mult [n:nat] : nat -> nat :=
[m:nat]Cases n of O => O
| (S p) => (plus m (mult p m)) end.
Hint eq_mult : core v62 := Resolve (f_equal2 nat nat nat mult).
Lemma mult_n_O : (n:nat) O=(mult n O).
Proof.
Induction n; Simpl; Auto.
Qed.
Hints Resolve mult_n_O : core v62.
Lemma mult_n_Sm : (n,m:nat) (plus (mult n m) n)=(mult n (S m)).
Proof.
Intros; Elim n; Simpl; Auto.
Intros p H; Case H; Elim plus_n_Sm; Apply (f_equal nat nat S).
Pattern 1 3 m; Elim m; Simpl; Auto.
Qed.
Hints Resolve mult_n_Sm : core v62.
Definition of the usual orders, the basic properties of le and lt can be found in files Le and Lt
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| An inductive definition to define the order |
Inductive le [n:nat] : nat -> Prop
:= le_n : (le n n)
| le_S : (m:nat)(le n m)->(le n (S m)).
Hint constr_le : core v62 := Constructors le.
Definition lt := [n,m:nat](le (S n) m).
Hints Unfold lt : core v62.
Definition ge := [n,m:nat](le m n).
Hints Unfold ge : core v62.
Definition gt := [n,m:nat](lt m n).
Hints Unfold gt : core v62.
| Pattern-Matching on natural numbers |
Theorem nat_case : (n:nat)(P:nat->Prop)(P O)->((m:nat)(P (S m)))->(P n).
Proof.
Induction n ; Auto.
Qed.
| Principle of double induction |
Theorem nat_double_ind : (R:nat->nat->Prop)
((n:nat)(R O n)) -> ((n:nat)(R (S n) O))
-> ((n,m:nat)(R n m)->(R (S n) (S m)))
-> (n,m:nat)(R n m).
Proof.
Induction n; Auto.
Induction m; Auto.
Qed.