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Example.v
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Example.v
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Require Import Patcher.Patch.
Require Import Arith NPeano.
Require Import List.
(* Simple difference in inductive proofs *)
(* ------------------------ *)
Theorem old1:
forall (n m p : nat),
n <= m ->
m <= p ->
n <= p + 1.
Proof.
intros. induction H0.
- auto with arith.
- constructor. auto.
Qed.
Theorem new1:
forall (n m p : nat),
n <= m ->
m <= p ->
n <= p.
Proof.
intros. induction H0.
- auto with arith.
- constructor. auto.
Qed.
(* ------------------------- *)
Theorem old2 :
forall (n m p : nat),
n <= m ->
m <= p ->
n <= S p.
Proof.
intros. induction H0.
- intuition.
- constructor. apply IHle.
Qed.
Theorem new2 :
forall (n m p : nat),
n <= m ->
m <= p ->
n <= p.
Proof.
intros. induction H0.
- intuition.
- constructor. apply IHle.
Qed.
(* ------------------------- *)
Theorem old3 :
forall (n m p : nat),
n <= m ->
m <= p ->
n < S p.
Proof.
intros. induction H0.
- intuition.
- constructor. apply IHle.
Qed.
Theorem new3 :
forall (n m p : nat),
n <= m ->
m <= p ->
n <= p.
Proof.
intros. induction H0.
- intuition.
- constructor. apply IHle.
Qed.
(* ------------------------- *)
Theorem old4 :
forall (n m p : nat),
n <= m ->
m <= p ->
n < p + 1.
Proof.
intros. induction H0.
- rewrite plus_comm. auto with arith.
- constructor. apply IHle.
Qed.
Theorem new4 :
forall (n m p : nat),
n <= m ->
m <= p ->
n < S p.
Proof.
intros. induction H0.
- auto with arith.
- constructor. apply IHle.
Qed.
(* -------------------------------- *)
Inductive ListSum : list nat -> nat -> Type :=
| ListSumNil :
ListSum nil 0
| ListSumCons :
forall (h : nat) (tl : list nat) (n : nat),
ListSum tl n ->
ListSum (h :: tl) (h + n).
Theorem old5 :
forall (n m : nat) (l1 l2 : list nat),
ListSum l1 n ->
ListSum (l1 ++ l2) (n + m) ->
ListSum (rev (rev l2)) m.
Proof.
intros. induction H.
- rewrite rev_involutive. apply H0.
- inversion H0. subst.
rewrite plus_assoc_reverse in H4.
assert (n0 = n + m).
+ eapply plus_reg_l; eauto.
+ subst. apply IHListSum in H2. apply H2.
Qed.
Theorem new5 :
forall (n m : nat) (l1 l2 : list nat),
ListSum l1 n ->
ListSum (l1 ++ l2) (n + m) ->
ListSum l2 m.
Proof.
intros. induction H.
- apply H0.
- inversion H0. subst.
rewrite plus_assoc_reverse in H4.
assert (n0 = n + m).
+ eapply plus_reg_l; eauto.
+ subst. apply IHListSum in H2. apply H2.
Qed.
(* -------------------------------- *)
Theorem old6:
forall l1 l2 : list nat,
length (rev (l1 ++ l2)) = (length (rev l1)) + (length (rev l2)).
Proof.
intros l1 l2.
induction l1 as [| n l1' IHl1'].
- simpl. reflexivity.
- repeat rewrite -> rev_length in *. simpl.
rewrite -> IHl1'. reflexivity.
Qed.
Theorem new6:
forall l1 l2 : list nat,
length (l1 ++ l2) = (length l1) + (length l2).
Proof.
intros l1 l2. induction l1 as [| n l1' IHl1'].
- reflexivity.
- simpl. rewrite -> IHl1'. reflexivity.
Qed.
(* -------------------------------- *)
Theorem old7 :
forall (A B : Type) (f : A -> B) (l : list A) (x : A),
In x l ->
In (f x) (rev (map f l)).
Proof.
intros A B f l x. rewrite <- in_rev.
induction l as [|x' l' IHl'].
- simpl. intros [].
- simpl. intros [H | H].
+ rewrite H. left. reflexivity.
+ right. apply IHl'. apply H.
Qed.
Theorem new7 :
forall (A B : Type) (f : A -> B) (l : list A) (x : A),
In x l ->
In (f x) (map f l).
Proof.
intros A B f l x.
induction l as [|x' l' IHl'].
- simpl. intros [].
- simpl. intros [H' | H'].
+ rewrite H'. left. reflexivity.
+ right. apply IHl'. apply H'.
Qed.
(* ----------------------------- *)
Definition old8 (n m n0 : nat) (H : eq m n) :=
Coq.Init.Logic.eq_ind_r
(fun (m : nat) => (le n (Coq.Init.Nat.add (Coq.Init.Nat.max n0 m) (S O))))
(Coq.Arith.Plus.le_plus_trans
n
(Coq.Init.Nat.max n0 n)
(S O)
(Coq.Arith.PeanoNat.Nat.le_max_r n0 n))
H.
Definition new8 (n m n0 : nat) (H : eq m n) :=
Coq.Init.Logic.eq_ind_r
(fun (m : nat) => (le n (Coq.Init.Nat.max n0 m)))
(Coq.Arith.PeanoNat.Nat.le_max_r n0 n)
H.
(* ----------------------------------- *)
(* Tool cannot support this yet, needs backup search:
(* ORIGINAL *)
Theorem old8:
forall b1 b2,
andb b1 b2 = true -> b1 = true \/ b2 = true.
Proof.
intros b1 b2. intros H. destruct b1; destruct b2.
- left. reflexivity.
- left. reflexivity.
- right. reflexivity.
- discriminate.
Qed.
(* MODIFIED *)
Theorem new8 :
forall b1 b2,
andb b1 b2 = true -> b1 = true /\ b2 = true.
Proof.
intros b1 b2. intros H. destruct b1; destruct b2; split; try reflexivity; discriminate.
Qed.
*)