Created
November 30, 2017 15:34
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Example of using `refine` to build up a proof term for induction principle.
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Inductive ev : nat -> Prop := | |
| ev_0 : ev 0 | |
| ev_SS : forall n, ev n -> ev (S (S n)). | |
Hint Constructors ev. | |
Inductive test : (nat -> nat) -> nat -> Prop := | |
| TestCase : forall f i, | |
(forall k, | |
0 <= k < f i -> | |
exists r, | |
ev r /\ test f (k + r)) -> | |
test f i. | |
Definition sample_f (n : nat) : nat := | |
match n with | |
| 2 => 0 | |
| 3 => 0 | |
| 6 => 2 | |
| _ => 6 | |
end. | |
Require Import Omega. | |
Example test_sample_f : test sample_f 6. | |
Proof. | |
apply TestCase. | |
intros. | |
destruct k. | |
- exists 2. split; eauto. | |
apply TestCase. | |
intros. | |
exists 2. split; eauto. | |
simpl in *. | |
omega. | |
- destruct k. | |
+ exists 2. split; eauto. | |
apply TestCase. | |
intros. | |
simpl in *. | |
omega. | |
+ simpl in *. | |
omega. | |
Qed. | |
Check test_ind. | |
(* test_ind | |
: forall P : (nat -> nat) -> nat -> Prop, | |
(forall (f : nat -> nat) (i : nat), | |
(forall k : nat, 0 <= k < f i -> exists r : nat, ev r /\ test f (k + r)) -> P f i) -> | |
forall (n : nat -> nat) (n0 : nat), test n n0 -> P n n0 *) | |
Lemma test_ind' : | |
forall P : (nat -> nat) -> nat -> Prop, | |
(forall (f : nat -> nat) (i : nat), | |
(forall k : nat, 0 <= k < f i -> | |
exists r : nat, | |
ev r /\ test f (k + r) /\ P f (k + r)) -> | |
P f i) -> | |
forall (f : nat -> nat) (n : nat), test f n -> P f n. | |
Proof. | |
intros P H. | |
refine (fix F f n t := | |
match t with | |
| TestCase f i Hrec => | |
H f i (fun k Hk => | |
match Hrec k Hk with | |
| ex_intro _ r Hr => | |
match Hr with | |
| conj Hev Hrest => | |
ex_intro _ r (conj Hev (conj Hrest (F f (k + r) Hrest))) | |
end | |
end) | |
end). | |
Qed. |
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