diff options
| -rw-r--r-- | mathcomp/ssrtest/elim.v | 72 |
1 files changed, 71 insertions, 1 deletions
diff --git a/mathcomp/ssrtest/elim.v b/mathcomp/ssrtest/elim.v index bc8701e..9f0f139 100644 --- a/mathcomp/ssrtest/elim.v +++ b/mathcomp/ssrtest/elim.v @@ -197,12 +197,82 @@ rewrite lognE -mem_primes; case: ifP => pi1p; last exact: dvd1n. by case: ifP => pr_p; [rewrite pi12 | rewrite if_same]. Qed. -Function plus (m n : nat) {struct n} : nat := +Fixpoint plus (m n : nat) {struct n} : nat := match n with | 0 => m | S p => S (plus m p) end. +Definition plus_equation : +forall m n : nat, + plus m n = + match n with + | 0 => m + | p.+1 => (plus m p).+1 + end +:= +fun m n : nat => +match + n as n0 + return + (forall m0 : nat, + plus m0 n0 = + match n0 with + | 0 => m0 + | p.+1 => (plus m0 p).+1 + end) +with +| 0 => @erefl nat +| n0.+1 => fun m0 : nat => erefl (plus m0 n0).+1 +end m. + +Definition plus_rect : +forall (m : nat) (P : nat -> nat -> Type), + (forall n : nat, n = 0 -> P 0 m) -> + (forall n p : nat, + n = p.+1 -> P p (plus m p) -> P p.+1 (plus m p).+1) -> + forall n : nat, P n (plus m n) +:= +fun (m : nat) (P : nat -> nat -> Type) + (f0 : forall n : nat, n = 0 -> P 0 m) + (f : forall n p : nat, + n = p.+1 -> P p (plus m p) -> P p.+1 (plus m p).+1) => +fix plus0 (n : nat) : P n (plus m n) := + eq_rect_r [eta P n] + (let f1 := f0 n in + let f2 := f n in + match + n as n0 + return + (n = n0 -> + (forall p : nat, + n0 = p.+1 -> P p (plus m p) -> P p.+1 (plus m p).+1) -> + (n0 = 0 -> P 0 m) -> + P n0 match n0 with + | 0 => m + | p.+1 => (plus m p).+1 + end) + with + | 0 => + fun (_ : n = 0) + (_ : forall p : nat, + 0 = p.+1 -> + P p (plus m p) -> P p.+1 (plus m p).+1) + (f4 : 0 = 0 -> P 0 m) => unkeyed (f4 (erefl 0)) + | n0.+1 => + fun (_ : n = n0.+1) + (f3 : forall p : nat, + n0.+1 = p.+1 -> + P p (plus m p) -> P p.+1 (plus m p).+1) + (_ : n0.+1 = 0 -> P 0 m) => + let f5 := + let p := n0 in + let H := erefl n0.+1 : n0.+1 = p.+1 in f3 p H in + unkeyed (let Hrec := plus0 n0 in f5 Hrec) + end (erefl n) f2 f1) (plus_equation m n). + +Definition plus_ind := plus_rect. + Lemma exF x y z: plus (plus x y) z = plus x (plus y z). elim/plus_ind: z / (plus _ z). match goal with |- forall n : nat, n = 0 -> plus x y = plus x (plus y 0) => idtac end. |