Proof not found for equivalence test on Artin_exercise_13_4_10

[egfiwhfe]

Desired proof term

The proof term that Canonical should generate, such that the goal is closed.

import Canonical
import Mathlib

open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators

/-
Natural language statement:
Prove that if a prime integer $p$ has the form $2^r+1$, then it actually has the form $2^{2^k}+1$.
-/

theorem dummy {p : ℕ} (hp : p.Prime) (h : ∃ r : ℕ, p = 2 ^ r + 1) : ∃ k : ℕ, p = 2 ^ (2 ^ k) + 1 := sorry

theorem exercise_13_4_10
    {p : ℕ} {hp : Nat.Prime p} (h : ∃ r : ℕ, p = 2 ^ r + 1) :
    ∃ (k : ℕ), p = 2 ^ (2 ^ k) + 1 :=
-- `canonical [congrFun, dummy]` couldn't find any proof even with a more generous timeout 
by exact dummy hp h

Canonical invocation

Please show the invocation of canonical used, with the +debug flag. Check the message in the InfoView containing the problem sent to Canonical, and ensure that the constant symbols in the desired proof term are present and do not have type *. You can add constant symbols to the canonical invocation with a list. For example:

canonical +debug [congrFun, dummy]

I failed to find a way to eliminate the * from debug output even putting dummy constant symbol in canonical's list argument.

(Destruct.2379 : *) →
(a.2380 : (a_w.2383 : Nat) → (a_h.2384 : Eq Nat p.2098 (HAdd_hAdd_0 (HPow_hPow_0 (Nat.succ (Nat.succ Nat.zero)) (HPow_hPow_0 (Nat.succ (Nat.succ Nat.zero)) a_w.2383)) (Nat.succ Nat.zero))) → Destruct.2379) →
(p.2098 : Nat := []) →
(hp.2099 : Nat.Prime p.2098 := []) →
(h_w.2100 : Nat := []) →
(h_h.2101 : Eq Nat p.2098 (HAdd_hAdd_0 (HPow_hPow_0 (Nat.succ (Nat.succ Nat.zero)) h_w.2100) (Nat.succ Nat.zero)) := []) →
(One_one_0 : Nat := []) →
(Exists.rec : (α.2649 : Sort) → (p.2650 : (a.2658 : α.2649) → Sort) → (motive.2651 : (t.2659 : Exists α.2649 (λ a.2662 ↦ p.2650 a.2662)) → Sort) → (intro.2652 : (w.2664 : α.2649) → (h.2665 : p.2650 w.2664) → motive.2651 (Exists.intro α.2649 (λ a.2672 ↦ p.2650 a.2672) w.2664 h.2665)) → (t.2653 : Exists α.2649 (λ a.2676 ↦ p.2650 a.2676)) → motive.2651 t.2653 := [Exists.rec α.2641 p.2642 motive.2643 intro.2644 (Exists.intro * * w.2645 h.2646) ↦ intro.2644 w.2645 h.2646]) →
(Exists.intro : (α.2605 : Sort) → (p.2606 : (a.2610 : α.2605) → Sort) → (w.2607 : α.2605) → (h.2608 : p.2606 w.2607) → Exists α.2605 (λ a.2614 ↦ p.2606 a.2614) := []) →
(Exists : (α.2596 : Sort) → (p.2597 : (a.2599 : α.2596) → Sort) → Sort := []) →
(dummy : (p.2585 : Nat) → (hp.2586 : Nat.Prime p.2585) → (h.2587 : Exists Nat (λ a.2679 ↦ Eq Nat p.2585 (HAdd_hAdd_0 (HPow_hPow_0 (Nat.succ (Nat.succ Nat.zero)) a.2679) (Nat.succ Nat.zero)))) → Exists Nat (λ a.2739 ↦ Eq Nat p.2585 (HAdd_hAdd_0 (HPow_hPow_0 (Nat.succ (Nat.succ Nat.zero)) (HPow_hPow_0 (Nat.succ (Nat.succ Nat.zero)) a.2739)) (Nat.succ Nat.zero))) := []) →
(Canonical.Pi.f : (A.2543 : Sort) → (B.2544 : (a.2548 : A.2543) → Sort) → (self.2545 : Canonical.Pi A.2543 (λ a.2551 ↦ B.2544 a.2551)) → (a.2546 : A.2543) → B.2544 a.2546 := [Canonical.Pi.f * * (Canonical.Pi.mk * * field) arg0 ↦ field arg0]) →
(Canonical.Pi.mk : (A.2522 : Sort) → (B.2523 : (a.2527 : A.2522) → Sort) → (f.2524 : (a.2528 : A.2522) → B.2523 a.2528) → Canonical.Pi A.2522 (λ a.2532 ↦ B.2523 a.2532) := []) →
(Canonical.Pi : (A.2513 : Sort) → (B.2514 : (a.2516 : A.2513) → Sort) → Sort := []) →
(congrFun : (α.2490 : Sort) → (β.2491 : (a.2499 : α.2490) → Sort) → (f.2492 : (x.2500 : α.2490) → β.2491 x.2500) → (g.2493 : (x.2502 : α.2490) → β.2491 x.2502) → (h.2494 : Eq (Canonical.Pi α.2490 (λ a.2554 ↦ β.2491 a.2554)) (Canonical.Pi.mk α.2490 (λ a.2560 ↦ β.2491 a.2560) (λ a.2562 ↦ f.2492 a.2562)) (Canonical.Pi.mk α.2490 (λ a.2568 ↦ β.2491 a.2568) (λ a.2570 ↦ g.2493 a.2570))) → (a.2495 : α.2490) → Eq (β.2491 a.2495) (f.2492 a.2495) (g.2493 a.2495) := []) →
(HPow_hPow_0 : (a.2368 : Nat) → (a.2369 : Nat) → Nat := [HPow_hPow_0 a.3423 Nat.zero ↦ One_one_0, HPow_hPow_0 a.3500 (Nat.succ Nat.zero) ↦ a.3500, HPow_hPow_0 One_one_0 n.4340 ↦ One_one_0]) →
(HAdd_hAdd_0 : (a.2309 : Nat) → (a.2310 : Nat) → Nat := [HAdd_hAdd_0 n.3534 Nat.zero ↦ n.3534, HAdd_hAdd_0 n.3566 (Nat.succ m.3567) ↦ Nat.succ (HAdd_hAdd_0 n.3566 m.3567), HAdd_hAdd_0 (Nat.succ n.3614) m.3615 ↦ Nat.succ (HAdd_hAdd_0 n.3614 m.3615), HAdd_hAdd_0 (HAdd_hAdd_0 n.3662 m.3663) k.3664 ↦ HAdd_hAdd_0 n.3662 (HAdd_hAdd_0 m.3663 k.3664), HAdd_hAdd_0 Nat.zero n.3753 ↦ n.3753]) →
(Eq.rec : (α.2238 : Sort) → (a.2239 : α.2238) → (motive.2240 : (a.2246 : α.2238) → (t.2247 : Eq α.2238 a.2239 a.2246) → Sort) → (refl.2241 : motive.2240 a.2239 (Eq.refl α.2238 a.2239)) → (a.2242 : α.2238) → (t.2243 : Eq α.2238 a.2239 a.2242) → motive.2240 a.2242 t.2243 := [Eq.rec α.2234 a.2235 motive.2236 refl.2237 * (Eq.refl * *) ↦ refl.2237]) →
(Eq.refl : (α.2211 : Sort) → (a.2212 : α.2211) → Eq α.2211 a.2212 a.2212 := []) →
(Irreducible_0 : (p.2202 : Nat) → Sort := []) →
(Nat.Prime : (p.2186 : Nat) → Sort := [Nat.Prime p.2183 ↦ Irreducible_0 p.2183]) →
(Nat.rec : (motive.2155 : (t.2162 : Nat) → Sort) → (zero.2156 : motive.2155 Nat.zero) → (succ.2157 : (n.2164 : Nat) → (n_ih.2165 : motive.2155 n.2164) → motive.2155 (Nat.succ n.2164)) → (t.2158 : Nat) → motive.2155 t.2158 := [Nat.rec motive.2131 zero.2132 succ.2133 Nat.zero ↦ zero.2132, Nat.rec motive.2145 zero.2146 succ.2147 (Nat.succ n.2148) ↦ succ.2147 n.2148 (Nat.rec (λ t.2170 ↦ motive.2145 t.2170) zero.2146 (λ n.2172 n_ih.2173 ↦ succ.2147 n.2172 n_ih.2173) n.2148)]) →
(Eq : (α.2206 : Sort) → (a.2207 : α.2206) → (a.2208 : α.2206) → Sort := [Eq Nat Nat.zero (Nat.succ *) ↦ False, Eq Nat (Nat.succ *) Nat.zero ↦ False, Eq Nat (Nat.succ u.3782) (Nat.succ v.3783) ↦ Eq Nat u.3782 v.3783]) →
(False : * := []) →
(Nat.succ : (n.2111 : Nat) → Nat := []) →
(Nat.zero : Nat := []) →
(Sort : Sort := []) →
(Nat : Sort := []) →
Destruct.2379

Observations

Let us know if you believe a particular aspect of the problem is responsible for the failure.

• This one is out of my knowledge