Theorems / novel_cat_03

Authored

Category theory · medium

Statement, exactly as the model and Lean see it

open CategoryTheory in
theorem lg_target {C : Type*} [Category C] {X Y Z : C} (f : X ⟶ Y) [Mono f] (g h : Z ⟶ X) (w : g ≫ f = h ≫ f) : g = h := by

Source: Written for LeanGraph, reference proof certified
Premises used by the reference proof: 1

Traces

Each configuration's full record: what was retrieved, what the model wrote, what Lean said back.

Verified0 model calls · 16 Lean checks · $0.00000

Attempts

attempt 1Wrong tactic

Proof

rfl

What Lean printed (this is all the model sees on repair)

5:2: error: Tactic `rfl` failed: The left-hand side
  g
is not definitionally equal to the right-hand side
  h

C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
X Y Z : C
f : X ⟶ Y
inst✝ : Mono f
g h : Z ⟶ X
w : g ≫ f = h ≫ f
⊢ g = h

attempt 2Other

Proof

decide

What Lean printed (this is all the model sees on repair)

5:2: error: Expected type must not contain free variables
  g = h

Hint: Use the `+revert` option to automatically clean up and revert free variables

attempt 3Valid but mathematically wrong

Proof

norm_num

What Lean printed (this is all the model sees on repair)

4:121: error: unsolved goals
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
X Y Z : C
f : X ⟶ Y
inst✝ : Mono f
g h : Z ⟶ X
w : g ≫ f = h ≫ f
⊢ g = h

attempt 4Wrong tactic
Proof
```
simp
```
What Lean printed (this is all the model sees on repair)
```
5:2: error: `simp` made no progress
```
attempt 5Wrong tactic
Proof
```
ring
```
What Lean printed (this is all the model sees on repair)
```
5:2: error: `ring_nf` made no progress on the goal
```

attempt 6Wrong tactic

Proof

linarith

What Lean printed (this is all the model sees on repair)

5:2: error: linarith failed to find a contradiction
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
X Y Z : C
f : X ⟶ Y
inst✝ : Mono f
g h : Z ⟶ X
w : g ≫ f = h ≫ f
⊢ False
failed

attempt 7Wrong tactic

Proof

nlinarith

What Lean printed (this is all the model sees on repair)

5:2: error: linarith failed to find a contradiction
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
X Y Z : C
f : X ⟶ Y
inst✝ : Mono f
g h : Z ⟶ X
w : g ≫ f = h ≫ f
⊢ False
failed

attempt 8Wrong tactic
Proof
```
positivity
```
What Lean printed (this is all the model sees on repair)
```
5:2: error: not a positivity goal
```

attempt 9Wrong tactic

Proof

omega

What Lean printed (this is all the model sees on repair)

5:2: error: omega could not prove the goal:
No usable constraints found. You may need to unfold definitions so `omega` can see linear arithmetic facts about `Nat` and `Int`, which may also involve multiplication, division, and modular remainder by constants.

attempt 10Wrong tactic
Proof
```
field_simp
```
What Lean printed (this is all the model sees on repair)
```
5:2: error: `field_simp` made no progress on the goal
```

attempt 11Wrong tactic

Proof

tauto

What Lean printed (this is all the model sees on repair)

5:2: error: Tactic `tauto` failed

C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
X Y Z : C
f : X ⟶ Y
inst✝ : Mono f
g h : Z ⟶ X
w : g ≫ f = h ≫ f
⊢ g = h

attempt 12Wrong tactic
Proof
```
simp_all
```
What Lean printed (this is all the model sees on repair)
```
5:2: error: simp_all made no progress
```

attempt 13Wrong tactic

Proof

aesop

What Lean printed (this is all the model sees on repair)

5:2: error: Tactic `aesop` failed, made no progress
Initial goal:
  C : Type u_1
  inst✝¹ : Category.{u_2, u_1} C
  X Y Z : C
  f : X ⟶ Y
  inst✝ : Mono f
  g h : Z ⟶ X
  w : g ≫ f = h ≫ f
  ⊢ g = h

attempt 14Other

Proof

grind

What Lean printed (this is all the model sees on repair)

5:2: error: `grind` failed
case grind
C : Type u_1
inst : Category.{u_2, u_1} C
X Y Z : C
f : X ⟶ Y
inst_1 : Mono f
g h : Z ⟶ X
w : g ≫ f = h ≫ f
h_1 : ¬g = h
⊢ False
[grind] Goal diagnostics
  [facts] Asserted facts
    [prop] Mono f
    [prop] g ≫ f = h ≫ f
    [prop] ¬g = h
  [eqc] True propositions
    [prop] Mono f
  [eqc] False propositions
    [prop] g = h
  [eqc] Equivalence classes
    [eqc] {g ≫ f, h ≫ f}
  [ematch] E-matching patterns
    [thm] Category.assoc: [@CategoryStruct.comp #8 _ #6 #5 #3 #2 (@CategoryStruct.comp _ _ #5 #4 #3 #1 #0)]
    [thm] Category.assoc: [@CategoryStruct

attempt 15Wrong tactic

Proof

norm_num <;> linarith

What Lean printed (this is all the model sees on repair)

5:15: error: linarith failed to find a contradiction
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
X Y Z : C
f : X ⟶ Y
inst✝ : Mono f
g h : Z ⟶ X
w : g ≫ f = h ≫ f
⊢ False
failed

attempt 16Verified
Proof
```
exact?
```

Verified proof

open CategoryTheory in
theorem lg_target {C : Type*} [Category C] {X Y Z : C} (f : X ⟶ Y) [Mono f] (g h : Z ⟶ X) (w : g ≫ f = h ≫ f) : g = h := by
  exact?

Axioms used: none

Reference proof

The proof we wrote and certified before any model ran.

exact (cancel_mono f).mp w