-- Mathlib/AlgebraicTopology/SimplexCategory/MorphismProperty.lean
lemma Truncated.morphismProperty_eq_top
    {d : ℕ} (W : MorphismProperty (Truncated d)) [W.IsMultiplicative]
    (δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)),
    W (SimplexCategory.δ (n := n) i : ⟨.mk n, by dsimp; omega⟩ ⟶
      ⟨.mk (n + 1), by dsimp; omega⟩))
    (σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)),
    W (SimplexCategory.σ (n := n) i : ⟨.mk (n + 1), by dsimp; omega⟩ ⟶
      ⟨.mk n, by dsimp; omega⟩)) :
    W = ⊤

Proof state:
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
c : ℕ
hc :
  ∀ (a : ℕ) (ha : a ≤ d) (b : ℕ) (hb : b ≤ d) (f : { obj := mk a, property := ha } ⟶ { obj := mk b, property := hb }),
    a + b = c → W f
b : ℕ
hb : b ≤ d
a : ℕ
ha : a + 1 ≤ d
f : { obj := mk (a + 1), property := ha } ⟶ { obj := mk b, property := hb }
h : a + 1 + b = c + 1
f' : mk (a + 1) ⟶ mk b := f
h₁ : Function.Surjective ⇑(Hom.toOrderHom f')
h₂ : ¬Function.Injective ⇑(Hom.toOrderHom f')
i : Fin (a + 1)
g' : mk a ⟶ mk b
hf' : f' = σ i ≫ g'
⊢ (mk a).len ≤ d

Tactic:
dsimp

Result:
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
c : ℕ
hc :
  ∀ (a : ℕ) (ha : a ≤ d) (b : ℕ) (hb : b ≤ d) (f : { obj := mk a, property := ha } ⟶ { obj := mk b, property := hb }),
    a + b = c → W f
b : ℕ
hb : b ≤ d
a : ℕ
ha : a + 1 ≤ d
f : { obj := mk (a + 1), property := ha } ⟶ { obj := mk b, property := hb }
h : a + 1 + b = c + 1
f' : mk (a + 1) ⟶ mk b := f
h₁ : Function.Surjective ⇑(Hom.toOrderHom f')
h₂ : ¬Function.Injective ⇑(Hom.toOrderHom f')
i : Fin (a + 1)
g' : mk a ⟶ mk b
hf' : f' = σ i ≫ g'
⊢ a ≤ d

Proof state:
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
⊢ W = ⊤

Tactic:
ext ⟨a, ha⟩ ⟨b, hb⟩ f

Result:
case h.mk.mk
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
a : SimplexCategory
ha : a.len ≤ d
b : SimplexCategory
hb : b.len ≤ d
f : { obj := a, property := ha } ⟶ { obj := b, property := hb }
⊢ W f ↔ ⊤ f

Proof state:
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
n : ℕ
hn : n < d
i : Fin (n + 1)
⊢ (mk n).len ≤ d

Tactic:
dsimp

Result:
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
n : ℕ
hn : n < d
i : Fin (n + 1)
⊢ n ≤ d

Proof state:
case neg.zero
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
c : ℕ
hc :
  ∀ (a : ℕ) (ha : a ≤ d) (b : ℕ) (hb : b ≤ d) (f : { obj := mk a, property := ha } ⟶ { obj := mk b, property := hb }),
    a + b = c → W f
b : ℕ
hb : b ≤ d
ha : 0 ≤ d
f : { obj := mk 0, property := ha } ⟶ { obj := mk b, property := hb }
h : 0 + b = c + 1
f' : mk 0 ⟶ mk b := f
h₁ : Function.Surjective ⇑(Hom.toOrderHom f')
h₂ : ¬Function.Injective ⇑(Hom.toOrderHom f')
⊢ W f

Tactic:
exact (h₂ (Function.injective_of_subsingleton (α := Fin 1) _)).elim

Result:
no goals

Proof state:
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
n : ℕ
hn : n < d
i : Fin (n + 2)
⊢ n + 1 ≤ d

Tactic:
omega

Result:
no goals

Proof state:
case h.mk.mk.h.h.zero
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
b : ℕ
hb : b ≤ d
ha : 0 ≤ d
f : { obj := mk 0, property := ha } ⟶ { obj := mk b, property := hb }
h : 0 + b = 0
⊢ W f

Tactic:
obtain rfl : b = 0 := by omega

Result:
case h.mk.mk.h.h.zero
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
ha hb : 0 ≤ d
f : { obj := mk 0, property := ha } ⟶ { obj := mk 0, property := hb }
h : 0 + 0 = 0
⊢ W f

Proof state:
case neg.succ
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
c : ℕ
hc :
  ∀ (a : ℕ) (ha : a ≤ d) (b : ℕ) (hb : b ≤ d) (f : { obj := mk a, property := ha } ⟶ { obj := mk b, property := hb }),
    a + b = c → W f
a : ℕ
ha : a ≤ d
b : ℕ
hb : b + 1 ≤ d
f : { obj := mk a, property := ha } ⟶ { obj := mk (b + 1), property := hb }
h : a + (b + 1) = c + 1
f' : mk a ⟶ mk (b + 1) := f
h₁ : ¬Function.Surjective ⇑(Hom.toOrderHom f')
⊢ W f

Tactic:
obtain ⟨i, g', hf'⟩ := eq_comp_δ_of_not_surjective _ h₁

Result:
case neg.succ.intro.intro
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
c : ℕ
hc :
  ∀ (a : ℕ) (ha : a ≤ d) (b : ℕ) (hb : b ≤ d) (f : { obj := mk a, property := ha } ⟶ { obj := mk b, property := hb }),
    a + b = c → W f
a : ℕ
ha : a ≤ d
b : ℕ
hb : b + 1 ≤ d
f : { obj := mk a, property := ha } ⟶ { obj := mk (b + 1), property := hb }
h : a + (b + 1) = c + 1
f' : mk a ⟶ mk (b + 1) := f
h₁ : ¬Function.Surjective ⇑(Hom.toOrderHom f')
i : Fin (b + 2)
g' : mk a ⟶ mk b
hf' : f' = g' ≫ δ i
⊢ W f

Proof state:
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
ha hb : 0 ≤ d
f : { obj := mk 0, property := ha } ⟶ { obj := mk 0, property := hb }
h : 0 + 0 = 0
⊢ f = 𝟙 { obj := mk 0, property := ha }

Tactic:
ext i : 3

Result:
case a.h.h
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
ha hb : 0 ≤ d
f : { obj := mk 0, property := ha } ⟶ { obj := mk 0, property := hb }
h : 0 + 0 = 0
i : Fin ({ obj := mk 0, property := ha }.obj.len + 1)
⊢ (Hom.toOrderHom f) i = (Hom.toOrderHom (𝟙 { obj := mk 0, property := ha })) i