Theorem: Real.Gamma_ne_zero
File: Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean

tactic: intro n
state_before: s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
state_after: s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
n : ℕ
⊢ -↑n < s → Gamma s ≠ 0

tactic: induction' n with _ n_ih generalizing s
state_before: s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
n : ℕ
⊢ -↑n < s → Gamma s ≠ 0
state_after: case zero
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ -↑0 < s → Gamma s ≠ 0

case succ
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ -↑(n✝ + 1) < s → Gamma s ≠ 0

tactic: apply this
state_before: s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ Gamma s ≠ 0
state_after: case a
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ -↑?n < s

case n
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ ℕ

tactic: swap
state_before: case a
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ -↑?n < s

case n
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ ℕ
state_after: case n
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ ℕ

case a
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ -↑?n < s

tactic: rw [neg_lt, Nat.cast_add, Nat.cast_one]
state_before: case a
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ -↑(⌊-s⌋₊ + 1) < s
state_after: case a
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ -s < ↑⌊-s⌋₊ + 1

tactic: exact Nat.lt_floor_add_one _
state_before: case a
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ -s < ↑⌊-s⌋₊ + 1
state_after: no goals

tactic: exact ⌊-s⌋₊ + 1
state_before: case n
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ ℕ
state_after: no goals

tactic: intro hs
state_before: case zero
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ -↑0 < s → Gamma s ≠ 0
state_after: case zero
s : ℝ
hs✝ : ∀ (m : ℕ), s ≠ -↑m
hs : -↑0 < s
⊢ Gamma s ≠ 0

tactic: refine (Gamma_pos_of_pos ?_).ne'
state_before: case zero
s : ℝ
hs✝ : ∀ (m : ℕ), s ≠ -↑m
hs : -↑0 < s
⊢ Gamma s ≠ 0
state_after: case zero
s : ℝ
hs✝ : ∀ (m : ℕ), s ≠ -↑m
hs : -↑0 < s
⊢ 0 < s

tactic: rwa [Nat.cast_zero, neg_zero] at hs
state_before: case zero
s : ℝ
hs✝ : ∀ (m : ℕ), s ≠ -↑m
hs : -↑0 < s
⊢ 0 < s
state_after: no goals

tactic: intro hs'
state_before: case succ
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ -↑(n✝ + 1) < s → Gamma s ≠ 0
state_after: case succ
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
⊢ Gamma s ≠ 0

tactic: rw [Gamma_add_one, mul_ne_zero_iff] at this
state_before: case succ
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
this : Gamma (s + 1) ≠ 0
⊢ Gamma s ≠ 0
state_after: case succ
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
this : s ≠ 0 ∧ Gamma s ≠ 0
⊢ Gamma s ≠ 0

case succ
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
this : Gamma (s + 1) ≠ 0
⊢ s ≠ 0

tactic: apply n_ih
state_before: n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
⊢ Gamma (s + 1) ≠ 0
state_after: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
⊢ ∀ (m : ℕ), s + 1 ≠ -↑m

case a
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
⊢ -↑n✝ < s + 1

tactic: intro m
state_before: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
⊢ ∀ (m : ℕ), s + 1 ≠ -↑m
state_after: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
m : ℕ
⊢ s + 1 ≠ -↑m

tactic: specialize hs (1 + m)
state_before: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
m : ℕ
⊢ s + 1 ≠ -↑m
state_after: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s ≠ -↑(1 + m)
⊢ s + 1 ≠ -↑m

tactic: contrapose! hs
state_before: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s ≠ -↑(1 + m)
⊢ s + 1 ≠ -↑m
state_after: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s + 1 = -↑m
⊢ s = -↑(1 + m)

tactic: rw [← eq_sub_iff_add_eq] at hs
state_before: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s + 1 = -↑m
⊢ s = -↑(1 + m)
state_after: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s = -↑m - 1
⊢ s = -↑(1 + m)

tactic: rw [hs]
state_before: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s = -↑m - 1
⊢ s = -↑(1 + m)
state_after: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s = -↑m - 1
⊢ -↑m - 1 = -↑(1 + m)

tactic: push_cast
state_before: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s = -↑m - 1
⊢ -↑m - 1 = -↑(1 + m)
state_after: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s = -↑m - 1
⊢ -↑m - 1 = -(1 + ↑m)

tactic: ring
state_before: case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s = -↑m - 1
⊢ -↑m - 1 = -(1 + ↑m)
state_after: no goals

tactic: rw [Nat.cast_add, Nat.cast_one, neg_add] at hs'
state_before: case a
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
⊢ -↑n✝ < s + 1
state_after: case a
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑n✝ + -1 < s
⊢ -↑n✝ < s + 1

tactic: linarith
state_before: case a
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑n✝ + -1 < s
⊢ -↑n✝ < s + 1
state_after: no goals

tactic: exact this.2
state_before: case succ
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
this : s ≠ 0 ∧ Gamma s ≠ 0
⊢ Gamma s ≠ 0
state_after: no goals

tactic: simpa using hs 0
state_before: case succ
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
hs' : -↑(n✝ + 1) < s
this : Gamma (s + 1) ≠ 0
⊢ s ≠ 0
state_after: no goals