Theorem: LO.FirstOrder.Completeness.chainU_val_fst_eq (Logic/FirstOrder/Completeness/SearchTree.lean)

State before:
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
s : ℕ
⊢ (chainU T Γ s).rank = s
Tactic:
induction' s with s ih <;> simp[SearchTree.rank]
State after:
case zero
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
⊢ (chainU T Γ 0).fst = 0

case succ
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
s : ℕ
ih : (chainU T Γ s).rank = s
⊢ (chainU T Γ (s + 1)).fst = s + 1

State before:
case zero
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
⊢ (chainU T Γ 0).fst = 0
Tactic:
exact rfl
State after:
no goals

State before:
case succ
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
s : ℕ
ih : (chainU T Γ s).rank = s
⊢ (chainU T Γ (s + 1)).fst = s + 1
Tactic:
simpa[ih] using SearchTree.rank_of_lt (chainU_spec nwf s)
State after:
no goals