Theorem: mem_stdinversions'' (Bruhat/Word.lean)

State before:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ (a, b) ∈ StdInversions g ↔ a < b ∧ g b < g a
Tactic:
rw [mem_stdinversions']
State after:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a

State before:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
Tactic:
by_cases h : a = b
State after:
case pos
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a

case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a

State before:
case pos
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a

case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
Tactic:
. simp [h]
State after:
case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a

State before:
case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
Tactic:
. have h' : g a ≠ g b := fun hg => h (g.injective hg)
  simp [h, h'.symm.le_iff_lt]
State after:
no goals

State before:
case pos
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
Tactic:
simp [h]
State after:
no goals

State before:
case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
Tactic:
have h' : g a ≠ g b := fun hg => h (g.injective hg)
State after:
case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
h' : g a ≠ g b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a

State before:
case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
h' : g a ≠ g b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
Tactic:
simp [h, h'.symm.le_iff_lt]
State after:
no goals