Theorem: Preorder.toLE_injective
File: Mathlib/Order/Basic.lean

tactic: cases h
state_before: ι : Type ?u.11187
α✝ : Type u
β : Type v
γ : Type w
π : ι → Type ?u.11198
r : α✝ → α✝ → Prop
α : Type u_1
A B : Preorder α
le✝¹ A_lt : α → α → Prop
le_refl✝¹ : ∀ (a : α), a ≤ a
le_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
A_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
le✝ B_lt : α → α → Prop
le_refl✝ : ∀ (a : α), a ≤ a
le_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
B_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
h : toLE = toLE
⊢ mk le_refl✝¹ le_trans✝¹ = mk le_refl✝ le_trans✝

tactic: have : A_lt = B_lt := by
  funext a b
  show (LT.mk A_lt).lt a b = (LT.mk B_lt).lt a b
  rw [A_iff, B_iff]
state_before: case refl
ι : Type ?u.11187
α✝ : Type u
β : Type v
γ : Type w
π : ι → Type ?u.11198
r : α✝ → α✝ → Prop
α : Type u_1
A B : Preorder α
le✝ A_lt : α → α → Prop
le_refl✝¹ : ∀ (a : α), a ≤ a
le_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
A_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
B_lt : α → α → Prop
le_refl✝ : ∀ (a : α), a ≤ a
le_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
B_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
⊢ mk le_refl✝¹ le_trans✝¹ = mk le_refl✝ le_trans✝

tactic: cases this
state_before: case refl
ι : Type ?u.11187
α✝ : Type u
β : Type v
γ : Type w
π : ι → Type ?u.11198
r : α✝ → α✝ → Prop
α : Type u_1
A B : Preorder α
le✝ A_lt : α → α → Prop
le_refl✝¹ : ∀ (a : α), a ≤ a
le_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
A_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
B_lt : α → α → Prop
le_refl✝ : ∀ (a : α), a ≤ a
le_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
B_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
this : A_lt = B_lt
⊢ mk le_refl✝¹ le_trans✝¹ = mk le_refl✝ le_trans✝

tactic: congr
state_before: case refl.refl
ι : Type ?u.11187
α✝ : Type u
β : Type v
γ : Type w
π : ι → Type ?u.11198
r : α✝ → α✝ → Prop
α : Type u_1
A B : Preorder α
le✝ A_lt : α → α → Prop
le_refl✝¹ : ∀ (a : α), a ≤ a
le_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
A_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
le_refl✝ : ∀ (a : α), a ≤ a
le_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
B_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
⊢ mk le_refl✝¹ le_trans✝¹ = mk le_refl✝ le_trans✝

tactic: funext a b
state_before: ι : Type ?u.11187
α✝ : Type u
β : Type v
γ : Type w
π : ι → Type ?u.11198
r : α✝ → α✝ → Prop
α : Type u_1
A B : Preorder α
le✝ A_lt : α → α → Prop
le_refl✝¹ : ∀ (a : α), a ≤ a
le_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
A_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
B_lt : α → α → Prop
le_refl✝ : ∀ (a : α), a ≤ a
le_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
B_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
⊢ A_lt = B_lt

tactic: show (LT.mk A_lt).lt a b = (LT.mk B_lt).lt a b
state_before: case h.h
ι : Type ?u.11187
α✝ : Type u
β : Type v
γ : Type w
π : ι → Type ?u.11198
r : α✝ → α✝ → Prop
α : Type u_1
A B : Preorder α
le✝ A_lt : α → α → Prop
le_refl✝¹ : ∀ (a : α), a ≤ a
le_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
A_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
B_lt : α → α → Prop
le_refl✝ : ∀ (a : α), a ≤ a
le_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
B_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
a b : α
⊢ A_lt a b = B_lt a b

tactic: rw [A_iff, B_iff]
state_before: case h.h
ι : Type ?u.11187
α✝ : Type u
β : Type v
γ : Type w
π : ι → Type ?u.11198
r : α✝ → α✝ → Prop
α : Type u_1
A B : Preorder α
le✝ A_lt : α → α → Prop
le_refl✝¹ : ∀ (a : α), a ≤ a
le_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
A_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
B_lt : α → α → Prop
le_refl✝ : ∀ (a : α), a ≤ a
le_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
B_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
a b : α
⊢ (a < b) = (a < b)