Theorem: Lean4Axiomatic.Integer.le_refl (Lean4Axiomatic/Integer/Order.lean)

State before:
ℕ : Type
inst✝⁷ : Natural ℕ
ℤ : Type
inst✝⁶ : Core ℤ
inst✝⁵ : Addition ℤ
inst✝⁴ : Multiplication ℤ
inst✝³ : Negation ℤ
inst✝² : Subtraction ℤ
inst✝¹ : Sign ℤ
inst✝ : Order ℤ
a : ℤ
⊢ a ≤ a
Tactic:
apply le_iff_add_nat.mpr
State after:
ℕ : Type
inst✝⁷ : Natural ℕ
ℤ : Type
inst✝⁶ : Core ℤ
inst✝⁵ : Addition ℤ
inst✝⁴ : Multiplication ℤ
inst✝³ : Negation ℤ
inst✝² : Subtraction ℤ
inst✝¹ : Sign ℤ
inst✝ : Order ℤ
a : ℤ
⊢ ∃ k, a ≃ a + coe k

State before:
ℕ : Type
inst✝⁷ : Natural ℕ
ℤ : Type
inst✝⁶ : Core ℤ
inst✝⁵ : Addition ℤ
inst✝⁴ : Multiplication ℤ
inst✝³ : Negation ℤ
inst✝² : Subtraction ℤ
inst✝¹ : Sign ℤ
inst✝ : Order ℤ
a : ℤ
⊢ ∃ k, a ≃ a + coe k
Tactic:
have : a ≃ a + coe (0 : ℕ) := calc
  a               ≃ _ := Rel.symm AA.identR
  a + 0           ≃ _ := Rel.refl
  a + coe (0 : ℕ) ≃ _ := Rel.refl
State after:
ℕ : Type
inst✝⁷ : Natural ℕ
ℤ : Type
inst✝⁶ : Core ℤ
inst✝⁵ : Addition ℤ
inst✝⁴ : Multiplication ℤ
inst✝³ : Negation ℤ
inst✝² : Subtraction ℤ
inst✝¹ : Sign ℤ
inst✝ : Order ℤ
a : ℤ
this : a ≃ a + coe 0
⊢ ∃ k, a ≃ a + coe k

State before:
ℕ : Type
inst✝⁷ : Natural ℕ
ℤ : Type
inst✝⁶ : Core ℤ
inst✝⁵ : Addition ℤ
inst✝⁴ : Multiplication ℤ
inst✝³ : Negation ℤ
inst✝² : Subtraction ℤ
inst✝¹ : Sign ℤ
inst✝ : Order ℤ
a : ℤ
this : a ≃ a + coe 0
⊢ ∃ k, a ≃ a + coe k
Tactic:
exact Exists.intro 0 ‹a ≃ a + coe (0 : ℕ)›
State after:
no goals