Theorem: Lean4Axiomatic.Rational.close_widen (Lean4Axiomatic/Rational/Metric.lean)

State before:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε ε' p q : ℚ
⊢ p ⊢ε⊣ q → ε' > ε → p ⊢ε'⊣ q
Tactic:
intro (_ : p ⊢ε⊣ q) (_ : ε' > ε)
State after:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε ε' p q : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : ε' > ε
⊢ p ⊢ε'⊣ q

State before:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε ε' p q : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : ε' > ε
⊢ p ⊢ε'⊣ q
Tactic:
have : dist p q ≤ ε := close_dist.mp ‹p ⊢ε⊣ q›
State after:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε ε' p q : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : ε' > ε
this : dist p q ≤ ε
⊢ p ⊢ε'⊣ q

State before:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε ε' p q : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : ε' > ε
this : dist p q ≤ ε
⊢ p ⊢ε'⊣ q
Tactic:
have : ε ≤ ε' := le_cases.mpr (Or.inl ‹ε < ε'›)
State after:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε ε' p q : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : ε' > ε
this✝ : dist p q ≤ ε
this : ε ≤ ε'
⊢ p ⊢ε'⊣ q

State before:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε ε' p q : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : ε' > ε
this✝ : dist p q ≤ ε
this : ε ≤ ε'
⊢ p ⊢ε'⊣ q
Tactic:
have : dist p q ≤ ε' := le_trans ‹dist p q ≤ ε› ‹ε ≤ ε'›
State after:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε ε' p q : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : ε' > ε
this✝¹ : dist p q ≤ ε
this✝ : ε ≤ ε'
this : dist p q ≤ ε'
⊢ p ⊢ε'⊣ q

State before:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε ε' p q : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : ε' > ε
this✝¹ : dist p q ≤ ε
this✝ : ε ≤ ε'
this : dist p q ≤ ε'
⊢ p ⊢ε'⊣ q
Tactic:
have : p ⊢ε'⊣ q := close_dist.mpr this
State after:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε ε' p q : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : ε' > ε
this✝² : dist p q ≤ ε
this✝¹ : ε ≤ ε'
this✝ : dist p q ≤ ε'
this : p ⊢ε'⊣ q
⊢ p ⊢ε'⊣ q

State before:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε ε' p q : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : ε' > ε
this✝² : dist p q ≤ ε
this✝¹ : ε ≤ ε'
this✝ : dist p q ≤ ε'
this : p ⊢ε'⊣ q
⊢ p ⊢ε'⊣ q
Tactic:
exact this
State after:
no goals