internlm/Lean-Github huggingface dataset:
Theorem: Lean4Axiomatic.Rational.close_subst_neg
Proof 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 p q : ℚ
x✝ : p ⊢ε⊣ q
this : dist p q ≤ ε
⊢ -p ⊢ε⊣ -q
Tactic:
have : dist (-p) (-q) ≤ ε := le_substL_eqv (eqv_symm dist_cancel_neg) this
Proof 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 p q : ℚ
x✝ : p ⊢ε⊣ q
this✝ : dist p q ≤ ε
this : dist (-p) (-q) ≤ ε
⊢ -p ⊢ε⊣ -q
Theorem: Lean4Axiomatic.Rational.close_subst_neg
Proof 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 p q : ℚ
x✝ : p ⊢ε⊣ q
this✝ : dist p q ≤ ε
this : dist (-p) (-q) ≤ ε
⊢ -p ⊢ε⊣ -q
Tactic:
have : -p ⊢ε⊣ -q := close_dist.mpr this
Proof 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 p q : ℚ
x✝ : p ⊢ε⊣ q
this✝¹ : dist p q ≤ ε
this✝ : dist (-p) (-q) ≤ ε
this : -p ⊢ε⊣ -q
⊢ -p ⊢ε⊣ -q
Theorem: Lean4Axiomatic.Rational.close_subst_neg
Proof 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 p q : ℚ
x✝ : p ⊢ε⊣ q
this✝¹ : dist p q ≤ ε
this✝ : dist (-p) (-q) ≤ ε
this : -p ⊢ε⊣ -q
⊢ -p ⊢ε⊣ -q
Tactic:
exact this
Proof state after:
no goals
Theorem: Lean4Axiomatic.Rational.close_substL_sub
Proof 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 p q r : ℚ
⊢ p ⊢ε⊣ q → p - r ⊢ε⊣ q - r
Tactic:
intro (_ : p ⊢ε⊣ q)
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
⊢ p - r ⊢ε⊣ q - r
Theorem: Lean4Axiomatic.Rational.close_substL_sub
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
⊢ p - r ⊢ε⊣ q - r
Tactic:
have : p + (-r) ⊢ε⊣ q + (-r) := close_substL_add ‹p ⊢ε⊣ q›
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this : p + -r ⊢ε⊣ q + -r
⊢ p - r ⊢ε⊣ q - r
Theorem: Lean4Axiomatic.Rational.close_substL_sub
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this : p + -r ⊢ε⊣ q + -r
⊢ p - r ⊢ε⊣ q - r
Tactic:
have : p - r ⊢ε⊣ q + (-r) := close_substL_eqv (eqv_symm sub_add_neg) this
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this✝ : p + -r ⊢ε⊣ q + -r
this : p - r ⊢ε⊣ q + -r
⊢ p - r ⊢ε⊣ q - r
Theorem: Lean4Axiomatic.Rational.close_substL_sub
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this✝ : p + -r ⊢ε⊣ q + -r
this : p - r ⊢ε⊣ q + -r
⊢ p - r ⊢ε⊣ q - r
Tactic:
have : p - r ⊢ε⊣ q - r := close_substR_eqv (eqv_symm sub_add_neg) this
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this✝¹ : p + -r ⊢ε⊣ q + -r
this✝ : p - r ⊢ε⊣ q + -r
this : p - r ⊢ε⊣ q - r
⊢ p - r ⊢ε⊣ q - r
Theorem: Lean4Axiomatic.Rational.close_substL_sub
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this✝¹ : p + -r ⊢ε⊣ q + -r
this✝ : p - r ⊢ε⊣ q + -r
this : p - r ⊢ε⊣ q - r
⊢ p - r ⊢ε⊣ q - r
Tactic:
exact this
Proof state after:
no goals
Theorem: Lean4Axiomatic.Rational.close_substR_sub
Proof 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 p q r : ℚ
⊢ p ⊢ε⊣ q → r - p ⊢ε⊣ r - q
Tactic:
intro (_ : p ⊢ε⊣ q)
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
⊢ r - p ⊢ε⊣ r - q
Theorem: Lean4Axiomatic.Rational.close_substR_sub
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
⊢ r - p ⊢ε⊣ r - q
Tactic:
have : -p ⊢ε⊣ -q := close_subst_neg ‹p ⊢ε⊣ q›
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this : -p ⊢ε⊣ -q
⊢ r - p ⊢ε⊣ r - q
Theorem: Lean4Axiomatic.Rational.close_substR_sub
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this : -p ⊢ε⊣ -q
⊢ r - p ⊢ε⊣ r - q
Tactic:
have : r + (-p) ⊢ε⊣ r + (-q) := close_substR_add this
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this✝ : -p ⊢ε⊣ -q
this : r + -p ⊢ε⊣ r + -q
⊢ r - p ⊢ε⊣ r - q
Theorem: Lean4Axiomatic.Rational.close_substR_sub
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this✝ : -p ⊢ε⊣ -q
this : r + -p ⊢ε⊣ r + -q
⊢ r - p ⊢ε⊣ r - q
Tactic:
have : r - p ⊢ε⊣ r + (-q) := close_substL_eqv (eqv_symm sub_add_neg) this
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this✝¹ : -p ⊢ε⊣ -q
this✝ : r + -p ⊢ε⊣ r + -q
this : r - p ⊢ε⊣ r + -q
⊢ r - p ⊢ε⊣ r - q
Theorem: Lean4Axiomatic.Rational.close_substR_sub
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this✝¹ : -p ⊢ε⊣ -q
this✝ : r + -p ⊢ε⊣ r + -q
this : r - p ⊢ε⊣ r + -q
⊢ r - p ⊢ε⊣ r - q
Tactic:
have : r - p ⊢ε⊣ r - q := close_substR_eqv (eqv_symm sub_add_neg) this
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this✝² : -p ⊢ε⊣ -q
this✝¹ : r + -p ⊢ε⊣ r + -q
this✝ : r - p ⊢ε⊣ r + -q
this : r - p ⊢ε⊣ r - q
⊢ r - p ⊢ε⊣ r - q
Theorem: Lean4Axiomatic.Rational.close_substR_sub
Proof 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 p q r : ℚ
x✝ : p ⊢ε⊣ q
this✝² : -p ⊢ε⊣ -q
this✝¹ : r + -p ⊢ε⊣ r + -q
this✝ : r - p ⊢ε⊣ r + -q
this : r - p ⊢ε⊣ r - q
⊢ r - p ⊢ε⊣ r - q
Tactic:
exact this
Proof state after:
no goals
Theorem: Lean4Axiomatic.Rational.close_add_pointwise
Proof 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 δ p q r s : ℚ
⊢ p ⊢ε⊣ q → r ⊢δ⊣ s → p + r ⊢ε + δ⊣ q + s
Tactic:
intro (_ : p ⊢ε⊣ q) (_ : r ⊢δ⊣ s)
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
⊢ p + r ⊢ε + δ⊣ q + s
Theorem: Lean4Axiomatic.Rational.close_add_pointwise
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
⊢ p + r ⊢ε + δ⊣ q + s
Tactic:
have : p + r ⊢ε⊣ q + r := close_substL_add ‹p ⊢ε⊣ q›
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
this : p + r ⊢ε⊣ q + r
⊢ p + r ⊢ε + δ⊣ q + s
Theorem: Lean4Axiomatic.Rational.close_add_pointwise
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
this : p + r ⊢ε⊣ q + r
⊢ p + r ⊢ε + δ⊣ q + s
Tactic:
have : q + r ⊢δ⊣ q + s := close_substR_add ‹r ⊢δ⊣ s›
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
this✝ : p + r ⊢ε⊣ q + r
this : q + r ⊢δ⊣ q + s
⊢ p + r ⊢ε + δ⊣ q + s
Theorem: Lean4Axiomatic.Rational.close_add_pointwise
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
this✝ : p + r ⊢ε⊣ q + r
this : q + r ⊢δ⊣ q + s
⊢ p + r ⊢ε + δ⊣ q + s
Tactic:
have : p + r ⊢ε+δ⊣ q + s := close_trans ‹p + r ⊢ε⊣ q + r› ‹q + r ⊢δ⊣ q + s›
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
this✝¹ : p + r ⊢ε⊣ q + r
this✝ : q + r ⊢δ⊣ q + s
this : p + r ⊢ε + δ⊣ q + s
⊢ p + r ⊢ε + δ⊣ q + s
Theorem: Lean4Axiomatic.Rational.close_add_pointwise
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
this✝¹ : p + r ⊢ε⊣ q + r
this✝ : q + r ⊢δ⊣ q + s
this : p + r ⊢ε + δ⊣ q + s
⊢ p + r ⊢ε + δ⊣ q + s
Tactic:
exact this
Proof state after:
no goals
Theorem: Lean4Axiomatic.Rational.close_sub_pointwise
Proof 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 δ p q r s : ℚ
⊢ p ⊢ε⊣ q → r ⊢δ⊣ s → p - r ⊢ε + δ⊣ q - s
Tactic:
intro (_ : p ⊢ε⊣ q) (_ : r ⊢δ⊣ s)
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
⊢ p - r ⊢ε + δ⊣ q - s
Theorem: Lean4Axiomatic.Rational.close_sub_pointwise
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
⊢ p - r ⊢ε + δ⊣ q - s
Tactic:
have : p - r ⊢ε⊣ q - r := close_substL_sub ‹p ⊢ε⊣ q›
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
this : p - r ⊢ε⊣ q - r
⊢ p - r ⊢ε + δ⊣ q - s
Theorem: Lean4Axiomatic.Rational.close_sub_pointwise
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
this : p - r ⊢ε⊣ q - r
⊢ p - r ⊢ε + δ⊣ q - s
Tactic:
have : q - r ⊢δ⊣ q - s := close_substR_sub ‹r ⊢δ⊣ s›
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
this✝ : p - r ⊢ε⊣ q - r
this : q - r ⊢δ⊣ q - s
⊢ p - r ⊢ε + δ⊣ q - s
Theorem: Lean4Axiomatic.Rational.close_sub_pointwise
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
this✝ : p - r ⊢ε⊣ q - r
this : q - r ⊢δ⊣ q - s
⊢ p - r ⊢ε + δ⊣ q - s
Tactic:
have : p - r ⊢ε+δ⊣ q - s := close_trans ‹p - r ⊢ε⊣ q - r› ‹q - r ⊢δ⊣ q - s›
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
this✝¹ : p - r ⊢ε⊣ q - r
this✝ : q - r ⊢δ⊣ q - s
this : p - r ⊢ε + δ⊣ q - s
⊢ p - r ⊢ε + δ⊣ q - s
Theorem: Lean4Axiomatic.Rational.close_sub_pointwise
Proof 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 δ p q r s : ℚ
x✝¹ : p ⊢ε⊣ q
x✝ : r ⊢δ⊣ s
this✝¹ : p - r ⊢ε⊣ q - r
this✝ : q - r ⊢δ⊣ q - s
this : p - r ⊢ε + δ⊣ q - s
⊢ p - r ⊢ε + δ⊣ q - s
Tactic:
exact this
Proof state after:
no goals
Theorem: Lean4Axiomatic.Rational.close_widen
Proof 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)
Proof 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
Theorem: Lean4Axiomatic.Rational.close_widen
Proof 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›
Proof 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
Theorem: Lean4Axiomatic.Rational.close_widen
Proof 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'›)
Proof 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
Theorem: Lean4Axiomatic.Rational.close_widen
Proof 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'›
Proof 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
Theorem: Lean4Axiomatic.Rational.close_widen
Proof 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
Proof 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
Theorem: Lean4Axiomatic.Rational.close_widen
Proof 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
Proof state after:
no goals
Theorem: Lean4Axiomatic.Rational.order_iff_min_max
Proof 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
p q r : ℚ
⊢ p ≤ q → (p ≤ r ∧ r ≤ q ↔ Ordered.min p q ≤ r ∧ r ≤ Ordered.max p q)
Tactic:
intro (_ : p ≤ q)
Proof 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
p q r : ℚ
x✝ : p ≤ q
⊢ p ≤ r ∧ r ≤ q ↔ Ordered.min p q ≤ r ∧ r ≤ Ordered.max p q