Theorem: Lean4Axiomatic.Rational.gt_sgn (Lean4Axiomatic/Rational/Order.lean)

State before:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
⊢ p > q ↔ sgn (p - q) ≃ 1
Tactic:
apply Iff.intro
State after:
case mp
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
⊢ p > q → sgn (p - q) ≃ 1

case mpr
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
⊢ sgn (p - q) ≃ 1 → p > q

State before:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
⊢ p > q → sgn (p - q) ≃ 1
Tactic:
case mp =>
  intro (_ : p > q)
  show sgn (p - q) ≃ 1
  have : q < p := ‹p > q›
  have : sgn (q - p) ≃ -1 := lt_sgn.mp this
  have : sgn (p - q) ≃ 1 := calc
    sgn (p - q)    ≃ _ := sgn_subst (eqv_symm neg_sub)
    sgn (-(q - p)) ≃ _ := sgn_compat_neg
    (-sgn (q - p)) ≃ _ := AA.subst₁ ‹sgn (q - p) ≃ -1›
    (-(-1))        ≃ _ := Integer.neg_involutive
    1              ≃ _ := Rel.refl
  exact this
State after:
no goals

State before:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
⊢ sgn (p - q) ≃ 1 → p > q
Tactic:
case mpr =>
  intro (_ : sgn (p - q) ≃ 1)
  show p > q
  have : sgn (q - p) ≃ -1 := calc
    sgn (q - p)    ≃ _ := sgn_subst (eqv_symm neg_sub)
    sgn (-(p - q)) ≃ _ := sgn_compat_neg
    (-sgn (p - q)) ≃ _ := AA.subst₁ ‹sgn (p - q) ≃ 1›
    (-1)           ≃ _ := Rel.refl
  have : q < p := lt_sgn.mpr this
  have : p > q := this
  exact this
State after:
no goals

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

State before:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : p > q
⊢ sgn (p - q) ≃ 1
Tactic:
have : q < p := ‹p > q›
State after:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : p > q
this : q < p
⊢ sgn (p - q) ≃ 1

State before:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : p > q
this : q < p
⊢ sgn (p - q) ≃ 1
Tactic:
have : sgn (q - p) ≃ -1 := lt_sgn.mp this
State after:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : p > q
this✝ : q < p
this : sgn (q - p) ≃ -1
⊢ sgn (p - q) ≃ 1

State before:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : p > q
this✝ : q < p
this : sgn (q - p) ≃ -1
⊢ sgn (p - q) ≃ 1
Tactic:
have : sgn (p - q) ≃ 1 := calc
  sgn (p - q)    ≃ _ := sgn_subst (eqv_symm neg_sub)
  sgn (-(q - p)) ≃ _ := sgn_compat_neg
  (-sgn (q - p)) ≃ _ := AA.subst₁ ‹sgn (q - p) ≃ -1›
  (-(-1))        ≃ _ := Integer.neg_involutive
  1              ≃ _ := Rel.refl
State after:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : p > q
this✝¹ : q < p
this✝ : sgn (q - p) ≃ -1
this : sgn (p - q) ≃ 1
⊢ sgn (p - q) ≃ 1

State before:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : p > q
this✝¹ : q < p
this✝ : sgn (q - p) ≃ -1
this : sgn (p - q) ≃ 1
⊢ sgn (p - q) ≃ 1
Tactic:
exact this
State after:
no goals

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

State before:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : sgn (p - q) ≃ 1
⊢ p > q
Tactic:
have : sgn (q - p) ≃ -1 := calc
  sgn (q - p)    ≃ _ := sgn_subst (eqv_symm neg_sub)
  sgn (-(p - q)) ≃ _ := sgn_compat_neg
  (-sgn (p - q)) ≃ _ := AA.subst₁ ‹sgn (p - q) ≃ 1›
  (-1)           ≃ _ := Rel.refl
State after:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : sgn (p - q) ≃ 1
this : sgn (q - p) ≃ -1
⊢ p > q

State before:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : sgn (p - q) ≃ 1
this : sgn (q - p) ≃ -1
⊢ p > q
Tactic:
have : q < p := lt_sgn.mpr this
State after:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : sgn (p - q) ≃ 1
this✝ : sgn (q - p) ≃ -1
this : q < p
⊢ p > q

State before:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : sgn (p - q) ≃ 1
this✝ : sgn (q - p) ≃ -1
this : q < p
⊢ p > q
Tactic:
have : p > q := this
State after:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : sgn (p - q) ≃ 1
this✝¹ : sgn (q - p) ≃ -1
this✝ : q < p
this : p > q
⊢ p > q

State before:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : sgn (p - q) ≃ 1
this✝¹ : sgn (q - p) ≃ -1
this✝ : q < p
this : p > q
⊢ p > q
Tactic:
exact this
State after:
no goals