url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.negative_iff_neg_positive | [477, 1] | [503, 15] | have : x ≃ -(-x) := Rational.eqv_symm Rational.neg_involutive | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
this✝¹ : sgn x ≃ -1
this✝ : sgn (-x) ≃ 1
this : Positive (-x)
⊢ ∃ y, Positive y ∧ x ≃ -y | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
this✝² : sgn x ≃ -1
this✝¹ : sgn (-x) ≃ 1
this✝ : Positive (-x)
this : x ≃ - -x
⊢ ∃ y, Positive y ∧ x ≃ -y | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
this✝¹ : sgn x ≃ -1
this✝ : sgn (-x) ≃ 1
this : Positive (-x)
⊢ ∃ y, Positive y ∧ x ≃ -y
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.negative_iff_neg_positive | [477, 1] | [503, 15] | exact Exists.intro (-x) (And.intro ‹Positive (-x)› ‹x ≃ -(-x)›) | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
this✝² : sgn x ≃ -1
this✝¹ : sgn (-x) ≃ 1
this✝ : Positive (-x)
this : x ≃ - -x
⊢ ∃ y, Positive y ∧ x ≃ -y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
this✝² : sgn x ≃ -1
this✝¹ : sgn (-x) ≃ 1
this✝ : Positive (-x)
this : x ≃ - -x
⊢ ∃ y, Positive y ∧ x ≃ -y
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.negative_iff_neg_positive | [477, 1] | [503, 15] | intro (Exists.intro (y : ℚ) (And.intro (_ : Positive y) (_ : x ≃ -y))) | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ (∃ y, Positive y ∧ x ≃ -y) → Negative x | ℚ : Type
inst✝ : Rational ℚ
x y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
⊢ Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ (∃ y, Positive y ∧ x ≃ -y) → Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.negative_iff_neg_positive | [477, 1] | [503, 15] | have : sgn y ≃ 1 := Rational.sgn_positive.mp ‹Positive y› | ℚ : Type
inst✝ : Rational ℚ
x y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
⊢ Negative x | ℚ : Type
inst✝ : Rational ℚ
x y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
this : sgn y ≃ 1
⊢ Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
⊢ Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.negative_iff_neg_positive | [477, 1] | [503, 15] | have : sgn x ≃ -1 := calc
sgn x ≃ _ := Rational.sgn_subst ‹x ≃ -y›
sgn (-y) ≃ _ := Rational.sgn_compat_neg
(-(sgn y)) ≃ _ := AA.subst₁ ‹sgn y ≃ 1›
(-1) ≃ _ := Rel.refl | ℚ : Type
inst✝ : Rational ℚ
x y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
this : sgn y ≃ 1
⊢ Negative x | ℚ : Type
inst✝ : Rational ℚ
x y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
this✝ : sgn y ≃ 1
this : sgn x ≃ -1
⊢ Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
this : sgn y ≃ 1
⊢ Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.negative_iff_neg_positive | [477, 1] | [503, 15] | have : Negative x := Rational.sgn_negative.mpr this | ℚ : Type
inst✝ : Rational ℚ
x y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
this✝ : sgn y ≃ 1
this : sgn x ≃ -1
⊢ Negative x | ℚ : Type
inst✝ : Rational ℚ
x y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
this✝¹ : sgn y ≃ 1
this✝ : sgn x ≃ -1
this : Negative x
⊢ Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
this✝ : sgn y ≃ 1
this : sgn x ≃ -1
⊢ Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.negative_iff_neg_positive | [477, 1] | [503, 15] | exact this | ℚ : Type
inst✝ : Rational ℚ
x y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
this✝¹ : sgn y ≃ 1
this✝ : sgn x ≃ -1
this : Negative x
⊢ Negative x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
this✝¹ : sgn y ≃ 1
this✝ : sgn x ≃ -1
this : Negative x
⊢ Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | apply Iff.intro | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Negative x ↔ AltNegative x | case mp
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Negative x → AltNegative x
case mpr
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ AltNegative x → Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Negative x ↔ AltNegative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | case mp =>
intro (_ : Negative x)
show AltNegative x
have (Exists.intro (y : ℚ) (And.intro (_ : Positive y) (_ : x ≃ -y))) :=
negative_iff_neg_positive.mp ‹Negative x›
have
(AltPositive.intro
(a : ℤ) (b : ℤ)
(_ : AP (Positive a)) (_ : AP (Positive b)) y_eqv_a_over_b)
:= alt_positive.mp ‹Positive y›
have : y ≃ a / b := y_eqv_a_over_b
have : x ≃ (-a) / b := calc
x ≃ _ := ‹x ≃ -y›
(-y) ≃ _ := Rational.neg_subst ‹y ≃ a / b›
(-((a : ℚ) / b)) ≃ _ := Rational.neg_scompatL_div
(-a : ℚ) / b ≃ _ := Rational.eqv_refl
exact AltNegative.mk this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Negative x → AltNegative x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Negative x → AltNegative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | case mpr =>
intro (_ : AltNegative x)
show Negative x
have
(AltNegative.intro
(a : ℤ) (b : ℤ)
(_ : AP (Positive a)) (_ : AP (Positive b)) x_eqv_neg_a_over_b)
:= ‹AltNegative x›
have : x ≃ (-a) / b := x_eqv_neg_a_over_b
have : AltPositive ((a : ℚ) / b) := AltPositive.mk Rational.eqv_refl
have : Positive ((a : ℚ) / b) := alt_positive.mpr this
have : x ≃ -((a : ℚ) / b) := calc
x ≃ _ := ‹x ≃ (-a) / b›
(-a : ℚ) / b ≃ _ := Rational.eqv_symm Rational.neg_scompatL_div
(-((a : ℚ) / b)) ≃ _ := Rational.eqv_refl
have : ∃ (y : ℚ), Positive y ∧ x ≃ -y :=
Exists.intro ((a : ℚ) / b) (And.intro ‹Positive ((a : ℚ) / b)› this)
have : Negative x := negative_iff_neg_positive.mpr this
exact this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ AltNegative x → Negative x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ AltNegative x → Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | intro (_ : Negative x) | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Negative x → AltNegative x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
⊢ AltNegative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Negative x → AltNegative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | have (Exists.intro (y : ℚ) (And.intro (_ : Positive y) (_ : x ≃ -y))) :=
negative_iff_neg_positive.mp ‹Negative x› | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
⊢ AltNegative x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
⊢ AltNegative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
⊢ AltNegative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | have
(AltPositive.intro
(a : ℤ) (b : ℤ)
(_ : AP (Positive a)) (_ : AP (Positive b)) y_eqv_a_over_b)
:= alt_positive.mp ‹Positive y› | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
⊢ AltNegative x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
y_eqv_a_over_b : y ≃ Rational.from_integer a / Rational.from_integer b
⊢ AltNegative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
⊢ AltNegative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | have : y ≃ a / b := y_eqv_a_over_b | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
y_eqv_a_over_b : y ≃ Rational.from_integer a / Rational.from_integer b
⊢ AltNegative x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
y_eqv_a_over_b this : y ≃ Rational.from_integer a / Rational.from_integer b
⊢ AltNegative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
y_eqv_a_over_b : y ≃ Rational.from_integer a / Rational.from_integer b
⊢ AltNegative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | have : x ≃ (-a) / b := calc
x ≃ _ := ‹x ≃ -y›
(-y) ≃ _ := Rational.neg_subst ‹y ≃ a / b›
(-((a : ℚ) / b)) ≃ _ := Rational.neg_scompatL_div
(-a : ℚ) / b ≃ _ := Rational.eqv_refl | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
y_eqv_a_over_b this : y ≃ Rational.from_integer a / Rational.from_integer b
⊢ AltNegative x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
y_eqv_a_over_b this✝ : y ≃ Rational.from_integer a / Rational.from_integer b
this : x ≃ -Rational.from_integer a / Rational.from_integer b
⊢ AltNegative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
y_eqv_a_over_b this : y ≃ Rational.from_integer a / Rational.from_integer b
⊢ AltNegative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | exact AltNegative.mk this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
y_eqv_a_over_b this✝ : y ≃ Rational.from_integer a / Rational.from_integer b
this : x ≃ -Rational.from_integer a / Rational.from_integer b
⊢ AltNegative x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
y : ℚ
left✝ : Positive y
right✝ : x ≃ -y
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
y_eqv_a_over_b this✝ : y ≃ Rational.from_integer a / Rational.from_integer b
this : x ≃ -Rational.from_integer a / Rational.from_integer b
⊢ AltNegative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | intro (_ : AltNegative x) | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ AltNegative x → Negative x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
⊢ Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ AltNegative x → Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | have
(AltNegative.intro
(a : ℤ) (b : ℤ)
(_ : AP (Positive a)) (_ : AP (Positive b)) x_eqv_neg_a_over_b)
:= ‹AltNegative x› | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
⊢ Negative x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b : x ≃ -Rational.from_integer a / Rational.from_integer b
⊢ Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
⊢ Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | have : x ≃ (-a) / b := x_eqv_neg_a_over_b | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b : x ≃ -Rational.from_integer a / Rational.from_integer b
⊢ Negative x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this : x ≃ -Rational.from_integer a / Rational.from_integer b
⊢ Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b : x ≃ -Rational.from_integer a / Rational.from_integer b
⊢ Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | have : AltPositive ((a : ℚ) / b) := AltPositive.mk Rational.eqv_refl | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this : x ≃ -Rational.from_integer a / Rational.from_integer b
⊢ Negative x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝ : x ≃ -Rational.from_integer a / Rational.from_integer b
this : AltPositive (Rational.from_integer a / Rational.from_integer b)
⊢ Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this : x ≃ -Rational.from_integer a / Rational.from_integer b
⊢ Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | have : Positive ((a : ℚ) / b) := alt_positive.mpr this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝ : x ≃ -Rational.from_integer a / Rational.from_integer b
this : AltPositive (Rational.from_integer a / Rational.from_integer b)
⊢ Negative x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝¹ : x ≃ -Rational.from_integer a / Rational.from_integer b
this✝ : AltPositive (Rational.from_integer a / Rational.from_integer b)
this : Positive (Rational.from_integer a / Rational.from_integer b)
⊢ Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝ : x ≃ -Rational.from_integer a / Rational.from_integer b
this : AltPositive (Rational.from_integer a / Rational.from_integer b)
⊢ Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | have : x ≃ -((a : ℚ) / b) := calc
x ≃ _ := ‹x ≃ (-a) / b›
(-a : ℚ) / b ≃ _ := Rational.eqv_symm Rational.neg_scompatL_div
(-((a : ℚ) / b)) ≃ _ := Rational.eqv_refl | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝¹ : x ≃ -Rational.from_integer a / Rational.from_integer b
this✝ : AltPositive (Rational.from_integer a / Rational.from_integer b)
this : Positive (Rational.from_integer a / Rational.from_integer b)
⊢ Negative x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝² : x ≃ -Rational.from_integer a / Rational.from_integer b
this✝¹ : AltPositive (Rational.from_integer a / Rational.from_integer b)
this✝ : Positive (Rational.from_integer a / Rational.from_integer b)
this : x ≃ -(Rational.from_integer a / Rational.from_integer b)
⊢ Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝¹ : x ≃ -Rational.from_integer a / Rational.from_integer b
this✝ : AltPositive (Rational.from_integer a / Rational.from_integer b)
this : Positive (Rational.from_integer a / Rational.from_integer b)
⊢ Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | have : ∃ (y : ℚ), Positive y ∧ x ≃ -y :=
Exists.intro ((a : ℚ) / b) (And.intro ‹Positive ((a : ℚ) / b)› this) | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝² : x ≃ -Rational.from_integer a / Rational.from_integer b
this✝¹ : AltPositive (Rational.from_integer a / Rational.from_integer b)
this✝ : Positive (Rational.from_integer a / Rational.from_integer b)
this : x ≃ -(Rational.from_integer a / Rational.from_integer b)
⊢ Negative x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝³ : x ≃ -Rational.from_integer a / Rational.from_integer b
this✝² : AltPositive (Rational.from_integer a / Rational.from_integer b)
this✝¹ : Positive (Rational.from_integer a / Rational.from_integer b)
this✝ : x ≃ -(Rational.from_integer a / Rational.from_integer b)
this : ∃ y, Positive y ∧ x ≃ -y
⊢ Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝² : x ≃ -Rational.from_integer a / Rational.from_integer b
this✝¹ : AltPositive (Rational.from_integer a / Rational.from_integer b)
this✝ : Positive (Rational.from_integer a / Rational.from_integer b)
this : x ≃ -(Rational.from_integer a / Rational.from_integer b)
⊢ Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | have : Negative x := negative_iff_neg_positive.mpr this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝³ : x ≃ -Rational.from_integer a / Rational.from_integer b
this✝² : AltPositive (Rational.from_integer a / Rational.from_integer b)
this✝¹ : Positive (Rational.from_integer a / Rational.from_integer b)
this✝ : x ≃ -(Rational.from_integer a / Rational.from_integer b)
this : ∃ y, Positive y ∧ x ≃ -y
⊢ Negative x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝⁴ : x ≃ -Rational.from_integer a / Rational.from_integer b
this✝³ : AltPositive (Rational.from_integer a / Rational.from_integer b)
this✝² : Positive (Rational.from_integer a / Rational.from_integer b)
this✝¹ : x ≃ -(Rational.from_integer a / Rational.from_integer b)
this✝ : ∃ y, Positive y ∧ x ≃ -y
this : Negative x
⊢ Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝³ : x ≃ -Rational.from_integer a / Rational.from_integer b
this✝² : AltPositive (Rational.from_integer a / Rational.from_integer b)
this✝¹ : Positive (Rational.from_integer a / Rational.from_integer b)
this✝ : x ≃ -(Rational.from_integer a / Rational.from_integer b)
this : ∃ y, Positive y ∧ x ≃ -y
⊢ Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_negative | [518, 1] | [555, 15] | exact this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝⁴ : x ≃ -Rational.from_integer a / Rational.from_integer b
this✝³ : AltPositive (Rational.from_integer a / Rational.from_integer b)
this✝² : Positive (Rational.from_integer a / Rational.from_integer b)
this✝¹ : x ≃ -(Rational.from_integer a / Rational.from_integer b)
this✝ : ∃ y, Positive y ∧ x ≃ -y
this : Negative x
⊢ Negative x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltNegative x
a b : ℤ
a_pos✝ : AP (Positive a)
b_pos✝ : AP (Positive b)
x_eqv_neg_a_over_b this✝⁴ : x ≃ -Rational.from_integer a / Rational.from_integer b
this✝³ : AltPositive (Rational.from_integer a / Rational.from_integer b)
this✝² : Positive (Rational.from_integer a / Rational.from_integer b)
this✝¹ : x ≃ -(Rational.from_integer a / Rational.from_integer b)
this✝ : ∃ y, Positive y ∧ x ≃ -y
this : Negative x
⊢ Negative x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | both_inclusions | [53, 1] | [53, 61] | aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) R R' →
(fun R R' => ∀ (x y : A), R x y → R' x y) R' R → (fun R R' => ∀ (x y : A), R x y ↔ R' x y) R R' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) R R' →
(fun R R' => ∀ (x y : A), R x y → R' x y) R' R → (fun R R' => ∀ (x y : A), R x y ↔ R' x y) R R'
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_left | [55, 1] | [55, 47] | aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y ↔ R' x y) R R' → (fun R R' => ∀ (x y : A), R x y → R' x y) R R' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y ↔ R' x y) R R' → (fun R R' => ∀ (x y : A), R x y → R' x y) R R'
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | equiv_right | [57, 1] | [57, 48] | aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y ↔ R' x y) R R' → (fun R R' => ∀ (x y : A), R x y → R' x y) R' R | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y ↔ R' x y) R R' → (fun R R' => ∀ (x y : A), R x y → R' x y) R' R
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_monotone | [60, 1] | [64, 27] | intros le x y tr | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) R R' →
(fun R R' => ∀ (x y : A), R x y → R' x y) (trans_clos R) (trans_clos R') | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
le : ∀ (x y : A), R x y → R' x y
x y : A
tr : trans_clos R x y
⊢ trans_clos R' x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) R R' →
(fun R R' => ∀ (x y : A), R x y → R' x y) (trans_clos R) (trans_clos R')
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_monotone | [60, 1] | [64, 27] | induction tr <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
le : ∀ (x y : A), R x y → R' x y
x y : A
tr : trans_clos R x y
⊢ trans_clos R' x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
le : ∀ (x y : A), R x y → R' x y
x y : A
tr : trans_clos R x y
⊢ trans_clos R' x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | sym_clos_monotone | [66, 1] | [70, 28] | intros le x y sym | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) R R' → (fun R R' => ∀ (x y : A), R x y → R' x y) (sym_clos R) (sym_clos R') | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
le : ∀ (x y : A), R x y → R' x y
x y : A
sym : sym_clos R x y
⊢ sym_clos R' x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) R R' → (fun R R' => ∀ (x y : A), R x y → R' x y) (sym_clos R) (sym_clos R')
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | sym_clos_monotone | [66, 1] | [70, 28] | induction sym <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
le : ∀ (x y : A), R x y → R' x y
x y : A
sym : sym_clos R x y
⊢ sym_clos R' x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
le : ∀ (x y : A), R x y → R' x y
x y : A
sym : sym_clos R x y
⊢ sym_clos R' x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_clos_monotone | [72, 1] | [76, 29] | intros le x y refl | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) R R' →
(fun R R' => ∀ (x y : A), R x y → R' x y) (refl_clos R) (refl_clos R') | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
le : ∀ (x y : A), R x y → R' x y
x y : A
refl : refl_clos R x y
⊢ refl_clos R' x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) R R' →
(fun R R' => ∀ (x y : A), R x y → R' x y) (refl_clos R) (refl_clos R')
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_clos_monotone | [72, 1] | [76, 29] | induction refl <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
le : ∀ (x y : A), R x y → R' x y
x y : A
refl : refl_clos R x y
⊢ refl_clos R' x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
le : ∀ (x y : A), R x y → R' x y
x y : A
refl : refl_clos R x y
⊢ refl_clos R' x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_transitive | [79, 1] | [82, 31] | intros tr_x_y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
⊢ trans_clos R x y → trans_clos R y z → trans_clos R x z | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
tr_x_y : trans_clos R x y
⊢ trans_clos R y z → trans_clos R x z | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
⊢ trans_clos R x y → trans_clos R y z → trans_clos R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_clos_transitive | [79, 1] | [82, 31] | induction tr_x_y <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
tr_x_y : trans_clos R x y
⊢ trans_clos R y z → trans_clos R x z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
tr_x_y : trans_clos R x y
⊢ trans_clos R y z → trans_clos R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_clos_transitive | [84, 1] | [88, 31] | intros tr_x_y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
⊢ refl_trans_clos R x y → refl_trans_clos R y z → refl_trans_clos R x z | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
tr_x_y : refl_trans_clos R x y
⊢ refl_trans_clos R y z → refl_trans_clos R x z | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
⊢ refl_trans_clos R x y → refl_trans_clos R y z → refl_trans_clos R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_clos_transitive | [84, 1] | [88, 31] | induction tr_x_y <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
tr_x_y : refl_trans_clos R x y
⊢ refl_trans_clos R y z → refl_trans_clos R x z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y z : A
tr_x_y : refl_trans_clos R x y
⊢ refl_trans_clos R y z → refl_trans_clos R x z
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | apply both_inclusions <;> intros x y hyp | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ (fun R R' => ∀ (x y : A), R x y ↔ R' x y) (refl_clos (trans_clos R)) (trans_clos (refl_clos R)) | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
hyp : refl_clos (trans_clos R) x y
⊢ trans_clos (refl_clos R) x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
hyp : trans_clos (refl_clos R) x y
⊢ refl_clos (trans_clos R) x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ (fun R R' => ∀ (x y : A), R x y ↔ R' x y) (refl_clos (trans_clos R)) (trans_clos (refl_clos R))
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | . cases hyp
. aesop
. apply trans_clos_monotone R <;> aesop | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
hyp : refl_clos (trans_clos R) x y
⊢ trans_clos (refl_clos R) x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
hyp : trans_clos (refl_clos R) x y
⊢ refl_clos (trans_clos R) x y | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
hyp : trans_clos (refl_clos R) x y
⊢ refl_clos (trans_clos R) x y | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
hyp : refl_clos (trans_clos R) x y
⊢ trans_clos (refl_clos R) x y
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
hyp : trans_clos (refl_clos R) x y
⊢ refl_clos (trans_clos R) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | . induction' hyp with _ _ rc _ _ _ rc step ih
. cases rc <;> aesop
. cases rc; trivial
apply refl_clos.base; aesop | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
hyp : trans_clos (refl_clos R) x y
⊢ refl_clos (trans_clos R) x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
hyp : trans_clos (refl_clos R) x y
⊢ refl_clos (trans_clos R) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | cases hyp | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
hyp : refl_clos (trans_clos R) x y
⊢ trans_clos (refl_clos R) x y | case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
⊢ trans_clos (refl_clos R) x x
case a.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
a✝ : trans_clos R x y
⊢ trans_clos (refl_clos R) x y | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
hyp : refl_clos (trans_clos R) x y
⊢ trans_clos (refl_clos R) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | . aesop | case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
⊢ trans_clos (refl_clos R) x x
case a.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
a✝ : trans_clos R x y
⊢ trans_clos (refl_clos R) x y | case a.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
a✝ : trans_clos R x y
⊢ trans_clos (refl_clos R) x y | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
⊢ trans_clos (refl_clos R) x x
case a.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
a✝ : trans_clos R x y
⊢ trans_clos (refl_clos R) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | . apply trans_clos_monotone R <;> aesop | case a.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
a✝ : trans_clos R x y
⊢ trans_clos (refl_clos R) x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
a✝ : trans_clos R x y
⊢ trans_clos (refl_clos R) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | aesop | case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
⊢ trans_clos (refl_clos R) x x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
⊢ trans_clos (refl_clos R) x x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | apply trans_clos_monotone R <;> aesop | case a.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
a✝ : trans_clos R x y
⊢ trans_clos (refl_clos R) x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
a✝ : trans_clos R x y
⊢ trans_clos (refl_clos R) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | induction' hyp with _ _ rc _ _ _ rc step ih | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
hyp : trans_clos (refl_clos R) x y
⊢ refl_clos (trans_clos R) x y | case a.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ b✝ : A
rc : refl_clos R a✝ b✝
⊢ refl_clos (trans_clos R) a✝ b✝
case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ b✝ c✝ : A
rc : refl_clos R a✝ b✝
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
⊢ refl_clos (trans_clos R) a✝ c✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
hyp : trans_clos (refl_clos R) x y
⊢ refl_clos (trans_clos R) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | . cases rc <;> aesop | case a.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ b✝ : A
rc : refl_clos R a✝ b✝
⊢ refl_clos (trans_clos R) a✝ b✝
case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ b✝ c✝ : A
rc : refl_clos R a✝ b✝
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
⊢ refl_clos (trans_clos R) a✝ c✝ | case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ b✝ c✝ : A
rc : refl_clos R a✝ b✝
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
⊢ refl_clos (trans_clos R) a✝ c✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ b✝ : A
rc : refl_clos R a✝ b✝
⊢ refl_clos (trans_clos R) a✝ b✝
case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ b✝ c✝ : A
rc : refl_clos R a✝ b✝
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
⊢ refl_clos (trans_clos R) a✝ c✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | . cases rc; trivial
apply refl_clos.base; aesop | case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ b✝ c✝ : A
rc : refl_clos R a✝ b✝
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
⊢ refl_clos (trans_clos R) a✝ c✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ b✝ c✝ : A
rc : refl_clos R a✝ b✝
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
⊢ refl_clos (trans_clos R) a✝ c✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | cases rc <;> aesop | case a.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ b✝ : A
rc : refl_clos R a✝ b✝
⊢ refl_clos (trans_clos R) a✝ b✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ b✝ : A
rc : refl_clos R a✝ b✝
⊢ refl_clos (trans_clos R) a✝ b✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | cases rc | case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ b✝ c✝ : A
rc : refl_clos R a✝ b✝
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
⊢ refl_clos (trans_clos R) a✝ c✝ | case a.step.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ c✝ : A
step : trans_clos (refl_clos R) a✝ c✝
ih : refl_clos (trans_clos R) a✝ c✝
⊢ refl_clos (trans_clos R) a✝ c✝
case a.step.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝¹ b✝ c✝ : A
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
a✝ : R a✝¹ b✝
⊢ refl_clos (trans_clos R) a✝¹ c✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ b✝ c✝ : A
rc : refl_clos R a✝ b✝
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
⊢ refl_clos (trans_clos R) a✝ c✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | trivial | case a.step.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ c✝ : A
step : trans_clos (refl_clos R) a✝ c✝
ih : refl_clos (trans_clos R) a✝ c✝
⊢ refl_clos (trans_clos R) a✝ c✝
case a.step.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝¹ b✝ c✝ : A
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
a✝ : R a✝¹ b✝
⊢ refl_clos (trans_clos R) a✝¹ c✝ | case a.step.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝¹ b✝ c✝ : A
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
a✝ : R a✝¹ b✝
⊢ refl_clos (trans_clos R) a✝¹ c✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝ c✝ : A
step : trans_clos (refl_clos R) a✝ c✝
ih : refl_clos (trans_clos R) a✝ c✝
⊢ refl_clos (trans_clos R) a✝ c✝
case a.step.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝¹ b✝ c✝ : A
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
a✝ : R a✝¹ b✝
⊢ refl_clos (trans_clos R) a✝¹ c✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | apply refl_clos.base | case a.step.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝¹ b✝ c✝ : A
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
a✝ : R a✝¹ b✝
⊢ refl_clos (trans_clos R) a✝¹ c✝ | case a.step.base.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝¹ b✝ c✝ : A
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
a✝ : R a✝¹ b✝
⊢ trans_clos R a✝¹ c✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.base
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝¹ b✝ c✝ : A
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
a✝ : R a✝¹ b✝
⊢ refl_clos (trans_clos R) a✝¹ c✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_refl | [91, 1] | [102, 36] | aesop | case a.step.base.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝¹ b✝ c✝ : A
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
a✝ : R a✝¹ b✝
⊢ trans_clos R a✝¹ c✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.step.base.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y a✝¹ b✝ c✝ : A
step : trans_clos (refl_clos R) b✝ c✝
ih : refl_clos (trans_clos R) b✝ c✝
a✝ : R a✝¹ b✝
⊢ trans_clos R a✝¹ c✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_sym_is_sym_refl | [104, 1] | [109, 55] | apply both_inclusions <;> intros x y hyp <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ (fun R R' => ∀ (x y : A), R x y ↔ R' x y) (refl_clos (sym_clos R)) (sym_clos (refl_clos R)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ (fun R R' => ∀ (x y : A), R x y ↔ R' x y) (refl_clos (sym_clos R)) (sym_clos (refl_clos R))
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_clos_monotone | [111, 1] | [115, 29] | intros le x y refl | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) R R' →
(fun R R' => ∀ (x y : A), R x y → R' x y) (refl_trans_clos R) (refl_trans_clos R') | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
le : ∀ (x y : A), R x y → R' x y
x y : A
refl : refl_trans_clos R x y
⊢ refl_trans_clos R' x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) R R' →
(fun R R' => ∀ (x y : A), R x y → R' x y) (refl_trans_clos R) (refl_trans_clos R')
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_clos_monotone | [111, 1] | [115, 29] | induction refl <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
le : ∀ (x y : A), R x y → R' x y
x y : A
refl : refl_trans_clos R x y
⊢ refl_trans_clos R' x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
R' : A → A → Prop
le : ∀ (x y : A), R x y → R' x y
x y : A
refl : refl_trans_clos R x y
⊢ refl_trans_clos R' x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | apply both_inclusions <;> intros | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ (fun R R' => ∀ (x y : A), R x y ↔ R' x y) (refl_clos (trans_clos (sym_clos R))) (trans_clos (sym_clos (refl_clos R))) | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ trans_clos (sym_clos (refl_clos R)) x✝ y✝
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ refl_clos (trans_clos (sym_clos R)) x✝ y✝ | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ (fun R R' => ∀ (x y : A), R x y ↔ R' x y) (refl_clos (trans_clos (sym_clos R))) (trans_clos (sym_clos (refl_clos R)))
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | . apply trans_clos_monotone
. intros
apply (refl_sym_is_sym_refl _ _ _).1
trivial
. apply (refl_trans_is_trans_refl _ _ _).1
trivial | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ trans_clos (sym_clos (refl_clos R)) x✝ y✝
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ refl_clos (trans_clos (sym_clos R)) x✝ y✝ | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ refl_clos (trans_clos (sym_clos R)) x✝ y✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ trans_clos (sym_clos (refl_clos R)) x✝ y✝
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ refl_clos (trans_clos (sym_clos R)) x✝ y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | . apply (refl_trans_is_trans_refl _ _ _).2
apply trans_clos_monotone
. intros x y red
apply (refl_sym_is_sym_refl _ _ _).2
trivial
. trivial | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ refl_clos (trans_clos (sym_clos R)) x✝ y✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ refl_clos (trans_clos (sym_clos R)) x✝ y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | apply trans_clos_monotone | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ trans_clos (sym_clos (refl_clos R)) x✝ y✝ | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ ∀ (x y : A), ?a.R✝ x y → sym_clos (refl_clos R) x y
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ trans_clos ?a.R✝ x✝ y✝
case a.R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ A → A → Prop | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ trans_clos (sym_clos (refl_clos R)) x✝ y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | . intros
apply (refl_sym_is_sym_refl _ _ _).1
trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ ∀ (x y : A), ?a.R✝ x y → sym_clos (refl_clos R) x y
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ trans_clos ?a.R✝ x✝ y✝
case a.R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ A → A → Prop | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ trans_clos (fun x y => refl_clos (sym_clos R) x y) x✝ y✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ ∀ (x y : A), ?a.R✝ x y → sym_clos (refl_clos R) x y
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ trans_clos ?a.R✝ x✝ y✝
case a.R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ A → A → Prop
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | . apply (refl_trans_is_trans_refl _ _ _).1
trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ trans_clos (fun x y => refl_clos (sym_clos R) x y) x✝ y✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ trans_clos (fun x y => refl_clos (sym_clos R) x y) x✝ y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | intros | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ ∀ (x y : A), ?a.R✝ x y → sym_clos (refl_clos R) x y | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝¹ y✝¹ : A
a✝¹ : refl_clos (trans_clos (sym_clos R)) x✝¹ y✝¹
x✝ y✝ : A
a✝ : ?a.R✝ x✝ y✝
⊢ sym_clos (refl_clos R) x✝ y✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ ∀ (x y : A), ?a.R✝ x y → sym_clos (refl_clos R) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | apply (refl_sym_is_sym_refl _ _ _).1 | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝¹ y✝¹ : A
a✝¹ : refl_clos (trans_clos (sym_clos R)) x✝¹ y✝¹
x✝ y✝ : A
a✝ : ?a.R✝ x✝ y✝
⊢ sym_clos (refl_clos R) x✝ y✝ | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝¹ y✝¹ : A
a✝¹ : refl_clos (trans_clos (sym_clos R)) x✝¹ y✝¹
x✝ y✝ : A
a✝ : ?a.R✝ x✝ y✝
⊢ refl_clos (sym_clos R) x✝ y✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝¹ y✝¹ : A
a✝¹ : refl_clos (trans_clos (sym_clos R)) x✝¹ y✝¹
x✝ y✝ : A
a✝ : ?a.R✝ x✝ y✝
⊢ sym_clos (refl_clos R) x✝ y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝¹ y✝¹ : A
a✝¹ : refl_clos (trans_clos (sym_clos R)) x✝¹ y✝¹
x✝ y✝ : A
a✝ : ?a.R✝ x✝ y✝
⊢ refl_clos (sym_clos R) x✝ y✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝¹ y✝¹ : A
a✝¹ : refl_clos (trans_clos (sym_clos R)) x✝¹ y✝¹
x✝ y✝ : A
a✝ : ?a.R✝ x✝ y✝
⊢ refl_clos (sym_clos R) x✝ y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | apply (refl_trans_is_trans_refl _ _ _).1 | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ trans_clos (fun x y => refl_clos (sym_clos R) x y) x✝ y✝ | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ refl_clos (trans_clos (sym_clos R)) x✝ y✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ trans_clos (fun x y => refl_clos (sym_clos R) x y) x✝ y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ refl_clos (trans_clos (sym_clos R)) x✝ y✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : refl_clos (trans_clos (sym_clos R)) x✝ y✝
⊢ refl_clos (trans_clos (sym_clos R)) x✝ y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | apply (refl_trans_is_trans_refl _ _ _).2 | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ refl_clos (trans_clos (sym_clos R)) x✝ y✝ | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ trans_clos (refl_clos (sym_clos R)) x✝ y✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ refl_clos (trans_clos (sym_clos R)) x✝ y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | apply trans_clos_monotone | case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ trans_clos (refl_clos (sym_clos R)) x✝ y✝ | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ ∀ (x y : A), ?a.R✝ x y → refl_clos (sym_clos R) x y
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ trans_clos ?a.R✝ x✝ y✝
case a.R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ A → A → Prop | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ trans_clos (refl_clos (sym_clos R)) x✝ y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | . intros x y red
apply (refl_sym_is_sym_refl _ _ _).2
trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ ∀ (x y : A), ?a.R✝ x y → refl_clos (sym_clos R) x y
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ trans_clos ?a.R✝ x✝ y✝
case a.R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ A → A → Prop | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ trans_clos (fun x y => sym_clos (refl_clos R) x y) x✝ y✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ ∀ (x y : A), ?a.R✝ x y → refl_clos (sym_clos R) x y
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ trans_clos ?a.R✝ x✝ y✝
case a.R
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ A → A → Prop
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | . trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ trans_clos (fun x y => sym_clos (refl_clos R) x y) x✝ y✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ trans_clos (fun x y => sym_clos (refl_clos R) x y) x✝ y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | intros x y red | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ ∀ (x y : A), ?a.R✝ x y → refl_clos (sym_clos R) x y | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
x y : A
red : ?a.R✝ x y
⊢ refl_clos (sym_clos R) x y | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ ∀ (x y : A), ?a.R✝ x y → refl_clos (sym_clos R) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | apply (refl_sym_is_sym_refl _ _ _).2 | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
x y : A
red : ?a.R✝ x y
⊢ refl_clos (sym_clos R) x y | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
x y : A
red : ?a.R✝ x y
⊢ sym_clos (refl_clos R) x y | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
x y : A
red : ?a.R✝ x y
⊢ refl_clos (sym_clos R) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
x y : A
red : ?a.R✝ x y
⊢ sym_clos (refl_clos R) x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
x y : A
red : ?a.R✝ x y
⊢ sym_clos (refl_clos R) x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_sym_is_trans_sym_refl | [117, 1] | [132, 18] | trivial | case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ trans_clos (fun x y => sym_clos (refl_clos R) x y) x✝ y✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x✝ y✝ : A
a✝ : trans_clos (sym_clos (refl_clos R)) x✝ y✝
⊢ trans_clos (fun x y => sym_clos (refl_clos R) x y) x✝ y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_is_refl_trans | [134, 1] | [137, 26] | intros red | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
⊢ trans_clos R x y → refl_trans_clos R x y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
red : trans_clos R x y
⊢ refl_trans_clos R x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
⊢ trans_clos R x y → refl_trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | trans_is_refl_trans | [134, 1] | [137, 26] | induction red <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
red : trans_clos R x y
⊢ refl_trans_clos R x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
red : trans_clos R x y
⊢ refl_trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_or_eq | [139, 1] | [142, 26] | intros red | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
⊢ refl_trans_clos R x y → x = y ∨ trans_clos R x y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
red : refl_trans_clos R x y
⊢ x = y ∨ trans_clos R x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
⊢ refl_trans_clos R x y → x = y ∨ trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_is_trans_or_eq | [139, 1] | [142, 26] | induction red <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
red : refl_trans_clos R x y
⊢ x = y ∨ trans_clos R x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
red : refl_trans_clos R x y
⊢ x = y ∨ trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_step_is_trans | [144, 1] | [148, 30] | intros red_x_z red_z_y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
⊢ R x z → refl_trans_clos R z y → trans_clos R x y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
red_x_z : R x z
red_z_y : refl_trans_clos R z y
⊢ trans_clos R x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
⊢ R x z → refl_trans_clos R z y → trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_step_is_trans | [144, 1] | [148, 30] | revert red_x_z x | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
red_x_z : R x z
red_z_y : refl_trans_clos R z y
⊢ trans_clos R x y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
z y : A
red_z_y : refl_trans_clos R z y
⊢ ∀ {x : A}, R x z → trans_clos R x y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x z y : A
red_x_z : R x z
red_z_y : refl_trans_clos R z y
⊢ trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | refl_trans_step_is_trans | [144, 1] | [148, 30] | induction red_z_y <;> aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
z y : A
red_z_y : refl_trans_clos R z y
⊢ ∀ {x : A}, R x z → trans_clos R x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
z y : A
red_z_y : refl_trans_clos R z y
⊢ ∀ {x : A}, R x z → trans_clos R x y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | sym_inv | [150, 1] | [151, 45] | aesop | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) ((fun R x y => R y x) R) (sym_clos R) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ (fun R R' => ∀ (x y : A), R x y → R' x y) ((fun R x y => R y x) R) (sym_clos R)
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normal_normalizing | [162, 1] | [170, 20] | simp [normal] | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
⊢ normal R x → normalizes R x | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
⊢ (∀ (x_1 : A), ¬R x x_1) → normalizes R x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
⊢ normal R x → normalizes R x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normal_normalizing | [162, 1] | [170, 20] | intros norm_x div_x | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
⊢ (∀ (x_1 : A), ¬R x x_1) → normalizes R x | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm_x : ∀ (x_1 : A), ¬R x x_1
div_x : diverge_from R x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
⊢ (∀ (x_1 : A), ¬R x x_1) → normalizes R x
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normal_normalizing | [162, 1] | [170, 20] | match div_x with
| Exists.intro xs (And.intro zero_case incr_case) =>
apply norm_x (xs 1)
rw [←zero_case]
apply incr_case | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm_x : ∀ (x_1 : A), ¬R x x_1
div_x : diverge_from R x
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm_x : ∀ (x_1 : A), ¬R x x_1
div_x : diverge_from R x
⊢ False
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normal_normalizing | [162, 1] | [170, 20] | apply norm_x (xs 1) | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm_x : ∀ (x_1 : A), ¬R x x_1
div_x : diverge_from R x
xs : ℕ → A
zero_case : xs 0 = x
incr_case : ∀ (n : ℕ), R (xs n) (xs (n + 1))
⊢ False | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm_x : ∀ (x_1 : A), ¬R x x_1
div_x : diverge_from R x
xs : ℕ → A
zero_case : xs 0 = x
incr_case : ∀ (n : ℕ), R (xs n) (xs (n + 1))
⊢ R x (xs 1) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm_x : ∀ (x_1 : A), ¬R x x_1
div_x : diverge_from R x
xs : ℕ → A
zero_case : xs 0 = x
incr_case : ∀ (n : ℕ), R (xs n) (xs (n + 1))
⊢ False
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normal_normalizing | [162, 1] | [170, 20] | rw [←zero_case] | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm_x : ∀ (x_1 : A), ¬R x x_1
div_x : diverge_from R x
xs : ℕ → A
zero_case : xs 0 = x
incr_case : ∀ (n : ℕ), R (xs n) (xs (n + 1))
⊢ R x (xs 1) | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm_x : ∀ (x_1 : A), ¬R x x_1
div_x : diverge_from R x
xs : ℕ → A
zero_case : xs 0 = x
incr_case : ∀ (n : ℕ), R (xs n) (xs (n + 1))
⊢ R (xs 0) (xs 1) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm_x : ∀ (x_1 : A), ¬R x x_1
div_x : diverge_from R x
xs : ℕ → A
zero_case : xs 0 = x
incr_case : ∀ (n : ℕ), R (xs n) (xs (n + 1))
⊢ R x (xs 1)
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normal_normalizing | [162, 1] | [170, 20] | apply incr_case | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm_x : ∀ (x_1 : A), ¬R x x_1
div_x : diverge_from R x
xs : ℕ → A
zero_case : xs 0 = x
incr_case : ∀ (n : ℕ), R (xs n) (xs (n + 1))
⊢ R (xs 0) (xs 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm_x : ∀ (x_1 : A), ¬R x x_1
div_x : diverge_from R x
xs : ℕ → A
zero_case : xs 0 = x
incr_case : ∀ (n : ℕ), R (xs n) (xs (n + 1))
⊢ R (xs 0) (xs 1)
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normal_red | [173, 1] | [179, 23] | intros norm red | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
⊢ normal R x → refl_trans_clos R x y → x = y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
norm : normal R x
red : refl_trans_clos R x y
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
⊢ normal R x → refl_trans_clos R x y → x = y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normal_red | [173, 1] | [179, 23] | cases red | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
norm : normal R x
red : refl_trans_clos R x y
⊢ x = y | case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm : normal R x
⊢ x = x
case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
norm : normal R x
b✝ : A
a✝¹ : R x b✝
a✝ : refl_trans_clos R b✝ y
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
norm : normal R x
red : refl_trans_clos R x y
⊢ x = y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normal_red | [173, 1] | [179, 23] | trivial | case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm : normal R x
⊢ x = x
case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
norm : normal R x
b✝ : A
a✝¹ : R x b✝
a✝ : refl_trans_clos R b✝ y
⊢ x = y | case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
norm : normal R x
b✝ : A
a✝¹ : R x b✝
a✝ : refl_trans_clos R b✝ y
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case refl
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x : A
norm : normal R x
⊢ x = x
case step
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y : A
norm : normal R x
b✝ : A
a✝¹ : R x b✝
a✝ : refl_trans_clos R b✝ y
⊢ x = y
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normal_red | [173, 1] | [179, 23] | case step y _ _ =>
by_contra
apply norm; exists y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y✝ : A
norm : normal R x
y : A
a✝¹ : R x y
a✝ : refl_trans_clos R y y✝
⊢ x = y✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y✝ : A
norm : normal R x
y : A
a✝¹ : R x y
a✝ : refl_trans_clos R y y✝
⊢ x = y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normal_red | [173, 1] | [179, 23] | by_contra | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y✝ : A
norm : normal R x
y : A
a✝¹ : R x y
a✝ : refl_trans_clos R y y✝
⊢ x = y✝ | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y✝ : A
norm : normal R x
y : A
a✝¹ : R x y
a✝ : refl_trans_clos R y y✝
x✝ : ¬x = y✝
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y✝ : A
norm : normal R x
y : A
a✝¹ : R x y
a✝ : refl_trans_clos R y y✝
⊢ x = y✝
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normal_red | [173, 1] | [179, 23] | apply norm | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y✝ : A
norm : normal R x
y : A
a✝¹ : R x y
a✝ : refl_trans_clos R y y✝
x✝ : ¬x = y✝
⊢ False | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y✝ : A
norm : normal R x
y : A
a✝¹ : R x y
a✝ : refl_trans_clos R y y✝
x✝ : ¬x = y✝
⊢ ∃ x', R x x' | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y✝ : A
norm : normal R x
y : A
a✝¹ : R x y
a✝ : refl_trans_clos R y y✝
x✝ : ¬x = y✝
⊢ False
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | normal_red | [173, 1] | [179, 23] | exists y | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y✝ : A
norm : normal R x
y : A
a✝¹ : R x y
a✝ : refl_trans_clos R y y✝
x✝ : ¬x = y✝
⊢ ∃ x', R x x' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
x y✝ : A
norm : normal R x
y : A
a✝¹ : R x y
a✝ : refl_trans_clos R y y✝
x✝ : ¬x = y✝
⊢ ∃ x', R x x'
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | exists_iff | [183, 1] | [193, 13] | intros T P Q equiv ex_t | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ ∀ (T : Sort u_1) (P Q : T → Prop), (∀ (t : T), Q t ↔ P t) → (∃ t, P t) → ∃ t, Q t | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
T : Sort u_1
P Q : T → Prop
equiv : ∀ (t : T), Q t ↔ P t
ex_t : ∃ t, P t
⊢ ∃ t, Q t | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
⊢ ∀ (T : Sort u_1) (P Q : T → Prop), (∀ (t : T), Q t ↔ P t) → (∃ t, P t) → ∃ t, Q t
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | exists_iff | [183, 1] | [193, 13] | match ex_t with
| Exists.intro w h =>
exists w
simp [equiv]
exact h | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
T : Sort u_1
P Q : T → Prop
equiv : ∀ (t : T), Q t ↔ P t
ex_t : ∃ t, P t
⊢ ∃ t, Q t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
T : Sort u_1
P Q : T → Prop
equiv : ∀ (t : T), Q t ↔ P t
ex_t : ∃ t, P t
⊢ ∃ t, Q t
TACTIC:
|
https://github.com/codyroux/traat-lean.git | f2babab84f81d4003446f476790022ac175d7236 | Traat/chapter1.lean | exists_iff | [183, 1] | [193, 13] | exists w | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
T : Sort u_1
P Q : T → Prop
equiv : ∀ (t : T), Q t ↔ P t
ex_t : ∃ t, P t
w : T
h : (fun t => P t) w
⊢ ∃ t, Q t | A : Type
R : A → A → Prop
inhabited_A : Nonempty A
T : Sort u_1
P Q : T → Prop
equiv : ∀ (t : T), Q t ↔ P t
ex_t : ∃ t, P t
w : T
h : (fun t => P t) w
⊢ Q w | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
T : Sort u_1
P Q : T → Prop
equiv : ∀ (t : T), Q t ↔ P t
ex_t : ∃ t, P t
w : T
h : (fun t => P t) w
⊢ ∃ t, Q t
TACTIC:
|
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