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/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.pos_iff_ex | [207, 1] | [230, 53] | exact Rel.trans ‹x ≃ 1 * coe n› AA.identL | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
x✝ : Positive x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ 1 * coe n
⊢ x ≃ coe n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
x✝ : Positive x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ 1 * coe n
⊢ x ≃ coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.pos_iff_ex | [207, 1] | [230, 53] | intro (Exists.intro (n : ℕ) (And.intro (_ : Positive n) (_ : x ≃ coe n))) | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
⊢ (∃ n, Positive n ∧ x ≃ coe n) → Positive x | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
n : ℕ
left✝ : Positive n
right✝ : x ≃ coe n
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
⊢ (∃ n, Positive n ∧ x ≃ coe n) → Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.pos_iff_ex | [207, 1] | [230, 53] | apply Impl.sign_props.positive_iff_sign_pos1.mpr | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
n : ℕ
left✝ : Positive n
right✝ : x ≃ coe n
⊢ Positive x | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
n : ℕ
left✝ : Positive n
right✝ : x ≃ coe n
⊢ Integer.NonzeroWithSign x 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
n : ℕ
left✝ : Positive n
right✝ : x ≃ coe n
⊢ Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.pos_iff_ex | [207, 1] | [230, 53] | apply Integer.NonzeroWithSign.intro n ‹Positive n› | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
n : ℕ
left✝ : Positive n
right✝ : x ≃ coe n
⊢ Integer.NonzeroWithSign x 1 | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
n : ℕ
left✝ : Positive n
right✝ : x ≃ coe n
⊢ x ≃ 1 * coe n | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
n : ℕ
left✝ : Positive n
right✝ : x ≃ coe n
⊢ Integer.NonzeroWithSign x 1
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.pos_iff_ex | [207, 1] | [230, 53] | exact Rel.trans ‹x ≃ coe n› (Rel.symm AA.identL) | x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
n : ℕ
left✝ : Positive n
right✝ : x ≃ coe n
⊢ x ≃ 1 * coe n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
mul_ident : AA.Identity (coe 1) fun x x_1 => x * x_1
n : ℕ
left✝ : Positive n
right✝ : x ≃ coe n
⊢ x ≃ 1 * coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.neg_iff_ex | [232, 1] | [254, 66] | apply Iff.intro | x : ℤ
⊢ Negative x ↔ ∃ n, Positive n ∧ x ≃ -coe n | case mp
x : ℤ
⊢ Negative x → ∃ n, Positive n ∧ x ≃ -coe n
case mpr
x : ℤ
⊢ (∃ n, Positive n ∧ x ≃ -coe n) → Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
⊢ Negative x ↔ ∃ n, Positive n ∧ x ≃ -coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.neg_iff_ex | [232, 1] | [254, 66] | case mp =>
intro (_ : Negative x)
show ∃ (n : ℕ), Positive n ∧ x ≃ -(coe n)
have (Integer.NonzeroWithSign.intro
(n : ℕ) (_ : Positive n) (_ : x ≃ -1 * coe n)) :=
Impl.sign_props.negative_iff_sign_neg1.mp ‹Negative x›
exists n
apply And.intro ‹Positive n›
show x ≃ -(coe n)
exact Rel.trans ‹x ≃ -1 * coe n› Integer.mul_neg_one | x : ℤ
⊢ Negative x → ∃ n, Positive n ∧ x ≃ -coe n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
⊢ Negative x → ∃ n, Positive n ∧ x ≃ -coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.neg_iff_ex | [232, 1] | [254, 66] | case mpr =>
intro
(Exists.intro (n : ℕ) (And.intro (_ : Positive n) (_ : x ≃ -(coe n))))
show Negative x
apply Impl.sign_props.negative_iff_sign_neg1.mpr
show Integer.NonzeroWithSign x (-1)
apply Integer.NonzeroWithSign.intro n ‹Positive n›
show x ≃ -1 * coe n
exact Rel.trans ‹x ≃ -(coe n)› (Rel.symm Integer.mul_neg_one) | x : ℤ
⊢ (∃ n, Positive n ∧ x ≃ -coe n) → Negative x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
⊢ (∃ n, Positive n ∧ x ≃ -coe n) → Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.neg_iff_ex | [232, 1] | [254, 66] | intro (_ : Negative x) | x : ℤ
⊢ Negative x → ∃ n, Positive n ∧ x ≃ -coe n | x : ℤ
x✝ : Negative x
⊢ ∃ n, Positive n ∧ x ≃ -coe n | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
⊢ Negative x → ∃ n, Positive n ∧ x ≃ -coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.neg_iff_ex | [232, 1] | [254, 66] | have (Integer.NonzeroWithSign.intro
(n : ℕ) (_ : Positive n) (_ : x ≃ -1 * coe n)) :=
Impl.sign_props.negative_iff_sign_neg1.mp ‹Negative x› | x : ℤ
x✝ : Negative x
⊢ ∃ n, Positive n ∧ x ≃ -coe n | x : ℤ
x✝ : Negative x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ -1 * coe n
⊢ ∃ n, Positive n ∧ x ≃ -coe n | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
x✝ : Negative x
⊢ ∃ n, Positive n ∧ x ≃ -coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.neg_iff_ex | [232, 1] | [254, 66] | exists n | x : ℤ
x✝ : Negative x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ -1 * coe n
⊢ ∃ n, Positive n ∧ x ≃ -coe n | x : ℤ
x✝ : Negative x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ -1 * coe n
⊢ Positive n ∧ x ≃ -coe n | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
x✝ : Negative x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ -1 * coe n
⊢ ∃ n, Positive n ∧ x ≃ -coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.neg_iff_ex | [232, 1] | [254, 66] | apply And.intro ‹Positive n› | x : ℤ
x✝ : Negative x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ -1 * coe n
⊢ Positive n ∧ x ≃ -coe n | x : ℤ
x✝ : Negative x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ -1 * coe n
⊢ x ≃ -coe n | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
x✝ : Negative x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ -1 * coe n
⊢ Positive n ∧ x ≃ -coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.neg_iff_ex | [232, 1] | [254, 66] | exact Rel.trans ‹x ≃ -1 * coe n› Integer.mul_neg_one | x : ℤ
x✝ : Negative x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ -1 * coe n
⊢ x ≃ -coe n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
x✝ : Negative x
n : ℕ
pos✝ : Positive n
eqv✝ : x ≃ -1 * coe n
⊢ x ≃ -coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.neg_iff_ex | [232, 1] | [254, 66] | intro
(Exists.intro (n : ℕ) (And.intro (_ : Positive n) (_ : x ≃ -(coe n)))) | x : ℤ
⊢ (∃ n, Positive n ∧ x ≃ -coe n) → Negative x | x : ℤ
n : ℕ
left✝ : Positive n
right✝ : x ≃ -coe n
⊢ Negative x | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
⊢ (∃ n, Positive n ∧ x ≃ -coe n) → Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.neg_iff_ex | [232, 1] | [254, 66] | apply Impl.sign_props.negative_iff_sign_neg1.mpr | x : ℤ
n : ℕ
left✝ : Positive n
right✝ : x ≃ -coe n
⊢ Negative x | x : ℤ
n : ℕ
left✝ : Positive n
right✝ : x ≃ -coe n
⊢ Integer.NonzeroWithSign x (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
n : ℕ
left✝ : Positive n
right✝ : x ≃ -coe n
⊢ Negative x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.neg_iff_ex | [232, 1] | [254, 66] | apply Integer.NonzeroWithSign.intro n ‹Positive n› | x : ℤ
n : ℕ
left✝ : Positive n
right✝ : x ≃ -coe n
⊢ Integer.NonzeroWithSign x (-1) | x : ℤ
n : ℕ
left✝ : Positive n
right✝ : x ≃ -coe n
⊢ x ≃ -1 * coe n | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
n : ℕ
left✝ : Positive n
right✝ : x ≃ -coe n
⊢ Integer.NonzeroWithSign x (-1)
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.neg_iff_ex | [232, 1] | [254, 66] | exact Rel.trans ‹x ≃ -(coe n)› (Rel.symm Integer.mul_neg_one) | x : ℤ
n : ℕ
left✝ : Positive n
right✝ : x ≃ -coe n
⊢ x ≃ -1 * coe n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
n : ℕ
left✝ : Positive n
right✝ : x ≃ -coe n
⊢ x ≃ -1 * coe n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.gt_iff_ge_neqv | [413, 1] | [428, 36] | apply Iff.intro | n m : ℤ
⊢ n > m ↔ n ≥ m ∧ n ≄ m | case mp
n m : ℤ
⊢ n > m → n ≥ m ∧ n ≄ m
case mpr
n m : ℤ
⊢ n ≥ m ∧ n ≄ m → n > m | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℤ
⊢ n > m ↔ n ≥ m ∧ n ≄ m
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.gt_iff_ge_neqv | [413, 1] | [428, 36] | case mp =>
intro (_ : n > m)
show n ≥ m ∧ n ≄ m
have (And.intro (_ : n ≥ m) (_ : m ≄ n)) :=
Integer.lt_iff_le_neqv.mp ‹n > m›
have : n ≄ m := Rel.symm ‹m ≄ n›
exact And.intro ‹n ≥ m› ‹n ≄ m› | n m : ℤ
⊢ n > m → n ≥ m ∧ n ≄ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℤ
⊢ n > m → n ≥ m ∧ n ≄ m
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.gt_iff_ge_neqv | [413, 1] | [428, 36] | case mpr =>
intro (And.intro (_ : n ≥ m) (_ : n ≄ m))
show n > m
apply Integer.lt_iff_le_neqv.mpr
show n ≥ m ∧ m ≄ n
have : m ≄ n := Rel.symm ‹n ≄ m›
exact And.intro ‹n ≥ m› ‹m ≄ n› | n m : ℤ
⊢ n ≥ m ∧ n ≄ m → n > m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℤ
⊢ n ≥ m ∧ n ≄ m → n > m
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.gt_iff_ge_neqv | [413, 1] | [428, 36] | intro (_ : n > m) | n m : ℤ
⊢ n > m → n ≥ m ∧ n ≄ m | n m : ℤ
x✝ : n > m
⊢ n ≥ m ∧ n ≄ m | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℤ
⊢ n > m → n ≥ m ∧ n ≄ m
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.gt_iff_ge_neqv | [413, 1] | [428, 36] | have (And.intro (_ : n ≥ m) (_ : m ≄ n)) :=
Integer.lt_iff_le_neqv.mp ‹n > m› | n m : ℤ
x✝ : n > m
⊢ n ≥ m ∧ n ≄ m | n m : ℤ
x✝ : n > m
left✝ : n ≥ m
right✝ : m ≄ n
⊢ n ≥ m ∧ n ≄ m | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℤ
x✝ : n > m
⊢ n ≥ m ∧ n ≄ m
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.gt_iff_ge_neqv | [413, 1] | [428, 36] | have : n ≄ m := Rel.symm ‹m ≄ n› | n m : ℤ
x✝ : n > m
left✝ : n ≥ m
right✝ : m ≄ n
⊢ n ≥ m ∧ n ≄ m | n m : ℤ
x✝ : n > m
left✝ : n ≥ m
right✝ : m ≄ n
this : n ≄ m
⊢ n ≥ m ∧ n ≄ m | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℤ
x✝ : n > m
left✝ : n ≥ m
right✝ : m ≄ n
⊢ n ≥ m ∧ n ≄ m
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.gt_iff_ge_neqv | [413, 1] | [428, 36] | exact And.intro ‹n ≥ m› ‹n ≄ m› | n m : ℤ
x✝ : n > m
left✝ : n ≥ m
right✝ : m ≄ n
this : n ≄ m
⊢ n ≥ m ∧ n ≄ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℤ
x✝ : n > m
left✝ : n ≥ m
right✝ : m ≄ n
this : n ≄ m
⊢ n ≥ m ∧ n ≄ m
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.gt_iff_ge_neqv | [413, 1] | [428, 36] | intro (And.intro (_ : n ≥ m) (_ : n ≄ m)) | n m : ℤ
⊢ n ≥ m ∧ n ≄ m → n > m | n m : ℤ
left✝ : n ≥ m
right✝ : n ≄ m
⊢ n > m | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℤ
⊢ n ≥ m ∧ n ≄ m → n > m
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.gt_iff_ge_neqv | [413, 1] | [428, 36] | apply Integer.lt_iff_le_neqv.mpr | n m : ℤ
left✝ : n ≥ m
right✝ : n ≄ m
⊢ n > m | n m : ℤ
left✝ : n ≥ m
right✝ : n ≄ m
⊢ m ≤ n ∧ m ≄ n | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℤ
left✝ : n ≥ m
right✝ : n ≄ m
⊢ n > m
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.gt_iff_ge_neqv | [413, 1] | [428, 36] | show n ≥ m ∧ m ≄ n | n m : ℤ
left✝ : n ≥ m
right✝ : n ≄ m
⊢ m ≤ n ∧ m ≄ n | n m : ℤ
left✝ : n ≥ m
right✝ : n ≄ m
⊢ n ≥ m ∧ m ≄ n | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℤ
left✝ : n ≥ m
right✝ : n ≄ m
⊢ m ≤ n ∧ m ≄ n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.gt_iff_ge_neqv | [413, 1] | [428, 36] | have : m ≄ n := Rel.symm ‹n ≄ m› | n m : ℤ
left✝ : n ≥ m
right✝ : n ≄ m
⊢ n ≥ m ∧ m ≄ n | n m : ℤ
left✝ : n ≥ m
right✝ : n ≄ m
this : m ≄ n
⊢ n ≥ m ∧ m ≄ n | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℤ
left✝ : n ≥ m
right✝ : n ≄ m
⊢ n ≥ m ∧ m ≄ n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec1Integers.lean | AnalysisI.Ch4.Sec1.gt_iff_ge_neqv | [413, 1] | [428, 36] | exact And.intro ‹n ≥ m› ‹m ≄ n› | n m : ℤ
left✝ : n ≥ m
right✝ : n ≄ m
this : m ≄ n
⊢ n ≥ m ∧ m ≄ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℤ
left✝ : n ≥ m
right✝ : n ≄ m
this : m ≄ n
⊢ n ≥ m ∧ m ≄ n
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.one_plus_m | [150, 1] | [155, 51] | calc
_ ≃ 1 + m := Rel.refl
_ ≃ step 0 + m := AA.substL Rel.refl
_ ≃ step (0 + m) := Natural.step_add
_ ≃ step m := AA.subst₁ Natural.zero_add | m : ℕ
⊢ 1 + m ≃ step m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
⊢ 1 + m ≃ step m
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | intro (_ : Positive q) | n q : ℕ
⊢ Positive q → Euclid n q | n q : ℕ
x✝ : Positive q
⊢ Euclid n q | Please generate a tactic in lean4 to solve the state.
STATE:
n q : ℕ
⊢ Positive q → Euclid n q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | have : 0 < q := Natural.lt_zero_pos.mp ‹Positive q› | n q : ℕ
x✝ : Positive q
⊢ Euclid n q | n q : ℕ
x✝ : Positive q
this : 0 < q
⊢ Euclid n q | Please generate a tactic in lean4 to solve the state.
STATE:
n q : ℕ
x✝ : Positive q
⊢ Euclid n q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | apply Natural.ind_on (motive := λ n => Euclid n q) n | n q : ℕ
x✝ : Positive q
this : 0 < q
⊢ Euclid n q | case zero
n q : ℕ
x✝ : Positive q
this : 0 < q
⊢ Euclid 0 q
case step
n q : ℕ
x✝ : Positive q
this : 0 < q
⊢ ∀ (m : ℕ), Euclid m q → Euclid (step m) q | Please generate a tactic in lean4 to solve the state.
STATE:
n q : ℕ
x✝ : Positive q
this : 0 < q
⊢ Euclid n q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | case zero =>
show Euclid 0 q
let m := 0
let r := 0
have r_bounded : r < q := ‹0 < q›
have n_divided : 0 ≃ m * q + r := calc
0 ≃ _ := Rel.symm Natural.zero_mul
0 * q ≃ _ := Rel.symm Natural.add_zero
0 * q + 0 ≃ _ := Rel.refl
m * q + r ≃ _ := Rel.refl
exact Euclid.intro m r r_bounded n_divided | n q : ℕ
x✝ : Positive q
this : 0 < q
⊢ Euclid 0 q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n q : ℕ
x✝ : Positive q
this : 0 < q
⊢ Euclid 0 q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | let m := 0 | n q : ℕ
x✝ : Positive q
this : 0 < q
⊢ Euclid 0 q | n q : ℕ
x✝ : Positive q
this : 0 < q
m : Nat := 0
⊢ Euclid 0 q | Please generate a tactic in lean4 to solve the state.
STATE:
n q : ℕ
x✝ : Positive q
this : 0 < q
⊢ Euclid 0 q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | let r := 0 | n q : ℕ
x✝ : Positive q
this : 0 < q
m : Nat := 0
⊢ Euclid 0 q | n q : ℕ
x✝ : Positive q
this : 0 < q
m : Nat := 0
r : Nat := 0
⊢ Euclid 0 q | Please generate a tactic in lean4 to solve the state.
STATE:
n q : ℕ
x✝ : Positive q
this : 0 < q
m : Nat := 0
⊢ Euclid 0 q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | have r_bounded : r < q := ‹0 < q› | n q : ℕ
x✝ : Positive q
this : 0 < q
m : Nat := 0
r : Nat := 0
⊢ Euclid 0 q | n q : ℕ
x✝ : Positive q
this : 0 < q
m : Nat := 0
r : Nat := 0
r_bounded : r < q
⊢ Euclid 0 q | Please generate a tactic in lean4 to solve the state.
STATE:
n q : ℕ
x✝ : Positive q
this : 0 < q
m : Nat := 0
r : Nat := 0
⊢ Euclid 0 q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | have n_divided : 0 ≃ m * q + r := calc
0 ≃ _ := Rel.symm Natural.zero_mul
0 * q ≃ _ := Rel.symm Natural.add_zero
0 * q + 0 ≃ _ := Rel.refl
m * q + r ≃ _ := Rel.refl | n q : ℕ
x✝ : Positive q
this : 0 < q
m : Nat := 0
r : Nat := 0
r_bounded : r < q
⊢ Euclid 0 q | n q : ℕ
x✝ : Positive q
this : 0 < q
m : Nat := 0
r : Nat := 0
r_bounded : r < q
n_divided : 0 ≃ m * q + r
⊢ Euclid 0 q | Please generate a tactic in lean4 to solve the state.
STATE:
n q : ℕ
x✝ : Positive q
this : 0 < q
m : Nat := 0
r : Nat := 0
r_bounded : r < q
⊢ Euclid 0 q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | exact Euclid.intro m r r_bounded n_divided | n q : ℕ
x✝ : Positive q
this : 0 < q
m : Nat := 0
r : Nat := 0
r_bounded : r < q
n_divided : 0 ≃ m * q + r
⊢ Euclid 0 q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n q : ℕ
x✝ : Positive q
this : 0 < q
m : Nat := 0
r : Nat := 0
r_bounded : r < q
n_divided : 0 ≃ m * q + r
⊢ Euclid 0 q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | intro n (ih : Euclid n q) | n q : ℕ
x✝ : Positive q
this : 0 < q
⊢ ∀ (m : ℕ), Euclid m q → Euclid (step m) q | n✝ q : ℕ
x✝ : Positive q
this : 0 < q
n : ℕ
ih : Euclid n q
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
n q : ℕ
x✝ : Positive q
this : 0 < q
⊢ ∀ (m : ℕ), Euclid m q → Euclid (step m) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | have ⟨m', r', (_ : r' < q), (_ : n ≃ m' * q + r')⟩ := ih | n✝ q : ℕ
x✝ : Positive q
this : 0 < q
n : ℕ
ih : Euclid n q
⊢ Euclid (step n) q | n✝ q : ℕ
x✝ : Positive q
this : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ q : ℕ
x✝ : Positive q
this : 0 < q
n : ℕ
ih : Euclid n q
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | have : step r' ≤ q := Natural.lt_step_le.mp ‹r' < q› | n✝ q : ℕ
x✝ : Positive q
this : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
⊢ Euclid (step n) q | n✝ q : ℕ
x✝ : Positive q
this✝ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this : step r' ≤ q
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ q : ℕ
x✝ : Positive q
this : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | have : step r' < q ∨ step r' ≃ q := Natural.le_split ‹step r' ≤ q› | n✝ q : ℕ
x✝ : Positive q
this✝ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this : step r' ≤ q
⊢ Euclid (step n) q | n✝ q : ℕ
x✝ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ q : ℕ
x✝ : Positive q
this✝ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this : step r' ≤ q
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | apply Or.elim ‹step r' < q ∨ step r' ≃ q› | n✝ q : ℕ
x✝ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
⊢ Euclid (step n) q | case left
n✝ q : ℕ
x✝ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
⊢ step r' < q → Euclid (step n) q
case right
n✝ q : ℕ
x✝ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
⊢ step r' ≃ q → Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ q : ℕ
x✝ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | intro (_ : step r' < q) | case left
n✝ q : ℕ
x✝ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
⊢ step r' < q → Euclid (step n) q | case left
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
case left
n✝ q : ℕ
x✝ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
⊢ step r' < q → Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | let m := m' | case left
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
⊢ Euclid (step n) q | case left
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
m : ℕ := m'
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
case left
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | let r := step r' | case left
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
m : ℕ := m'
⊢ Euclid (step n) q | case left
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
m : ℕ := m'
r : ℕ := step r'
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
case left
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
m : ℕ := m'
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | have : r < q := ‹step r' < q› | case left
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
m : ℕ := m'
r : ℕ := step r'
⊢ Euclid (step n) q | case left
n✝ q : ℕ
x✝¹ : Positive q
this✝² : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝¹ : step r' ≤ q
this✝ : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
m : ℕ := m'
r : ℕ := step r'
this : r < q
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
case left
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
m : ℕ := m'
r : ℕ := step r'
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | have : step n ≃ m * q + r := calc
step n ≃ _ := AA.subst₁ ‹n ≃ m' * q + r'›
step (m' * q + r') ≃ _ := Rel.symm Natural.add_step
m' * q + step r' ≃ _ := Rel.refl
m * q + r ≃ _ := Rel.refl | case left
n✝ q : ℕ
x✝¹ : Positive q
this✝² : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝¹ : step r' ≤ q
this✝ : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
m : ℕ := m'
r : ℕ := step r'
this : r < q
⊢ Euclid (step n) q | case left
n✝ q : ℕ
x✝¹ : Positive q
this✝³ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝² : step r' ≤ q
this✝¹ : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
m : ℕ := m'
r : ℕ := step r'
this✝ : r < q
this : step n ≃ m * q + r
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
case left
n✝ q : ℕ
x✝¹ : Positive q
this✝² : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝¹ : step r' ≤ q
this✝ : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
m : ℕ := m'
r : ℕ := step r'
this : r < q
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | exact ⟨m, r, ‹r < q›, ‹step n ≃ m * q + r›⟩ | case left
n✝ q : ℕ
x✝¹ : Positive q
this✝³ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝² : step r' ≤ q
this✝¹ : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
m : ℕ := m'
r : ℕ := step r'
this✝ : r < q
this : step n ≃ m * q + r
⊢ Euclid (step n) q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
n✝ q : ℕ
x✝¹ : Positive q
this✝³ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝² : step r' ≤ q
this✝¹ : step r' < q ∨ step r' ≃ q
x✝ : step r' < q
m : ℕ := m'
r : ℕ := step r'
this✝ : r < q
this : step n ≃ m * q + r
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | intro (_ : step r' ≃ q) | case right
n✝ q : ℕ
x✝ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
⊢ step r' ≃ q → Euclid (step n) q | case right
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
case right
n✝ q : ℕ
x✝ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
⊢ step r' ≃ q → Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | let m := step m' | case right
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
⊢ Euclid (step n) q | case right
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
m : ℕ := step m'
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
case right
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | let r := 0 | case right
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
m : ℕ := step m'
⊢ Euclid (step n) q | case right
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
m : ℕ := step m'
r : Nat := 0
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
case right
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
m : ℕ := step m'
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | have : r < q := ‹0 < q› | case right
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
m : ℕ := step m'
r : Nat := 0
⊢ Euclid (step n) q | case right
n✝ q : ℕ
x✝¹ : Positive q
this✝² : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝¹ : step r' ≤ q
this✝ : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
m : ℕ := step m'
r : Nat := 0
this : r < q
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
case right
n✝ q : ℕ
x✝¹ : Positive q
this✝¹ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝ : step r' ≤ q
this : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
m : ℕ := step m'
r : Nat := 0
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | have : step n ≃ m * q + r := calc
step n ≃ _ := AA.subst₁ ‹n ≃ m' * q + r'›
step (m' * q + r') ≃ _ := Rel.symm Natural.add_step
m' * q + step r' ≃ _ := AA.substR ‹step r' ≃ q›
m' * q + q ≃ _ := Rel.symm Natural.step_mul
step m' * q ≃ _ := Rel.symm Natural.add_zero
step m' * q + 0 ≃ _ := Rel.refl
m * q + r ≃ _ := Rel.refl | case right
n✝ q : ℕ
x✝¹ : Positive q
this✝² : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝¹ : step r' ≤ q
this✝ : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
m : ℕ := step m'
r : Nat := 0
this : r < q
⊢ Euclid (step n) q | case right
n✝ q : ℕ
x✝¹ : Positive q
this✝³ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝² : step r' ≤ q
this✝¹ : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
m : ℕ := step m'
r : Nat := 0
this✝ : r < q
this : step n ≃ m * q + r
⊢ Euclid (step n) q | Please generate a tactic in lean4 to solve the state.
STATE:
case right
n✝ q : ℕ
x✝¹ : Positive q
this✝² : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝¹ : step r' ≤ q
this✝ : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
m : ℕ := step m'
r : Nat := 0
this : r < q
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch2.lean | AnalysisI.Ch2.euclidean_algorithm | [517, 1] | [564, 50] | exact ⟨m, r, ‹r < q›, ‹step n ≃ m * q + r›⟩ | case right
n✝ q : ℕ
x✝¹ : Positive q
this✝³ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝² : step r' ≤ q
this✝¹ : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
m : ℕ := step m'
r : Nat := 0
this✝ : r < q
this : step n ≃ m * q + r
⊢ Euclid (step n) q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
n✝ q : ℕ
x✝¹ : Positive q
this✝³ : 0 < q
n : ℕ
ih : Euclid n q
m' r' : ℕ
r_bounded✝ : r' < q
n_divided✝ : n ≃ m' * q + r'
this✝² : step r' ≤ q
this✝¹ : step r' < q ∨ step r' ≃ q
x✝ : step r' ≃ q
m : ℕ := step m'
r : Nat := 0
this✝ : r < q
this : step n ≃ m * q + r
⊢ Euclid (step n) q
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | apply Iff.intro | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Positive x ↔ AltPositive x | case mp
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Positive x → AltPositive x
case mpr
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ AltPositive x → Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Positive x ↔ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | case mp =>
intro (_ : Positive x)
show AltPositive x
have (Rational.AsRatio.intro a b (_ : Integer.Nonzero b) x_eqv) :=
Rational.as_ratio x
have : x ≃ a / b := x_eqv
have : sgn a * sgn b ≃ 1 := calc
sgn a * sgn b ≃ _ := Rel.symm Rational.sgn_div_integers
sgn ((a : ℚ) / b) ≃ _ := Rational.sgn_subst (Rel.symm ‹x ≃ a / b›)
sgn x ≃ _ := Rational.sgn_positive.mp ‹Positive x›
1 ≃ _ := Rel.refl
have (And.intro (_ : Integer.Sqrt1 (sgn b)) (_ : sgn a ≃ sgn b)) :=
Integer.mul_sqrt1_eqv.mp this
have : sgn b ≃ 1 ∨ sgn b ≃ -1 :=
Integer.sqrt1_cases.mp ‹Integer.Sqrt1 (sgn b)›
match this with
| Or.inl (_ : sgn b ≃ 1) =>
have : sgn a ≃ 1 := Rel.trans ‹sgn a ≃ sgn b› ‹sgn b ≃ 1›
have : AP (Positive a) := AP.mk (Integer.sgn_positive.mpr ‹sgn a ≃ 1›)
have : AP (Positive b) := AP.mk (Integer.sgn_positive.mpr ‹sgn b ≃ 1›)
exact AltPositive.mk ‹x ≃ a / b›
| Or.inr (_ : sgn b ≃ -1) =>
have : sgn (-b) ≃ 1 := calc
sgn (-b) ≃ _ := Integer.sgn_compat_neg
(-(sgn b)) ≃ _ := AA.subst₁ ‹sgn b ≃ -1›
(-(-1)) ≃ _ := Integer.neg_involutive
1 ≃ _ := Rel.refl
have : sgn (-a) ≃ 1 := calc
sgn (-a) ≃ _ := Integer.sgn_compat_neg
(-(sgn a)) ≃ _ := AA.subst₁ ‹sgn a ≃ sgn b›
(-(sgn b)) ≃ _ := Rel.symm Integer.sgn_compat_neg
sgn (-b) ≃ _ := ‹sgn (-b) ≃ 1›
1 ≃ _ := Rel.refl
have : Positive (-a) := Integer.sgn_positive.mpr ‹sgn (-a) ≃ 1›
have : Positive (-b) := Integer.sgn_positive.mpr ‹sgn (-b) ≃ 1›
have : AP (Positive (-a)) := AP.mk ‹Positive (-a)›
have : AP (Positive (-b)) := AP.mk ‹Positive (-b)›
have neg_eqv {z : ℤ} : -(z : ℚ) ≃ ((-z : ℤ) : ℚ) :=
Rational.eqv_symm Rational.neg_compat_from_integer
have : x ≃ ((-a : ℤ) : ℚ) / ((-b : ℤ) : ℚ) := calc
x ≃ _ := ‹x ≃ a / b›
(a : ℚ) / b ≃ _ := Rel.symm Rational.div_neg_cancel
(-(a : ℚ)) / (-(b : ℚ)) ≃ _ := Rational.div_substL neg_eqv
((-a : ℤ) : ℚ) / (-b : ℚ) ≃ _ := Rational.div_substR neg_eqv
((-a : ℤ) : ℚ) / ((-b : ℤ) : ℚ) ≃ _ := Rational.eqv_refl
exact AltPositive.mk this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Positive x → AltPositive x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Positive x → AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | case mpr =>
intro (_ : AltPositive x)
show Positive x
have (AltPositive.intro (a : ℤ) (b : ℤ) a_pos b_pos x_eqv) :=
‹AltPositive x›
have : AP (Positive a) := a_pos
have : sgn a ≃ 1 := Integer.sgn_positive.mp this.ev
have : AP (Positive b) := b_pos
have : sgn b ≃ 1 := Integer.sgn_positive.mp this.ev
have : x ≃ a / b := x_eqv
have : sgn a ≃ sgn b := Rel.trans ‹sgn a ≃ 1› (Rel.symm ‹sgn b ≃ 1›)
have : Integer.Nonzero b := Integer.nonzero_from_positive_inst
have : Integer.Sqrt1 (sgn b) := Integer.sgn_nonzero.mp this
have : sgn a * sgn b ≃ 1 :=
Integer.mul_sqrt1_eqv.mpr (And.intro this ‹sgn a ≃ sgn b›)
have : sgn x ≃ 1 := calc
sgn x ≃ _ := Rational.sgn_subst ‹x ≃ a / b›
sgn ((a : ℚ) / (b : ℚ)) ≃ _ := Rational.sgn_div_integers
sgn a * sgn b ≃ _ := ‹sgn a * sgn b ≃ 1›
1 ≃ _ := Rel.refl
have : Positive x := Rational.sgn_positive.mpr this
exact this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ AltPositive x → Positive x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ AltPositive x → Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | intro (_ : Positive x) | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Positive x → AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Positive x → AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have (Rational.AsRatio.intro a b (_ : Integer.Nonzero b) x_eqv) :=
Rational.as_ratio x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : x ≃ a / b := x_eqv | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this : x ≃ Rational.from_integer a / Rational.from_integer b
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : sgn a * sgn b ≃ 1 := calc
sgn a * sgn b ≃ _ := Rel.symm Rational.sgn_div_integers
sgn ((a : ℚ) / b) ≃ _ := Rational.sgn_subst (Rel.symm ‹x ≃ a / b›)
sgn x ≃ _ := Rational.sgn_positive.mp ‹Positive x›
1 ≃ _ := Rel.refl | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this : x ≃ Rational.from_integer a / Rational.from_integer b
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : x ≃ Rational.from_integer a / Rational.from_integer b
this : sgn a * sgn b ≃ 1
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this : x ≃ Rational.from_integer a / Rational.from_integer b
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have (And.intro (_ : Integer.Sqrt1 (sgn b)) (_ : sgn a ≃ sgn b)) :=
Integer.mul_sqrt1_eqv.mp this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : x ≃ Rational.from_integer a / Rational.from_integer b
this : sgn a * sgn b ≃ 1
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : x ≃ Rational.from_integer a / Rational.from_integer b
this : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : x ≃ Rational.from_integer a / Rational.from_integer b
this : sgn a * sgn b ≃ 1
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : sgn b ≃ 1 ∨ sgn b ≃ -1 :=
Integer.sqrt1_cases.mp ‹Integer.Sqrt1 (sgn b)› | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : x ≃ Rational.from_integer a / Rational.from_integer b
this : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this : sgn b ≃ 1 ∨ sgn b ≃ -1
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : x ≃ Rational.from_integer a / Rational.from_integer b
this : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | match this with
| Or.inl (_ : sgn b ≃ 1) =>
have : sgn a ≃ 1 := Rel.trans ‹sgn a ≃ sgn b› ‹sgn b ≃ 1›
have : AP (Positive a) := AP.mk (Integer.sgn_positive.mpr ‹sgn a ≃ 1›)
have : AP (Positive b) := AP.mk (Integer.sgn_positive.mpr ‹sgn b ≃ 1›)
exact AltPositive.mk ‹x ≃ a / b›
| Or.inr (_ : sgn b ≃ -1) =>
have : sgn (-b) ≃ 1 := calc
sgn (-b) ≃ _ := Integer.sgn_compat_neg
(-(sgn b)) ≃ _ := AA.subst₁ ‹sgn b ≃ -1›
(-(-1)) ≃ _ := Integer.neg_involutive
1 ≃ _ := Rel.refl
have : sgn (-a) ≃ 1 := calc
sgn (-a) ≃ _ := Integer.sgn_compat_neg
(-(sgn a)) ≃ _ := AA.subst₁ ‹sgn a ≃ sgn b›
(-(sgn b)) ≃ _ := Rel.symm Integer.sgn_compat_neg
sgn (-b) ≃ _ := ‹sgn (-b) ≃ 1›
1 ≃ _ := Rel.refl
have : Positive (-a) := Integer.sgn_positive.mpr ‹sgn (-a) ≃ 1›
have : Positive (-b) := Integer.sgn_positive.mpr ‹sgn (-b) ≃ 1›
have : AP (Positive (-a)) := AP.mk ‹Positive (-a)›
have : AP (Positive (-b)) := AP.mk ‹Positive (-b)›
have neg_eqv {z : ℤ} : -(z : ℚ) ≃ ((-z : ℤ) : ℚ) :=
Rational.eqv_symm Rational.neg_compat_from_integer
have : x ≃ ((-a : ℤ) : ℚ) / ((-b : ℤ) : ℚ) := calc
x ≃ _ := ‹x ≃ a / b›
(a : ℚ) / b ≃ _ := Rel.symm Rational.div_neg_cancel
(-(a : ℚ)) / (-(b : ℚ)) ≃ _ := Rational.div_substL neg_eqv
((-a : ℤ) : ℚ) / (-b : ℚ) ≃ _ := Rational.div_substR neg_eqv
((-a : ℤ) : ℚ) / ((-b : ℤ) : ℚ) ≃ _ := Rational.eqv_refl
exact AltPositive.mk this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this : sgn b ≃ 1 ∨ sgn b ≃ -1
⊢ AltPositive x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this : sgn b ≃ 1 ∨ sgn b ≃ -1
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : sgn a ≃ 1 := Rel.trans ‹sgn a ≃ sgn b› ‹sgn b ≃ 1› | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ 1
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ 1
this : sgn a ≃ 1
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ 1
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : AP (Positive a) := AP.mk (Integer.sgn_positive.mpr ‹sgn a ≃ 1›) | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ 1
this : sgn a ≃ 1
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝¹ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ 1
this✝ : sgn a ≃ 1
this : AP (Positive a)
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ 1
this : sgn a ≃ 1
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : AP (Positive b) := AP.mk (Integer.sgn_positive.mpr ‹sgn b ≃ 1›) | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝¹ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ 1
this✝ : sgn a ≃ 1
this : AP (Positive a)
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝² : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ 1
this✝¹ : sgn a ≃ 1
this✝ : AP (Positive a)
this : AP (Positive b)
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝¹ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ 1
this✝ : sgn a ≃ 1
this : AP (Positive a)
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | exact AltPositive.mk ‹x ≃ a / b› | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝² : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ 1
this✝¹ : sgn a ≃ 1
this✝ : AP (Positive a)
this : AP (Positive b)
⊢ AltPositive x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝² : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ 1
this✝¹ : sgn a ≃ 1
this✝ : AP (Positive a)
this : AP (Positive b)
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : sgn (-b) ≃ 1 := calc
sgn (-b) ≃ _ := Integer.sgn_compat_neg
(-(sgn b)) ≃ _ := AA.subst₁ ‹sgn b ≃ -1›
(-(-1)) ≃ _ := Integer.neg_involutive
1 ≃ _ := Rel.refl | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this : sgn (-b) ≃ 1
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : sgn (-a) ≃ 1 := calc
sgn (-a) ≃ _ := Integer.sgn_compat_neg
(-(sgn a)) ≃ _ := AA.subst₁ ‹sgn a ≃ sgn b›
(-(sgn b)) ≃ _ := Rel.symm Integer.sgn_compat_neg
sgn (-b) ≃ _ := ‹sgn (-b) ≃ 1›
1 ≃ _ := Rel.refl | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this : sgn (-b) ≃ 1
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝¹ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝ : sgn (-b) ≃ 1
this : sgn (-a) ≃ 1
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this : sgn (-b) ≃ 1
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : Positive (-a) := Integer.sgn_positive.mpr ‹sgn (-a) ≃ 1› | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝¹ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝ : sgn (-b) ≃ 1
this : sgn (-a) ≃ 1
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝² : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝¹ : sgn (-b) ≃ 1
this✝ : sgn (-a) ≃ 1
this : Positive (-a)
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝¹ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝ : sgn (-b) ≃ 1
this : sgn (-a) ≃ 1
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : Positive (-b) := Integer.sgn_positive.mpr ‹sgn (-b) ≃ 1› | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝² : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝¹ : sgn (-b) ≃ 1
this✝ : sgn (-a) ≃ 1
this : Positive (-a)
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁵ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝³ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝² : sgn (-b) ≃ 1
this✝¹ : sgn (-a) ≃ 1
this✝ : Positive (-a)
this : Positive (-b)
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝² : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝¹ : sgn (-b) ≃ 1
this✝ : sgn (-a) ≃ 1
this : Positive (-a)
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : AP (Positive (-a)) := AP.mk ‹Positive (-a)› | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁵ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝³ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝² : sgn (-b) ≃ 1
this✝¹ : sgn (-a) ≃ 1
this✝ : Positive (-a)
this : Positive (-b)
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁶ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁵ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝⁴ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝³ : sgn (-b) ≃ 1
this✝² : sgn (-a) ≃ 1
this✝¹ : Positive (-a)
this✝ : Positive (-b)
this : AP (Positive (-a))
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁵ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝³ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝² : sgn (-b) ≃ 1
this✝¹ : sgn (-a) ≃ 1
this✝ : Positive (-a)
this : Positive (-b)
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : AP (Positive (-b)) := AP.mk ‹Positive (-b)› | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁶ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁵ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝⁴ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝³ : sgn (-b) ≃ 1
this✝² : sgn (-a) ≃ 1
this✝¹ : Positive (-a)
this✝ : Positive (-b)
this : AP (Positive (-a))
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁷ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁶ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝⁵ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝⁴ : sgn (-b) ≃ 1
this✝³ : sgn (-a) ≃ 1
this✝² : Positive (-a)
this✝¹ : Positive (-b)
this✝ : AP (Positive (-a))
this : AP (Positive (-b))
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁶ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁵ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝⁴ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝³ : sgn (-b) ≃ 1
this✝² : sgn (-a) ≃ 1
this✝¹ : Positive (-a)
this✝ : Positive (-b)
this : AP (Positive (-a))
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have neg_eqv {z : ℤ} : -(z : ℚ) ≃ ((-z : ℤ) : ℚ) :=
Rational.eqv_symm Rational.neg_compat_from_integer | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁷ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁶ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝⁵ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝⁴ : sgn (-b) ≃ 1
this✝³ : sgn (-a) ≃ 1
this✝² : Positive (-a)
this✝¹ : Positive (-b)
this✝ : AP (Positive (-a))
this : AP (Positive (-b))
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁷ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁶ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝⁵ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝⁴ : sgn (-b) ≃ 1
this✝³ : sgn (-a) ≃ 1
this✝² : Positive (-a)
this✝¹ : Positive (-b)
this✝ : AP (Positive (-a))
this : AP (Positive (-b))
neg_eqv : ∀ {z : ℤ}, -Rational.from_integer z ≃ Rational.from_integer (-z)
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁷ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁶ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝⁵ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝⁴ : sgn (-b) ≃ 1
this✝³ : sgn (-a) ≃ 1
this✝² : Positive (-a)
this✝¹ : Positive (-b)
this✝ : AP (Positive (-a))
this : AP (Positive (-b))
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : x ≃ ((-a : ℤ) : ℚ) / ((-b : ℤ) : ℚ) := calc
x ≃ _ := ‹x ≃ a / b›
(a : ℚ) / b ≃ _ := Rel.symm Rational.div_neg_cancel
(-(a : ℚ)) / (-(b : ℚ)) ≃ _ := Rational.div_substL neg_eqv
((-a : ℤ) : ℚ) / (-b : ℚ) ≃ _ := Rational.div_substR neg_eqv
((-a : ℤ) : ℚ) / ((-b : ℤ) : ℚ) ≃ _ := Rational.eqv_refl | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁷ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁶ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝⁵ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝⁴ : sgn (-b) ≃ 1
this✝³ : sgn (-a) ≃ 1
this✝² : Positive (-a)
this✝¹ : Positive (-b)
this✝ : AP (Positive (-a))
this : AP (Positive (-b))
neg_eqv : ∀ {z : ℤ}, -Rational.from_integer z ≃ Rational.from_integer (-z)
⊢ AltPositive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁸ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁷ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝⁶ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝⁵ : sgn (-b) ≃ 1
this✝⁴ : sgn (-a) ≃ 1
this✝³ : Positive (-a)
this✝² : Positive (-b)
this✝¹ : AP (Positive (-a))
this✝ : AP (Positive (-b))
neg_eqv : ∀ {z : ℤ}, -Rational.from_integer z ≃ Rational.from_integer (-z)
this : x ≃ Rational.from_integer (-a) / Rational.from_integer (-b)
⊢ AltPositive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁷ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁶ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝⁵ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝⁴ : sgn (-b) ≃ 1
this✝³ : sgn (-a) ≃ 1
this✝² : Positive (-a)
this✝¹ : Positive (-b)
this✝ : AP (Positive (-a))
this : AP (Positive (-b))
neg_eqv : ∀ {z : ℤ}, -Rational.from_integer z ≃ Rational.from_integer (-z)
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | exact AltPositive.mk this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁸ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁷ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝⁶ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝⁵ : sgn (-b) ≃ 1
this✝⁴ : sgn (-a) ≃ 1
this✝³ : Positive (-a)
this✝² : Positive (-b)
this✝¹ : AP (Positive (-a))
this✝ : AP (Positive (-b))
neg_eqv : ∀ {z : ℤ}, -Rational.from_integer z ≃ Rational.from_integer (-z)
this : x ≃ Rational.from_integer (-a) / Rational.from_integer (-b)
⊢ AltPositive x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Positive x
a b : Integer.Impl.Difference ℕ
b_nonzero✝ : Nonzero b
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁸ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁷ : sgn a * sgn b ≃ 1
left✝ : Integer.Sqrt1 (sgn b)
right✝ : sgn a ≃ sgn b
this✝⁶ : sgn b ≃ 1 ∨ sgn b ≃ -1
h✝ : sgn b ≃ -1
this✝⁵ : sgn (-b) ≃ 1
this✝⁴ : sgn (-a) ≃ 1
this✝³ : Positive (-a)
this✝² : Positive (-b)
this✝¹ : AP (Positive (-a))
this✝ : AP (Positive (-b))
neg_eqv : ∀ {z : ℤ}, -Rational.from_integer z ≃ Rational.from_integer (-z)
this : x ≃ Rational.from_integer (-a) / Rational.from_integer (-b)
⊢ AltPositive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | intro (_ : AltPositive x) | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ AltPositive x → Positive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ AltPositive x → Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have (AltPositive.intro (a : ℤ) (b : ℤ) a_pos b_pos x_eqv) :=
‹AltPositive x› | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
⊢ Positive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
⊢ Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : AP (Positive a) := a_pos | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
⊢ Positive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this : AP (Positive a)
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
⊢ Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : sgn a ≃ 1 := Integer.sgn_positive.mp this.ev | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this : AP (Positive a)
⊢ Positive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : AP (Positive a)
this : sgn a ≃ 1
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this : AP (Positive a)
⊢ Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : AP (Positive b) := b_pos | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : AP (Positive a)
this : sgn a ≃ 1
⊢ Positive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : AP (Positive a)
this✝ : sgn a ≃ 1
this : AP (Positive b)
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : AP (Positive a)
this : sgn a ≃ 1
⊢ Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : sgn b ≃ 1 := Integer.sgn_positive.mp this.ev | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : AP (Positive a)
this✝ : sgn a ≃ 1
this : AP (Positive b)
⊢ Positive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : AP (Positive a)
this✝¹ : sgn a ≃ 1
this✝ : AP (Positive b)
this : sgn b ≃ 1
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : AP (Positive a)
this✝ : sgn a ≃ 1
this : AP (Positive b)
⊢ Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : x ≃ a / b := x_eqv | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : AP (Positive a)
this✝¹ : sgn a ≃ 1
this✝ : AP (Positive b)
this : sgn b ≃ 1
⊢ Positive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : AP (Positive a)
this✝² : sgn a ≃ 1
this✝¹ : AP (Positive b)
this✝ : sgn b ≃ 1
this : x ≃ Rational.from_integer a / Rational.from_integer b
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : AP (Positive a)
this✝¹ : sgn a ≃ 1
this✝ : AP (Positive b)
this : sgn b ≃ 1
⊢ Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : sgn a ≃ sgn b := Rel.trans ‹sgn a ≃ 1› (Rel.symm ‹sgn b ≃ 1›) | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : AP (Positive a)
this✝² : sgn a ≃ 1
this✝¹ : AP (Positive b)
this✝ : sgn b ≃ 1
this : x ≃ Rational.from_integer a / Rational.from_integer b
⊢ Positive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : AP (Positive a)
this✝³ : sgn a ≃ 1
this✝² : AP (Positive b)
this✝¹ : sgn b ≃ 1
this✝ : x ≃ Rational.from_integer a / Rational.from_integer b
this : sgn a ≃ sgn b
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : AP (Positive a)
this✝² : sgn a ≃ 1
this✝¹ : AP (Positive b)
this✝ : sgn b ≃ 1
this : x ≃ Rational.from_integer a / Rational.from_integer b
⊢ Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : Integer.Nonzero b := Integer.nonzero_from_positive_inst | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : AP (Positive a)
this✝³ : sgn a ≃ 1
this✝² : AP (Positive b)
this✝¹ : sgn b ≃ 1
this✝ : x ≃ Rational.from_integer a / Rational.from_integer b
this : sgn a ≃ sgn b
⊢ Positive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁵ : AP (Positive a)
this✝⁴ : sgn a ≃ 1
this✝³ : AP (Positive b)
this✝² : sgn b ≃ 1
this✝¹ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : sgn a ≃ sgn b
this : Nonzero b
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : AP (Positive a)
this✝³ : sgn a ≃ 1
this✝² : AP (Positive b)
this✝¹ : sgn b ≃ 1
this✝ : x ≃ Rational.from_integer a / Rational.from_integer b
this : sgn a ≃ sgn b
⊢ Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : Integer.Sqrt1 (sgn b) := Integer.sgn_nonzero.mp this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁵ : AP (Positive a)
this✝⁴ : sgn a ≃ 1
this✝³ : AP (Positive b)
this✝² : sgn b ≃ 1
this✝¹ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : sgn a ≃ sgn b
this : Nonzero b
⊢ Positive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁶ : AP (Positive a)
this✝⁵ : sgn a ≃ 1
this✝⁴ : AP (Positive b)
this✝³ : sgn b ≃ 1
this✝² : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : sgn a ≃ sgn b
this✝ : Nonzero b
this : Integer.Sqrt1 (sgn b)
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁵ : AP (Positive a)
this✝⁴ : sgn a ≃ 1
this✝³ : AP (Positive b)
this✝² : sgn b ≃ 1
this✝¹ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝ : sgn a ≃ sgn b
this : Nonzero b
⊢ Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : sgn a * sgn b ≃ 1 :=
Integer.mul_sqrt1_eqv.mpr (And.intro this ‹sgn a ≃ sgn b›) | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁶ : AP (Positive a)
this✝⁵ : sgn a ≃ 1
this✝⁴ : AP (Positive b)
this✝³ : sgn b ≃ 1
this✝² : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : sgn a ≃ sgn b
this✝ : Nonzero b
this : Integer.Sqrt1 (sgn b)
⊢ Positive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁷ : AP (Positive a)
this✝⁶ : sgn a ≃ 1
this✝⁵ : AP (Positive b)
this✝⁴ : sgn b ≃ 1
this✝³ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : sgn a ≃ sgn b
this✝¹ : Nonzero b
this✝ : Integer.Sqrt1 (sgn b)
this : sgn a * sgn b ≃ 1
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁶ : AP (Positive a)
this✝⁵ : sgn a ≃ 1
this✝⁴ : AP (Positive b)
this✝³ : sgn b ≃ 1
this✝² : x ≃ Rational.from_integer a / Rational.from_integer b
this✝¹ : sgn a ≃ sgn b
this✝ : Nonzero b
this : Integer.Sqrt1 (sgn b)
⊢ Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : sgn x ≃ 1 := calc
sgn x ≃ _ := Rational.sgn_subst ‹x ≃ a / b›
sgn ((a : ℚ) / (b : ℚ)) ≃ _ := Rational.sgn_div_integers
sgn a * sgn b ≃ _ := ‹sgn a * sgn b ≃ 1›
1 ≃ _ := Rel.refl | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁷ : AP (Positive a)
this✝⁶ : sgn a ≃ 1
this✝⁵ : AP (Positive b)
this✝⁴ : sgn b ≃ 1
this✝³ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : sgn a ≃ sgn b
this✝¹ : Nonzero b
this✝ : Integer.Sqrt1 (sgn b)
this : sgn a * sgn b ≃ 1
⊢ Positive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁸ : AP (Positive a)
this✝⁷ : sgn a ≃ 1
this✝⁶ : AP (Positive b)
this✝⁵ : sgn b ≃ 1
this✝⁴ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : sgn a ≃ sgn b
this✝² : Nonzero b
this✝¹ : Integer.Sqrt1 (sgn b)
this✝ : sgn a * sgn b ≃ 1
this : sgn x ≃ 1
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁷ : AP (Positive a)
this✝⁶ : sgn a ≃ 1
this✝⁵ : AP (Positive b)
this✝⁴ : sgn b ≃ 1
this✝³ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝² : sgn a ≃ sgn b
this✝¹ : Nonzero b
this✝ : Integer.Sqrt1 (sgn b)
this : sgn a * sgn b ≃ 1
⊢ Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | have : Positive x := Rational.sgn_positive.mpr this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁸ : AP (Positive a)
this✝⁷ : sgn a ≃ 1
this✝⁶ : AP (Positive b)
this✝⁵ : sgn b ≃ 1
this✝⁴ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : sgn a ≃ sgn b
this✝² : Nonzero b
this✝¹ : Integer.Sqrt1 (sgn b)
this✝ : sgn a * sgn b ≃ 1
this : sgn x ≃ 1
⊢ Positive x | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁹ : AP (Positive a)
this✝⁸ : sgn a ≃ 1
this✝⁷ : AP (Positive b)
this✝⁶ : sgn b ≃ 1
this✝⁵ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : sgn a ≃ sgn b
this✝³ : Nonzero b
this✝² : Integer.Sqrt1 (sgn b)
this✝¹ : sgn a * sgn b ≃ 1
this✝ : sgn x ≃ 1
this : Positive x
⊢ Positive x | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁸ : AP (Positive a)
this✝⁷ : sgn a ≃ 1
this✝⁶ : AP (Positive b)
this✝⁵ : sgn b ≃ 1
this✝⁴ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝³ : sgn a ≃ sgn b
this✝² : Nonzero b
this✝¹ : Integer.Sqrt1 (sgn b)
this✝ : sgn a * sgn b ≃ 1
this : sgn x ≃ 1
⊢ Positive x
TACTIC:
|
https://github.com/cruhland/lean4-analysis.git | e48553c40a65962fa08f8a86a9001433bf422a62 | Lean4Analysis/Ch4/Ch4Sec2Rationals.lean | AnalysisI.Ch4.Sec2.alt_positive | [401, 1] | [473, 15] | exact this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁹ : AP (Positive a)
this✝⁸ : sgn a ≃ 1
this✝⁷ : AP (Positive b)
this✝⁶ : sgn b ≃ 1
this✝⁵ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : sgn a ≃ sgn b
this✝³ : Nonzero b
this✝² : Integer.Sqrt1 (sgn b)
this✝¹ : sgn a * sgn b ≃ 1
this✝ : sgn x ≃ 1
this : Positive x
⊢ Positive x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : AltPositive x
a b : ℤ
a_pos : AP (Positive a)
b_pos : AP (Positive b)
x_eqv : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁹ : AP (Positive a)
this✝⁸ : sgn a ≃ 1
this✝⁷ : AP (Positive b)
this✝⁶ : sgn b ≃ 1
this✝⁵ : x ≃ Rational.from_integer a / Rational.from_integer b
this✝⁴ : sgn a ≃ sgn b
this✝³ : Nonzero b
this✝² : Integer.Sqrt1 (sgn b)
this✝¹ : sgn a * sgn b ≃ 1
this✝ : sgn x ≃ 1
this : Positive x
⊢ Positive 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] | apply Iff.intro | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Negative x ↔ ∃ y, Positive y ∧ x ≃ -y | case mp
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Negative x → ∃ y, Positive y ∧ x ≃ -y
case mpr
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ (∃ 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
⊢ Negative 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] | case mp =>
intro (_ : Negative x)
show ∃ (y : ℚ), Positive y ∧ x ≃ -y
have : sgn x ≃ -1 := Rational.sgn_negative.mp ‹Negative x›
have : sgn (-x) ≃ 1 := calc
sgn (-x) ≃ _ := Rational.sgn_compat_neg
(-sgn x) ≃ _ := AA.subst₁ ‹sgn x ≃ -1›
(-(-1)) ≃ _ := Integer.neg_involutive
1 ≃ _ := Rel.refl
have : Positive (-x) := Rational.sgn_positive.mpr ‹sgn (-x) ≃ 1›
have : x ≃ -(-x) := Rational.eqv_symm Rational.neg_involutive
exact Exists.intro (-x) (And.intro ‹Positive (-x)› ‹x ≃ -(-x)›) | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Negative 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
⊢ Negative 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] | case mpr =>
intro (Exists.intro (y : ℚ) (And.intro (_ : Positive y) (_ : x ≃ -y)))
show Negative x
have : sgn y ≃ 1 := Rational.sgn_positive.mp ‹Positive y›
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
have : Negative x := Rational.sgn_negative.mpr this
exact this | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ (∃ y, Positive y ∧ x ≃ -y) → Negative x | no goals | 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] | intro (_ : Negative x) | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Negative x → ∃ y, Positive y ∧ x ≃ -y | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
⊢ ∃ y, Positive y ∧ x ≃ -y | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
⊢ Negative 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] | have : sgn x ≃ -1 := Rational.sgn_negative.mp ‹Negative x› | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
⊢ ∃ y, Positive y ∧ x ≃ -y | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
this : sgn x ≃ -1
⊢ ∃ 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
⊢ ∃ 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] | have : sgn (-x) ≃ 1 := calc
sgn (-x) ≃ _ := Rational.sgn_compat_neg
(-sgn x) ≃ _ := AA.subst₁ ‹sgn x ≃ -1›
(-(-1)) ≃ _ := Integer.neg_involutive
1 ≃ _ := Rel.refl | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
this : sgn x ≃ -1
⊢ ∃ y, Positive y ∧ x ≃ -y | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
this✝ : sgn x ≃ -1
this : sgn (-x) ≃ 1
⊢ ∃ 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 : sgn x ≃ -1
⊢ ∃ 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] | have : Positive (-x) := Rational.sgn_positive.mpr ‹sgn (-x) ≃ 1› | ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
this✝ : sgn x ≃ -1
this : sgn (-x) ≃ 1
⊢ ∃ y, Positive y ∧ x ≃ -y | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
ℚ : Type
inst✝ : Rational ℚ
x : ℚ
x✝ : Negative x
this✝ : sgn x ≃ -1
this : sgn (-x) ≃ 1
⊢ ∃ y, Positive y ∧ x ≃ -y
TACTIC:
|
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