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/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ |a - b| > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ |a - b| > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | change |a - b| / 2 > 0 | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ ε > 0 | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ |a - b| / 2 > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ ε > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | linarith | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ |a - b| / 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ |a - b| / 2 > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ |s N - a| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ |s N - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ |s N - b| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ |s N - b| < ε
TA... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ |a - b| < |a - b| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| <... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S04_Conjunction_and_Iff.lean | C03S04.aux | [107, 1] | [109, 17] | sorry | x y : ℝ
h : x ^ 2 + y ^ 2 = 0
⊢ x ^ 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : x ^ 2 + y ^ 2 = 0
⊢ x ^ 2 = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S04_Conjunction_and_Iff.lean | C03S04.not_monotone_iff | [127, 1] | [130, 6] | rw [Monotone] | f : ℝ → ℝ
⊢ ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y | f : ℝ → ℝ
⊢ (¬∀ ⦃a b : ℝ⦄, a ≤ b → f a ≤ f b) ↔ ∃ x y, x ≤ y ∧ f x > f y | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
⊢ ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S04_Conjunction_and_Iff.lean | C03S04.not_monotone_iff | [127, 1] | [130, 6] | push_neg | f : ℝ → ℝ
⊢ (¬∀ ⦃a b : ℝ⦄, a ≤ b → f a ≤ f b) ↔ ∃ x y, x ≤ y ∧ f x > f y | f : ℝ → ℝ
⊢ (∃ a b, a ≤ b ∧ f b < f a) ↔ ∃ x y, x ≤ y ∧ f x > f y | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
⊢ (¬∀ ⦃a b : ℝ⦄, a ≤ b → f a ≤ f b) ↔ ∃ x y, x ≤ y ∧ f x > f y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S04_Conjunction_and_Iff.lean | C03S04.not_monotone_iff | [127, 1] | [130, 6] | rfl | f : ℝ → ℝ
⊢ (∃ a b, a ≤ b ∧ f b < f a) ↔ ∃ x y, x ≤ y ∧ f x > f y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
⊢ (∃ a b, a ≤ b ∧ f b < f a) ↔ ∃ x y, x ≤ y ∧ f x > f y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean | absorb1 | [48, 1] | [49, 8] | sorry | α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) = x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean | absorb2 | [51, 1] | [52, 8] | sorry | α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y = x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_neg_cancel_right | [8, 1] | [9, 42] | rw [add_assoc, add_right_neg, add_zero] | R : Type u_1
inst✝ : Ring R
a b : R
⊢ a + b + -b = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b : R
⊢ a + b + -b = a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_left_cancel | [11, 1] | [12, 57] | rw [← neg_add_cancel_left a b, h, neg_add_cancel_left] | R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = a + c
⊢ b = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = a + c
⊢ b = c
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_right_cancel | [14, 1] | [15, 59] | rw [← add_neg_cancel_right a b, h, add_neg_cancel_right] | R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = c + b
⊢ a = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = c + b
⊢ a = c
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.zero_mul | [17, 1] | [19, 25] | have h : 0 * a + 0 * a = 0 * a + 0 := by rw [← add_mul, add_zero, add_zero] | R : Type u_1
inst✝ : Ring R
a : R
⊢ 0 * a = 0 | R : Type u_1
inst✝ : Ring R
a : R
h : 0 * a + 0 * a = 0 * a + 0
⊢ 0 * a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ 0 * a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.zero_mul | [17, 1] | [19, 25] | rw [add_left_cancel h] | R : Type u_1
inst✝ : Ring R
a : R
h : 0 * a + 0 * a = 0 * a + 0
⊢ 0 * a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
h : 0 * a + 0 * a = 0 * a + 0
⊢ 0 * a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.zero_mul | [17, 1] | [19, 25] | rw [← add_mul, add_zero, add_zero] | R : Type u_1
inst✝ : Ring R
a : R
⊢ 0 * a + 0 * a = 0 * a + 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ 0 * a + 0 * a = 0 * a + 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_eq_of_add_eq_zero | [21, 1] | [22, 46] | rw [← neg_add_cancel_left a b, h, add_zero] | R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ -a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ -a = b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.eq_neg_of_add_eq_zero | [24, 1] | [27, 19] | symm | R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ a = -b | R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ -b = a | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ a = -b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.eq_neg_of_add_eq_zero | [24, 1] | [27, 19] | apply neg_eq_of_add_eq_zero | R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ -b = a | case h
R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ b + a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ -b = a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.eq_neg_of_add_eq_zero | [24, 1] | [27, 19] | rw [add_comm, h] | case h
R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ b + a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ b + a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_zero | [29, 1] | [31, 16] | apply neg_eq_of_add_eq_zero | R : Type u_1
inst✝ : Ring R
⊢ -0 = 0 | case h
R : Type u_1
inst✝ : Ring R
⊢ 0 + 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
⊢ -0 = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_zero | [29, 1] | [31, 16] | rw [add_zero] | case h
R : Type u_1
inst✝ : Ring R
⊢ 0 + 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝ : Ring R
⊢ 0 + 0 = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_neg | [33, 1] | [35, 20] | apply neg_eq_of_add_eq_zero | R : Type u_1
inst✝ : Ring R
a : R
⊢ - -a = a | case h
R : Type u_1
inst✝ : Ring R
a : R
⊢ -a + a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ - -a = a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_neg | [33, 1] | [35, 20] | rw [add_left_neg] | case h
R : Type u_1
inst✝ : Ring R
a : R
⊢ -a + a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝ : Ring R
a : R
⊢ -a + a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.self_sub | [42, 1] | [43, 37] | rw [sub_eq_add_neg, add_right_neg] | R : Type u_1
inst✝ : Ring R
a : R
⊢ a - a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ a - a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.one_add_one_eq_two | [45, 1] | [46, 11] | norm_num | R : Type u_1
inst✝ : Ring R
⊢ 1 + 1 = 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
⊢ 1 + 1 = 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.two_mul | [48, 1] | [49, 46] | rw [← one_add_one_eq_two, add_mul, one_mul] | R : Type u_1
inst✝ : Ring R
a : R
⊢ 2 * a = a + a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ 2 * a = a + a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_right_inv | [58, 1] | [61, 47] | have h : (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1 := by
rw [mul_assoc, ← mul_assoc a⁻¹ a, mul_left_inv, one_mul, mul_left_inv] | G : Type u_1
inst✝ : Group G
a : G
⊢ a * a⁻¹ = 1 | G : Type u_1
inst✝ : Group G
a : G
h : (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1
⊢ a * a⁻¹ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
a : G
⊢ a * a⁻¹ = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_right_inv | [58, 1] | [61, 47] | rw [← h, ← mul_assoc, mul_left_inv, one_mul] | G : Type u_1
inst✝ : Group G
a : G
h : (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1
⊢ a * a⁻¹ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
a : G
h : (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1
⊢ a * a⁻¹ = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_right_inv | [58, 1] | [61, 47] | rw [mul_assoc, ← mul_assoc a⁻¹ a, mul_left_inv, one_mul, mul_left_inv] | G : Type u_1
inst✝ : Group G
a : G
⊢ (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
a : G
⊢ (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_one | [63, 1] | [64, 61] | rw [← mul_left_inv a, ← mul_assoc, mul_right_inv, one_mul] | G : Type u_1
inst✝ : Group G
a : G
⊢ a * 1 = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
a : G
⊢ a * 1 = a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_inv_rev | [66, 1] | [68, 52] | rw [← one_mul (b⁻¹ * a⁻¹), ← mul_left_inv (a * b), mul_assoc, mul_assoc, ← mul_assoc b b⁻¹,
mul_right_inv, one_mul, mul_right_inv, mul_one] | G : Type u_1
inst✝ : Group G
a b : G
⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
a b : G
⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact1 | [38, 1] | [43, 11] | have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 | a b c d e : ℝ
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 | case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact1 | [38, 1] | [43, 11] | calc
a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring
_ ≥ 0 := by apply pow_two_nonneg | case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 | a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact1 | [38, 1] | [43, 11] | linarith | a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact1 | [38, 1] | [43, 11] | ring | a b c d e : ℝ
⊢ a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
⊢ a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact1 | [38, 1] | [43, 11] | apply pow_two_nonneg | a b c d e : ℝ
⊢ (a - b) ^ 2 ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
⊢ (a - b) ^ 2 ≥ 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact2 | [45, 1] | [50, 11] | have h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2 | a b c d e : ℝ
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 | case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact2 | [45, 1] | [50, 11] | calc
a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2 := by ring
_ ≥ 0 := by apply pow_two_nonneg | case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 | a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact2 | [45, 1] | [50, 11] | linarith | a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact2 | [45, 1] | [50, 11] | ring | a b c d e : ℝ
⊢ a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
⊢ a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact2 | [45, 1] | [50, 11] | apply pow_two_nonneg | a b c d e : ℝ
⊢ (a + b) ^ 2 ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
⊢ (a + b) ^ 2 ≥ 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S04_More_on_Order_and_Divisibility.lean | C02S04.aux | [45, 1] | [46, 8] | sorry | a b c d : ℝ
⊢ min a b + c ≤ min (a + c) (b + c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d : ℝ
⊢ min a b + c ≤ min (a + c) (b + c)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb1 | [64, 1] | [69, 20] | apply le_antisymm | α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) = x | case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) ≤ x
case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊓ (x ⊔ y) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) = x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb1 | [64, 1] | [69, 20] | apply le_inf | case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊓ (x ⊔ y) | case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x
case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊔ y | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊓ (x ⊔ y)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb1 | [64, 1] | [69, 20] | apply le_sup_left | case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊔ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊔ y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb1 | [64, 1] | [69, 20] | apply inf_le_left | case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) ≤ x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb1 | [64, 1] | [69, 20] | apply le_refl | case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb2 | [71, 1] | [76, 20] | apply le_antisymm | α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y = x | case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y ≤ x
case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊔ x ⊓ y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y = x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb2 | [71, 1] | [76, 20] | apply le_sup_left | case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊔ x ⊓ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊔ x ⊓ y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb2 | [71, 1] | [76, 20] | apply sup_le | case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y ≤ x | case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x
case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ y ≤ x | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y ≤ x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb2 | [71, 1] | [76, 20] | apply inf_le_left | case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ y ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ y ≤ x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb2 | [71, 1] | [76, 20] | apply le_refl | case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | aux1 | [108, 1] | [110, 26] | rw [← sub_self a, sub_eq_add_neg, sub_eq_add_neg, add_comm, add_comm b] | R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : a ≤ b
⊢ 0 ≤ b - a | R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : a ≤ b
⊢ -a + a ≤ -a + b | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : a ≤ b
⊢ 0 ≤ b - a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | aux1 | [108, 1] | [110, 26] | apply add_le_add_left h | R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : a ≤ b
⊢ -a + a ≤ -a + b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : a ≤ b
⊢ -a + a ≤ -a + b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | aux2 | [112, 1] | [114, 26] | rw [← add_zero a, ← sub_add_cancel b a, add_comm (b - a)] | R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : 0 ≤ b - a
⊢ a ≤ b | R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : 0 ≤ b - a
⊢ a + 0 ≤ a + (b - a) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : 0 ≤ b - a
⊢ a ≤ b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | aux2 | [112, 1] | [114, 26] | apply add_le_add_left h | R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : 0 ≤ b - a
⊢ a + 0 ≤ a + (b - a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : 0 ≤ b - a
⊢ a + 0 ≤ a + (b - a)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [105, 1] | [111, 7] | rcases sosx with ⟨a, b, xeq⟩ | α : Type u_1
inst✝ : CommRing α
x y : α
sosx : SumOfSquares x
sosy : SumOfSquares y
⊢ SumOfSquares (x * y) | case intro.intro
α : Type u_1
inst✝ : CommRing α
x y : α
sosy : SumOfSquares y
a b : α
xeq : x = a ^ 2 + b ^ 2
⊢ SumOfSquares (x * y) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : CommRing α
x y : α
sosx : SumOfSquares x
sosy : SumOfSquares y
⊢ SumOfSquares (x * y)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [105, 1] | [111, 7] | rcases sosy with ⟨c, d, yeq⟩ | case intro.intro
α : Type u_1
inst✝ : CommRing α
x y : α
sosy : SumOfSquares y
a b : α
xeq : x = a ^ 2 + b ^ 2
⊢ SumOfSquares (x * y) | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares (x * y) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝ : CommRing α
x y : α
sosy : SumOfSquares y
a b : α
xeq : x = a ^ 2 + b ^ 2
⊢ SumOfSquares (x * y)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [105, 1] | [111, 7] | rw [xeq, yeq] | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares (x * y) | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares (x * y)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [105, 1] | [111, 7] | use a * c - b * d, a * d + b * c | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2)) | case h
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2))
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [105, 1] | [111, 7] | ring | case h
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul' | [113, 1] | [118, 7] | rcases sosx with ⟨a, b, rfl⟩ | α : Type u_1
inst✝ : CommRing α
x y : α
sosx : SumOfSquares x
sosy : SumOfSquares y
⊢ SumOfSquares (x * y) | case intro.intro
α : Type u_1
inst✝ : CommRing α
y : α
sosy : SumOfSquares y
a b : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * y) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : CommRing α
x y : α
sosx : SumOfSquares x
sosy : SumOfSquares y
⊢ SumOfSquares (x * y)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul' | [113, 1] | [118, 7] | rcases sosy with ⟨c, d, rfl⟩ | case intro.intro
α : Type u_1
inst✝ : CommRing α
y : α
sosy : SumOfSquares y
a b : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * y) | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝ : CommRing α
y : α
sosy : SumOfSquares y
a b : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * y)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul' | [113, 1] | [118, 7] | use a * c - b * d, a * d + b * c | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2)) | case h
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2))
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul' | [113, 1] | [118, 7] | ring | case h
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | have : x ∈ g '' univ := by
contrapose! hx
rw [sbSet, mem_iUnion]
use 0
rw [sbAux, mem_diff]
exact ⟨mem_univ _, hx⟩ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
⊢ g (invFun g x) = x | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ g (invFun g x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
⊢ g (invFun g x) = x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | have : ∃ y, g y = x := by
simp at this
assumption | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ g (invFun g x) = x | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this✝ : x ∈ g '' univ
this : ∃ y, g y = x
⊢ g (invFun g x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ g (invFun g x) = x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | exact invFun_eq this | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this✝ : x ∈ g '' univ
this : ∃ y, g y = x
⊢ g (invFun g x) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this✝ : x ∈ g '' univ
this : ∃ y, g y = x
⊢ g (invFun g x) = x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | contrapose! hx | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
⊢ x ∈ g '' univ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbSet f g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
⊢ x ∈ g '' univ
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | rw [sbSet, mem_iUnion] | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbSet f g | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ ∃ i, x ∈ sbAux f g i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbSet f g
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | use 0 | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ ∃ i, x ∈ sbAux f g i | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbAux f g 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ ∃ i, x ∈ sbAux f g i
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | rw [sbAux, mem_diff] | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbAux f g 0 | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ univ ∧ x ∉ g '' univ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbAux f g 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | exact ⟨mem_univ _, hx⟩ | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ univ ∧ x ∉ g '' univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ univ ∧ x ∉ g '' univ
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | simp at this | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ ∃ y, g y = x | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : ∃ y, g y = x
⊢ ∃ y, g y = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ ∃ y, g y = x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | assumption | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : ∃ y, g y = x
⊢ ∃ y, g y = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : ∃ y, g y = x
⊢ ∃ y, g y = x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | set A := sbSet f g with A_def | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
⊢ Injective (sbFun f g) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Injective (sbFun f g) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
⊢ Injective (sbFun f g)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | set h := sbFun f g with h_def | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Injective (sbFun f g) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Injective h | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Injective (sbFun f g)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | intro x₁ x₂ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Injective h | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
⊢ h x₁ = h x₂ → x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Injective h
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | intro (hxeq : h x₁ = h x₂) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
⊢ h x₁ = h x₂ → x₁ = x₂ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : h x₁ = h x₂
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
⊢ h x₁ = h x₂ → x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | simp only [h_def, sbFun, ← A_def] at hxeq | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : h x₁ = h x₂
⊢ x₁ = x₂ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : h x₁ = h x₂
⊢ x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | by_cases xA : x₁ ∈ A ∨ x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
⊢ x₁ = x₂ | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
⊢ x₁ = x₂
case neg
α :... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invF... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | push_neg at xA | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : ¬(x₁ ∈ A ∨ x₂ ∈ A)
⊢ x₁ = x₂ | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∉ A ∧ x₂ ∉ A
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | rw [if_neg xA.1, if_neg xA.2] at hxeq | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∉ A ∧ x₂ ∉ A
⊢ x₁ = x₂ | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : invFun g x₁ = invFun g x₂
xA : x₁ ∉ A ∧ x₂ ∉ A
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | rw [← sb_right_inv f g xA.1, hxeq, sb_right_inv f g xA.2] | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : invFun g x₁ = invFun g x₂
xA : x₁ ∉ A ∧ x₂ ∉ A
⊢ x₁ = x₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : invFun g x₁ = invFun g x₂
xA : x₁ ∉ A ∧ x₂ ∉ A
⊢ x₁ = x₂
TACT... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
⊢ x₁ = x₂ | case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | have x₂A : x₂ ∈ A := by
apply not_imp_self.mp
intro (x₂nA : x₂ ∉ A)
rw [if_pos x₁A, if_neg x₂nA] at hxeq
rw [A_def, sbSet, mem_iUnion] at x₁A
have x₂eq : x₂ = g (f x₁) := by
rw [hxeq, sb_right_inv f g x₂nA]
rcases x₁A with ⟨n, hn⟩
rw [A_def, sbSet, mem_iUnion]
use n + 1
simp [sbAux]
exact ⟨x₁, h... | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₁ = x₂ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂A : x₂ ∈ A
⊢ x₁ =... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invF... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | rw [if_pos x₁A, if_pos x₂A] at hxeq | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂A : x₂ ∈ A
⊢ x₁ =... | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = f x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂A : x₂ ∈ A
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invF... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | exact hf hxeq | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = f x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂A : x₂ ∈ A
⊢ x₁ = x₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = f x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂A : x₂ ∈ A
⊢ x₁ = x₂
T... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | symm | case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ :... | case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | apply this hxeq.symm xA.symm (xA.resolve_left x₁A) | case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | apply not_imp_self.mp | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₂ ∉ A → x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invF... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | intro (x₂nA : x₂ ∉ A) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₂ ∉ A → x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invF... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | rw [if_pos x₁A, if_neg x₂nA] at hxeq | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ... | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invF... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | rw [A_def, sbSet, mem_iUnion] at x₁A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | have x₂eq : x₂ = g (f x₁) := by
rw [hxeq, sb_right_inv f g x₂nA] | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
⊢ x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | rcases x₁A with ⟨n, hn⟩ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
⊢ x₂ ∈ A | case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | rw [A_def, sbSet, mem_iUnion] | case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ x₂ ∈ A | case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ ∃ i, x₂ ∈ sbAux... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq ... |
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