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/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rw [← zsmul_zero ⌊x⌋, h, h0, Int.cast_one, mul_one] | case refine_1
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1
x : R
⊢ f x = f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1
x : R
⊢ f x = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rcases exists_between (zero_lt_one' R) with ⟨c, hc⟩ | case refine_2
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
⊢ f = 0 | case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | replace h0 : ⌊f c⌋ = 0 := by
specialize h c c
have h1 : ⌊c⌋ = 0 := Int.floor_eq_zero_iff.mpr ⟨hc.1.le, hc.2⟩
rw [h1, zero_zsmul, h0, zero_eq_mul] at h
exact h.elim (λ h ↦ h ▸ Int.floor_zero) Int.cast_eq_zero.mp | case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
⊢ f = 0 | case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | replace h0 : f (-c) = 0 := by
specialize h (-1) c
have h1 : ⌊(-1 : R)⌋ = -1 := by
rw [← Int.cast_one, ← Int.cast_neg, Int.floor_intCast]
rwa [h1, neg_one_zsmul, h0, Int.cast_zero, mul_zero] at h | case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
⊢ f = 0 | case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | funext y | case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
⊢ f = 0 | case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
⊢ f y = 0 y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | specialize h (-c) (-y) | case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
⊢ f y = 0 y | case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
⊢ f y = 0 y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
⊢ f y = 0 y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | have h1 : ⌊-c⌋ = -1 := by
rw [Int.floor_eq_iff, Int.cast_neg, Int.cast_one, neg_add_self]
exact ⟨neg_le_neg hc.2.le, neg_lt_zero.mpr hc.1⟩ | case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
⊢ f y = 0 y | case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
h1 : ⌊-c⌋ = -1
⊢ f y = 0 y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
⊢ f y = 0 y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rwa [h0, zero_mul, h1, neg_one_zsmul, neg_neg] at h | case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
h1 : ⌊-c⌋ = -1
⊢ f y = 0 y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
h1 : ⌊-c⌋ = -1
⊢ f y = 0 y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | specialize h c c | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
⊢ ⌊f c⌋ = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f (⌊c⌋ • c) = f c * ↑⌊f c⌋
⊢ ⌊f c⌋ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
⊢ ⌊f c⌋ = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | have h1 : ⌊c⌋ = 0 := Int.floor_eq_zero_iff.mpr ⟨hc.1.le, hc.2⟩ | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f (⌊c⌋ • c) = f c * ↑⌊f c⌋
⊢ ⌊f c⌋ = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f (⌊c⌋ • c) = f c * ↑⌊f c⌋
h1 : ⌊c⌋ = 0
⊢ ⌊f c⌋ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f (⌊c⌋ • c) = f c * ↑⌊f c⌋
⊢ ⌊f c⌋ = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rw [h1, zero_zsmul, h0, zero_eq_mul] at h | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f (⌊c⌋ • c) = f c * ↑⌊f c⌋
h1 : ⌊c⌋ = 0
⊢ ⌊f c⌋ = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f c = 0 ∨ ↑⌊f c⌋ = 0
h1 : ⌊c⌋ = 0
⊢ ⌊f c⌋ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f (⌊c⌋ • c) = f c * ↑⌊f c⌋
h1 : ⌊c⌋ = 0
⊢ ⌊f c⌋ = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | exact h.elim (λ h ↦ h ▸ Int.floor_zero) Int.cast_eq_zero.mp | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f c = 0 ∨ ↑⌊f c⌋ = 0
h1 : ⌊c⌋ = 0
⊢ ⌊f c⌋ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f c = 0 ∨ ↑⌊f c⌋ = 0
h1 : ⌊c⌋ = 0
⊢ ⌊f c⌋ = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | specialize h (-1) c | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
⊢ f (-c) = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
h : f (⌊-1⌋ • c) = f (-1) * ↑⌊f c⌋
⊢ f (-c) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
⊢ f (-c) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | have h1 : ⌊(-1 : R)⌋ = -1 := by
rw [← Int.cast_one, ← Int.cast_neg, Int.floor_intCast] | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
h : f (⌊-1⌋ • c) = f (-1) * ↑⌊f c⌋
⊢ f (-c) = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
h : f (⌊-1⌋ • c) = f (-1) * ↑⌊f c⌋
h1 : ⌊-1⌋ = -1
⊢ f (-c) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
h : f (⌊-1⌋ • c) = f (-1) * ↑⌊f c⌋
⊢ f (-c) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rwa [h1, neg_one_zsmul, h0, Int.cast_zero, mul_zero] at h | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
h : f (⌊-1⌋ • c) = f (-1) * ↑⌊f c⌋
h1 : ⌊-1⌋ = -1
⊢ f (-c) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
h : f (⌊-1⌋ • c) = f (-1) * ↑⌊f c⌋
h1 : ⌊-1⌋ = -1
⊢ f (-c) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rw [← Int.cast_one, ← Int.cast_neg, Int.floor_intCast] | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
h : f (⌊-1⌋ • c) = f (-1) * ↑⌊f c⌋
⊢ ⌊-1⌋ = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
h : f (⌊-1⌋ • c) = f (-1) * ↑⌊f c⌋
⊢ ⌊-1⌋ = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rw [Int.floor_eq_iff, Int.cast_neg, Int.cast_one, neg_add_self] | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
⊢ ⌊-c⌋ = -1 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
⊢ -1 ≤ -c ∧ -c < 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
⊢ ⌊-c⌋ = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | exact ⟨neg_le_neg hc.2.le, neg_lt_zero.mpr hc.1⟩ | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
⊢ -1 ≤ -c ∧ -c < 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
⊢ -1 ≤ -c ∧ -c < 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rw [h0, h, Int.cast_one, mul_one] | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h✝ : (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0
x✝² : ∃ C, ⌊C⌋ = 1 ∧ f = fun x => C
x✝¹ x✝ : R
C : S
h : ⌊C⌋ = 1
h0 : f = fun x => C
⊢ f (⌊x✝¹⌋ • x✝) = f x✝¹ * ↑⌊f x✝⌋ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h✝ : (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0
x✝² : ∃ C, ⌊C⌋ = 1 ∧ f = fun x => C
x✝¹ x✝ : R
C : S
h : ⌊C⌋ = 1
h0 : f = fun x => C
⊢ f (⌊x✝¹⌋ • x✝) = f x✝¹ * ↑⌊f x✝⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.sum_neg_one_pow_Ico | [37, 1] | [46, 74] | rw [add_zero, Ico_self] | a : ℕ
⊢ ∑ x ∈ Ico a (a + 0), (-1) ^ x = bif Nat.bodd 0 then (-1) ^ a else 0 | a : ℕ
⊢ ∑ x ∈ ∅, (-1) ^ x = bif Nat.bodd 0 then (-1) ^ a else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
⊢ ∑ x ∈ Ico a (a + 0), (-1) ^ x = bif Nat.bodd 0 then (-1) ^ a else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.sum_neg_one_pow_Ico | [37, 1] | [46, 74] | rfl | a : ℕ
⊢ ∑ x ∈ ∅, (-1) ^ x = bif Nat.bodd 0 then (-1) ^ a else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
⊢ ∑ x ∈ ∅, (-1) ^ x = bif Nat.bodd 0 then (-1) ^ a else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.sum_neg_one_pow_Ico | [37, 1] | [46, 74] | rw [Nat.Ico_succ_singleton, sum_singleton] | a : ℕ
⊢ ∑ x ∈ Ico a (a + 1), (-1) ^ x = bif Nat.bodd 1 then (-1) ^ a else 0 | a : ℕ
⊢ (-1) ^ a = bif Nat.bodd 1 then (-1) ^ a else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
⊢ ∑ x ∈ Ico a (a + 1), (-1) ^ x = bif Nat.bodd 1 then (-1) ^ a else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.sum_neg_one_pow_Ico | [37, 1] | [46, 74] | rfl | a : ℕ
⊢ (-1) ^ a = bif Nat.bodd 1 then (-1) ^ a else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
⊢ (-1) ^ a = bif Nat.bodd 1 then (-1) ^ a else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.sum_neg_one_pow_Ico | [37, 1] | [46, 74] | have h : a ≤ a + n := a.le_add_right n | a n : ℕ
⊢ ∑ x ∈ Ico a (a + (n + 2)), (-1) ^ x = bif (n + 2).bodd then (-1) ^ a else 0 | a n : ℕ
h : a ≤ a + n
⊢ ∑ x ∈ Ico a (a + (n + 2)), (-1) ^ x = bif (n + 2).bodd then (-1) ^ a else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a n : ℕ
⊢ ∑ x ∈ Ico a (a + (n + 2)), (-1) ^ x = bif (n + 2).bodd then (-1) ^ a else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.sum_neg_one_pow_Ico | [37, 1] | [46, 74] | rw [← add_assoc, sum_Ico_succ_top (h.trans (a + n).le_succ),
pow_succ', neg_one_mul, sum_Ico_succ_top h, sum_neg_one_pow_Ico a n,
add_neg_cancel_right, Nat.bodd_add, Nat.bodd_two, Bool.xor_false] | a n : ℕ
h : a ≤ a + n
⊢ ∑ x ∈ Ico a (a + (n + 2)), (-1) ^ x = bif (n + 2).bodd then (-1) ^ a else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a n : ℕ
h : a ≤ a + n
⊢ ∑ x ∈ Ico a (a + (n + 2)), (-1) ^ x = bif (n + 2).bodd then (-1) ^ a else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | rw [weight, latticeRect, sum_product] | q : (ℕ × ℕ) × ℕ × ℕ
⊢ weight (latticeRect q) = bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | q : (ℕ × ℕ) × ℕ × ℕ
⊢ ∑ x ∈ Ico q.1.1 (q.1.1 + q.2.1), ∑ y ∈ Ico q.1.2 (q.1.2 + q.2.2), (-1) ^ (x, y).1 * (-1) ^ (x, y).2 =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
q : (ℕ × ℕ) × ℕ × ℕ
⊢ weight (latticeRect q) = bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | simp_rw [← mul_sum] | q : (ℕ × ℕ) × ℕ × ℕ
⊢ ∑ x ∈ Ico q.1.1 (q.1.1 + q.2.1), ∑ y ∈ Ico q.1.2 (q.1.2 + q.2.2), (-1) ^ (x, y).1 * (-1) ^ (x, y).2 =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | q : (ℕ × ℕ) × ℕ × ℕ
⊢ ∑ x ∈ Ico q.1.1 (q.1.1 + q.2.1), (-1) ^ x * ∑ i ∈ Ico q.1.2 (q.1.2 + q.2.2), (-1) ^ i =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ∑ x ∈ Ico q.1.1 (q.1.1 + q.2.1), ∑ y ∈ Ico q.1.2 (q.1.2 + q.2.2), (-1) ^ (x, y).1 * (-1) ^ (x, y).2 =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | rw [← sum_mul, sum_neg_one_pow_Ico, sum_neg_one_pow_Ico] | q : (ℕ × ℕ) × ℕ × ℕ
⊢ ∑ x ∈ Ico q.1.1 (q.1.1 + q.2.1), (-1) ^ x * ∑ i ∈ Ico q.1.2 (q.1.2 + q.2.2), (-1) ^ i =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif q.2.1.bodd then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ∑ x ∈ Ico q.1.1 (q.1.1 + q.2.1), (-1) ^ x * ∑ i ∈ Ico q.1.2 (q.1.2 + q.2.2), (-1) ^ i =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | cases q.2.1.bodd | q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif q.2.1.bodd then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | case false
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif false then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif false && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
case true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif true then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif true && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif q.2.1.bodd then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | exact zero_mul _ | case false
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif false then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif false && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
case true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif true then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif true && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | case true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif true then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif true && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case false
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif false then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif false && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
case true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif true then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif true && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | cases q.2.2.bodd | case true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif true then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif true && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | case true.false
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif true then (-1) ^ q.1.1 else 0) * bif false then (-1) ^ q.1.2 else 0) =
bif true && false then (-1) ^ (q.1.1 + q.1.2) else 0
case true.true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif true then (-1) ^ q.1.1 else 0) * bif true then (-1) ^ q.1.2 else 0) =
bif true && true then (-1) ^ (q.1.1 + q.1.2) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif true then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif true && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | exacts [mul_zero _, (pow_add _ _ _).symm] | case true.false
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif true then (-1) ^ q.1.1 else 0) * bif false then (-1) ^ q.1.2 else 0) =
bif true && false then (-1) ^ (q.1.1 + q.1.2) else 0
case true.true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif true then (-1) ^ q.1.1 else 0) * bif true then (-1) ^ q.1.2 else 0) =
bif true && true then (-1) ^ (q.1.1 + q.1.2) else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case true.false
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif true then (-1) ^ q.1.1 else 0) * bif false then (-1) ^ q.1.2 else 0) =
bif true && false then (-1) ^ (q.1.1 + q.1.2) else 0
case true.true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ ((bif true then (-1) ^ q.1.1 else 0) * bif true then (-1) ^ q.1.2 else 0) =
bif true && true then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight_pos_imp | [63, 1] | [72, 69] | rw [latticeRect_weight, ← Bool.and_eq_true] | q : (ℕ × ℕ) × ℕ × ℕ
⊢ 0 < weight (latticeRect q) → (q.2.1.bodd = true ∧ q.2.2.bodd = true) ∧ (q.1.1 + q.1.2).bodd = false | q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0) →
(q.2.1.bodd && q.2.2.bodd) = true ∧ (q.1.1 + q.1.2).bodd = false | Please generate a tactic in lean4 to solve the state.
STATE:
q : (ℕ × ℕ) × ℕ × ℕ
⊢ 0 < weight (latticeRect q) → (q.2.1.bodd = true ∧ q.2.2.bodd = true) ∧ (q.1.1 + q.1.2).bodd = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight_pos_imp | [63, 1] | [72, 69] | cases q.2.1.bodd && q.2.2.bodd | q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0) →
(q.2.1.bodd && q.2.2.bodd) = true ∧ (q.1.1 + q.1.2).bodd = false | case false
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif false then (-1) ^ (q.1.1 + q.1.2) else 0) → false = true ∧ (q.1.1 + q.1.2).bodd = false
case true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif true then (-1) ^ (q.1.1 + q.1.2) else 0) → true = true ∧ (q.1.1 + q.1.2).bodd = false | Please generate a tactic in lean4 to solve the state.
STATE:
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0) →
(q.2.1.bodd && q.2.2.bodd) = true ∧ (q.1.1 + q.1.2).bodd = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight_pos_imp | [63, 1] | [72, 69] | exact λ h ↦ absurd h (le_refl 0).not_lt | case false
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif false then (-1) ^ (q.1.1 + q.1.2) else 0) → false = true ∧ (q.1.1 + q.1.2).bodd = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case false
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif false then (-1) ^ (q.1.1 + q.1.2) else 0) → false = true ∧ (q.1.1 + q.1.2).bodd = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight_pos_imp | [63, 1] | [72, 69] | rw [neg_one_pow_eq_pow_mod_two (R := ℤ), Nat.mod_two_of_bodd] | case true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif true then (-1) ^ (q.1.1 + q.1.2) else 0) → true = true ∧ (q.1.1 + q.1.2).bodd = false | case true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif true then (-1) ^ bif (q.1.1 + q.1.2).bodd then 1 else 0 else 0) → true = true ∧ (q.1.1 + q.1.2).bodd = false | Please generate a tactic in lean4 to solve the state.
STATE:
case true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif true then (-1) ^ (q.1.1 + q.1.2) else 0) → true = true ∧ (q.1.1 + q.1.2).bodd = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight_pos_imp | [63, 1] | [72, 69] | cases (q.1.1 + q.1.2).bodd | case true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif true then (-1) ^ bif (q.1.1 + q.1.2).bodd then 1 else 0 else 0) → true = true ∧ (q.1.1 + q.1.2).bodd = false | case true.false
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif true then (-1) ^ bif false then 1 else 0 else 0) → true = true ∧ false = false
case true.true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif true then (-1) ^ bif true then 1 else 0 else 0) → true = true ∧ true = false | Please generate a tactic in lean4 to solve the state.
STATE:
case true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif true then (-1) ^ bif (q.1.1 + q.1.2).bodd then 1 else 0 else 0) → true = true ∧ (q.1.1 + q.1.2).bodd = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight_pos_imp | [63, 1] | [72, 69] | exacts [λ _ ↦ ⟨rfl, rfl⟩, λ h ↦ absurd neg_one_lt_zero h.not_lt] | case true.false
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif true then (-1) ^ bif false then 1 else 0 else 0) → true = true ∧ false = false
case true.true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif true then (-1) ^ bif true then 1 else 0 else 0) → true = true ∧ true = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case true.false
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif true then (-1) ^ bif false then 1 else 0 else 0) → true = true ∧ false = false
case true.true
q : (ℕ × ℕ) × ℕ × ℕ
⊢ (0 < bif true then (-1) ^ bif true then 1 else 0 else 0) → true = true ∧ true = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | suffices ∃ i ∈ I, 0 < weight (latticeRect (Q i))
from this.elim λ i h3 ↦ ⟨i, h3.1, latticeRect_weight_pos_imp h3.2⟩ | ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : latticeRect ((0, 0), m, n) = I.disjiUnion (latticeRect ∘ Q) h
⊢ ∃ i ∈ I, ((Q i).2.1.bodd = true ∧ (Q i).2.2.bodd = true) ∧ ((Q i).1.1 + (Q i).1.2).bodd = false | ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : latticeRect ((0, 0), m, n) = I.disjiUnion (latticeRect ∘ Q) h
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i)) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : latticeRect ((0, 0), m, n) = I.disjiUnion (latticeRect ∘ Q) h
⊢ ∃ i ∈ I, ((Q i).2.1.bodd = true ∧ (Q i).2.2.bodd = true) ∧ ((Q i).1.1 + (Q i).1.2).bodd = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | apply_fun weight at h1 | ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : latticeRect ((0, 0), m, n) = I.disjiUnion (latticeRect ∘ Q) h
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i)) | ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : weight (latticeRect ((0, 0), m, n)) = weight (I.disjiUnion (latticeRect ∘ Q) h)
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i)) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : latticeRect ((0, 0), m, n) = I.disjiUnion (latticeRect ∘ Q) h
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | rw [latticeRect_weight] at h1 | ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : weight (latticeRect ((0, 0), m, n)) = weight (I.disjiUnion (latticeRect ∘ Q) h)
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i)) | ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 :
(bif ((0, 0), m, n).2.1.bodd && ((0, 0), m, n).2.2.bodd then (-1) ^ (((0, 0), m, n).1.1 + ((0, 0), m, n).1.2)
else 0) =
weight (I.disjiUnion (latticeRect ∘ Q) h)
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i)) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : weight (latticeRect ((0, 0), m, n)) = weight (I.disjiUnion (latticeRect ∘ Q) h)
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | simp only at h1 | ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 :
(bif ((0, 0), m, n).2.1.bodd && ((0, 0), m, n).2.2.bodd then (-1) ^ (((0, 0), m, n).1.1 + ((0, 0), m, n).1.2)
else 0) =
weight (I.disjiUnion (latticeRect ∘ Q) h)
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i)) | ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : (bif m.bodd && n.bodd then (-1) ^ (0 + 0) else 0) = weight (I.disjiUnion (latticeRect ∘ Q) h)
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i)) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 :
(bif ((0, 0), m, n).2.1.bodd && ((0, 0), m, n).2.2.bodd then (-1) ^ (((0, 0), m, n).1.1 + ((0, 0), m, n).1.2)
else 0) =
weight (I.disjiUnion (latticeRect ∘ Q) h)
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | rw [disjiUnion_weight_eq, Function.comp, h0.1, h0.2, add_zero] at h1 | ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : (bif m.bodd && n.bodd then (-1) ^ (0 + 0) else 0) = weight (I.disjiUnion (latticeRect ∘ Q) h)
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i)) | ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = ∑ i ∈ I, weight ((fun x => latticeRect (Q x)) i)
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i)) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : (bif m.bodd && n.bodd then (-1) ^ (0 + 0) else 0) = weight (I.disjiUnion (latticeRect ∘ Q) h)
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | apply exists_lt_of_sum_lt | ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = ∑ i ∈ I, weight ((fun x => latticeRect (Q x)) i)
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i)) | case Hlt
ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = ∑ i ∈ I, weight ((fun x => latticeRect (Q x)) i)
⊢ ∑ i ∈ I, 0 < ∑ i ∈ I, weight (latticeRect (Q i)) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = ∑ i ∈ I, weight ((fun x => latticeRect (Q x)) i)
⊢ ∃ i ∈ I, 0 < weight (latticeRect (Q i))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | rw [sum_const_zero, ← h1] | case Hlt
ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = ∑ i ∈ I, weight ((fun x => latticeRect (Q x)) i)
⊢ ∑ i ∈ I, 0 < ∑ i ∈ I, weight (latticeRect (Q i)) | case Hlt
ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = ∑ i ∈ I, weight ((fun x => latticeRect (Q x)) i)
⊢ 0 < bif true && true then (-1) ^ 0 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case Hlt
ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = ∑ i ∈ I, weight ((fun x => latticeRect (Q x)) i)
⊢ ∑ i ∈ I, 0 < ∑ i ∈ I, weight (latticeRect (Q i))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | exact Int.zero_lt_one | case Hlt
ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = ∑ i ∈ I, weight ((fun x => latticeRect (Q x)) i)
⊢ 0 < bif true && true then (-1) ^ 0 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case Hlt
ι : Type u_1
Q : ι → (ℕ × ℕ) × ℕ × ℕ
m n : ℕ
I : Finset ι
h : (↑I).PairwiseDisjoint (latticeRect ∘ Q)
h0 : m.bodd = true ∧ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = ∑ i ∈ I, weight ((fun x => latticeRect (Q x)) i)
⊢ 0 < bif true && true then (-1) ^ 0 else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | by_cases h1 : c = 0 | c m : ℤ
h : 2 * c ∣ m - 3
h0 : 2 * c ∣ f m - 3
⊢ 2 * (2 * c) ∣ m - 3 | case pos
c m : ℤ
h : 2 * c ∣ m - 3
h0 : 2 * c ∣ f m - 3
h1 : c = 0
⊢ 2 * (2 * c) ∣ m - 3
case neg
c m : ℤ
h : 2 * c ∣ m - 3
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
⊢ 2 * (2 * c) ∣ m - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
c m : ℤ
h : 2 * c ∣ m - 3
h0 : 2 * c ∣ f m - 3
⊢ 2 * (2 * c) ∣ m - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | rwa [h1, mul_zero, ← h1] | case pos
c m : ℤ
h : 2 * c ∣ m - 3
h0 : 2 * c ∣ f m - 3
h1 : c = 0
⊢ 2 * (2 * c) ∣ m - 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c m : ℤ
h : 2 * c ∣ m - 3
h0 : 2 * c ∣ f m - 3
h1 : c = 0
⊢ 2 * (2 * c) ∣ m - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | rcases h with ⟨d, h⟩ | case neg
c m : ℤ
h : 2 * c ∣ m - 3
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
⊢ 2 * (2 * c) ∣ m - 3 | case neg.intro
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
⊢ 2 * (2 * c) ∣ m - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c m : ℤ
h : 2 * c ∣ m - 3
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
⊢ 2 * (2 * c) ∣ m - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | rw [h, mul_comm] | case neg.intro
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
⊢ 2 * (2 * c) ∣ m - 3 | case neg.intro
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
⊢ 2 * c * 2 ∣ 2 * c * d | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
⊢ 2 * (2 * c) ∣ m - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | apply mul_dvd_mul_left | case neg.intro
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
⊢ 2 * c * 2 ∣ 2 * c * d | case neg.intro.h
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
⊢ 2 ∣ d | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
⊢ 2 * c * 2 ∣ 2 * c * d
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | have X : (2 : ℤ) ≠ 0 := two_ne_zero | case neg.intro.h
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
⊢ 2 ∣ d | case neg.intro.h
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
X : 2 ≠ 0
⊢ 2 ∣ d | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.h
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
⊢ 2 ∣ d
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | have X0 : (3 / 2 : ℤ) = 1 := rfl | case neg.intro.h
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
X : 2 ≠ 0
⊢ 2 ∣ d | case neg.intro.h
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
X : 2 ≠ 0
X0 : 3 / 2 = 1
⊢ 2 ∣ d | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.h
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
X : 2 ≠ 0
⊢ 2 ∣ d
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | rw [f, eq_add_of_sub_eq h, add_mul, add_sub_assoc, mul_assoc, ← mul_sub_one,
dvd_add_right ⟨_, rfl⟩, mul_assoc, add_comm, Int.add_mul_ediv_left _ _ X,
X0, add_sub_cancel_left, ← two_add_one_eq_three, add_one_mul (α := ℤ),
← mul_assoc, dvd_add_right ⟨_, rfl⟩, mul_comm] at h0 | case neg.intro.h
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
X : 2 ≠ 0
X0 : 3 / 2 = 1
⊢ 2 ∣ d | case neg.intro.h
c m : ℤ
h1 : ¬c = 0
d : ℤ
h0 : c * 2 ∣ c * d
h : m - 3 = 2 * c * d
X : 2 ≠ 0
X0 : 3 / 2 = 1
⊢ 2 ∣ d | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.h
c m : ℤ
h0 : 2 * c ∣ f m - 3
h1 : ¬c = 0
d : ℤ
h : m - 3 = 2 * c * d
X : 2 ≠ 0
X0 : 3 / 2 = 1
⊢ 2 ∣ d
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | exact Int.dvd_of_mul_dvd_mul_left h1 h0 | case neg.intro.h
c m : ℤ
h1 : ¬c = 0
d : ℤ
h0 : c * 2 ∣ c * d
h : m - 3 = 2 * c * d
X : 2 ≠ 0
X0 : 3 / 2 = 1
⊢ 2 ∣ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.h
c m : ℤ
h1 : ¬c = 0
d : ℤ
h0 : c * 2 ∣ c * d
h : m - 3 = 2 * c * d
X : 2 ≠ 0
X0 : 3 / 2 = 1
⊢ 2 ∣ d
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [iff_not_comm, not_exists] | M : ℤ
⊢ (∃ k, 2 ∣ f^[k] M) ↔ M ≠ 3 | M : ℤ
⊢ M = 3 ↔ ∀ (x : ℕ), ¬2 ∣ f^[x] M | Please generate a tactic in lean4 to solve the state.
STATE:
M : ℤ
⊢ (∃ k, 2 ∣ f^[k] M) ↔ M ≠ 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | refine ⟨λ h n ↦ ?_, λ h ↦ ?_⟩ | M : ℤ
⊢ M = 3 ↔ ∀ (x : ℕ), ¬2 ∣ f^[x] M | case refine_1
M : ℤ
h : M = 3
n : ℕ
⊢ ¬2 ∣ f^[n] M
case refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
⊢ M = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
M : ℤ
⊢ M = 3 ↔ ∀ (x : ℕ), ¬2 ∣ f^[x] M
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | suffices h0 : ∀ n k, 2 ^ (n + 1) ∣ f^[k] M - 3 by
let K := (M - 3).natAbs
refine eq_of_sub_eq_zero <| Int.eq_zero_of_abs_lt_dvd (h0 K 0) <| ?_
rw [← Int.natCast_natAbs, ← Nat.cast_ofNat (n := 2), ← Int.natCast_pow]
exact Int.ofNat_lt.mpr (K.lt_succ_self.trans K.succ.lt_two_pow) | case refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
⊢ M = 3 | case refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
⊢ ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
⊢ M = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | refine Nat.rec (λ k ↦ ?_) (λ n h0 k ↦ ?_) | case refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
⊢ ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3 | case refine_2.refine_1
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
k : ℕ
⊢ 2 ^ (Nat.zero + 1) ∣ f^[k] M - 3
case refine_2.refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 ^ (n.succ + 1) ∣ f^[k] M - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
⊢ ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | have h0 : f 3 = 3 := rfl | case refine_1
M : ℤ
h : M = 3
n : ℕ
⊢ ¬2 ∣ f^[n] M | case refine_1
M : ℤ
h : M = 3
n : ℕ
h0 : f 3 = 3
⊢ ¬2 ∣ f^[n] M | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
M : ℤ
h : M = 3
n : ℕ
⊢ ¬2 ∣ f^[n] M
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [h, Function.iterate_fixed h0, ← two_add_one_eq_three] | case refine_1
M : ℤ
h : M = 3
n : ℕ
h0 : f 3 = 3
⊢ ¬2 ∣ f^[n] M | case refine_1
M : ℤ
h : M = 3
n : ℕ
h0 : f 3 = 3
⊢ ¬2 ∣ 2 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
M : ℤ
h : M = 3
n : ℕ
h0 : f 3 = 3
⊢ ¬2 ∣ f^[n] M
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | exact Int.two_not_dvd_two_mul_add_one 1 | case refine_1
M : ℤ
h : M = 3
n : ℕ
h0 : f 3 = 3
⊢ ¬2 ∣ 2 + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
M : ℤ
h : M = 3
n : ℕ
h0 : f 3 = 3
⊢ ¬2 ∣ 2 + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | let K := (M - 3).natAbs | M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
h0 : ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
⊢ M = 3 | M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
h0 : ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
K : ℕ := (M - 3).natAbs
⊢ M = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
h0 : ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
⊢ M = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | refine eq_of_sub_eq_zero <| Int.eq_zero_of_abs_lt_dvd (h0 K 0) <| ?_ | M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
h0 : ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
K : ℕ := (M - 3).natAbs
⊢ M = 3 | M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
h0 : ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
K : ℕ := (M - 3).natAbs
⊢ |M - 3| < 2 ^ (K + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
h0 : ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
K : ℕ := (M - 3).natAbs
⊢ M = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [← Int.natCast_natAbs, ← Nat.cast_ofNat (n := 2), ← Int.natCast_pow] | M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
h0 : ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
K : ℕ := (M - 3).natAbs
⊢ |M - 3| < 2 ^ (K + 1) | M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
h0 : ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
K : ℕ := (M - 3).natAbs
⊢ ↑(M - 3).natAbs < ↑(OfNat.ofNat 2 ^ (K + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
h0 : ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
K : ℕ := (M - 3).natAbs
⊢ |M - 3| < 2 ^ (K + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | exact Int.ofNat_lt.mpr (K.lt_succ_self.trans K.succ.lt_two_pow) | M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
h0 : ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
K : ℕ := (M - 3).natAbs
⊢ ↑(M - 3).natAbs < ↑(OfNat.ofNat 2 ^ (K + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
h0 : ∀ (n k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
K : ℕ := (M - 3).natAbs
⊢ ↑(M - 3).natAbs < ↑(OfNat.ofNat 2 ^ (K + 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [Int.dvd_iff_emod_eq_zero, Nat.zero_add, pow_one,
← Int.even_iff, Int.even_sub', Int.odd_iff_not_even,
Int.even_iff, ← Int.dvd_iff_emod_eq_zero] | case refine_2.refine_1
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
k : ℕ
⊢ 2 ^ (Nat.zero + 1) ∣ f^[k] M - 3 | case refine_2.refine_1
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
k : ℕ
⊢ ¬2 ∣ f^[k] M ↔ Odd 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_1
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
k : ℕ
⊢ 2 ^ (Nat.zero + 1) ∣ f^[k] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | exact iff_of_true (h k) (Int.odd_iff.mpr rfl) | case refine_2.refine_1
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
k : ℕ
⊢ ¬2 ∣ f^[k] M ↔ Odd 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_1
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
k : ℕ
⊢ ¬2 ∣ f^[k] M ↔ Odd 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [pow_succ', pow_succ'] | case refine_2.refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 ^ (n.succ + 1) ∣ f^[k] M - 3 | case refine_2.refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 * (2 * 2 ^ n) ∣ f^[k] M - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 ^ (n.succ + 1) ∣ f^[k] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | refine main_claim ?_ ?_ | case refine_2.refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 * (2 * 2 ^ n) ∣ f^[k] M - 3 | case refine_2.refine_2.refine_1
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 * 2 ^ n ∣ f^[k] M - 3
case refine_2.refine_2.refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 * 2 ^ n ∣ f (f^[k] M) - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 * (2 * 2 ^ n) ∣ f^[k] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [← pow_succ'] | case refine_2.refine_2.refine_1
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 * 2 ^ n ∣ f^[k] M - 3 | case refine_2.refine_2.refine_1
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 ^ (n + 1) ∣ f^[k] M - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_1
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 * 2 ^ n ∣ f^[k] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | exact h0 k | case refine_2.refine_2.refine_1
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 ^ (n + 1) ∣ f^[k] M - 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_1
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 ^ (n + 1) ∣ f^[k] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [← pow_succ', ← f.iterate_succ_apply'] | case refine_2.refine_2.refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 * 2 ^ n ∣ f (f^[k] M) - 3 | case refine_2.refine_2.refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 ^ (n + 1) ∣ f^[k.succ] M - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 * 2 ^ n ∣ f (f^[k] M) - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | exact h0 (k + 1) | case refine_2.refine_2.refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 ^ (n + 1) ∣ f^[k.succ] M - 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_2
M : ℤ
h : ∀ (x : ℕ), ¬2 ∣ f^[x] M
n : ℕ
h0 : ∀ (k : ℕ), 2 ^ (n + 1) ∣ f^[k] M - 3
k : ℕ
⊢ 2 ^ (n + 1) ∣ f^[k.succ] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2008/A3/A3.lean | IMOSL.IMO2008A3.final_solution_part_2_general | [79, 1] | [94, 42] | rwa [prod_lex_lt_iff, Nat.succ_lt_succ_iff, Nat.succ_inj', ← prod_lex_lt_iff] | β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p q : Lex (ℕ × β)
h2 : p < q
⊢ (p.1.succ, p.2) < (q.1.succ, q.2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p q : Lex (ℕ × β)
h2 : p < q
⊢ (p.1.succ, p.2) < (q.1.succ, q.2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2008/A3/A3.lean | IMOSL.IMO2008A3.final_solution_part_2_general | [79, 1] | [94, 42] | rw [prod_lex_lt_iff] at h2 ⊢ | β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p q : Lex (ℕ × β)
h2 : p < q
⊢ (p.1, φ^[3 ^ p.1] p.2) < (q.1, φ^[3 ^ q.1] q.2) | β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p q : Lex (ℕ × β)
h2 : p.1 < q.1 ∨ p.1 = q.1 ∧ p.2 < q.2
⊢ (p.1, φ^[3 ^ p.1] p.2).1 < (q.1, φ^[3 ^ q.1] q.2).1 ∨
(p.1, φ^[3 ^ p.1] p.2).1 = (q.1, φ^[3 ^ q.1] q.2).1 ∧ (p.1, φ^[3 ^ p.1] p.2).2 < (q.1, φ^[3 ^ q.1] q.2).2 | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p q : Lex (ℕ × β)
h2 : p < q
⊢ (p.1, φ^[3 ^ p.1] p.2) < (q.1, φ^[3 ^ q.1] q.2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2008/A3/A3.lean | IMOSL.IMO2008A3.final_solution_part_2_general | [79, 1] | [94, 42] | refine h2.imp_right λ h3 ↦ ⟨h3.1, (h.iterate _ h3.2).trans_eq ?_⟩ | β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p q : Lex (ℕ × β)
h2 : p.1 < q.1 ∨ p.1 = q.1 ∧ p.2 < q.2
⊢ (p.1, φ^[3 ^ p.1] p.2).1 < (q.1, φ^[3 ^ q.1] q.2).1 ∨
(p.1, φ^[3 ^ p.1] p.2).1 = (q.1, φ^[3 ^ q.1] q.2).1 ∧ (p.1, φ^[3 ^ p.1] p.2).2 < (q.1, φ^[3 ^ q.1] q.2).2 | β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p q : Lex (ℕ × β)
h2 : p.1 < q.1 ∨ p.1 = q.1 ∧ p.2 < q.2
h3 : p.1 = q.1 ∧ p.2 < q.2
⊢ φ^[3 ^ p.1] q.2 = (q.1, φ^[3 ^ q.1] q.2).2 | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p q : Lex (ℕ × β)
h2 : p.1 < q.1 ∨ p.1 = q.1 ∧ p.2 < q.2
⊢ (p.1, φ^[3 ^ p.1] p.2).1 < (q.1, φ^[3 ^ q.1] q.2).1 ∨
(p.1, φ^[3 ^ p.1] p.2).1 = (q.1, φ^[3 ^ q.1] q.2).1 ∧ (p.1, φ^[3 ^ p.1] p.2).2 < (q.1, φ^[3 ^ q.1] q.2).2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2008/A3/A3.lean | IMOSL.IMO2008A3.final_solution_part_2_general | [79, 1] | [94, 42] | rw [← h3.1] | β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p q : Lex (ℕ × β)
h2 : p.1 < q.1 ∨ p.1 = q.1 ∧ p.2 < q.2
h3 : p.1 = q.1 ∧ p.2 < q.2
⊢ φ^[3 ^ p.1] q.2 = (q.1, φ^[3 ^ q.1] q.2).2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p q : Lex (ℕ × β)
h2 : p.1 < q.1 ∨ p.1 = q.1 ∧ p.2 < q.2
h3 : p.1 = q.1 ∧ p.2 < q.2
⊢ φ^[3 ^ p.1] q.2 = (q.1, φ^[3 ^ q.1] q.2).2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2008/A3/A3.lean | IMOSL.IMO2008A3.final_solution_part_2_general | [79, 1] | [94, 42] | refine prod_lex_lt_iff.mpr <| Or.inr <| ⟨rfl, ?_⟩ | β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p : Lex (ℕ × β)
⊢ (((p.1, φ^[3 ^ p.1] p.2).1, φ^[3 ^ (p.1, φ^[3 ^ p.1] p.2).1] (p.1, φ^[3 ^ p.1] p.2).2).1.succ,
((p.1, φ^[3 ^ p.1] p.2).1, φ^[3 ^ (p.1, φ^[3 ^ p.1] p.2).1] (p.1, φ^[3 ^ p.1] p.2).2).2) <
((p.1.succ, p.2).1, φ^[3 ^ (p.1.succ, p.2).1] (p.1.succ, p.2).2) | β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p : Lex (ℕ × β)
⊢ (((p.1, φ^[3 ^ p.1] p.2).1, φ^[3 ^ (p.1, φ^[3 ^ p.1] p.2).1] (p.1, φ^[3 ^ p.1] p.2).2).1.succ,
((p.1, φ^[3 ^ p.1] p.2).1, φ^[3 ^ (p.1, φ^[3 ^ p.1] p.2).1] (p.1, φ^[3 ^ p.1] p.2).2).2).2 <
((p.1.succ, p.2).1, φ^[3 ^ (p.1.succ, p.2).1] (p.1.succ, p.2).2).2 | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p : Lex (ℕ × β)
⊢ (((p.1, φ^[3 ^ p.1] p.2).1, φ^[3 ^ (p.1, φ^[3 ^ p.1] p.2).1] (p.1, φ^[3 ^ p.1] p.2).2).1.succ,
((p.1, φ^[3 ^ p.1] p.2).1, φ^[3 ^ (p.1, φ^[3 ^ p.1] p.2).1] (p.1, φ^[3 ^ p.1] p.2).2).2) <
((p.1.succ, p.2).1, φ^[3 ^ (p.1.succ, p.2).1] (p.1.succ, p.2).2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2008/A3/A3.lean | IMOSL.IMO2008A3.final_solution_part_2_general | [79, 1] | [94, 42] | rw [← Function.iterate_add_apply, ← two_mul, pow_succ'] | β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p : Lex (ℕ × β)
⊢ (((p.1, φ^[3 ^ p.1] p.2).1, φ^[3 ^ (p.1, φ^[3 ^ p.1] p.2).1] (p.1, φ^[3 ^ p.1] p.2).2).1.succ,
((p.1, φ^[3 ^ p.1] p.2).1, φ^[3 ^ (p.1, φ^[3 ^ p.1] p.2).1] (p.1, φ^[3 ^ p.1] p.2).2).2).2 <
((p.1.succ, p.2).1, φ^[3 ^ (p.1.succ, p.2).1] (p.1.succ, p.2).2).2 | β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p : Lex (ℕ × β)
⊢ (((p.1, φ^[3 ^ p.1] p.2).1, φ^[2 * 3 ^ (p.1, φ^[3 ^ p.1] p.2).1] p.2).1.succ,
((p.1, φ^[3 ^ p.1] p.2).1, φ^[2 * 3 ^ (p.1, φ^[3 ^ p.1] p.2).1] p.2).2).2 <
((p.1.succ, p.2).1, φ^[3 * 3 ^ p.1] (p.1.succ, p.2).2).2 | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p : Lex (ℕ × β)
⊢ (((p.1, φ^[3 ^ p.1] p.2).1, φ^[3 ^ (p.1, φ^[3 ^ p.1] p.2).1] (p.1, φ^[3 ^ p.1] p.2).2).1.succ,
((p.1, φ^[3 ^ p.1] p.2).1, φ^[3 ^ (p.1, φ^[3 ^ p.1] p.2).1] (p.1, φ^[3 ^ p.1] p.2).2).2).2 <
((p.1.succ, p.2).1, φ^[3 ^ (p.1.succ, p.2).1] (p.1.succ, p.2).2).2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2008/A3/A3.lean | IMOSL.IMO2008A3.final_solution_part_2_general | [79, 1] | [94, 42] | exact h.strictMono_iterate_of_lt_map (h0 p.2)
(Nat.mul_lt_mul_of_pos_right (Nat.lt_succ_self 2)
(pow_pos (Nat.succ_pos 2) _)) | β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p : Lex (ℕ × β)
⊢ (((p.1, φ^[3 ^ p.1] p.2).1, φ^[2 * 3 ^ (p.1, φ^[3 ^ p.1] p.2).1] p.2).1.succ,
((p.1, φ^[3 ^ p.1] p.2).1, φ^[2 * 3 ^ (p.1, φ^[3 ^ p.1] p.2).1] p.2).2).2 <
((p.1.succ, p.2).1, φ^[3 * 3 ^ p.1] (p.1.succ, p.2).2).2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type u_1
φ : β → β
inst✝ : Preorder β
h : StrictMono φ
h0 : ∀ (x : β), x < φ x
p : Lex (ℕ × β)
⊢ (((p.1, φ^[3 ^ p.1] p.2).1, φ^[2 * 3 ^ (p.1, φ^[3 ^ p.1] p.2).1] p.2).1.succ,
((p.1, φ^[3 ^ p.1] p.2).1, φ^[2 * 3 ^ (p.1, φ^[3 ^ p.1] p.2).1] p.2).2).2 <
((p.1.succ, p.2).1, φ^[3 * 3 ^ p.1] (p.1.succ, p.2).2).2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | have h0 : ⌊f 0⌋ = 1 ∨ f 0 = 0 := by
have h0 := h 0 0
rw [zsmul_zero, ← sub_eq_zero, ← mul_one_sub, mul_eq_zero] at h0
exact h0.symm.imp_left λ h0 ↦ Int.cast_eq_one.mp (eq_of_sub_eq_zero h0).symm | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1 ∨ f 0 = 0
⊢ (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | revert h0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1 ∨ f 0 = 0
⊢ (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 → (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1 ∨ f 0 = 0
⊢ (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | refine Or.imp (λ h0 ↦ ?_) (λ h0 ↦ ?_) | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 → (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0 | case refine_1
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1
⊢ ∃ C, ⌊C⌋ = 1 ∧ f = fun x => C
case refine_2
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 → (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | have h0 := h 0 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f (⌊0⌋ • 0) = f 0 * ↑⌊f 0⌋
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rw [zsmul_zero, ← sub_eq_zero, ← mul_one_sub, mul_eq_zero] at h0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f (⌊0⌋ • 0) = f 0 * ↑⌊f 0⌋
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0 ∨ 1 - ↑⌊f 0⌋ = 0
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f (⌊0⌋ • 0) = f 0 * ↑⌊f 0⌋
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | exact h0.symm.imp_left λ h0 ↦ Int.cast_eq_one.mp (eq_of_sub_eq_zero h0).symm | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0 ∨ 1 - ↑⌊f 0⌋ = 0
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0 ∨ 1 - ↑⌊f 0⌋ = 0
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | refine ⟨f 0, h0, funext λ x ↦ ?_⟩ | case refine_1
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1
⊢ ∃ C, ⌊C⌋ = 1 ∧ f = fun x => C | case refine_1
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1
x : R
⊢ f x = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1
⊢ ∃ C, ⌊C⌋ = 1 ∧ f = fun x => C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rw [← zsmul_zero ⌊x⌋, h, h0, Int.cast_one, mul_one] | case refine_1
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1
x : R
⊢ f x = f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1
x : R
⊢ f x = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rcases exists_between (zero_lt_one' R) with ⟨c, hc⟩ | case refine_2
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
⊢ f = 0 | case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | replace h0 : ⌊f c⌋ = 0 := by
specialize h c c
have h1 : ⌊c⌋ = 0 := Int.floor_eq_zero_iff.mpr ⟨hc.1.le, hc.2⟩
rw [h1, zero_zsmul, h0, zero_eq_mul] at h
exact h.elim (λ h ↦ h ▸ Int.floor_zero) Int.cast_eq_zero.mp | case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
⊢ f = 0 | case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | replace h0 : f (-c) = 0 := by
specialize h (-1) c
have h1 : ⌊(-1 : R)⌋ = -1 := by
rw [← Int.cast_one, ← Int.cast_neg, Int.floor_intCast]
rwa [h1, neg_one_zsmul, h0, Int.cast_zero, mul_zero] at h | case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
⊢ f = 0 | case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : ⌊f c⌋ = 0
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | funext y | case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
⊢ f = 0 | case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
⊢ f y = 0 y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | specialize h (-c) (-y) | case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
⊢ f y = 0 y | case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
⊢ f y = 0 y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
⊢ f y = 0 y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | have h1 : ⌊-c⌋ = -1 := by
rw [Int.floor_eq_iff, Int.cast_neg, Int.cast_one, neg_add_self]
exact ⟨neg_le_neg hc.2.le, neg_lt_zero.mpr hc.1⟩ | case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
⊢ f y = 0 y | case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
h1 : ⌊-c⌋ = -1
⊢ f y = 0 y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
⊢ f y = 0 y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rwa [h0, zero_mul, h1, neg_one_zsmul, neg_neg] at h | case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
h1 : ⌊-c⌋ = -1
⊢ f y = 0 y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.h
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
c : R
hc : 0 < c ∧ c < 1
h0 : f (-c) = 0
y : R
h : f (⌊-c⌋ • -y) = f (-c) * ↑⌊f (-y)⌋
h1 : ⌊-c⌋ = -1
⊢ f y = 0 y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | specialize h c c | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
⊢ ⌊f c⌋ = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f (⌊c⌋ • c) = f c * ↑⌊f c⌋
⊢ ⌊f c⌋ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
⊢ ⌊f c⌋ = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | have h1 : ⌊c⌋ = 0 := Int.floor_eq_zero_iff.mpr ⟨hc.1.le, hc.2⟩ | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f (⌊c⌋ • c) = f c * ↑⌊f c⌋
⊢ ⌊f c⌋ = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f (⌊c⌋ • c) = f c * ↑⌊f c⌋
h1 : ⌊c⌋ = 0
⊢ ⌊f c⌋ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f (⌊c⌋ • c) = f c * ↑⌊f c⌋
⊢ ⌊f c⌋ = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rw [h1, zero_zsmul, h0, zero_eq_mul] at h | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f (⌊c⌋ • c) = f c * ↑⌊f c⌋
h1 : ⌊c⌋ = 0
⊢ ⌊f c⌋ = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f c = 0 ∨ ↑⌊f c⌋ = 0
h1 : ⌊c⌋ = 0
⊢ ⌊f c⌋ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f (⌊c⌋ • c) = f c * ↑⌊f c⌋
h1 : ⌊c⌋ = 0
⊢ ⌊f c⌋ = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | exact h.elim (λ h ↦ h ▸ Int.floor_zero) Int.cast_eq_zero.mp | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f c = 0 ∨ ↑⌊f c⌋ = 0
h1 : ⌊c⌋ = 0
⊢ ⌊f c⌋ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h0 : f 0 = 0
c : R
hc : 0 < c ∧ c < 1
h : f c = 0 ∨ ↑⌊f c⌋ = 0
h1 : ⌊c⌋ = 0
⊢ ⌊f c⌋ = 0
TACTIC:
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