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/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | have h3 : ∀ n, f n ≠ n := λ n h3 ↦ h2 n <| Function.iterate_fixed h3 2 | case intro
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
h2 : ∀ (n : ℕ), f (f n) ≠ n
⊢ f = Nat.succ ∨ f = answer2 | case intro
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
h2 : ∀ (n : ℕ), f (f n) ≠ n
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rcases h1 with ⟨h1, h4, (rfl | rfl) | h5⟩ | case intro
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : (f (f 0) = 0 ∨ f (f 0) = (f 0).succ) ∨ f (f 0) = Nat.succ 0
h4 : (f 0 = 0 ∨ f 0 = (f 0).succ) ∨ f 0 = Nat.succ 0
⊢ f = Nat.succ ∨ f = answer2
case intro.intro.intro.inl.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : (f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ) ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : (f (f 0).succ = 0 ∨ f (f 0).succ = (f 0).succ) ∨ f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2
case intro.intro.intro.inr
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h1 : (f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ
h4 : (f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ
h5 : a = a.succ
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [singleton_subset_iff, mem_union,
mem_singleton, mem_insert, mem_singleton] at h1 | case intro
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ {a} ⊆ {0, (f 0).succ} ∪ {a.succ}
⊢ f = Nat.succ ∨ f = answer2 | case intro
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ {a} ⊆ {0, (f 0).succ} ∪ {a.succ}
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | apply absurd (iter_four_eq_add_four h n) | f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
n : ℕ
h2 : f (f n) = n
⊢ False | f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
n : ℕ
h2 : f (f n) = n
⊢ ¬f^[4] n = n + 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
n : ℕ
h2 : f (f n) = n
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | change f (f (f (f n))) ≠ n + 4 | f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
n : ℕ
h2 : f (f n) = n
⊢ ¬f^[4] n = n + 4 | f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
n : ℕ
h2 : f (f n) = n
⊢ f (f (f (f n))) ≠ n + 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
n : ℕ
h2 : f (f n) = n
⊢ ¬f^[4] n = n + 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h2, h2, self_ne_add_right] | f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
n : ℕ
h2 : f (f n) = n
⊢ f (f (f (f n))) ≠ n + 4 | f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
n : ℕ
h2 : f (f n) = n
⊢ 4 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
n : ℕ
h2 : f (f n) = n
⊢ f (f (f (f n))) ≠ n + 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | exact Nat.succ_ne_zero 3 | f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
n : ℕ
h2 : f (f n) = n
⊢ 4 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ) ∧
((f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ) ∧ ((a = 0 ∨ a = (f 0).succ) ∨ a = a.succ)
n : ℕ
h2 : f (f n) = n
⊢ 4 ≠ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [or_iff_right (h3 0), or_iff_right (f 0).succ_ne_self.symm] at h4 | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : (f (f 0) = 0 ∨ f (f 0) = (f 0).succ) ∨ f (f 0) = Nat.succ 0
h4 : (f 0 = 0 ∨ f 0 = (f 0).succ) ∨ f 0 = Nat.succ 0
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : (f (f 0) = 0 ∨ f (f 0) = (f 0).succ) ∨ f (f 0) = Nat.succ 0
h4 : f 0 = Nat.succ 0
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : (f (f 0) = 0 ∨ f (f 0) = (f 0).succ) ∨ f (f 0) = Nat.succ 0
h4 : (f 0 = 0 ∨ f 0 = (f 0).succ) ∨ f 0 = Nat.succ 0
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [or_iff_right (h2 0), h4, or_iff_left (h3 1)] at h1 | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : (f (f 0) = 0 ∨ f (f 0) = (f 0).succ) ∨ f (f 0) = Nat.succ 0
h4 : f 0 = Nat.succ 0
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : (f (f 0) = 0 ∨ f (f 0) = (f 0).succ) ∨ f (f 0) = Nat.succ 0
h4 : f 0 = Nat.succ 0
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | have h5 : f (f (f 0)) = f 1 + 1 := h 0 | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | have h6 : f (f (f (f 0))) = 4 := iter_four_zero_eq_four h | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
h6 : f (f (f (f 0))) = 4
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h5, h1] at h6 | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
h6 : f (f (f (f 0))) = 4
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
h6 : f (f (f (f 0))) = 4
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h4, h1] at h5 | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | left | case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inl.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
⊢ f = Nat.succ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | refine funext (add_four_induction h4 h1 h5 h6 ?_) | case intro.intro.intro.inl.inl.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
⊢ f = Nat.succ | case intro.intro.intro.inl.inl.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
⊢ ∀ (n : ℕ), f n = n.succ → f (n + 4) = (n + 4).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
⊢ f = Nat.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | intro n h7 | case intro.intro.intro.inl.inl.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
⊢ ∀ (n : ℕ), f n = n.succ → f (n + 4) = (n + 4).succ | case intro.intro.intro.inl.inl.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
n : ℕ
h7 : f n = n.succ
⊢ f (n + 4) = (n + 4).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
⊢ ∀ (n : ℕ), f n = n.succ → f (n + 4) = (n + 4).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [map_add_four h, h7] | case intro.intro.intro.inl.inl.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
n : ℕ
h7 : f n = n.succ
⊢ f (n + 4) = (n + 4).succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
n : ℕ
h7 : f n = n.succ
⊢ f (n + 4) = (n + 4).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [or_iff_left (h3 _)] at h4 | case intro.intro.intro.inl.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : (f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ) ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : (f (f 0).succ = 0 ∨ f (f 0).succ = (f 0).succ) ∨ f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : (f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ) ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0 ∨ f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : (f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ) ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : (f (f 0).succ = 0 ∨ f (f 0).succ = (f 0).succ) ∨ f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [or_iff_left (h2 _)] at h1 | case intro.intro.intro.inl.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : (f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ) ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0 ∨ f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0 ∨ f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : (f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ) ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0 ∨ f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rcases h4 with h4 | h4 | case intro.intro.intro.inl.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0 ∨ f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inr.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0
⊢ f = Nat.succ ∨ f = answer2
case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0 ∨ f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h4, or_iff_left (h3 _)] at h1 | case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | have h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1 := h (f 0).succ | case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | have h6 : f (f (f (f (f 0 + 1)))) = f 0 + 5 := iter_four_eq_add_four h _ | case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
h6 : f (f (f (f (f 0 + 1)))) = f 0 + 5
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h5, h1, zero_add] at h6 | case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
h6 : f (f (f (f (f 0 + 1)))) = f 0 + 5
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
h6 : f 1 = f 0 + 5
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
h6 : f (f (f (f (f 0 + 1)))) = f 0 + 5
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h4, h1, zero_add] at h5 | case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
h6 : f 1 = f 0 + 5
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f 0 = 1
h6 : f 1 = f 0 + 5
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
h6 : f 1 = f 0 + 5
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h5] at h1 h4 h6 | case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f 0 = 1
h6 : f 1 = f 0 + 5
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f 0 = 1
h6 : f 1 = f 0 + 5
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | right | case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inr.inr.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
⊢ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | refine funext (add_four_induction h5 h6 h4 h1 ?_) | case intro.intro.intro.inl.inr.inr.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
⊢ f = answer2 | case intro.intro.intro.inl.inr.inr.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
⊢ ∀ (n : ℕ), f n = answer2 n → f (n + 4) = answer2 (n + 4) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
⊢ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | intro n h7 | case intro.intro.intro.inl.inr.inr.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
⊢ ∀ (n : ℕ), f n = answer2 n → f (n + 4) = answer2 (n + 4) | case intro.intro.intro.inl.inr.inr.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
n : ℕ
h7 : f n = answer2 n
⊢ f (n + 4) = answer2 (n + 4) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
⊢ ∀ (n : ℕ), f n = answer2 n → f (n + 4) = answer2 (n + 4)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [map_add_four h, h7, answer2] | case intro.intro.intro.inl.inr.inr.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
n : ℕ
h7 : f n = answer2 n
⊢ f (n + 4) = answer2 (n + 4) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr.h
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
n : ℕ
h7 : f n = answer2 n
⊢ f (n + 4) = answer2 (n + 4)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h4, or_iff_right (h3 0)] at h1 | case intro.intro.intro.inl.inr.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0
⊢ f = Nat.succ ∨ f = answer2 | case intro.intro.intro.inl.inr.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f 0 = (f 0).succ.succ
h4 : f (f 0).succ = 0
⊢ f = Nat.succ ∨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 ∨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | exact absurd h1 (Nat.lt_succ.mpr (f 0).le_succ).ne | case intro.intro.intro.inl.inr.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f 0 = (f 0).succ.succ
h4 : f (f 0).succ = 0
⊢ f = Nat.succ ∨ f = answer2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inl
f : ℕ → ℕ
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f 0 = (f 0).succ.succ
h4 : f (f 0).succ = 0
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | exact absurd h5 a.lt_succ_self.ne | case intro.intro.intro.inr
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h1 : (f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ
h4 : (f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ
h5 : a = a.succ
⊢ f = Nat.succ ∨ f = answer2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inr
f : ℕ → ℕ
h : good f
a : ℕ
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h2 : ∀ (n : ℕ), f (f n) ≠ n
h3 : ∀ (n : ℕ), f n ≠ n
h1 : (f (f a) = 0 ∨ f (f a) = (f 0).succ) ∨ f (f a) = a.succ
h4 : (f a = 0 ∨ f a = (f 0).succ) ∨ f a = a.succ
h5 : a = a.succ
⊢ f = Nat.succ ∨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/SeqMax.lean | IMOSL.Extra.exists_map_eq_seqMax | [40, 1] | [46, 70] | rcases le_total (seqMax f n) (f n.succ) with h | h | α : Type u_1
inst✝ : LinearOrder α
f : ℕ → α
n : ℕ
⊢ ∃ k, k ≤ n + 1 ∧ f k = seqMax f (n + 1) | case inl
α : Type u_1
inst✝ : LinearOrder α
f : ℕ → α
n : ℕ
h : seqMax f n ≤ f n.succ
⊢ ∃ k, k ≤ n + 1 ∧ f k = seqMax f (n + 1)
case inr
α : Type u_1
inst✝ : LinearOrder α
f : ℕ → α
n : ℕ
h : f n.succ ≤ seqMax f n
⊢ ∃ k, k ≤ n + 1 ∧ f k = seqMax f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
f : ℕ → α
n : ℕ
⊢ ∃ k, k ≤ n + 1 ∧ f k = seqMax f (n + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/SeqMax.lean | IMOSL.Extra.exists_map_eq_seqMax | [40, 1] | [46, 70] | exact ⟨n + 1, le_refl (n + 1), (max_eq_right h).symm⟩ | case inl
α : Type u_1
inst✝ : LinearOrder α
f : ℕ → α
n : ℕ
h : seqMax f n ≤ f n.succ
⊢ ∃ k, k ≤ n + 1 ∧ f k = seqMax f (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝ : LinearOrder α
f : ℕ → α
n : ℕ
h : seqMax f n ≤ f n.succ
⊢ ∃ k, k ≤ n + 1 ∧ f k = seqMax f (n + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/SeqMax.lean | IMOSL.Extra.exists_map_eq_seqMax | [40, 1] | [46, 70] | rcases exists_map_eq_seqMax n with ⟨k, h0, h1⟩ | case inr
α : Type u_1
inst✝ : LinearOrder α
f : ℕ → α
n : ℕ
h : f n.succ ≤ seqMax f n
⊢ ∃ k, k ≤ n + 1 ∧ f k = seqMax f (n + 1) | case inr.intro.intro
α : Type u_1
inst✝ : LinearOrder α
f : ℕ → α
n : ℕ
h : f n.succ ≤ seqMax f n
k : ℕ
h0 : k ≤ n
h1 : f k = seqMax f n
⊢ ∃ k, k ≤ n + 1 ∧ f k = seqMax f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
f : ℕ → α
n : ℕ
h : f n.succ ≤ seqMax f n
⊢ ∃ k, k ≤ n + 1 ∧ f k = seqMax f (n + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/SeqMax.lean | IMOSL.Extra.exists_map_eq_seqMax | [40, 1] | [46, 70] | exact ⟨k, n.le_succ.trans' h0, h1.trans (max_eq_left h).symm⟩ | case inr.intro.intro
α : Type u_1
inst✝ : LinearOrder α
f : ℕ → α
n : ℕ
h : f n.succ ≤ seqMax f n
k : ℕ
h0 : k ≤ n
h1 : f k = seqMax f n
⊢ ∃ k, k ≤ n + 1 ∧ f k = seqMax f (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro
α : Type u_1
inst✝ : LinearOrder α
f : ℕ → α
n : ℕ
h : f n.succ ≤ seqMax f n
k : ℕ
h0 : k ≤ n
h1 : f k = seqMax f n
⊢ ∃ k, k ≤ n + 1 ∧ f k = seqMax f (n + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq1 | [30, 1] | [32, 87] | have h0 := hf.is_good (x + 1) (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (-x) = f (x + 1) * f (-1) + f x | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f ((x + 1) * -1 + 1) = f (x + 1) * f (-1) + f (x + 1 + -1)
⊢ f (-x) = f (x + 1) * f (-1) + f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (-x) = f (x + 1) * f (-1) + f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq1 | [30, 1] | [32, 87] | rwa [add_neg_cancel_right, mul_neg_one (x + 1), neg_add, neg_add_cancel_right] at h0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f ((x + 1) * -1 + 1) = f (x + 1) * f (-1) + f (x + 1 + -1)
⊢ f (-x) = f (x + 1) * f (-1) + f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f ((x + 1) * -1 + 1) = f (x + 1) * f (-1) + f (x + 1 + -1)
⊢ f (-x) = f (x + 1) * f (-1) + f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq2 | [35, 1] | [40, 38] | replace hf := Eq1 hf | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (-x) = -f (x + 2) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
⊢ f (-x) = -f (x + 2) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (-x) = -f (x + 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq2 | [35, 1] | [40, 38] | have h0 := hf (-(x + 1)) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
⊢ f (-x) = -f (x + 2) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
h0 : f (- -(x + 1)) = f (-(x + 1) + 1) * f (-1) + f (-(x + 1))
⊢ f (-x) = -f (x + 2) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
⊢ f (-x) = -f (x + 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq2 | [35, 1] | [40, 38] | rw [neg_neg, hf (x + 1), neg_add, neg_add_cancel_right, ← add_assoc, self_eq_add_left,
← add_mul, mul_eq_zero, or_iff_left h, add_assoc, one_add_one_eq_two] at h0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
h0 : f (- -(x + 1)) = f (-(x + 1) + 1) * f (-1) + f (-(x + 1))
⊢ f (-x) = -f (x + 2) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
h0 : f (-x) + f (x + 2) = 0
⊢ f (-x) = -f (x + 2) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
h0 : f (- -(x + 1)) = f (-(x + 1) + 1) * f (-1) + f (-(x + 1))
⊢ f (-x) = -f (x + 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq2 | [35, 1] | [40, 38] | exact eq_neg_of_add_eq_zero_left h0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
h0 : f (-x) + f (x + 2) = 0
⊢ f (-x) = -f (x + 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
h0 : f (-x) + f (x + 2) = 0
⊢ f (-x) = -f (x + 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.map_two | [42, 1] | [43, 66] | rw [← zero_add 2, ← neg_inj, ← Eq2 hf h, neg_zero, hf.map_zero] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f 2 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq3 | [46, 1] | [47, 54] | rw [neg_add, ← hf.is_good, neg_mul_neg, hf.is_good] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x y : R
⊢ f (-x) * f (-y) + f (-(x + y)) = f x * f y + f (x + y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x y : R
⊢ f (-x) * f (-y) + f (-(x + y)) = f x * f y + f (x + y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq4 | [50, 1] | [51, 79] | rw [hf.is_good, map_two hf h, one_mul, add_comm 2, Eq2 hf h, sub_neg_eq_add] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (2 * x + 1) = f x - f (-x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (2 * x + 1) = f x - f (-x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | have h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x)) := by
have h0 : x * -x = -x * x := by rw [mul_neg, neg_mul]
rw [sub_mul, mul_add, mul_add, map_commute_of_commute hf.is_good h0,
add_comm, add_sub_add_left_eq_sub] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | have h1 := Eq3 hf x x | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1 : f (-x) * f (-x) + f (-(x + x)) = f x * f x + f (x + x)
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | rw [add_comm, ← sub_eq_sub_iff_add_eq_add, h0, Eq1 hf, add_sub_cancel_right,
← two_mul, Eq4 hf h, ← sub_eq_zero, ← mul_sub, mul_eq_zero] at h1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1 : f (-x) * f (-x) + f (-(x + x)) = f x * f x + f (x + x)
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1 : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1 : f (-x) * f (-x) + f (-(x + x)) = f x * f x + f (x + x)
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | refine h1.imp (λ h1 ↦ ?_) (λ h1 ↦ (eq_of_sub_eq_zero h1).symm) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1 : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1✝ : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
h1 : f x - f (-x) = 0
⊢ f (x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1 : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | rw [Eq1 hf, sub_add_cancel_right, neg_eq_zero, mul_eq_zero] at h1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1✝ : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
h1 : f x - f (-x) = 0
⊢ f (x + 1) = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1✝ : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
h1 : f (x + 1) = 0 ∨ f (-1) = 0
⊢ f (x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1✝ : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
h1 : f x - f (-x) = 0
⊢ f (x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | exact h1.resolve_right h | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1✝ : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
h1 : f (x + 1) = 0 ∨ f (-1) = 0
⊢ f (x + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1✝ : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
h1 : f (x + 1) = 0 ∨ f (-1) = 0
⊢ f (x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | have h0 : x * -x = -x * x := by rw [mul_neg, neg_mul] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x)) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : x * -x = -x * x
⊢ f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | rw [sub_mul, mul_add, mul_add, map_commute_of_commute hf.is_good h0,
add_comm, add_sub_add_left_eq_sub] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : x * -x = -x * x
⊢ f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : x * -x = -x * x
⊢ f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | rw [mul_neg, neg_mul] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ x * -x = -x * x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ x * -x = -x * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | have h1 := Eq1 hf x | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f (x + 1) * f (-1) + f x
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | rw [h0, zero_mul, zero_add] at h1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f (x + 1) * f (-1) + f x
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f (x + 1) * f (-1) + f x
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | have h2 := Eq3 hf x (-(x + 1)) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : f (-x) * f (- -(x + 1)) + f (-(x + -(x + 1))) = f x * f (-(x + 1)) + f (x + -(x + 1))
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | rw [neg_neg, h0, mul_zero, zero_add, ← sub_eq_add_neg,
sub_add_cancel_left, neg_neg, hf.map_one] at h2 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : f (-x) * f (- -(x + 1)) + f (-(x + -(x + 1))) = f x * f (-(x + 1)) + f (x + -(x + 1))
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : f (-x) * f (- -(x + 1)) + f (-(x + -(x + 1))) = f x * f (-(x + 1)) + f (x + -(x + 1))
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | have h3 := Eq3 hf x (x + 1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
h3 : f (-x) * f (-(x + 1)) + f (-(x + (x + 1))) = f x * f (x + 1) + f (x + (x + 1))
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | rw [h0, mul_zero, zero_add, ← add_assoc, ← two_mul,
Eq4 hf h, h1, sub_self, h2, add_right_inj] at h3 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
h3 : f (-x) * f (-(x + 1)) + f (-(x + (x + 1))) = f x * f (x + 1) + f (x + (x + 1))
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
h3 : f (-(2 * x + 1)) = f (-1)
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
h3 : f (-x) * f (-(x + 1)) + f (-(x + (x + 1))) = f x * f (x + 1) + f (x + (x + 1))
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | replace h2 := Eq4 hf h (-x - 1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
h3 : f (-(2 * x + 1)) = f (-1)
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h3 : f (-(2 * x + 1)) = f (-1)
h2 : f (2 * (-x - 1) + 1) = f (-x - 1) - f (-(-x - 1))
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
h3 : f (-(2 * x + 1)) = f (-1)
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | rw [two_mul, add_assoc, sub_add_cancel, ← add_sub_right_comm, ← two_mul, mul_neg,
← neg_add', h3, eq_comm, Eq1 hf, sub_add_cancel_right, sub_add_cancel,
neg_eq_iff_add_eq_zero, h1, ← add_one_mul (f x), mul_eq_zero] at h2 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h3 : f (-(2 * x + 1)) = f (-1)
h2 : f (2 * (-x - 1) + 1) = f (-x - 1) - f (-(-x - 1))
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h3 : f (-(2 * x + 1)) = f (-1)
h2 : f x + 1 = 0 ∨ f (-1) = 0
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h3 : f (-(2 * x + 1)) = f (-1)
h2 : f (2 * (-x - 1) + 1) = f (-x - 1) - f (-(-x - 1))
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | exact eq_neg_of_add_eq_zero_left (h2.resolve_right h) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h3 : f (-(2 * x + 1)) = f (-1)
h2 : f x + 1 = 0 ∨ f (-1) = 0
⊢ f x = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h3 : f (-(2 * x + 1)) = f (-1)
h2 : f x + 1 = 0 ∨ f (-1) = 0
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.map_neg_one_cases | [88, 1] | [95, 78] | rw [← sub_eq_zero (b := 1), eq_neg_iff_add_eq_zero, ← mul_eq_zero,
mul_sub_one, add_mul, two_mul, add_sub_assoc, add_sub_add_left_eq_sub] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f (-1) = -2 ∨ f (-1) = 1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f (-1) = -2 ∨ f (-1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.map_neg_one_cases | [88, 1] | [95, 78] | have h0 := Eq5 hf h (-1 + -1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1 + 1) = 0 ∨ f (-1 + -1) + f (-(-1 + -1)) = f (-1)
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.map_neg_one_cases | [88, 1] | [95, 78] | rw [neg_add_cancel_right, or_iff_right h, ← eq_sub_iff_add_eq] at h0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1 + 1) = 0 ∨ f (-1 + -1) + f (-(-1 + -1)) = f (-1)
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1) = f (-1) - f (-(-1 + -1))
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1 + 1) = 0 ∨ f (-1 + -1) + f (-(-1 + -1)) = f (-1)
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.map_neg_one_cases | [88, 1] | [95, 78] | have h1 := hf.is_good (-1) (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1) = f (-1) - f (-(-1 + -1))
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1) = f (-1) - f (-(-1 + -1))
h1 : f (-1 * -1 + 1) = f (-1) * f (-1) + f (-1 + -1)
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1) = f (-1) - f (-(-1 + -1))
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.map_neg_one_cases | [88, 1] | [95, 78] | rwa [h0, neg_mul_neg, one_mul, ← neg_add, neg_neg, one_add_one_eq_two, map_two hf h,
eq_comm, ← sub_eq_zero, add_sub_assoc, sub_sub, one_add_one_eq_two] at h1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1) = f (-1) - f (-(-1 + -1))
h1 : f (-1 * -1 + 1) = f (-1) * f (-1) + f (-1 + -1)
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1) = f (-1) - f (-(-1 + -1))
h1 : f (-1 * -1 + 1) = f (-1) * f (-1) + f (-1 + -1)
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Subcase11_solution | [101, 1] | [110, 88] | rcases Eq5 hf h x with h2 | h2 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
⊢ f (x + 1) = f x + 1 | case inl
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : f (x + 1) = 0
⊢ f (x + 1) = f x + 1
case inr
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : f x + f (-x) = f (-1)
⊢ f (x + 1) = f x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Subcase11_solution | [101, 1] | [110, 88] | rw [Eq6 hf h h2, h2, neg_add_self] | case inl
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : f (x + 1) = 0
⊢ f (x + 1) = f x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : f (x + 1) = 0
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Subcase11_solution | [101, 1] | [110, 88] | rw [Eq1 hf, h0, add_left_comm, ← mul_two, mul_neg, ← neg_mul, ← add_mul,
eq_neg_iff_add_eq_zero, ← add_one_mul _ (2 : S), mul_eq_zero, add_assoc] at h2 | case inr
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : f x + f (-x) = f (-1)
⊢ f (x + 1) = f x + 1 | case inr
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : -f (x + 1) + (f x + 1) = 0 ∨ 2 = 0
⊢ f (x + 1) = f x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : f x + f (-x) = f (-1)
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Subcase11_solution | [101, 1] | [110, 88] | exact neg_add_eq_zero.mp (h2.resolve_right (neg_ne_zero.mp (h0.symm.trans_ne h))) | case inr
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : -f (x + 1) + (f x + 1) = 0 ∨ 2 = 0
⊢ f (x + 1) = f x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : -f (x + 1) + (f x + 1) = 0 ∨ 2 = 0
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.map_neg_one_ne_zero | [126, 1] | [127, 77] | rw [← mul_one (3 : S), ← hf.map_neg_one, h, mul_zero] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : f (-1) = 0
⊢ 3 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : f (-1) = 0
⊢ 3 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | have h0 (x) : f (-x) = f (x + 1) + f x := by
rw [Eq1 hf.toNontrivialGood, hf.map_neg_one, mul_one] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
⊢ x = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
⊢ x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | have h1 : f (-x) = f x := by rw [h0, h, zero_add] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
⊢ x = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
⊢ x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | refine hf.period_imp_zero λ y ↦ ?_ | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
⊢ x = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
y : R
⊢ f (y + x) = f y | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
⊢ x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | have h2 := Eq3 hf.toNontrivialGood x (y - 1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
y : R
⊢ f (y + x) = f y | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
y : R
h2 : f (-x) * f (-(y - 1)) + f (-(x + (y - 1))) = f x * f (y - 1) + f (x + (y - 1))
⊢ f (y + x) = f y | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
y : R
⊢ f (y + x) = f y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | rwa [h1, Case1.Eq6 hf.toNontrivialGood (map_neg_one_ne_zero hf) h, neg_one_mul,
neg_one_mul, h0, h0 (x + _), add_assoc, sub_add_cancel, ← add_assoc, add_left_inj,
neg_add_rev, add_assoc, add_right_eq_self, neg_add_eq_zero, eq_comm, add_comm] at h2 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
y : R
h2 : f (-x) * f (-(y - 1)) + f (-(x + (y - 1))) = f x * f (y - 1) + f (x + (y - 1))
⊢ f (y + x) = f y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
y : R
h2 : f (-x) * f (-(y - 1)) + f (-(x + (y - 1))) = f x * f (y - 1) + f (x + (y - 1))
⊢ f (y + x) = f y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | rw [Eq1 hf.toNontrivialGood, hf.map_neg_one, mul_one] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : f (x✝ + 1) = 0
x : R
⊢ f (-x) = f (x + 1) + f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : f (x✝ + 1) = 0
x : R
⊢ f (-x) = f (x + 1) + f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | rw [h0, h, zero_add] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
⊢ f (-x) = f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
⊢ f (-x) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.triple_sum_eq_zero | [144, 1] | [147, 68] | have h := Eq1 hf.toNontrivialGood x | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ f x + f (x + 1) + f (x + 2) = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (-x) = f (x + 1) * f (-1) + f x
⊢ f x + f (x + 1) + f (x + 2) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ f x + f (x + 1) + f (x + 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.triple_sum_eq_zero | [144, 1] | [147, 68] | rwa [Case1.Eq2 hf.toNontrivialGood (map_neg_one_ne_zero hf), hf.map_neg_one,
mul_one, neg_eq_iff_add_eq_zero, add_comm, add_comm (f _)] at h | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (-x) = f (x + 1) * f (-1) + f x
⊢ f x + f (x + 1) + f (x + 2) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (-x) = f (x + 1) * f (-1) + f x
⊢ f x + f (x + 1) + f (x + 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.Rchar | [149, 1] | [154, 69] | refine hf.period_imp_zero λ x ↦ ?_ | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
⊢ 3 = 0 | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ f (x + 3) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
⊢ 3 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.Rchar | [149, 1] | [154, 69] | have h := triple_sum_eq_zero hf | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ f (x + 3) = f x | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), f x + f (x + 1) + f (x + 2) = 0
⊢ f (x + 3) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ f (x + 3) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.Rchar | [149, 1] | [154, 69] | have h0 := h (x + 1) | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), f x + f (x + 1) + f (x + 2) = 0
⊢ f (x + 3) = f x | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), f x + f (x + 1) + f (x + 2) = 0
h0 : f (x + 1) + f (x + 1 + 1) + f (x + 1 + 2) = 0
⊢ f (x + 3) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), f x + f (x + 1) + f (x + 2) = 0
⊢ f (x + 3) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.Rchar | [149, 1] | [154, 69] | rwa [add_assoc x, one_add_one_eq_two, ← add_rotate, ← h x, add_left_inj,
add_left_inj, add_assoc, add_comm 1, two_add_one_eq_three] at h0 | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), f x + f (x + 1) + f (x + 2) = 0
h0 : f (x + 1) + f (x + 1 + 1) + f (x + 1 + 2) = 0
⊢ f (x + 3) = f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), f x + f (x + 1) + f (x + 2) = 0
h0 : f (x + 1) + f (x + 1 + 1) + f (x + 1 + 2) = 0
⊢ f (x + 3) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | have h := eq_zero_or_map_neg_add_self hf | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ x = 0 ∨ x = 1 ∨ x = -1 | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
⊢ x = 0 ∨ x = 1 ∨ x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ x = 0 ∨ x = 1 ∨ x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | refine (h x).imp_right λ h0 ↦ (h (x - 1)).imp eq_of_sub_eq_zero
λ h1 ↦ eq_neg_of_add_eq_zero_left <| (h (x + 1)).resolve_right λ h2 ↦ ?_ | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
⊢ x = 0 ∨ x = 1 ∨ x = -1 | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
⊢ x = 0 ∨ x = 1 ∨ x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | replace h (x) : f x + f (x + 1) + f (x - 1) = 0 := by
have h3 : (2 : R) = -1 := by
rw [eq_neg_iff_add_eq_zero, two_add_one_eq_three, Rchar hf]
rw [sub_eq_add_neg, ← h3, triple_sum_eq_zero hf] | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
⊢ False | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
h : ∀ (x : R), f x + f (x + 1) + f (x - 1) = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | rw [add_zero, ← neg_add', add_right_comm (f _), add_add_add_comm,
add_add_add_comm (f _), h0, h2, neg_add_eq_sub, ← neg_sub x, h1,
one_add_one_eq_two, two_add_one_eq_three] at h | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
h : f x + f (x + 1) + f (x - 1) + (f (-x) + f (-x + 1) + f (-x - 1)) = 0 + 0
⊢ False | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
h : 3 = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
h : f x + f (x + 1) + f (x - 1) + (f (-x) + f (-x + 1) + f (-x - 1)) = 0 + 0
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | exact hf.Schar h | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
h : 3 = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
h : 3 = 0
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | have h3 : (2 : R) = -1 := by
rw [eq_neg_iff_add_eq_zero, two_add_one_eq_three, Rchar hf] | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
⊢ f x + f (x + 1) + f (x - 1) = 0 | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
h3 : 2 = -1
⊢ f x + f (x + 1) + f (x - 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
⊢ f x + f (x + 1) + f (x - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | rw [sub_eq_add_neg, ← h3, triple_sum_eq_zero hf] | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
h3 : 2 = -1
⊢ f x + f (x + 1) + f (x - 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
h3 : 2 = -1
⊢ f x + f (x + 1) + f (x - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | rw [eq_neg_iff_add_eq_zero, two_add_one_eq_three, Rchar hf] | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
⊢ 2 = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
⊢ 2 = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.solution | [173, 1] | [186, 78] | rw [h, neg_zero, ← h, hf.map_one] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : 1 = 0
⊢ f (-1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : 1 = 0
⊢ f (-1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.solution | [173, 1] | [186, 78] | change f 0 = ((-1 : ℤ) : S) | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : Function.Bijective 𝔽₃.cast
⊢ f 𝔽₃.𝔽₃0.cast = ↑(𝔽₃Map1 𝔽₃.𝔽₃0) | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : Function.Bijective 𝔽₃.cast
⊢ f 0 = ↑(-1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : Function.Bijective 𝔽₃.cast
⊢ f 𝔽₃.𝔽₃0.cast = ↑(𝔽₃Map1 𝔽₃.𝔽₃0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.solution | [173, 1] | [186, 78] | rw [hf.map_zero, Int.cast_neg, Int.cast_one] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : Function.Bijective 𝔽₃.cast
⊢ f 0 = ↑(-1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : Function.Bijective 𝔽₃.cast
⊢ f 0 = ↑(-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.solution | [194, 1] | [203, 51] | rwa [← two_add_one_eq_three, Ne,
← neg_eq_iff_add_eq_zero, ← h1, eq_comm] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) ≠ 0
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : ¬f (-1) = -2
h1 : f (-1) = 1
⊢ 3 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) ≠ 0
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : ¬f (-1) = -2
h1 : f (-1) = 1
⊢ 3 ≠ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.add_one_iterate | [32, 1] | [34, 81] | rw [iterate_succ_apply', add_one_iterate n a, add_assoc] | n : ℕ
a : ℤ
⊢ (fun x => x + 1)^[n + 1] a = a + ↑(n + 1) | n : ℕ
a : ℤ
⊢ a + (↑n + 1) = a + ↑(n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
a : ℤ
⊢ (fun x => x + 1)^[n + 1] a = a + ↑(n + 1)
TACTIC:
|
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