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https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.f_lt_f_of_g
[68, 1]
[75, 29]
rw [Rat.div_lt_div_iff_mul_lt_mul (X a) (X b), ← Nat.cast_mul, ← Nat.cast_mul]
case succ.succ a b : β„• h : g (a + 1) * (b + 1) < g (b + 1) * (a + 1) X : βˆ€ (n : β„•), 0 < ↑n.succ ⊒ ↑↑(g (a + 1)) / ↑↑(a + 1) < ↑↑(g (b + 1)) / ↑↑(b + 1)
case succ.succ a b : β„• h : g (a + 1) * (b + 1) < g (b + 1) * (a + 1) X : βˆ€ (n : β„•), 0 < ↑n.succ ⊒ ↑(g (a + 1) * b.succ) < ↑(g (b + 1) * a.succ)
Please generate a tactic in lean4 to solve the state. STATE: case succ.succ a b : β„• h : g (a + 1) * (b + 1) < g (b + 1) * (a + 1) X : βˆ€ (n : β„•), 0 < ↑n.succ ⊒ ↑↑(g (a + 1)) / ↑↑(a + 1) < ↑↑(g (b + 1)) / ↑↑(b + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.f_lt_f_of_g
[68, 1]
[75, 29]
exact Int.ofNat_lt.mpr h
case succ.succ a b : β„• h : g (a + 1) * (b + 1) < g (b + 1) * (a + 1) X : βˆ€ (n : β„•), 0 < ↑n.succ ⊒ ↑(g (a + 1) * b.succ) < ↑(g (b + 1) * a.succ)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case succ.succ a b : β„• h : g (a + 1) * (b + 1) < g (b + 1) * (a + 1) X : βˆ€ (n : β„•), 0 < ↑n.succ ⊒ ↑(g (a + 1) * b.succ) < ↑(g (b + 1) * a.succ) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.f_self_lt_f_succ_of_divisors_card
[77, 1]
[84, 58]
apply f_lt_f_of_g
n : β„• h : n β‰  0 h0 : βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card ⊒ f n < f n.succ
case h n : β„• h : n β‰  0 h0 : βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card ⊒ g n * n.succ < g n.succ * n
Please generate a tactic in lean4 to solve the state. STATE: n : β„• h : n β‰  0 h0 : βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card ⊒ f n < f n.succ TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.f_self_lt_f_succ_of_divisors_card
[77, 1]
[84, 58]
rw [Nat.mul_succ, g_succ, add_mul, add_lt_add_iff_left, g_eq_sum_divisors_card]
case h n : β„• h : n β‰  0 h0 : βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card ⊒ g n * n.succ < g n.succ * n
case h n : β„• h : n β‰  0 h0 : βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card ⊒ βˆ‘ k ∈ range n, k.succ.divisors.card < n.succ.divisors.card * n
Please generate a tactic in lean4 to solve the state. STATE: case h n : β„• h : n β‰  0 h0 : βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card ⊒ g n * n.succ < g n.succ * n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.f_self_lt_f_succ_of_divisors_card
[77, 1]
[84, 58]
calc _ < (range n).sum Ξ» _ ↦ n.succ.divisors.card := sum_lt_sum_of_nonempty (nonempty_range_iff.mpr h) (Ξ» k h1 ↦ h0 k (mem_range.mp h1)) _ = n.succ.divisors.card * n := by rw [sum_const, card_range, smul_eq_mul, mul_comm]
case h n : β„• h : n β‰  0 h0 : βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card ⊒ βˆ‘ k ∈ range n, k.succ.divisors.card < n.succ.divisors.card * n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h n : β„• h : n β‰  0 h0 : βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card ⊒ βˆ‘ k ∈ range n, k.succ.divisors.card < n.succ.divisors.card * n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.f_self_lt_f_succ_of_divisors_card
[77, 1]
[84, 58]
rw [sum_const, card_range, smul_eq_mul, mul_comm]
n : β„• h : n β‰  0 h0 : βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card ⊒ βˆ‘ x ∈ range n, n.succ.divisors.card = n.succ.divisors.card * n
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• h : n β‰  0 h0 : βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card ⊒ βˆ‘ x ∈ range n, n.succ.divisors.card = n.succ.divisors.card * n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.final_solution_part1
[87, 1]
[98, 83]
obtain ⟨n, h0, h1⟩ : βˆƒ n : β„•, N < n ∧ βˆ€ k : β„•, k < n β†’ k.succ.divisors.card < n.succ.divisors.card := by obtain ⟨K, h0⟩ : βˆƒ K : β„•, βˆ€ k : β„•, k ≀ N β†’ k.succ.divisors.card ≀ K := ⟨Extra.seqMax (Ξ» n ↦ n.succ.divisors.card) N, Ξ» _ ↦ Extra.le_seqMax_of_le (Ξ» n ↦ n.succ.divisors.card)⟩ have h1 := exists_lt_card_divisor_succ K refine ⟨Nat.find h1, (Nat.lt_find_iff h1 _).mpr Ξ» k h2 ↦ (h0 k h2).not_lt, Ξ» k h2 ↦ (le_of_not_lt (Nat.find_min h1 h2)).trans_lt (Nat.find_spec h1)⟩
N : β„• ⊒ βˆƒ b ∈ {n | f n < f n.succ}, N < b
case intro.intro N n : β„• h0 : N < n h1 : βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card ⊒ βˆƒ b ∈ {n | f n < f n.succ}, N < b
Please generate a tactic in lean4 to solve the state. STATE: N : β„• ⊒ βˆƒ b ∈ {n | f n < f n.succ}, N < b TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.final_solution_part1
[87, 1]
[98, 83]
exact ⟨n, f_self_lt_f_succ_of_divisors_card (Nat.not_eq_zero_of_lt h0) h1, h0⟩
case intro.intro N n : β„• h0 : N < n h1 : βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card ⊒ βˆƒ b ∈ {n | f n < f n.succ}, N < b
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro N n : β„• h0 : N < n h1 : βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card ⊒ βˆƒ b ∈ {n | f n < f n.succ}, N < b TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.final_solution_part1
[87, 1]
[98, 83]
obtain ⟨K, h0⟩ : βˆƒ K : β„•, βˆ€ k : β„•, k ≀ N β†’ k.succ.divisors.card ≀ K := ⟨Extra.seqMax (Ξ» n ↦ n.succ.divisors.card) N, Ξ» _ ↦ Extra.le_seqMax_of_le (Ξ» n ↦ n.succ.divisors.card)⟩
N : β„• ⊒ βˆƒ n, N < n ∧ βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card
case intro N K : β„• h0 : βˆ€ k ≀ N, k.succ.divisors.card ≀ K ⊒ βˆƒ n, N < n ∧ βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card
Please generate a tactic in lean4 to solve the state. STATE: N : β„• ⊒ βˆƒ n, N < n ∧ βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.final_solution_part1
[87, 1]
[98, 83]
have h1 := exists_lt_card_divisor_succ K
case intro N K : β„• h0 : βˆ€ k ≀ N, k.succ.divisors.card ≀ K ⊒ βˆƒ n, N < n ∧ βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card
case intro N K : β„• h0 : βˆ€ k ≀ N, k.succ.divisors.card ≀ K h1 : βˆƒ n, K < n.succ.divisors.card ⊒ βˆƒ n, N < n ∧ βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card
Please generate a tactic in lean4 to solve the state. STATE: case intro N K : β„• h0 : βˆ€ k ≀ N, k.succ.divisors.card ≀ K ⊒ βˆƒ n, N < n ∧ βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.final_solution_part1
[87, 1]
[98, 83]
refine ⟨Nat.find h1, (Nat.lt_find_iff h1 _).mpr Ξ» k h2 ↦ (h0 k h2).not_lt, Ξ» k h2 ↦ (le_of_not_lt (Nat.find_min h1 h2)).trans_lt (Nat.find_spec h1)⟩
case intro N K : β„• h0 : βˆ€ k ≀ N, k.succ.divisors.card ≀ K h1 : βˆƒ n, K < n.succ.divisors.card ⊒ βˆƒ n, N < n ∧ βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro N K : β„• h0 : βˆ€ k ≀ N, k.succ.divisors.card ≀ K h1 : βˆƒ n, K < n.succ.divisors.card ⊒ βˆƒ n, N < n ∧ βˆ€ k < n, k.succ.divisors.card < n.succ.divisors.card TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.f_succ_lt_self_of_succ_prime_large
[100, 1]
[104, 25]
apply f_lt_f_of_g
n : β„• h : 6 ≀ n h0 : n.succ.Prime ⊒ f n.succ < f n
case h n : β„• h : 6 ≀ n h0 : n.succ.Prime ⊒ g n.succ * n < g n * n.succ
Please generate a tactic in lean4 to solve the state. STATE: n : β„• h : 6 ≀ n h0 : n.succ.Prime ⊒ f n.succ < f n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.f_succ_lt_self_of_succ_prime_large
[100, 1]
[104, 25]
rw [g_succ, card_divisors_prime h0, add_mul, Nat.mul_succ, add_lt_add_iff_left]
case h n : β„• h : 6 ≀ n h0 : n.succ.Prime ⊒ g n.succ * n < g n * n.succ
case h n : β„• h : 6 ≀ n h0 : n.succ.Prime ⊒ 2 * n < g n
Please generate a tactic in lean4 to solve the state. STATE: case h n : β„• h : 6 ≀ n h0 : n.succ.Prime ⊒ g n.succ * n < g n * n.succ TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.f_succ_lt_self_of_succ_prime_large
[100, 1]
[104, 25]
exact two_mul_lt_g n h
case h n : β„• h : 6 ≀ n h0 : n.succ.Prime ⊒ 2 * n < g n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h n : β„• h : 6 ≀ n h0 : n.succ.Prime ⊒ 2 * n < g n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.final_solution_part2
[107, 1]
[114, 59]
obtain ⟨n, h, h0, h1⟩ : βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n := by rcases (max 6 (N + 1) + 1).exists_infinite_primes with ⟨_ | n, h, h0⟩ exact absurd h0 Nat.not_prime_zero rw [Nat.add_le_add_iff_right, max_le_iff] at h exact ⟨n, h.1, h0, h.2⟩
N : β„• ⊒ βˆƒ b ∈ {n | f n.succ < f n}, N < b
case intro.intro.intro N n : β„• h : 6 ≀ n h0 : n.succ.Prime h1 : N < n ⊒ βˆƒ b ∈ {n | f n.succ < f n}, N < b
Please generate a tactic in lean4 to solve the state. STATE: N : β„• ⊒ βˆƒ b ∈ {n | f n.succ < f n}, N < b TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.final_solution_part2
[107, 1]
[114, 59]
exact ⟨n, f_succ_lt_self_of_succ_prime_large h h0, h1⟩
case intro.intro.intro N n : β„• h : 6 ≀ n h0 : n.succ.Prime h1 : N < n ⊒ βˆƒ b ∈ {n | f n.succ < f n}, N < b
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro N n : β„• h : 6 ≀ n h0 : n.succ.Prime h1 : N < n ⊒ βˆƒ b ∈ {n | f n.succ < f n}, N < b TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.final_solution_part2
[107, 1]
[114, 59]
rcases (max 6 (N + 1) + 1).exists_infinite_primes with ⟨_ | n, h, h0⟩
N : β„• ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n
case intro.zero.intro N : β„• h : max 6 (N + 1) + 1 ≀ 0 h0 : Nat.Prime 0 ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n case intro.succ.intro N n : β„• h : max 6 (N + 1) + 1 ≀ n + 1 h0 : (n + 1).Prime ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n
Please generate a tactic in lean4 to solve the state. STATE: N : β„• ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.final_solution_part2
[107, 1]
[114, 59]
exact absurd h0 Nat.not_prime_zero
case intro.zero.intro N : β„• h : max 6 (N + 1) + 1 ≀ 0 h0 : Nat.Prime 0 ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n case intro.succ.intro N n : β„• h : max 6 (N + 1) + 1 ≀ n + 1 h0 : (n + 1).Prime ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n
case intro.succ.intro N n : β„• h : max 6 (N + 1) + 1 ≀ n + 1 h0 : (n + 1).Prime ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n
Please generate a tactic in lean4 to solve the state. STATE: case intro.zero.intro N : β„• h : max 6 (N + 1) + 1 ≀ 0 h0 : Nat.Prime 0 ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n case intro.succ.intro N n : β„• h : max 6 (N + 1) + 1 ≀ n + 1 h0 : (n + 1).Prime ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.final_solution_part2
[107, 1]
[114, 59]
rw [Nat.add_le_add_iff_right, max_le_iff] at h
case intro.succ.intro N n : β„• h : max 6 (N + 1) + 1 ≀ n + 1 h0 : (n + 1).Prime ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n
case intro.succ.intro N n : β„• h : 6 ≀ n ∧ N + 1 ≀ n h0 : (n + 1).Prime ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n
Please generate a tactic in lean4 to solve the state. STATE: case intro.succ.intro N n : β„• h : max 6 (N + 1) + 1 ≀ n + 1 h0 : (n + 1).Prime ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2006/N3/N3.lean
IMOSL.IMO2006N3.final_solution_part2
[107, 1]
[114, 59]
exact ⟨n, h.1, h0, h.2⟩
case intro.succ.intro N n : β„• h : 6 ≀ n ∧ N + 1 ≀ n h0 : (n + 1).Prime ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.succ.intro N n : β„• h : 6 ≀ n ∧ N + 1 ≀ n h0 : (n + 1).Prime ⊒ βˆƒ n, 6 ≀ n ∧ n.succ.Prime ∧ N < n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/F4.lean
IMOSL.IMO2012A5.𝔽₄.add_mul
[229, 11]
[230, 58]
rw [𝔽₄.mul_comm, 𝔽₄.mul_add, z.mul_comm, z.mul_comm]
x y z : 𝔽₄ ⊒ (x + y) * z = x * z + y * z
no goals
Please generate a tactic in lean4 to solve the state. STATE: x y z : 𝔽₄ ⊒ (x + y) * z = x * z + y * z TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_add_cancel_left
[29, 1]
[30, 47]
rw [← add_assoc, add_self_eq_zero, zero_add]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y : M ⊒ x + (x + y) = y
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y : M ⊒ x + (x + y) = y TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_add_cancel_right
[32, 1]
[33, 45]
rw [add_assoc, add_self_eq_zero, add_zero]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y : M ⊒ x + y + y = x
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y : M ⊒ x + y + y = x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_add_add_cancel_middle
[35, 1]
[36, 41]
rw [← add_assoc, add_add_cancel_right]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + y + (y + z) = x + z
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + y + (y + z) = x + z TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_comm
[38, 1]
[40, 59]
rw [← add_add_cancel_right (x + y) (y + x), add_add_add_cancel_middle, add_self_eq_zero, zero_add]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y : M ⊒ x + y = y + x
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y : M ⊒ x + y = y + x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_add_cancel_middle₁
[42, 1]
[43, 38]
rw [add_comm, add_add_cancel_right]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y : M ⊒ x + (y + x) = y
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y : M ⊒ x + (y + x) = y TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_add_cancel_middleβ‚‚
[45, 1]
[46, 37]
rw [add_comm, add_add_cancel_left]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y : M ⊒ x + y + x = y
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y : M ⊒ x + y + x = y TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_right_inj
[48, 1]
[49, 86]
rw [← add_add_cancel_left x y, h, add_add_cancel_left]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M h : x + y = x + z ⊒ y = z
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M h : x + y = x + z ⊒ y = z TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_left_inj
[51, 1]
[52, 43]
rw [add_comm, add_comm y, add_right_inj]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + z = y + z ↔ x = y
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + z = y + z ↔ x = y TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_eq_iff_eq_add
[54, 1]
[55, 53]
rw [← add_left_inj (z := y), add_add_cancel_right]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + y = z ↔ x = z + y
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + y = z ↔ x = z + y TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_eq_iff_eq_add'
[57, 1]
[58, 35]
rw [add_eq_iff_eq_add, add_comm]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + y = z ↔ x = y + z
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + y = z ↔ x = y + z TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_eq_iff_eq_add''
[60, 1]
[61, 36]
rw [add_comm, add_eq_iff_eq_add']
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + y = z ↔ y = x + z
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + y = z ↔ y = x + z TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_eq_iff_eq_add'''
[63, 1]
[64, 37]
rw [add_eq_iff_eq_add'', add_comm]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + y = z ↔ y = z + x
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + y = z ↔ y = z + x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_eq_zero_iff_eq
[66, 1]
[67, 35]
rw [add_eq_iff_eq_add, zero_add]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y : M ⊒ x + y = 0 ↔ x = y
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y : M ⊒ x + y = 0 ↔ x = y TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_add_add_cancel_left
[69, 1]
[70, 43]
rw [← add_assoc, add_add_cancel_middleβ‚‚]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + y + (x + z) = y + z
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + y + (x + z) = y + z TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.add_add_add_cancel_right
[72, 1]
[73, 41]
rw [add_assoc, add_add_cancel_middle₁]
M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + z + (y + z) = x + y
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type u_1 inst✝¹ : AddMonoid M inst✝ : CharTwo M x y z : M ⊒ x + z + (y + z) = x + y TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Defs.lean
IMOSL.Extra.CharTwo.sub_eq_add
[93, 1]
[94, 35]
rw [sub_eq_add_neg, neg_eq_self]
G : Type u_1 inst✝¹ : AddGroup G inst✝ : CharTwo G x y : G ⊒ x - y = x + y
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝¹ : AddGroup G inst✝ : CharTwo G x y : G ⊒ x - y = x + y TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.targetSum_general_lower_bound
[43, 1]
[50, 42]
rw [map_cons, prod_cons, targetSum, pow_add, length_cons, Nat.factorial_succ, mul_mul_mul_comm]
a : β„• l : List β„• ⊒ (map Nat.succ (a :: l)).prod ≀ 2 ^ targetSum (a :: l) * (a :: l).length.factorial
a : β„• l : List β„• ⊒ a.succ * (map Nat.succ l).prod ≀ 2 ^ (a / l.length.succ) * (l.length + 1) * (2 ^ targetSum l * l.length.factorial)
Please generate a tactic in lean4 to solve the state. STATE: a : β„• l : List β„• ⊒ (map Nat.succ (a :: l)).prod ≀ 2 ^ targetSum (a :: l) * (a :: l).length.factorial TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.targetSum_general_lower_bound
[43, 1]
[50, 42]
exact Nat.mul_le_mul (succ_le_mul_two_pow_div a l.length.succ_pos) (targetSum_general_lower_bound l)
a : β„• l : List β„• ⊒ a.succ * (map Nat.succ l).prod ≀ 2 ^ (a / l.length.succ) * (l.length + 1) * (2 ^ targetSum l * l.length.factorial)
no goals
Please generate a tactic in lean4 to solve the state. STATE: a : β„• l : List β„• ⊒ a.succ * (map Nat.succ l).prod ≀ 2 ^ (a / l.length.succ) * (l.length + 1) * (2 ^ targetSum l * l.length.factorial) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.targetSum_perm_iota_n_lower_bound
[52, 1]
[57, 72]
have h0 := targetSum_general_lower_bound l
l : List β„• n : β„• h : l ~ iota n ⊒ n.size ≀ targetSum l
l : List β„• n : β„• h : l ~ iota n h0 : (map Nat.succ l).prod ≀ 2 ^ targetSum l * l.length.factorial ⊒ n.size ≀ targetSum l
Please generate a tactic in lean4 to solve the state. STATE: l : List β„• n : β„• h : l ~ iota n ⊒ n.size ≀ targetSum l TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.targetSum_perm_iota_n_lower_bound
[52, 1]
[57, 72]
rw [(h.map Nat.succ).prod_eq, h.length_eq, prod_map_succ_iota, length_iota] at h0
l : List β„• n : β„• h : l ~ iota n h0 : (map Nat.succ l).prod ≀ 2 ^ targetSum l * l.length.factorial ⊒ n.size ≀ targetSum l
l : List β„• n : β„• h : l ~ iota n h0 : (n + 1).factorial ≀ 2 ^ targetSum l * n.factorial ⊒ n.size ≀ targetSum l
Please generate a tactic in lean4 to solve the state. STATE: l : List β„• n : β„• h : l ~ iota n h0 : (map Nat.succ l).prod ≀ 2 ^ targetSum l * l.length.factorial ⊒ n.size ≀ targetSum l TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.targetSum_perm_iota_n_lower_bound
[52, 1]
[57, 72]
exact Nat.size_le.mpr (Nat.le_of_mul_le_mul_right h0 n.factorial_pos)
l : List β„• n : β„• h : l ~ iota n h0 : (n + 1).factorial ≀ 2 ^ targetSum l * n.factorial ⊒ n.size ≀ targetSum l
no goals
Please generate a tactic in lean4 to solve the state. STATE: l : List β„• n : β„• h : l ~ iota n h0 : (n + 1).factorial ≀ 2 ^ targetSum l * n.factorial ⊒ n.size ≀ targetSum l TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.iota_map_add_append_iota_eq_iota
[81, 1]
[85, 76]
rw [iota_succ, map_cons, cons_append]
n k : β„• ⊒ map n.add (iota (k + 1)) ++ iota n = iota (n + (k + 1))
n k : β„• ⊒ n.add (k + 1) :: (map n.add (iota k) ++ iota n) = iota (n + (k + 1))
Please generate a tactic in lean4 to solve the state. STATE: n k : β„• ⊒ map n.add (iota (k + 1)) ++ iota n = iota (n + (k + 1)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.iota_map_add_append_iota_eq_iota
[81, 1]
[85, 76]
exact congr_argβ‚‚ _ rfl (iota_map_add_append_iota_eq_iota n k)
n k : β„• ⊒ n.add (k + 1) :: (map n.add (iota k) ++ iota n) = iota (n + (k + 1))
no goals
Please generate a tactic in lean4 to solve the state. STATE: n k : β„• ⊒ n.add (k + 1) :: (map n.add (iota k) ++ iota n) = iota (n + (k + 1)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.lowerBoundMk_perm_iota
[87, 1]
[99, 59]
rw [lowerBoundMk_bit0_succ, Nat.bit_false, Nat.bit0_val]
b : Bool n k : β„• h : lowerBoundMk (k + 1) ~ iota (k + 1) ⊒ lowerBoundMk (Nat.bit false (k + 1)) ~ iota (Nat.bit false (k + 1))
b : Bool n k : β„• h : lowerBoundMk (k + 1) ~ iota (k + 1) ⊒ map (k + 1).add (iota k) ++ (2 * k + 2) :: lowerBoundMk (k + 1) ~ iota (2 * (k + 1))
Please generate a tactic in lean4 to solve the state. STATE: b : Bool n k : β„• h : lowerBoundMk (k + 1) ~ iota (k + 1) ⊒ lowerBoundMk (Nat.bit false (k + 1)) ~ iota (Nat.bit false (k + 1)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.lowerBoundMk_perm_iota
[87, 1]
[99, 59]
refine perm_middle.trans (((h.append_left _).trans ?_).cons _)
b : Bool n k : β„• h : lowerBoundMk (k + 1) ~ iota (k + 1) ⊒ map (k + 1).add (iota k) ++ (2 * k + 2) :: lowerBoundMk (k + 1) ~ iota (2 * (k + 1))
b : Bool n k : β„• h : lowerBoundMk (k + 1) ~ iota (k + 1) ⊒ map (k + 1).add (iota k) ++ iota (k + 1) ~ iota (Nat.mul 2 k + 1)
Please generate a tactic in lean4 to solve the state. STATE: b : Bool n k : β„• h : lowerBoundMk (k + 1) ~ iota (k + 1) ⊒ map (k + 1).add (iota k) ++ (2 * k + 2) :: lowerBoundMk (k + 1) ~ iota (2 * (k + 1)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.lowerBoundMk_perm_iota
[87, 1]
[99, 59]
rw [iota_map_add_append_iota_eq_iota, add_right_comm, ← two_mul]
b : Bool n k : β„• h : lowerBoundMk (k + 1) ~ iota (k + 1) ⊒ map (k + 1).add (iota k) ++ iota (k + 1) ~ iota (Nat.mul 2 k + 1)
b : Bool n k : β„• h : lowerBoundMk (k + 1) ~ iota (k + 1) ⊒ iota (2 * k + 1) ~ iota (Nat.mul 2 k + 1)
Please generate a tactic in lean4 to solve the state. STATE: b : Bool n k : β„• h : lowerBoundMk (k + 1) ~ iota (k + 1) ⊒ map (k + 1).add (iota k) ++ iota (k + 1) ~ iota (Nat.mul 2 k + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.lowerBoundMk_perm_iota
[87, 1]
[99, 59]
exact Perm.refl _
b : Bool n k : β„• h : lowerBoundMk (k + 1) ~ iota (k + 1) ⊒ iota (2 * k + 1) ~ iota (Nat.mul 2 k + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: b : Bool n k : β„• h : lowerBoundMk (k + 1) ~ iota (k + 1) ⊒ iota (2 * k + 1) ~ iota (Nat.mul 2 k + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.lowerBoundMk_perm_iota
[87, 1]
[99, 59]
rw [lowerBoundMk_bit1, Nat.bit_true, Nat.bit1_val]
b : Bool n✝ n : β„• h : lowerBoundMk n ~ iota n ⊒ lowerBoundMk (Nat.bit true n) ~ iota (Nat.bit true n)
b : Bool n✝ n : β„• h : lowerBoundMk n ~ iota n ⊒ map n.add (iota n) ++ (2 * n + 1) :: lowerBoundMk n ~ iota (2 * n + 1)
Please generate a tactic in lean4 to solve the state. STATE: b : Bool n✝ n : β„• h : lowerBoundMk n ~ iota n ⊒ lowerBoundMk (Nat.bit true n) ~ iota (Nat.bit true n) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.lowerBoundMk_perm_iota
[87, 1]
[99, 59]
refine perm_middle.trans (((h.append_left _).trans ?_).cons _)
b : Bool n✝ n : β„• h : lowerBoundMk n ~ iota n ⊒ map n.add (iota n) ++ (2 * n + 1) :: lowerBoundMk n ~ iota (2 * n + 1)
b : Bool n✝ n : β„• h : lowerBoundMk n ~ iota n ⊒ map n.add (iota n) ++ iota n ~ iota (2 * n)
Please generate a tactic in lean4 to solve the state. STATE: b : Bool n✝ n : β„• h : lowerBoundMk n ~ iota n ⊒ map n.add (iota n) ++ (2 * n + 1) :: lowerBoundMk n ~ iota (2 * n + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.lowerBoundMk_perm_iota
[87, 1]
[99, 59]
rw [iota_map_add_append_iota_eq_iota, ← two_mul]
b : Bool n✝ n : β„• h : lowerBoundMk n ~ iota n ⊒ map n.add (iota n) ++ iota n ~ iota (2 * n)
no goals
Please generate a tactic in lean4 to solve the state. STATE: b : Bool n✝ n : β„• h : lowerBoundMk n ~ iota n ⊒ map n.add (iota n) ++ iota n ~ iota (2 * n) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.targetSum_map_add_iota_length_succ
[104, 1]
[112, 50]
rw [iota_succ, map_cons, cons_append, targetSum, length_cons, targetSum_map_add_iota_length_succ h k, Nat.add_eq_right]
n : β„• l : List β„• h : l.length = n.succ k : β„• ⊒ targetSum (map n.add (iota (k + 1)) ++ l) = targetSum l
n : β„• l : List β„• h : l.length = n.succ k : β„• ⊒ n.add (k + 1) / (map n.add (iota k) ++ l).length.succ = 0
Please generate a tactic in lean4 to solve the state. STATE: n : β„• l : List β„• h : l.length = n.succ k : β„• ⊒ targetSum (map n.add (iota (k + 1)) ++ l) = targetSum l TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.targetSum_map_add_iota_length_succ
[104, 1]
[112, 50]
refine Nat.div_eq_of_lt ?_
n : β„• l : List β„• h : l.length = n.succ k : β„• ⊒ n.add (k + 1) / (map n.add (iota k) ++ l).length.succ = 0
n : β„• l : List β„• h : l.length = n.succ k : β„• ⊒ n.add (k + 1) < (map n.add (iota k) ++ l).length.succ
Please generate a tactic in lean4 to solve the state. STATE: n : β„• l : List β„• h : l.length = n.succ k : β„• ⊒ n.add (k + 1) / (map n.add (iota k) ++ l).length.succ = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.targetSum_map_add_iota_length_succ
[104, 1]
[112, 50]
rw [length_append, length_map, length_iota, h]
n : β„• l : List β„• h : l.length = n.succ k : β„• ⊒ n.add (k + 1) < (map n.add (iota k) ++ l).length.succ
n : β„• l : List β„• h : l.length = n.succ k : β„• ⊒ n.add (k + 1) < (k + n.succ).succ
Please generate a tactic in lean4 to solve the state. STATE: n : β„• l : List β„• h : l.length = n.succ k : β„• ⊒ n.add (k + 1) < (map n.add (iota k) ++ l).length.succ TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.lowerBoundMk_targetSum
[114, 1]
[137, 78]
have h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ := congr_arg Nat.succ (lowerBoundMk_length (k + 1))
X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n k : β„• h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size ⊒ targetSum (lowerBoundMk (Nat.bit false (k + 1))) = (Nat.bit false (k + 1)).size
X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n k : β„• h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ ⊒ targetSum (lowerBoundMk (Nat.bit false (k + 1))) = (Nat.bit false (k + 1)).size
Please generate a tactic in lean4 to solve the state. STATE: X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n k : β„• h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size ⊒ targetSum (lowerBoundMk (Nat.bit false (k + 1))) = (Nat.bit false (k + 1)).size TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.lowerBoundMk_targetSum
[114, 1]
[137, 78]
rw [lowerBoundMk_bit0_succ, Nat.bit_false, Nat.size_bit0 k.succ_ne_zero, (k + 1).size.succ_eq_add_one, targetSum_map_add_iota_length_succ h0, targetSum, h0, h, add_comm, Nat.add_right_inj]
X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n k : β„• h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ ⊒ targetSum (lowerBoundMk (Nat.bit false (k + 1))) = (Nat.bit false (k + 1)).size
X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n k : β„• h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ ⊒ (2 * k + 2) / (k + 1).succ = 1
Please generate a tactic in lean4 to solve the state. STATE: X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n k : β„• h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ ⊒ targetSum (lowerBoundMk (Nat.bit false (k + 1))) = (Nat.bit false (k + 1)).size TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.lowerBoundMk_targetSum
[114, 1]
[137, 78]
exact Nat.div_eq_of_lt_le ((one_mul _).trans_le <| Nat.add_le_add_right (X0 _) 2) (Nat.mul_lt_mul_of_pos_left (k + 1).lt_succ_self X)
X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n k : β„• h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ ⊒ (2 * k + 2) / (k + 1).succ = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n k : β„• h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ ⊒ (2 * k + 2) / (k + 1).succ = 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.lowerBoundMk_targetSum
[114, 1]
[137, 78]
have h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ := congr_arg Nat.succ (lowerBoundMk_length n)
X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n✝ n : β„• h : targetSum (lowerBoundMk n) = n.size ⊒ targetSum (lowerBoundMk (Nat.bit true n)) = (Nat.bit true n).size
X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n✝ n : β„• h : targetSum (lowerBoundMk n) = n.size h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ ⊒ targetSum (lowerBoundMk (Nat.bit true n)) = (Nat.bit true n).size
Please generate a tactic in lean4 to solve the state. STATE: X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n✝ n : β„• h : targetSum (lowerBoundMk n) = n.size ⊒ targetSum (lowerBoundMk (Nat.bit true n)) = (Nat.bit true n).size TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.lowerBoundMk_targetSum
[114, 1]
[137, 78]
rw [lowerBoundMk_bit1, Nat.bit_true, Nat.size_bit1, targetSum_map_add_iota_length_succ h0, targetSum, h0, h, n.size.succ_eq_add_one, add_comm, Nat.add_right_inj]
X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n✝ n : β„• h : targetSum (lowerBoundMk n) = n.size h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ ⊒ targetSum (lowerBoundMk (Nat.bit true n)) = (Nat.bit true n).size
X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n✝ n : β„• h : targetSum (lowerBoundMk n) = n.size h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ ⊒ (2 * n + 1) / n.succ = 1
Please generate a tactic in lean4 to solve the state. STATE: X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n✝ n : β„• h : targetSum (lowerBoundMk n) = n.size h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ ⊒ targetSum (lowerBoundMk (Nat.bit true n)) = (Nat.bit true n).size TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/A3/A3.lean
IMOSL.IMO2021A3.lowerBoundMk_targetSum
[114, 1]
[137, 78]
exact Nat.div_eq_of_lt_le ((one_mul _).trans_le <| Nat.succ_le_succ (X0 _)) (Nat.le_refl _)
X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n✝ n : β„• h : targetSum (lowerBoundMk n) = n.size h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ ⊒ (2 * n + 1) / n.succ = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: X : 0 < Nat.succ 1 X0 : βˆ€ (m : β„•), m ≀ 2 * m b : Bool n✝ n : β„• h : targetSum (lowerBoundMk n) = n.size h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ ⊒ (2 * n + 1) / n.succ = 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
have h0 : βˆ€ t x : R, f (f t) - f (f x) ≀ (f t - x) * f x := Ξ» t x ↦ by rw [sub_le_iff_le_add] apply (h _ _).trans_eq' rw [add_sub_cancel]
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
replace h0 : βˆ€ t x : R, 0 ≀ (f t - x) * f x + (f x - t) * f t := Ξ» t x ↦ by rw [← sub_self (f (f t)), ← sub_add_sub_cancel _ (f (f x))] exact add_le_add (h0 t x) (h0 x t)
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), 0 ≀ (f t - x) * f x + (f x - t) * f t ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
replace h0 : βˆ€ x : R, x * f x ≀ 0 := Ξ» x ↦ by have h1 := h0 x (f x + f x) rwa [sub_add_cancel_left, sub_mul, neg_mul, mul_comm, ← add_sub_assoc, neg_add_self, zero_sub, neg_nonneg] at h1
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), 0 ≀ (f t - x) * f x + (f x - t) * f t ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), 0 ≀ (f t - x) * f x + (f x - t) * f t ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
have h1 : βˆ€ x : R, f x ≀ f (f x) := Ξ» x ↦ by have h1 := h x 0 rwa [add_zero, zero_mul, zero_add] at h1
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 h1 : βˆ€ (x : R), f x ≀ f (f x) ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
replace h1 : βˆ€ x : R, f x ≀ 0 := Ξ» x ↦ le_of_not_lt Ξ» h2 ↦ (h0 (f x)).not_lt <| mul_pos h2 (h2.trans_le (h1 x))
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 h1 : βˆ€ (x : R), f x ≀ f (f x) ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 h1 : βˆ€ (x : R), f x ≀ 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 h1 : βˆ€ (x : R), f x ≀ f (f x) ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
replace h0 : βˆ€ x : R, x < 0 β†’ f x = 0 := Ξ» x h2 ↦ (h1 x).antisymm (nonneg_of_mul_nonpos_right (h0 x) h2)
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 h1 : βˆ€ (x : R), f x ≀ 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 h1 : βˆ€ (x : R), f x ≀ 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
intros x h2
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 x : R h2 : x ≀ 0 ⊒ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
rcases h2.lt_or_eq with h2 | rfl
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 x : R h2 : x ≀ 0 ⊒ f x = 0
case inl R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 x : R h2✝ : x ≀ 0 h2 : x < 0 ⊒ f x = 0 case inr R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 h2 : 0 ≀ 0 ⊒ f 0 = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 x : R h2 : x ≀ 0 ⊒ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
rw [sub_le_iff_le_add]
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) - f (f x) ≀ (f t - x) * f x
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) ≀ (f t - x) * f x + f (f x)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) - f (f x) ≀ (f t - x) * f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
apply (h _ _).trans_eq'
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) ≀ (f t - x) * f x + f (f x)
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) = f (x + (f t - x))
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) ≀ (f t - x) * f x + f (f x) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
rw [add_sub_cancel]
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) = f (x + (f t - x))
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) = f (x + (f t - x)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
rw [← sub_self (f (f t)), ← sub_add_sub_cancel _ (f (f x))]
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x t x : R ⊒ 0 ≀ (f t - x) * f x + (f x - t) * f t
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x t x : R ⊒ f (f t) - f (f x) + (f (f x) - f (f t)) ≀ (f t - x) * f x + (f x - t) * f t
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x t x : R ⊒ 0 ≀ (f t - x) * f x + (f x - t) * f t TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
exact add_le_add (h0 t x) (h0 x t)
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x t x : R ⊒ f (f t) - f (f x) + (f (f x) - f (f t)) ≀ (f t - x) * f x + (f x - t) * f t
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x t x : R ⊒ f (f t) - f (f x) + (f (f x) - f (f t)) ≀ (f t - x) * f x + (f x - t) * f t TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
have h1 := h0 x (f x + f x)
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), 0 ≀ (f t - x) * f x + (f x - t) * f t x : R ⊒ x * f x ≀ 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), 0 ≀ (f t - x) * f x + (f x - t) * f t x : R h1 : 0 ≀ (f x - (f x + f x)) * f (f x + f x) + (f (f x + f x) - x) * f x ⊒ x * f x ≀ 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), 0 ≀ (f t - x) * f x + (f x - t) * f t x : R ⊒ x * f x ≀ 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
rwa [sub_add_cancel_left, sub_mul, neg_mul, mul_comm, ← add_sub_assoc, neg_add_self, zero_sub, neg_nonneg] at h1
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), 0 ≀ (f t - x) * f x + (f x - t) * f t x : R h1 : 0 ≀ (f x - (f x + f x)) * f (f x + f x) + (f (f x + f x) - x) * f x ⊒ x * f x ≀ 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), 0 ≀ (f t - x) * f x + (f x - t) * f t x : R h1 : 0 ≀ (f x - (f x + f x)) * f (f x + f x) + (f (f x + f x) - x) * f x ⊒ x * f x ≀ 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
have h1 := h x 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 x : R ⊒ f x ≀ f (f x)
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 x : R h1 : f (x + 0) ≀ 0 * f x + f (f x) ⊒ f x ≀ f (f x)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 x : R ⊒ f x ≀ f (f x) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
rwa [add_zero, zero_mul, zero_add] at h1
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 x : R h1 : f (x + 0) ≀ 0 * f x + f (f x) ⊒ f x ≀ f (f x)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 x : R h1 : f (x + 0) ≀ 0 * f x + f (f x) ⊒ f x ≀ f (f x) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
exact h0 x h2
case inl R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 x : R h2✝ : x ≀ 0 h2 : x < 0 ⊒ f x = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inl R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 x : R h2✝ : x ≀ 0 h2 : x < 0 ⊒ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
apply (h1 0).antisymm
case inr R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 h2 : 0 ≀ 0 ⊒ f 0 = 0
case inr R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 h2 : 0 ≀ 0 ⊒ 0 ≀ f 0
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 h2 : 0 ≀ 0 ⊒ f 0 = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
specialize h (-1) 0
case inr R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 h2 : 0 ≀ 0 ⊒ 0 ≀ f 0
case inr R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 h2 : 0 ≀ 0 h : f (-1 + 0) ≀ 0 * f (-1) + f (f (-1)) ⊒ 0 ≀ f 0
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 h2 : 0 ≀ 0 ⊒ 0 ≀ f 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
rwa [add_zero, zero_mul, zero_add, h0 _ neg_one_lt_zero] at h
case inr R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 h2 : 0 ≀ 0 h : f (-1 + 0) ≀ 0 * f (-1) + f (f (-1)) ⊒ 0 ≀ f 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 h2 : 0 ≀ 0 h : f (-1 + 0) ≀ 0 * f (-1) + f (f (-1)) ⊒ 0 ≀ f 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/F2e.lean
IMOSL.IMO2012A5.𝔽₂Ρ.add_mul
[229, 11]
[230, 60]
rw [𝔽₂Ρ.mul_comm, 𝔽₂Ρ.mul_add, z.mul_comm, z.mul_comm]
x y z : 𝔽₂Ρ ⊒ (x + y) * z = x * z + y * z
no goals
Please generate a tactic in lean4 to solve the state. STATE: x y z : 𝔽₂Ρ ⊒ (x + y) * z = x * z + y * z TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/F2e.lean
IMOSL.IMO2012A5.𝔽₂Ρ.castRingHom_injective
[290, 1]
[297, 74]
rw [← one_mul r, h1, zero_mul]
R : Type u_1 inst✝¹ : NonAssocSemiring R inst✝ : CharTwo R r : R h : r * r = 0 h0 : r β‰  0 h1 : 1 = 0 ⊒ r = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : NonAssocSemiring R inst✝ : CharTwo R r : R h : r * r = 0 h0 : r β‰  0 h1 : 1 = 0 ⊒ r = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/F2e.lean
IMOSL.IMO2012A5.𝔽₂Ρ.castRingHom_injective
[290, 1]
[297, 74]
rwa [add_eq_zero_iff_eq.mp h2, one_mul] at h
R : Type u_1 inst✝¹ : NonAssocSemiring R inst✝ : CharTwo R r : R h : r * r = 0 h0 : r β‰  0 h1 : 1 β‰  0 x : 𝔽₂Ρ h2 : (castRingHom h) Y = 0 ⊒ 1 = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : NonAssocSemiring R inst✝ : CharTwo R r : R h : r * r = 0 h0 : r β‰  0 h1 : 1 β‰  0 x : 𝔽₂Ρ h2 : (castRingHom h) Y = 0 ⊒ 1 = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Hom.lean
IMOSL.Extra.CharTwo.pullback_of_inj
[22, 1]
[23, 71]
rw [Ο†.map_add, Ο†.map_zero, CharTwo.add_self_eq_zero]
R : Type u_1 R' : Type u_2 inst✝² : AddMonoid R inst✝¹ : CharTwo R inst✝ : AddMonoid R' Ο† : R' β†’+ R h : Function.Injective ⇑φ x : R' ⊒ Ο† (x + x) = Ο† 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 R' : Type u_2 inst✝² : AddMonoid R inst✝¹ : CharTwo R inst✝ : AddMonoid R' Ο† : R' β†’+ R h : Function.Injective ⇑φ x : R' ⊒ Ο† (x + x) = Ο† 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/CharTwo/Hom.lean
IMOSL.Extra.CharTwo.forward_of_surj
[25, 1]
[26, 86]
rw [← h0, ← Ο†.map_add, add_self_eq_zero, Ο†.map_zero]
R : Type u_2 R' : Type u_1 inst✝² : AddMonoid R inst✝¹ : CharTwo R inst✝ : AddMonoid R' Ο† : R β†’+ R' h : Function.Surjective ⇑φ x : R' c : R h0 : Ο† c = x ⊒ x + x = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 R' : Type u_1 inst✝² : AddMonoid R inst✝¹ : CharTwo R inst✝ : AddMonoid R' Ο† : R β†’+ R' h : Function.Surjective ⇑φ x : R' c : R h0 : Ο† c = x ⊒ x + x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.XpowMul_nat_add
[131, 1]
[133, 80]
unfold XpowMul
m n : β„• P : 𝔽₂X ⊒ XpowMul (m + n) P = XpowMul n (XpowMul m P)
m n : β„• P : 𝔽₂X ⊒ { toFinset := Finset.image (fun k => k + (m + n)) P.toFinset } = { toFinset := Finset.image (fun k => k + n) { toFinset := Finset.image (fun k => k + m) P.toFinset }.toFinset }
Please generate a tactic in lean4 to solve the state. STATE: m n : β„• P : 𝔽₂X ⊒ XpowMul (m + n) P = XpowMul n (XpowMul m P) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.XpowMul_nat_add
[131, 1]
[133, 80]
rw [𝔽₂X.ext_iff, eq_comm, Finset.image_image, comp_add_right]
m n : β„• P : 𝔽₂X ⊒ { toFinset := Finset.image (fun k => k + (m + n)) P.toFinset } = { toFinset := Finset.image (fun k => k + n) { toFinset := Finset.image (fun k => k + m) P.toFinset }.toFinset }
no goals
Please generate a tactic in lean4 to solve the state. STATE: m n : β„• P : 𝔽₂X ⊒ { toFinset := Finset.image (fun k => k + (m + n)) P.toFinset } = { toFinset := Finset.image (fun k => k + n) { toFinset := Finset.image (fun k => k + m) P.toFinset }.toFinset } TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.XpowMul_sum
[139, 1]
[142, 71]
rw [Finset.sum_insert h, Finset.sum_insert h, XpowMul_𝔽₂X_add, h0]
ΞΉ : Type u_1 n : β„• inst✝ : DecidableEq ΞΉ f : ΞΉ β†’ 𝔽₂X S✝ : Finset ΞΉ i : ΞΉ S : Finset ΞΉ h : i βˆ‰ S h0 : XpowMul n (S.sum f) = βˆ‘ i ∈ S, XpowMul n (f i) ⊒ XpowMul n ((insert i S).sum f) = βˆ‘ i ∈ insert i S, XpowMul n (f i)
no goals
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 n : β„• inst✝ : DecidableEq ΞΉ f : ΞΉ β†’ 𝔽₂X S✝ : Finset ΞΉ i : ΞΉ S : Finset ΞΉ h : i βˆ‰ S h0 : XpowMul n (S.sum f) = βˆ‘ i ∈ S, XpowMul n (f i) ⊒ XpowMul n ((insert i S).sum f) = βˆ‘ i ∈ insert i S, XpowMul n (f i) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.sum_Xpow_eq_ofFinset
[144, 1]
[147, 55]
rw [Finset.sum_insert h, h0]
i : β„• S : Finset β„• h : i βˆ‰ S h0 : S.sum Xpow = ofFinset S ⊒ (insert i S).sum Xpow = ofFinset (insert i S)
i : β„• S : Finset β„• h : i βˆ‰ S h0 : S.sum Xpow = ofFinset S ⊒ Xpow i + ofFinset S = ofFinset (insert i S)
Please generate a tactic in lean4 to solve the state. STATE: i : β„• S : Finset β„• h : i βˆ‰ S h0 : S.sum Xpow = ofFinset S ⊒ (insert i S).sum Xpow = ofFinset (insert i S) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.sum_Xpow_eq_ofFinset
[144, 1]
[147, 55]
exact 𝔽₂X.ext _ _ (symmDiff_singleton_eq_insert h)
i : β„• S : Finset β„• h : i βˆ‰ S h0 : S.sum Xpow = ofFinset S ⊒ Xpow i + ofFinset S = ofFinset (insert i S)
no goals
Please generate a tactic in lean4 to solve the state. STATE: i : β„• S : Finset β„• h : i βˆ‰ S h0 : S.sum Xpow = ofFinset S ⊒ Xpow i + ofFinset S = ofFinset (insert i S) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.Xpow_add
[188, 1]
[189, 53]
rw [← XpowMul_eq_Xpow_mul, XpowMul_Xpow, add_comm]
k m : β„• ⊒ Xpow (k + m) = Xpow k * Xpow m
no goals
Please generate a tactic in lean4 to solve the state. STATE: k m : β„• ⊒ Xpow (k + m) = Xpow k * Xpow m TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.XpowMul_eq_mul_Xpow
[191, 1]
[194, 59]
rw [XpowMul_Xpow, Xpow_add]
n k : β„• ⊒ XpowMul n (Xpow k) = Xpow k * Xpow n
no goals
Please generate a tactic in lean4 to solve the state. STATE: n k : β„• ⊒ XpowMul n (Xpow k) = Xpow k * Xpow n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.XpowMul_eq_mul_Xpow
[191, 1]
[194, 59]
rw [XpowMul_𝔽₂X_add, 𝔽₂X.add_mul, h, h0]
n : β„• P✝ Q✝ : 𝔽₂X h : XpowMul n P✝ = P✝ * Xpow n h0 : XpowMul n Q✝ = Q✝ * Xpow n ⊒ XpowMul n (P✝ + Q✝) = (P✝ + Q✝) * Xpow n
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• P✝ Q✝ : 𝔽₂X h : XpowMul n P✝ = P✝ * Xpow n h0 : XpowMul n Q✝ = Q✝ * Xpow n ⊒ XpowMul n (P✝ + Q✝) = (P✝ + Q✝) * Xpow n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.mul_comm
[196, 1]
[199, 55]
rw [← XpowMul_eq_Xpow_mul, XpowMul_eq_mul_Xpow]
P : 𝔽₂X n : β„• ⊒ P * Xpow n = Xpow n * P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P : 𝔽₂X n : β„• ⊒ P * Xpow n = Xpow n * P TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.mul_comm
[196, 1]
[199, 55]
rw [𝔽₂X.add_mul, 𝔽₂X.mul_add, h, h0]
P P✝ Q✝ : 𝔽₂X h : P * P✝ = P✝ * P h0 : P * Q✝ = Q✝ * P ⊒ P * (P✝ + Q✝) = (P✝ + Q✝) * P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P P✝ Q✝ : 𝔽₂X h : P * P✝ = P✝ * P h0 : P * Q✝ = Q✝ * P ⊒ P * (P✝ + Q✝) = (P✝ + Q✝) * P TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.mul_XpowMul_left
[201, 1]
[205, 72]
rw [← XpowMul_eq_mul_Xpow, ← XpowMul_eq_mul_Xpow, ← XpowMul_nat_add, ← XpowMul_nat_add, Nat.add_comm]
n : β„• P : 𝔽₂X k : β„• ⊒ XpowMul n P * Xpow k = XpowMul n (P * Xpow k)
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• P : 𝔽₂X k : β„• ⊒ XpowMul n P * Xpow k = XpowMul n (P * Xpow k) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.mul_XpowMul_left
[201, 1]
[205, 72]
rw [𝔽₂X.mul_add, 𝔽₂X.mul_add, h, h0, XpowMul_𝔽₂X_add]
n : β„• P P✝ Q✝ : 𝔽₂X h : XpowMul n P * P✝ = XpowMul n (P * P✝) h0 : XpowMul n P * Q✝ = XpowMul n (P * Q✝) ⊒ XpowMul n P * (P✝ + Q✝) = XpowMul n (P * (P✝ + Q✝))
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• P P✝ Q✝ : 𝔽₂X h : XpowMul n P * P✝ = XpowMul n (P * P✝) h0 : XpowMul n P * Q✝ = XpowMul n (P * Q✝) ⊒ XpowMul n P * (P✝ + Q✝) = XpowMul n (P * (P✝ + Q✝)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.mul_XpowMul_right
[207, 1]
[208, 48]
rw [P.mul_comm, mul_XpowMul_left, Q.mul_comm]
n : β„• P Q : 𝔽₂X ⊒ P * XpowMul n Q = XpowMul n (P * Q)
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• P Q : 𝔽₂X ⊒ P * XpowMul n Q = XpowMul n (P * Q) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.mul_assoc
[210, 1]
[213, 68]
rw [← XpowMul_eq_mul_Xpow, ← XpowMul_eq_mul_Xpow, mul_XpowMul_right]
P Q : 𝔽₂X n : β„• ⊒ P * Q * Xpow n = P * (Q * Xpow n)
no goals
Please generate a tactic in lean4 to solve the state. STATE: P Q : 𝔽₂X n : β„• ⊒ P * Q * Xpow n = P * (Q * Xpow n) TACTIC: