run large_stringclasses 1
value | benchmark large_stringclasses 1
value | problem_id int64 0 346 | k_index int64 0 7 | is_best_in_problem bool 2
classes | statement large_stringclasses 347
values | original_proof large_stringclasses 347
values | candidate_body large_stringlengths 8 7.19k | len_orig int64 9 1.95k | len_new int64 2 1.85k | d_len int64 -692 729 | ratio float64 0.62 90 | is_shorter bool 2
classes | tier_orig float64 4 476 | tier_new float64 1 606 | d_tier float64 -180 188 | tier_collapse bool 2
classes | verified null |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
additive | minif2f | 337 | 4 | false | theorem numbertheory_2dvd4expn (n : β) (hβ : n β 0) : 2 β£ 4 ^ n | by
have h_main : 2 β£ 4 ^ n := by
apply dvd_pow
Β· -- Prove that 2 divides 4
norm_num
Β· -- Prove that n β 0
exact hβ
exact h_main | by
apply dvd_pow
Β· -- Prove that 2 divides 4
norm_num
Β· -- Prove that n β 0
exact hβ | 34 | 22 | 12 | 1.5455 | true | 13.5 | 10 | 3.5 | false | null |
additive | minif2f | 337 | 5 | false | theorem numbertheory_2dvd4expn (n : β) (hβ : n β 0) : 2 β£ 4 ^ n | by
have h_main : 2 β£ 4 ^ n := by
apply dvd_pow
Β· -- Prove that 2 divides 4
norm_num
Β· -- Prove that n β 0
exact hβ
exact h_main | by
have h_main : 2 β£ 4 ^ n := by
apply dvd_pow
Β· -- Prove that 2 divides 4
norm_num
Β· -- Prove that n β 0
exact hβ
exact h_main | 34 | 34 | 0 | 1 | false | 13.5 | 13.5 | 0 | false | null |
additive | minif2f | 337 | 6 | false | theorem numbertheory_2dvd4expn (n : β) (hβ : n β 0) : 2 β£ 4 ^ n | by
have h_main : 2 β£ 4 ^ n := by
apply dvd_pow
Β· -- Prove that 2 divides 4
norm_num
Β· -- Prove that n β 0
exact hβ
exact h_main | by
apply dvd_pow
Β· -- Prove that 2 divides 4
norm_num
Β· -- Prove that n β 0
exact hβ | 34 | 22 | 12 | 1.5455 | true | 13.5 | 10 | 3.5 | false | null |
additive | minif2f | 337 | 7 | false | theorem numbertheory_2dvd4expn (n : β) (hβ : n β 0) : 2 β£ 4 ^ n | by
have h_main : 2 β£ 4 ^ n := by
apply dvd_pow
Β· -- Prove that 2 divides 4
norm_num
Β· -- Prove that n β 0
exact hβ
exact h_main | by
apply dvd_pow
Β· -- Prove that 2 divides 4
norm_num
Β· -- Prove that n β 0
exact hβ | 34 | 22 | 12 | 1.5455 | true | 13.5 | 10 | 3.5 | false | null |
additive | minif2f | 338 | 0 | false | theorem numbertheory_4x3m7y3neq2003
(x y : β€) :
4 * x^3 - 7 * y^3 β 2003 | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | 479 | 479 | 0 | 1 | false | 170.5 | 170.5 | 0 | false | null |
additive | minif2f | 338 | 1 | false | theorem numbertheory_4x3m7y3neq2003
(x y : β€) :
4 * x^3 - 7 * y^3 β 2003 | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | 479 | 479 | 0 | 1 | false | 170.5 | 170.5 | 0 | false | null |
additive | minif2f | 338 | 2 | false | theorem numbertheory_4x3m7y3neq2003
(x y : β€) :
4 * x^3 - 7 * y^3 β 2003 | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | 479 | 479 | 0 | 1 | false | 170.5 | 170.5 | 0 | false | null |
additive | minif2f | 338 | 3 | false | theorem numbertheory_4x3m7y3neq2003
(x y : β€) :
4 * x^3 - 7 * y^3 β 2003 | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | 479 | 479 | 0 | 1 | false | 170.5 | 170.5 | 0 | false | null |
additive | minif2f | 338 | 4 | false | theorem numbertheory_4x3m7y3neq2003
(x y : β€) :
4 * x^3 - 7 * y^3 β 2003 | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | 479 | 479 | 0 | 1 | false | 170.5 | 170.5 | 0 | false | null |
additive | minif2f | 338 | 5 | false | theorem numbertheory_4x3m7y3neq2003
(x y : β€) :
4 * x^3 - 7 * y^3 β 2003 | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | 479 | 479 | 0 | 1 | false | 170.5 | 170.5 | 0 | false | null |
additive | minif2f | 338 | 6 | false | theorem numbertheory_4x3m7y3neq2003
(x y : β€) :
4 * x^3 - 7 * y^3 β 2003 | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | 479 | 479 | 0 | 1 | false | 170.5 | 170.5 | 0 | false | null |
additive | minif2f | 338 | 7 | false | theorem numbertheory_4x3m7y3neq2003
(x y : β€) :
4 * x^3 - 7 * y^3 β 2003 | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | by
have h_main : 4 * x ^ 3 - 7 * y ^ 3 = 2003 β False := by
intro h
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = 2003 % 7 := by
rw [h]
have hβ : (4 * x ^ 3 - 7 * y ^ 3) % 7 = (4 * x ^ 3) % 7 := by
have hβ : (7 * y ^ 3 : β€) % 7 = 0 := by
norm_num [Int.mul_emod, pow_three]
have hβ : (4 ... | 479 | 479 | 0 | 1 | false | 170.5 | 170.5 | 0 | false | null |
additive | minif2f | 339 | 0 | true | theorem numbertheory_aoddbdiv4asqpbsqmod8eq1
(a : β€)
(b : β€)
(hβ : Odd a)
(hβ : 4 β£ b)
(hβ : b >= 0) :
(a^2 + b^2) % 8 = 1 | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | 661 | 627 | 34 | 1.0542 | true | 158 | 154.5 | 3.5 | false | null |
additive | minif2f | 339 | 1 | false | theorem numbertheory_aoddbdiv4asqpbsqmod8eq1
(a : β€)
(b : β€)
(hβ : Odd a)
(hβ : 4 β£ b)
(hβ : b >= 0) :
(a^2 + b^2) % 8 = 1 | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | 661 | 627 | 34 | 1.0542 | true | 158 | 154.5 | 3.5 | false | null |
additive | minif2f | 339 | 2 | false | theorem numbertheory_aoddbdiv4asqpbsqmod8eq1
(a : β€)
(b : β€)
(hβ : Odd a)
(hβ : 4 β£ b)
(hβ : b >= 0) :
(a^2 + b^2) % 8 = 1 | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | 661 | 661 | 0 | 1 | false | 158 | 158 | 0 | false | null |
additive | minif2f | 339 | 3 | false | theorem numbertheory_aoddbdiv4asqpbsqmod8eq1
(a : β€)
(b : β€)
(hβ : Odd a)
(hβ : 4 β£ b)
(hβ : b >= 0) :
(a^2 + b^2) % 8 = 1 | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | 661 | 661 | 0 | 1 | false | 158 | 158 | 0 | false | null |
additive | minif2f | 339 | 4 | false | theorem numbertheory_aoddbdiv4asqpbsqmod8eq1
(a : β€)
(b : β€)
(hβ : Odd a)
(hβ : 4 β£ b)
(hβ : b >= 0) :
(a^2 + b^2) % 8 = 1 | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | 661 | 661 | 0 | 1 | false | 158 | 158 | 0 | false | null |
additive | minif2f | 339 | 5 | false | theorem numbertheory_aoddbdiv4asqpbsqmod8eq1
(a : β€)
(b : β€)
(hβ : Odd a)
(hβ : 4 β£ b)
(hβ : b >= 0) :
(a^2 + b^2) % 8 = 1 | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
hav... | 661 | 627 | 34 | 1.0542 | true | 158 | 154.5 | 3.5 | false | null |
additive | minif2f | 339 | 6 | false | theorem numbertheory_aoddbdiv4asqpbsqmod8eq1
(a : β€)
(b : β€)
(hβ : Odd a)
(hβ : 4 β£ b)
(hβ : b >= 0) :
(a^2 + b^2) % 8 = 1 | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | 661 | 627 | 34 | 1.0542 | true | 158 | 154.5 | 3.5 | false | null |
additive | minif2f | 339 | 7 | false | theorem numbertheory_aoddbdiv4asqpbsqmod8eq1
(a : β€)
(b : β€)
(hβ : Odd a)
(hβ : 4 β£ b)
(hβ : b >= 0) :
(a^2 + b^2) % 8 = 1 | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | by
have h_a_sq_mod_8 : a^2 % 8 = 1 := by
have hβ : β k : β€, a = 2 * k + 1 := by
cases' hβ with k hk
exact β¨k, by linarithβ©
obtain β¨k, hkβ© := hβ
have hβ : a ^ 2 = 4 * (k * (k + 1)) + 1 := by
rw [hk]
ring_nf
<;> nlinarith
have hβ
: (a ^ 2) % 8 = 1 := by
have hβ : (4 :... | 661 | 661 | 0 | 1 | false | 158 | 158 | 0 | false | null |
additive | minif2f | 340 | 0 | false | theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : β) (hβ : 0 < n β§ 0 < k) (hβ : k β€ n) :
Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
have hβ
: n - 1 + 1 = n := by
omega
exact hβ
exact hβ
have hβ : k - 1 + 1 = k := by
ha... | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
omega
exact hβ
have hβ : k - 1 + 1 = k := by
have hβ : 0 < k := hβ.2
have hβ
: k - 1 + 1 = k := by... | 406 | 350 | 56 | 1.16 | true | 72.5 | 63.5 | 9 | false | null |
additive | minif2f | 340 | 1 | false | theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : β) (hβ : 0 < n β§ 0 < k) (hβ : k β€ n) :
Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
have hβ
: n - 1 + 1 = n := by
omega
exact hβ
exact hβ
have hβ : k - 1 + 1 = k := by
ha... | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
have hβ
: n - 1 + 1 = n := by
omega
exact hβ
exact hβ
have hβ : k - 1 + 1 = k := by
ha... | 406 | 406 | 0 | 1 | false | 72.5 | 72.5 | 0 | false | null |
additive | minif2f | 340 | 2 | false | theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : β) (hβ : 0 < n β§ 0 < k) (hβ : k β€ n) :
Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
have hβ
: n - 1 + 1 = n := by
omega
exact hβ
exact hβ
have hβ : k - 1 + 1 = k := by
ha... | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
have hβ
: n - 1 + 1 = n := by
omega
exact hβ
exact hβ
have hβ : k - 1 + 1 = k := by
ha... | 406 | 406 | 0 | 1 | false | 72.5 | 72.5 | 0 | false | null |
additive | minif2f | 340 | 3 | false | theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : β) (hβ : 0 < n β§ 0 < k) (hβ : k β€ n) :
Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
have hβ
: n - 1 + 1 = n := by
omega
exact hβ
exact hβ
have hβ : k - 1 + 1 = k := by
ha... | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
have hβ
: n - 1 + 1 = n := by
omega
exact hβ
exact hβ
have hβ : k - 1 + 1 = k := by
ha... | 406 | 378 | 28 | 1.0741 | true | 72.5 | 70.5 | 2 | false | null |
additive | minif2f | 340 | 4 | false | theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : β) (hβ : 0 < n β§ 0 < k) (hβ : k β€ n) :
Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
have hβ
: n - 1 + 1 = n := by
omega
exact hβ
exact hβ
have hβ : k - 1 + 1 = k := by
ha... | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
omega
exact hβ
have hβ : k - 1 + 1 = k := by
have hβ : 0 < k := hβ.2
have hβ
: k - 1 + 1 = k := b... | 406 | 342 | 64 | 1.1871 | true | 72.5 | 61.5 | 11 | false | null |
additive | minif2f | 340 | 5 | false | theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : β) (hβ : 0 < n β§ 0 < k) (hβ : k β€ n) :
Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
have hβ
: n - 1 + 1 = n := by
omega
exact hβ
exact hβ
have hβ : k - 1 + 1 = k := by
ha... | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
have hβ
: n - 1 + 1 = n := by
omega
exact hβ
exact hβ
have hβ : k - 1 + 1 = k := by
ha... | 406 | 378 | 28 | 1.0741 | true | 72.5 | 70.5 | 2 | false | null |
additive | minif2f | 340 | 6 | false | theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : β) (hβ : 0 < n β§ 0 < k) (hβ : k β€ n) :
Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
have hβ
: n - 1 + 1 = n := by
omega
exact hβ
exact hβ
have hβ : k - 1 + 1 = k := by
ha... | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
have hβ
: n - 1 + 1 = n := by
omega
exact hβ
exact hβ
have hβ : k - 1 + 1 = k := by
h... | 406 | 378 | 28 | 1.0741 | true | 72.5 | 70.5 | 2 | false | null |
additive | minif2f | 340 | 7 | true | theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : β) (hβ : 0 < n β§ 0 < k) (hβ : k β€ n) :
Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) | by
have h_main : Nat.choose n k = Nat.choose (n - 1) (k - 1) + Nat.choose (n - 1) k := by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
have hβ
: n - 1 + 1 = n := by
omega
exact hβ
exact hβ
have hβ : k - 1 + 1 = k := by
ha... | by
have hβ : n - 1 + 1 = n := by
have hβ : 0 < n := hβ.1
have hβ : n - 1 + 1 = n := by
omega
exact hβ
have hβ : k - 1 + 1 = k := by
have hβ : 0 < k := hβ.2
have hβ
: k - 1 + 1 = k := by
omega
exact hβ
have hβ : Nat.choose (n - 1 + 1) (k - 1 + 1) = Nat.choose (n - 1) (k - 1) + N... | 406 | 215 | 191 | 1.8884 | true | 72.5 | 48.5 | 24 | false | null |
additive | minif2f | 341 | 0 | false | theorem numbertheory_notequiv2i2jasqbsqdiv8 :
Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | 342 | 342 | 0 | 1 | false | 50.5 | 50.5 | 0 | false | null |
additive | minif2f | 341 | 1 | false | theorem numbertheory_notequiv2i2jasqbsqdiv8 :
Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | 342 | 342 | 0 | 1 | false | 50.5 | 50.5 | 0 | false | null |
additive | minif2f | 341 | 2 | false | theorem numbertheory_notequiv2i2jasqbsqdiv8 :
Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | 342 | 342 | 0 | 1 | false | 50.5 | 50.5 | 0 | false | null |
additive | minif2f | 341 | 3 | false | theorem numbertheory_notequiv2i2jasqbsqdiv8 :
Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | 342 | 342 | 0 | 1 | false | 50.5 | 50.5 | 0 | false | null |
additive | minif2f | 341 | 4 | false | theorem numbertheory_notequiv2i2jasqbsqdiv8 :
Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | 342 | 342 | 0 | 1 | false | 50.5 | 50.5 | 0 | false | null |
additive | minif2f | 341 | 5 | false | theorem numbertheory_notequiv2i2jasqbsqdiv8 :
Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | 342 | 342 | 0 | 1 | false | 50.5 | 50.5 | 0 | false | null |
additive | minif2f | 341 | 6 | false | theorem numbertheory_notequiv2i2jasqbsqdiv8 :
Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | 342 | 342 | 0 | 1 | false | 50.5 | 50.5 | 0 | false | null |
additive | minif2f | 341 | 7 | false | theorem numbertheory_notequiv2i2jasqbsqdiv8 :
Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | by
have h_main : Β¬ (β a b : β€, (β i j, a = 2*i β§ b=2*j) β (β k, a^2 + b^2 = 8*k)) := by
intro h
have hβ : (β i j : β€, (2 : β€) = 2 * i β§ (4 : β€) = 2 * j) := by
refine' β¨1, 2, _β©
norm_num
have hβ : Β¬ (β k : β€, (2 : β€)^2 + (4 : β€)^2 = 8 * k) := by
intro hβ
rcases hβ with β¨k, hkβ©
... | 342 | 342 | 0 | 1 | false | 50.5 | 50.5 | 0 | false | null |
additive | minif2f | 342 | 0 | true | theorem numbertheory_prmdvsneqnsqmodpeq0 (n : β€) (p : β) (hβ : Nat.Prime p) :
βp β£ n β n ^ 2 % p = 0 | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
-- If p divides n, then p^2 divides n^2
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
-- Since p^2 divides n^2, p also divides n^2
exact... | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
exact dvd_trans (by norm_cast <;> exact dvd_pow_self _ (by
have hβ := hβ.two_le
omega)... | 569 | 392 | 177 | 1.4515 | true | 117 | 93 | 24 | false | null |
additive | minif2f | 342 | 1 | false | theorem numbertheory_prmdvsneqnsqmodpeq0 (n : β€) (p : β) (hβ : Nat.Prime p) :
βp β£ n β n ^ 2 % p = 0 | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
-- If p divides n, then p^2 divides n^2
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
-- Since p^2 divides n^2, p also divides n^2
exact... | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
-- If p divides n, then p^2 divides n^2
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
-- Since p^2 divides n^2, p also divides n^2
exact... | 569 | 569 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 342 | 2 | false | theorem numbertheory_prmdvsneqnsqmodpeq0 (n : β€) (p : β) (hβ : Nat.Prime p) :
βp β£ n β n ^ 2 % p = 0 | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
-- If p divides n, then p^2 divides n^2
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
-- Since p^2 divides n^2, p also divides n^2
exact... | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
exact dvd_trans (by norm_cast <;> exact dvd_pow_self _ (by
have hβ := hβ.two_le
omega)... | 569 | 392 | 177 | 1.4515 | true | 117 | 93 | 24 | false | null |
additive | minif2f | 342 | 3 | false | theorem numbertheory_prmdvsneqnsqmodpeq0 (n : β€) (p : β) (hβ : Nat.Prime p) :
βp β£ n β n ^ 2 % p = 0 | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
-- If p divides n, then p^2 divides n^2
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
-- Since p^2 divides n^2, p also divides n^2
exact... | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
have hβ : (p : β€) ^ 2 β£ n ^ 2 := hβ
have hβ
: (p : β€) β£ n ^ 2 := by
exact dvd_trans (by ... | 569 | 424 | 145 | 1.342 | true | 117 | 98.5 | 18.5 | false | null |
additive | minif2f | 342 | 4 | false | theorem numbertheory_prmdvsneqnsqmodpeq0 (n : β€) (p : β) (hβ : Nat.Prime p) :
βp β£ n β n ^ 2 % p = 0 | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
-- If p divides n, then p^2 divides n^2
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
-- Since p^2 divides n^2, p also divides n^2
exact... | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
-- If p divides n, then p^2 divides n^2
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
-- Since p^2 divides n^2, p also divides n^2
exact... | 569 | 569 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 342 | 5 | false | theorem numbertheory_prmdvsneqnsqmodpeq0 (n : β€) (p : β) (hβ : Nat.Prime p) :
βp β£ n β n ^ 2 % p = 0 | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
-- If p divides n, then p^2 divides n^2
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
-- Since p^2 divides n^2, p also divides n^2
exact... | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
-- If p divides n, then p^2 divides n^2
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
-- Since p^2 divides n^2, p also divides n^2
exact... | 569 | 569 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 342 | 6 | false | theorem numbertheory_prmdvsneqnsqmodpeq0 (n : β€) (p : β) (hβ : Nat.Prime p) :
βp β£ n β n ^ 2 % p = 0 | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
-- If p divides n, then p^2 divides n^2
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
-- Since p^2 divides n^2, p also divides n^2
exact... | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
exact dvd_trans (by norm_cast <;> exact dvd_pow_self _ (by
have hβ := hβ.two_le
omega)... | 569 | 392 | 177 | 1.4515 | true | 117 | 93 | 24 | false | null |
additive | minif2f | 342 | 7 | false | theorem numbertheory_prmdvsneqnsqmodpeq0 (n : β€) (p : β) (hβ : Nat.Prime p) :
βp β£ n β n ^ 2 % p = 0 | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
-- If p divides n, then p^2 divides n^2
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
-- Since p^2 divides n^2, p also divides n^2
exact... | by
have h_forward : (βp : β€) β£ n β n ^ 2 % p = 0 := by
intro h
have hβ : (p : β€) β£ n := h
have hβ : (p : β€) ^ 2 β£ n ^ 2 := by
exact pow_dvd_pow_of_dvd hβ 2
have hβ : (p : β€) β£ n ^ 2 := by
exact dvd_trans (by norm_cast <;> exact dvd_pow_self _ (by
have hβ := hβ.two_le
omega)... | 569 | 392 | 177 | 1.4515 | true | 117 | 93 | 24 | false | null |
additive | minif2f | 343 | 0 | false | theorem numbertheory_sqmod3in01d (a : β€) : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
r... | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
... | 372 | 372 | 0 | 1 | false | 95.5 | 95.5 | 0 | false | null |
additive | minif2f | 343 | 1 | false | theorem numbertheory_sqmod3in01d (a : β€) : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
r... | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
rcas... | 372 | 372 | 0 | 1 | false | 95.5 | 95.5 | 0 | false | null |
additive | minif2f | 343 | 2 | false | theorem numbertheory_sqmod3in01d (a : β€) : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
r... | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
rcas... | 372 | 372 | 0 | 1 | false | 95.5 | 95.5 | 0 | false | null |
additive | minif2f | 343 | 3 | false | theorem numbertheory_sqmod3in01d (a : β€) : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
r... | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
rcas... | 372 | 372 | 0 | 1 | false | 95.5 | 95.5 | 0 | false | null |
additive | minif2f | 343 | 4 | true | theorem numbertheory_sqmod3in01d (a : β€) : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
r... | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
rcases hβ with (hβ | hβ | hβ)
Β· -- Case: a β‘ 0 mod 3
have hβ : ... | 372 | 349 | 23 | 1.0659 | true | 95.5 | 88 | 7.5 | false | null |
additive | minif2f | 343 | 5 | false | theorem numbertheory_sqmod3in01d (a : β€) : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
r... | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
rcas... | 372 | 372 | 0 | 1 | false | 95.5 | 95.5 | 0 | false | null |
additive | minif2f | 343 | 6 | false | theorem numbertheory_sqmod3in01d (a : β€) : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
r... | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
rcases hβ with (hβ | hβ | hβ)
Β· -- Case: a β‘ 0 mod 3
have hβ ... | 372 | 349 | 23 | 1.0659 | true | 95.5 | 88 | 7.5 | false | null |
additive | minif2f | 343 | 7 | false | theorem numbertheory_sqmod3in01d (a : β€) : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
have hβ : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := h_main
r... | by
have h_main : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have h : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
have : a % 3 = 0 β¨ a % 3 = 1 β¨ a % 3 = 2 := by
omega
tauto
tauto
have h_final : a ^ 2 % 3 = 0 β¨ a ^ 2 % 3 = 1 := by
rcases h_main with (hβ | hβ | hβ)
Β· -- Case: a β‘ 0 mod 3
... | 372 | 350 | 22 | 1.0629 | true | 95.5 | 93.5 | 2 | false | null |
additive | minif2f | 344 | 0 | false | theorem numbertheory_sqmod4in01d (a : β€) : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 | by
have h_main : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h with (h | h | h | h)
Β· -- Case: a β‘ 0 mod 4
have hβ : a ^ 2 % 4 = 0 := by
have hβ : a % 4 = 0 := h
have hβ : a ^ 2 % 4 = 0 := by
have : a %... | by
have h_main : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h with (h | h | h | h)
Β· -- Case: a β‘ 0 mod 4
have hβ : a ^ 2 % 4 = 0 := by
have hβ : a % 4 = 0 := h
have hβ : a ^ 2 % 4 = 0 := by
have : a %... | 380 | 380 | 0 | 1 | false | 89 | 89 | 0 | false | null |
additive | minif2f | 344 | 1 | true | theorem numbertheory_sqmod4in01d (a : β€) : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 | by
have h_main : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h with (h | h | h | h)
Β· -- Case: a β‘ 0 mod 4
have hβ : a ^ 2 % 4 = 0 := by
have hβ : a % 4 = 0 := h
have hβ : a ^ 2 % 4 = 0 := by
have : a %... | by
have h : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h1 : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h1 with (h1 | h1 | h1 | h1) <;>
(
simp [pow_two, Int.add_emod, Int.mul_emod, h1, Int.sub_emod, Int.ofNat]
)
exact h | 380 | 81 | 299 | 4.6914 | true | 89 | 21 | 68 | true | null |
additive | minif2f | 344 | 2 | false | theorem numbertheory_sqmod4in01d (a : β€) : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 | by
have h_main : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h with (h | h | h | h)
Β· -- Case: a β‘ 0 mod 4
have hβ : a ^ 2 % 4 = 0 := by
have hβ : a % 4 = 0 := h
have hβ : a ^ 2 % 4 = 0 := by
have : a %... | by
have h_main : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h with (h | h | h | h)
Β· -- Case: a β‘ 0 mod 4
have hβ : a ^ 2 % 4 = 0 := by
have hβ : a % 4 = 0 := h
have hβ : a ^ 2 % 4 = 0 := by
have : a ... | 380 | 380 | 0 | 1 | false | 89 | 89 | 0 | false | null |
additive | minif2f | 344 | 3 | false | theorem numbertheory_sqmod4in01d (a : β€) : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 | by
have h_main : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h with (h | h | h | h)
Β· -- Case: a β‘ 0 mod 4
have hβ : a ^ 2 % 4 = 0 := by
have hβ : a % 4 = 0 := h
have hβ : a ^ 2 % 4 = 0 := by
have : a %... | by
have h_main : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h with (h | h | h | h)
Β· -- Case: a β‘ 0 mod 4
have hβ : a ^ 2 % 4 = 0 := by
have hβ : a % 4 = 0 := h
have hβ : a ^ 2 % 4 = 0 := by
have : a ... | 380 | 380 | 0 | 1 | false | 89 | 89 | 0 | false | null |
additive | minif2f | 344 | 4 | false | theorem numbertheory_sqmod4in01d (a : β€) : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 | by
have h_main : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h with (h | h | h | h)
Β· -- Case: a β‘ 0 mod 4
have hβ : a ^ 2 % 4 = 0 := by
have hβ : a % 4 = 0 := h
have hβ : a ^ 2 % 4 = 0 := by
have : a %... | by
have h : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h1 : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h1 with (h1 | h1 | h1 | h1)
Β· -- Case a % 4 = 0
have h2 : a ^ 2 % 4 = 0 := by
have h3 : a % 4 = 0 := h1
simp [pow_two, Int.mul_emod, h3]
exact Or.inl... | 380 | 232 | 148 | 1.6379 | true | 89 | 65 | 24 | true | null |
additive | minif2f | 344 | 5 | false | theorem numbertheory_sqmod4in01d (a : β€) : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 | by
have h_main : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h with (h | h | h | h)
Β· -- Case: a β‘ 0 mod 4
have hβ : a ^ 2 % 4 = 0 := by
have hβ : a % 4 = 0 := h
have hβ : a ^ 2 % 4 = 0 := by
have : a %... | by
have h : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h1 : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h1 with (h1 | h1 | h1 | h1)
Β· -- Case where a β‘ 0 [MOD 4]
have h2 : a ^ 2 % 4 = 0 := by
have h3 : a % 4 = 0 := h1
have h4 : a ^ 2 % 4 = 0 := by
s... | 380 | 300 | 80 | 1.2667 | true | 89 | 79 | 10 | true | null |
additive | minif2f | 344 | 6 | false | theorem numbertheory_sqmod4in01d (a : β€) : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 | by
have h_main : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h with (h | h | h | h)
Β· -- Case: a β‘ 0 mod 4
have hβ : a ^ 2 % 4 = 0 := by
have hβ : a % 4 = 0 := h
have hβ : a ^ 2 % 4 = 0 := by
have : a %... | by
have h_main : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h with (h | h | h | h)
Β· -- Case: a β‘ 0 mod 4
have hβ : a ^ 2 % 4 = 0 := by
have hβ : a % 4 = 0 := h
have hβ : a ^ 2 % 4 = 0 := by
have : a %... | 380 | 380 | 0 | 1 | false | 89 | 89 | 0 | false | null |
additive | minif2f | 344 | 7 | false | theorem numbertheory_sqmod4in01d (a : β€) : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 | by
have h_main : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h with (h | h | h | h)
Β· -- Case: a β‘ 0 mod 4
have hβ : a ^ 2 % 4 = 0 := by
have hβ : a % 4 = 0 := h
have hβ : a ^ 2 % 4 = 0 := by
have : a %... | by
have h_main : a ^ 2 % 4 = 0 β¨ a ^ 2 % 4 = 1 := by
have h : a % 4 = 0 β¨ a % 4 = 1 β¨ a % 4 = 2 β¨ a % 4 = 3 := by
omega
rcases h with (h | h | h | h)
Β· -- Case: a β‘ 0 mod 4
have hβ : a ^ 2 % 4 = 0 := by
have hβ : a % 4 = 0 := h
have hβ : a ^ 2 % 4 = 0 := by
have : a %... | 380 | 380 | 0 | 1 | false | 89 | 89 | 0 | false | null |
additive | minif2f | 345 | 0 | false | theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : β) (hβ : 0 < n) :
(β k in Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | 1,106 | 1,057 | 49 | 1.0464 | true | 231.5 | 222 | 9.5 | false | null |
additive | minif2f | 345 | 1 | false | theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : β) (hβ : 0 < n) :
(β k in Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | 1,106 | 1,099 | 7 | 1.0064 | true | 231.5 | 225.5 | 6 | false | null |
additive | minif2f | 345 | 2 | false | theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : β) (hβ : 0 < n) :
(β k in Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | 1,106 | 1,070 | 36 | 1.0336 | true | 231.5 | 228 | 3.5 | false | null |
additive | minif2f | 345 | 3 | false | theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : β) (hβ : 0 < n) :
(β k in Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | 1,106 | 1,099 | 7 | 1.0064 | true | 231.5 | 225.5 | 6 | false | null |
additive | minif2f | 345 | 4 | false | theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : β) (hβ : 0 < n) :
(β k in Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | 1,106 | 1,070 | 36 | 1.0336 | true | 231.5 | 228 | 3.5 | false | null |
additive | minif2f | 345 | 5 | true | theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : β) (hβ : 0 < n) :
(β k in Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | 1,106 | 1,025 | 81 | 1.079 | true | 231.5 | 224 | 7.5 | false | null |
additive | minif2f | 345 | 6 | false | theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : β) (hβ : 0 < n) :
(β k in Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | 1,106 | 1,062 | 44 | 1.0414 | true | 231.5 | 227.5 | 4 | false | null |
additive | minif2f | 345 | 7 | false | theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : β) (hβ : 0 < n) :
(β k in Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | by
have hβ : (β k in Finset.Icc 1 n, k * Nat.choose n k) = β k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by
apply Finset.sum_congr rfl
intro k hk
have hβ : k β Finset.Icc 1 n := hk
have hβ : 1 β€ k := by
simp [Finset.mem_Icc] at hβ
linarith
have hβ : k β€ n := by
simp [F... | 1,106 | 1,063 | 43 | 1.0405 | true | 231.5 | 222 | 9.5 | false | null |
additive | minif2f | 346 | 0 | false | theorem numbertheory_x5neqy2p4
(x y : β€) :
x^5 β y^2 + 4 | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | 623 | 623 | 0 | 1 | false | 69 | 69 | 0 | false | null |
additive | minif2f | 346 | 1 | false | theorem numbertheory_x5neqy2p4
(x y : β€) :
x^5 β y^2 + 4 | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | 623 | 623 | 0 | 1 | false | 69 | 69 | 0 | false | null |
additive | minif2f | 346 | 2 | false | theorem numbertheory_x5neqy2p4
(x y : β€) :
x^5 β y^2 + 4 | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | 623 | 623 | 0 | 1 | false | 69 | 69 | 0 | false | null |
additive | minif2f | 346 | 3 | false | theorem numbertheory_x5neqy2p4
(x y : β€) :
x^5 β y^2 + 4 | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | 623 | 623 | 0 | 1 | false | 69 | 69 | 0 | false | null |
additive | minif2f | 346 | 4 | true | theorem numbertheory_x5neqy2p4
(x y : β€) :
x^5 β y^2 + 4 | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€) % 11 = 4 β¨ (x : β€) % 11 = 5 β¨ (x : β€) % 11 = 6 β¨ (x : β€) % ... | 623 | 565 | 58 | 1.1027 | true | 69 | 61 | 8 | false | null |
additive | minif2f | 346 | 5 | false | theorem numbertheory_x5neqy2p4
(x y : β€) :
x^5 β y^2 + 4 | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | 623 | 623 | 0 | 1 | false | 69 | 69 | 0 | false | null |
additive | minif2f | 346 | 6 | false | theorem numbertheory_x5neqy2p4
(x y : β€) :
x^5 β y^2 + 4 | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€) % 11 = 4 β¨ (x : β€) % 11 = 5 β¨ (x : β€) % 11 = 6 β¨ (x : β€) % ... | 623 | 569 | 54 | 1.0949 | true | 69 | 59 | 10 | false | null |
additive | minif2f | 346 | 7 | false | theorem numbertheory_x5neqy2p4
(x y : β€) :
x^5 β y^2 + 4 | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | by
have h_main : β (x y : β€), x^5 = y^2 + 4 β False := by
intro x y h
have hβ := h
-- Reduce the equation modulo 11
have hβ : (x^5 : β€) % 11 = (y^2 + 4 : β€) % 11 := by
rw [hβ]
-- Compute x^5 % 11
have hβ : (x : β€) % 11 = 0 β¨ (x : β€) % 11 = 1 β¨ (x : β€) % 11 = 2 β¨ (x : β€) % 11 = 3 β¨ (x : β€... | 623 | 623 | 0 | 1 | false | 69 | 69 | 0 | false | null |
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