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 | 25 | 0 | false | theorem algebra_bleqa_apbon2msqrtableqambsqon8b
(a b : β)
(hβ : 0 < a β§ 0 < b)
(hβ : b β€ a) :
(a + b) / 2 - Real.sqrt (a * b) β€ (a - b)^2 / (8 * b) | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by po... | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by po... | 855 | 855 | 0 | 1 | false | 143 | 143 | 0 | false | null |
additive | minif2f | 25 | 1 | false | theorem algebra_bleqa_apbon2msqrtableqambsqon8b
(a b : β)
(hβ : 0 < a β§ 0 < b)
(hβ : b β€ a) :
(a + b) / 2 - Real.sqrt (a * b) β€ (a - b)^2 / (8 * b) | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by po... | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by po... | 855 | 855 | 0 | 1 | false | 143 | 143 | 0 | false | null |
additive | minif2f | 25 | 2 | false | theorem algebra_bleqa_apbon2msqrtableqambsqon8b
(a b : β)
(hβ : 0 < a β§ 0 < b)
(hβ : b β€ a) :
(a + b) / 2 - Real.sqrt (a * b) β€ (a - b)^2 / (8 * b) | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by po... | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by positivity
... | 855 | 831 | 24 | 1.0289 | true | 143 | 141 | 2 | false | null |
additive | minif2f | 25 | 3 | true | theorem algebra_bleqa_apbon2msqrtableqambsqon8b
(a b : β)
(hβ : 0 < a β§ 0 < b)
(hβ : b β€ a) :
(a + b) / 2 - Real.sqrt (a * b) β€ (a - b)^2 / (8 * b) | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by po... | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by positivity
... | 855 | 827 | 28 | 1.0339 | true | 143 | 134.5 | 8.5 | false | null |
additive | minif2f | 25 | 4 | false | theorem algebra_bleqa_apbon2msqrtableqambsqon8b
(a b : β)
(hβ : 0 < a β§ 0 < b)
(hβ : b β€ a) :
(a + b) / 2 - Real.sqrt (a * b) β€ (a - b)^2 / (8 * b) | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by po... | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by po... | 855 | 855 | 0 | 1 | false | 143 | 143 | 0 | false | null |
additive | minif2f | 25 | 5 | false | theorem algebra_bleqa_apbon2msqrtableqambsqon8b
(a b : β)
(hβ : 0 < a β§ 0 < b)
(hβ : b β€ a) :
(a + b) / 2 - Real.sqrt (a * b) β€ (a - b)^2 / (8 * b) | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by po... | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by positivity
... | 855 | 827 | 28 | 1.0339 | true | 143 | 134.5 | 8.5 | false | null |
additive | minif2f | 25 | 6 | false | theorem algebra_bleqa_apbon2msqrtableqambsqon8b
(a b : β)
(hβ : 0 < a β§ 0 < b)
(hβ : b β€ a) :
(a + b) / 2 - Real.sqrt (a * b) β€ (a - b)^2 / (8 * b) | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by po... | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by po... | 855 | 855 | 0 | 1 | false | 143 | 143 | 0 | false | null |
additive | minif2f | 25 | 7 | false | theorem algebra_bleqa_apbon2msqrtableqambsqon8b
(a b : β)
(hβ : 0 < a β§ 0 < b)
(hβ : b β€ a) :
(a + b) / 2 - Real.sqrt (a * b) β€ (a - b)^2 / (8 * b) | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by po... | by
have hβ : 0 < b := by linarith
have hβ : 0 < a := by linarith
have hβ : 0 β€ Real.sqrt a := Real.sqrt_nonneg a
have hβ
: 0 β€ Real.sqrt b := Real.sqrt_nonneg b
have hβ : (a + b) / 2 - Real.sqrt (a * b) = (Real.sqrt a - Real.sqrt b)^2 / 2 := by
have hββ : 0 β€ Real.sqrt a * Real.sqrt b := by po... | 855 | 855 | 0 | 1 | false | 143 | 143 | 0 | false | null |
additive | minif2f | 26 | 0 | true | theorem algebra_manipexpr_2erprsqpesqeqnrpnesq (e r : β) :
2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 | by
have h_main : 2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 := by
ring_nf
<;> simp [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;> ring_nf
<;> simp_all [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;>
(try
{
... | by
ring_nf | 121 | 2 | 119 | 60.5 | true | 67 | 2 | 65 | false | null |
additive | minif2f | 26 | 1 | false | theorem algebra_manipexpr_2erprsqpesqeqnrpnesq (e r : β) :
2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 | by
have h_main : 2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 := by
ring_nf
<;> simp [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;> ring_nf
<;> simp_all [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;>
(try
{
... | by
ring_nf | 121 | 2 | 119 | 60.5 | true | 67 | 2 | 65 | false | null |
additive | minif2f | 26 | 2 | false | theorem algebra_manipexpr_2erprsqpesqeqnrpnesq (e r : β) :
2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 | by
have h_main : 2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 := by
ring_nf
<;> simp [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;> ring_nf
<;> simp_all [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;>
(try
{
... | by
ring_nf | 121 | 2 | 119 | 60.5 | true | 67 | 2 | 65 | false | null |
additive | minif2f | 26 | 3 | false | theorem algebra_manipexpr_2erprsqpesqeqnrpnesq (e r : β) :
2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 | by
have h_main : 2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 := by
ring_nf
<;> simp [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;> ring_nf
<;> simp_all [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;>
(try
{
... | by
ring_nf | 121 | 2 | 119 | 60.5 | true | 67 | 2 | 65 | false | null |
additive | minif2f | 26 | 4 | false | theorem algebra_manipexpr_2erprsqpesqeqnrpnesq (e r : β) :
2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 | by
have h_main : 2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 := by
ring_nf
<;> simp [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;> ring_nf
<;> simp_all [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;>
(try
{
... | by
ring_nf | 121 | 2 | 119 | 60.5 | true | 67 | 2 | 65 | false | null |
additive | minif2f | 26 | 5 | false | theorem algebra_manipexpr_2erprsqpesqeqnrpnesq (e r : β) :
2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 | by
have h_main : 2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 := by
ring_nf
<;> simp [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;> ring_nf
<;> simp_all [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;>
(try
{
... | by
ring_nf | 121 | 2 | 119 | 60.5 | true | 67 | 2 | 65 | false | null |
additive | minif2f | 26 | 6 | false | theorem algebra_manipexpr_2erprsqpesqeqnrpnesq (e r : β) :
2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 | by
have h_main : 2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 := by
ring_nf
<;> simp [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;> ring_nf
<;> simp_all [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;>
(try
{
... | by
ring_nf | 121 | 2 | 119 | 60.5 | true | 67 | 2 | 65 | false | null |
additive | minif2f | 26 | 7 | false | theorem algebra_manipexpr_2erprsqpesqeqnrpnesq (e r : β) :
2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 | by
have h_main : 2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 := by
ring_nf
<;> simp [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;> ring_nf
<;> simp_all [Complex.ext_iff, pow_two, mul_comm, mul_assoc, mul_left_comm]
<;> norm_num
<;>
(try
{
... | by
ring_nf | 121 | 2 | 119 | 60.5 | true | 67 | 2 | 65 | false | null |
additive | minif2f | 27 | 0 | false | theorem algebra_manipexpr_apbeq2cceqiacpbceqm2 (a b c : β) (hβ : a + b = 2 * c)
(hβ : c = Complex.I) : a * c + b * c = -2 | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := ... | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := by rw ... | 216 | 216 | 0 | 1 | false | 51.5 | 51.5 | 0 | false | null |
additive | minif2f | 27 | 1 | false | theorem algebra_manipexpr_apbeq2cceqiacpbceqm2 (a b c : β) (hβ : a + b = 2 * c)
(hβ : c = Complex.I) : a * c + b * c = -2 | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := ... | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := ... | 216 | 216 | 0 | 1 | false | 51.5 | 51.5 | 0 | false | null |
additive | minif2f | 27 | 2 | false | theorem algebra_manipexpr_apbeq2cceqiacpbceqm2 (a b c : β) (hβ : a + b = 2 * c)
(hβ : c = Complex.I) : a * c + b * c = -2 | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := ... | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := by rw ... | 216 | 216 | 0 | 1 | false | 51.5 | 51.5 | 0 | false | null |
additive | minif2f | 27 | 3 | false | theorem algebra_manipexpr_apbeq2cceqiacpbceqm2 (a b c : β) (hβ : a + b = 2 * c)
(hβ : c = Complex.I) : a * c + b * c = -2 | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := ... | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := ... | 216 | 216 | 0 | 1 | false | 51.5 | 51.5 | 0 | false | null |
additive | minif2f | 27 | 4 | false | theorem algebra_manipexpr_apbeq2cceqiacpbceqm2 (a b c : β) (hβ : a + b = 2 * c)
(hβ : c = Complex.I) : a * c + b * c = -2 | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := ... | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := ... | 216 | 216 | 0 | 1 | false | 51.5 | 51.5 | 0 | false | null |
additive | minif2f | 27 | 5 | false | theorem algebra_manipexpr_apbeq2cceqiacpbceqm2 (a b c : β) (hβ : a + b = 2 * c)
(hβ : c = Complex.I) : a * c + b * c = -2 | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := ... | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := ... | 216 | 216 | 0 | 1 | false | 51.5 | 51.5 | 0 | false | null |
additive | minif2f | 27 | 6 | false | theorem algebra_manipexpr_apbeq2cceqiacpbceqm2 (a b c : β) (hβ : a + b = 2 * c)
(hβ : c = Complex.I) : a * c + b * c = -2 | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := ... | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := ... | 216 | 216 | 0 | 1 | false | 51.5 | 51.5 | 0 | false | null |
additive | minif2f | 27 | 7 | false | theorem algebra_manipexpr_apbeq2cceqiacpbceqm2 (a b c : β) (hβ : a + b = 2 * c)
(hβ : c = Complex.I) : a * c + b * c = -2 | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := ... | by
have hβ : a + b = 2 * Complex.I := by
rw [hβ] at hβ
exact hβ
have hβ : a * c + b * c = (a + b) * c := by
calc
a * c + b * c = (a + b) * c := by ring
_ = (a + b) * c := by rfl
have hβ : a * c + b * c = (2 * Complex.I) * Complex.I := by
calc
a * c + b * c = (a + b) * c := by rw ... | 216 | 216 | 0 | 1 | false | 51.5 | 51.5 | 0 | false | null |
additive | minif2f | 28 | 0 | false | theorem algebra_others_exirrpowirrrat :
β a b, Irrational a β§ Irrational b β§ Β¬ Irrational (a^b) | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a ^ b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2... | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a^b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2 a... | 675 | 675 | 0 | 1 | false | 99 | 99 | 0 | false | null |
additive | minif2f | 28 | 1 | false | theorem algebra_others_exirrpowirrrat :
β a b, Irrational a β§ Irrational b β§ Β¬ Irrational (a^b) | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a ^ b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2... | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a^b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2 a... | 675 | 675 | 0 | 1 | false | 99 | 99 | 0 | false | null |
additive | minif2f | 28 | 2 | false | theorem algebra_others_exirrpowirrrat :
β a b, Irrational a β§ Irrational b β§ Β¬ Irrational (a^b) | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a ^ b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2... | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a^b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2 a... | 675 | 675 | 0 | 1 | false | 99 | 99 | 0 | false | null |
additive | minif2f | 28 | 3 | false | theorem algebra_others_exirrpowirrrat :
β a b, Irrational a β§ Irrational b β§ Β¬ Irrational (a^b) | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a ^ b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2... | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a ^ b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2... | 675 | 675 | 0 | 1 | false | 99 | 99 | 0 | false | null |
additive | minif2f | 28 | 4 | false | theorem algebra_others_exirrpowirrrat :
β a b, Irrational a β§ Irrational b β§ Β¬ Irrational (a^b) | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a ^ b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2... | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a^b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2 a... | 675 | 675 | 0 | 1 | false | 99 | 99 | 0 | false | null |
additive | minif2f | 28 | 5 | false | theorem algebra_others_exirrpowirrrat :
β a b, Irrational a β§ Irrational b β§ Β¬ Irrational (a^b) | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a ^ b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2... | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a^b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2 a... | 675 | 675 | 0 | 1 | false | 99 | 99 | 0 | false | null |
additive | minif2f | 28 | 6 | false | theorem algebra_others_exirrpowirrrat :
β a b, Irrational a β§ Irrational b β§ Β¬ Irrational (a^b) | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a ^ b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2... | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a^b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2 a... | 675 | 675 | 0 | 1 | false | 99 | 99 | 0 | false | null |
additive | minif2f | 28 | 7 | false | theorem algebra_others_exirrpowirrrat :
β a b, Irrational a β§ Irrational b β§ Β¬ Irrational (a^b) | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a ^ b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2... | by
have h_main : β (a b : β), Irrational a β§ Irrational b β§ Β¬Irrational (a^b) := by
by_cases h : β (q : β), (Real.sqrt 2 : β) ^ (Real.sqrt 2 : β) = q
Β· -- Case: (β2)^β2 is rational
have hβ : Irrational (Real.sqrt 2) := by
apply Nat.Prime.irrational_sqrt
<;> norm_num
-- Use a = β2 a... | 675 | 675 | 0 | 1 | false | 99 | 99 | 0 | false | null |
additive | minif2f | 29 | 0 | false | theorem algebra_sqineq_2at2pclta2c2p41pc (a c : β) :
2 * a * (2 + c) β€ a ^ 2 + c ^ 2 + 4 * (1 + c) | by
nlinarith [sq_nonneg (a - (2 + c))] | by
nlinarith [sq_nonneg (a - (2 + c))] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 29 | 1 | false | theorem algebra_sqineq_2at2pclta2c2p41pc (a c : β) :
2 * a * (2 + c) β€ a ^ 2 + c ^ 2 + 4 * (1 + c) | by
nlinarith [sq_nonneg (a - (2 + c))] | by
nlinarith [sq_nonneg (a - (2 + c))] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 29 | 2 | false | theorem algebra_sqineq_2at2pclta2c2p41pc (a c : β) :
2 * a * (2 + c) β€ a ^ 2 + c ^ 2 + 4 * (1 + c) | by
nlinarith [sq_nonneg (a - (2 + c))] | by
nlinarith [sq_nonneg (a - (2 + c))] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 29 | 3 | false | theorem algebra_sqineq_2at2pclta2c2p41pc (a c : β) :
2 * a * (2 + c) β€ a ^ 2 + c ^ 2 + 4 * (1 + c) | by
nlinarith [sq_nonneg (a - (2 + c))] | by
nlinarith [sq_nonneg (a - (2 + c))] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 29 | 4 | false | theorem algebra_sqineq_2at2pclta2c2p41pc (a c : β) :
2 * a * (2 + c) β€ a ^ 2 + c ^ 2 + 4 * (1 + c) | by
nlinarith [sq_nonneg (a - (2 + c))] | by
nlinarith [sq_nonneg (a - (2 + c))] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 29 | 5 | false | theorem algebra_sqineq_2at2pclta2c2p41pc (a c : β) :
2 * a * (2 + c) β€ a ^ 2 + c ^ 2 + 4 * (1 + c) | by
nlinarith [sq_nonneg (a - (2 + c))] | by
nlinarith [sq_nonneg (a - (2 + c))] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 29 | 6 | false | theorem algebra_sqineq_2at2pclta2c2p41pc (a c : β) :
2 * a * (2 + c) β€ a ^ 2 + c ^ 2 + 4 * (1 + c) | by
nlinarith [sq_nonneg (a - (2 + c))] | by
nlinarith [sq_nonneg (a - (2 + c))] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 29 | 7 | false | theorem algebra_sqineq_2at2pclta2c2p41pc (a c : β) :
2 * a * (2 + c) β€ a ^ 2 + c ^ 2 + 4 * (1 + c) | by
nlinarith [sq_nonneg (a - (2 + c))] | by
nlinarith [sq_nonneg (a - (2 + c))] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 30 | 0 | false | theorem algebra_sqineq_2unitcircatblt1 (a b : β) (hβ : a ^ 2 + b ^ 2 = 2) : a * b β€ 1 | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | 37 | 37 | 0 | 1 | false | 9.5 | 9.5 | 0 | false | null |
additive | minif2f | 30 | 1 | false | theorem algebra_sqineq_2unitcircatblt1 (a b : β) (hβ : a ^ 2 + b ^ 2 = 2) : a * b β€ 1 | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | 37 | 37 | 0 | 1 | false | 9.5 | 9.5 | 0 | false | null |
additive | minif2f | 30 | 2 | false | theorem algebra_sqineq_2unitcircatblt1 (a b : β) (hβ : a ^ 2 + b ^ 2 = 2) : a * b β€ 1 | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | 37 | 37 | 0 | 1 | false | 9.5 | 9.5 | 0 | false | null |
additive | minif2f | 30 | 3 | true | theorem algebra_sqineq_2unitcircatblt1 (a b : β) (hβ : a ^ 2 + b ^ 2 = 2) : a * b β€ 1 | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)] | 37 | 25 | 12 | 1.48 | true | 9.5 | 6 | 3.5 | false | null |
additive | minif2f | 30 | 4 | false | theorem algebra_sqineq_2unitcircatblt1 (a b : β) (hβ : a ^ 2 + b ^ 2 = 2) : a * b β€ 1 | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | 37 | 37 | 0 | 1 | false | 9.5 | 9.5 | 0 | false | null |
additive | minif2f | 30 | 5 | false | theorem algebra_sqineq_2unitcircatblt1 (a b : β) (hβ : a ^ 2 + b ^ 2 = 2) : a * b β€ 1 | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | 37 | 37 | 0 | 1 | false | 9.5 | 9.5 | 0 | false | null |
additive | minif2f | 30 | 6 | false | theorem algebra_sqineq_2unitcircatblt1 (a b : β) (hβ : a ^ 2 + b ^ 2 = 2) : a * b β€ 1 | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)] | 37 | 25 | 12 | 1.48 | true | 9.5 | 6 | 3.5 | false | null |
additive | minif2f | 30 | 7 | false | theorem algebra_sqineq_2unitcircatblt1 (a b : β) (hβ : a ^ 2 + b ^ 2 = 2) : a * b β€ 1 | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | by
have h_main : a * b β€ 1 := by
have hβ : (a - b) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a + b)]
exact h_main | 37 | 37 | 0 | 1 | false | 9.5 | 9.5 | 0 | false | null |
additive | minif2f | 31 | 0 | false | theorem algebra_sqineq_36azm9asqle36zsq (z a : β) : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | 172 | 172 | 0 | 1 | false | 30.5 | 30.5 | 0 | false | null |
additive | minif2f | 31 | 1 | false | theorem algebra_sqineq_36azm9asqle36zsq (z a : β) : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | 172 | 172 | 0 | 1 | false | 30.5 | 30.5 | 0 | false | null |
additive | minif2f | 31 | 2 | false | theorem algebra_sqineq_36azm9asqle36zsq (z a : β) : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | 172 | 172 | 0 | 1 | false | 30.5 | 30.5 | 0 | false | null |
additive | minif2f | 31 | 3 | false | theorem algebra_sqineq_36azm9asqle36zsq (z a : β) : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36 * ... | 172 | 162 | 10 | 1.0617 | true | 30.5 | 28.5 | 2 | false | null |
additive | minif2f | 31 | 4 | false | theorem algebra_sqineq_36azm9asqle36zsq (z a : β) : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | 172 | 172 | 0 | 1 | false | 30.5 | 30.5 | 0 | false | null |
additive | minif2f | 31 | 5 | false | theorem algebra_sqineq_36azm9asqle36zsq (z a : β) : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | 172 | 162 | 10 | 1.0617 | true | 30.5 | 28.5 | 2 | false | null |
additive | minif2f | 31 | 6 | false | theorem algebra_sqineq_36azm9asqle36zsq (z a : β) : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | 172 | 162 | 10 | 1.0617 | true | 30.5 | 28.5 | 2 | false | null |
additive | minif2f | 31 | 7 | true | theorem algebra_sqineq_36azm9asqle36zsq (z a : β) : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36... | by
have h_main : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 β₯ 0 := by
have h1 : 36 * z ^ 2 - 36 * (a * z) + 9 * a ^ 2 = (6 * z - 3 * a) ^ 2 := by
ring_nf
<;>
linarith
rw [h1]
nlinarith [sq_nonneg (6 * z - 3 * a)]
have h_final : 36 * (a * z) - 9 * a ^ 2 β€ 36 * z ^ 2 := by
have h2 : 36 * ... | 172 | 136 | 36 | 1.2647 | true | 30.5 | 25 | 5.5 | false | null |
additive | minif2f | 32 | 0 | false | theorem algebra_sqineq_4bap1lt4bsqpap1sq (a b : β) : 4 * b * (a + 1) β€ 4 * b ^ 2 + (a + 1) ^ 2 | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 32 | 1 | false | theorem algebra_sqineq_4bap1lt4bsqpap1sq (a b : β) : 4 * b * (a + 1) β€ 4 * b ^ 2 + (a + 1) ^ 2 | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 32 | 2 | false | theorem algebra_sqineq_4bap1lt4bsqpap1sq (a b : β) : 4 * b * (a + 1) β€ 4 * b ^ 2 + (a + 1) ^ 2 | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 32 | 3 | false | theorem algebra_sqineq_4bap1lt4bsqpap1sq (a b : β) : 4 * b * (a + 1) β€ 4 * b ^ 2 + (a + 1) ^ 2 | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 32 | 4 | false | theorem algebra_sqineq_4bap1lt4bsqpap1sq (a b : β) : 4 * b * (a + 1) β€ 4 * b ^ 2 + (a + 1) ^ 2 | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 32 | 5 | false | theorem algebra_sqineq_4bap1lt4bsqpap1sq (a b : β) : 4 * b * (a + 1) β€ 4 * b ^ 2 + (a + 1) ^ 2 | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 32 | 6 | false | theorem algebra_sqineq_4bap1lt4bsqpap1sq (a b : β) : 4 * b * (a + 1) β€ 4 * b ^ 2 + (a + 1) ^ 2 | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 32 | 7 | false | theorem algebra_sqineq_4bap1lt4bsqpap1sq (a b : β) : 4 * b * (a + 1) β€ 4 * b ^ 2 + (a + 1) ^ 2 | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | by
nlinarith [sq_nonneg (a + 1 - 2 * b)] | 14 | 14 | 0 | 1 | false | 4 | 4 | 0 | false | null |
additive | minif2f | 33 | 0 | false | theorem algebra_sqineq_at2malt1
(a : β) :
a * (2 - a) β€ 1 | by
have h_main : a * (2 - a) β€ 1 := by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
-- Use the fact that (a - 1)^2 β₯ 0 to prove the inequality
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ
exact ... | by
have h_main : a * (2 - a) β€ 1 := by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ
exact h_main | 101 | 82 | 19 | 1.2317 | true | 20.5 | 18.5 | 2 | false | null |
additive | minif2f | 33 | 1 | true | theorem algebra_sqineq_at2malt1
(a : β) :
a * (2 - a) β€ 1 | by
have h_main : a * (2 - a) β€ 1 := by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
-- Use the fact that (a - 1)^2 β₯ 0 to prove the inequality
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ
exact ... | by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ | 101 | 66 | 35 | 1.5303 | true | 20.5 | 15 | 5.5 | false | null |
additive | minif2f | 33 | 2 | false | theorem algebra_sqineq_at2malt1
(a : β) :
a * (2 - a) β€ 1 | by
have h_main : a * (2 - a) β€ 1 := by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
-- Use the fact that (a - 1)^2 β₯ 0 to prove the inequality
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ
exact ... | by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ | 101 | 66 | 35 | 1.5303 | true | 20.5 | 15 | 5.5 | false | null |
additive | minif2f | 33 | 3 | false | theorem algebra_sqineq_at2malt1
(a : β) :
a * (2 - a) β€ 1 | by
have h_main : a * (2 - a) β€ 1 := by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
-- Use the fact that (a - 1)^2 β₯ 0 to prove the inequality
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ
exact ... | by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ | 101 | 66 | 35 | 1.5303 | true | 20.5 | 15 | 5.5 | false | null |
additive | minif2f | 33 | 4 | false | theorem algebra_sqineq_at2malt1
(a : β) :
a * (2 - a) β€ 1 | by
have h_main : a * (2 - a) β€ 1 := by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
-- Use the fact that (a - 1)^2 β₯ 0 to prove the inequality
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ
exact ... | by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ | 101 | 66 | 35 | 1.5303 | true | 20.5 | 15 | 5.5 | false | null |
additive | minif2f | 33 | 5 | false | theorem algebra_sqineq_at2malt1
(a : β) :
a * (2 - a) β€ 1 | by
have h_main : a * (2 - a) β€ 1 := by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
-- Use the fact that (a - 1)^2 β₯ 0 to prove the inequality
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ
exact ... | by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ | 101 | 66 | 35 | 1.5303 | true | 20.5 | 15 | 5.5 | false | null |
additive | minif2f | 33 | 6 | false | theorem algebra_sqineq_at2malt1
(a : β) :
a * (2 - a) β€ 1 | by
have h_main : a * (2 - a) β€ 1 := by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
-- Use the fact that (a - 1)^2 β₯ 0 to prove the inequality
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ
exact ... | by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ | 101 | 66 | 35 | 1.5303 | true | 20.5 | 15 | 5.5 | false | null |
additive | minif2f | 33 | 7 | false | theorem algebra_sqineq_at2malt1
(a : β) :
a * (2 - a) β€ 1 | by
have h_main : a * (2 - a) β€ 1 := by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
-- Use the fact that (a - 1)^2 β₯ 0 to prove the inequality
have hβ : (a - 1) ^ 2 β₯ 0 := by nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ
exact ... | by
have hβ : a * (2 - a) β€ 1 := by
have hβ : a * (2 - a) = 2 * a - a ^ 2 := by
ring
rw [hβ]
have hβ : 2 * a - a ^ 2 β€ 1 := by
have hβ : (a - 1) ^ 2 β₯ 0 := by
nlinarith
nlinarith [sq_nonneg (a - 1)]
exact hβ
exact hβ | 101 | 82 | 19 | 1.2317 | true | 20.5 | 22.5 | -2 | false | null |
additive | minif2f | 34 | 0 | true | theorem algebra_sqineq_unitcircatbpabsamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + |a - b| β€ 1 | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * b + b ^ 2 := by ring
_ = (a ^ 2 + b ^ 2) - 2 * a * b... | 348 | 330 | 18 | 1.0545 | true | 70 | 66.5 | 3.5 | false | null |
additive | minif2f | 34 | 1 | false | theorem algebra_sqineq_unitcircatbpabsamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + |a - b| β€ 1 | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | 348 | 348 | 0 | 1 | false | 70 | 70 | 0 | false | null |
additive | minif2f | 34 | 2 | false | theorem algebra_sqineq_unitcircatbpabsamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + |a - b| β€ 1 | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | 348 | 348 | 0 | 1 | false | 70 | 70 | 0 | false | null |
additive | minif2f | 34 | 3 | false | theorem algebra_sqineq_unitcircatbpabsamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + |a - b| β€ 1 | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | 348 | 348 | 0 | 1 | false | 70 | 70 | 0 | false | null |
additive | minif2f | 34 | 4 | false | theorem algebra_sqineq_unitcircatbpabsamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + |a - b| β€ 1 | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | 348 | 348 | 0 | 1 | false | 70 | 70 | 0 | false | null |
additive | minif2f | 34 | 5 | false | theorem algebra_sqineq_unitcircatbpabsamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + |a - b| β€ 1 | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | 348 | 348 | 0 | 1 | false | 70 | 70 | 0 | false | null |
additive | minif2f | 34 | 6 | false | theorem algebra_sqineq_unitcircatbpabsamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + |a - b| β€ 1 | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | 348 | 348 | 0 | 1 | false | 70 | 70 | 0 | false | null |
additive | minif2f | 34 | 7 | false | theorem algebra_sqineq_unitcircatbpabsamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + |a - b| β€ 1 | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | by
have h_main : a * b + |a - b| β€ 1 := by
cases' le_total a b with h h
Β· -- Case 1: a β€ b
have hβ : |a - b| = b - a := by
rw [abs_of_nonpos (by linarith)]
<;> linarith
rw [hβ]
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
calc
(a - b) ^ 2 = a ^ 2 - 2 * a * ... | 348 | 348 | 0 | 1 | false | 70 | 70 | 0 | false | null |
additive | minif2f | 35 | 0 | false | theorem algebra_sqineq_unitcircatbpamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + (a - b) β€ 1 | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | 240 | 240 | 0 | 1 | false | 42 | 42 | 0 | false | null |
additive | minif2f | 35 | 1 | false | theorem algebra_sqineq_unitcircatbpamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + (a - b) β€ 1 | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | 240 | 240 | 0 | 1 | false | 42 | 42 | 0 | false | null |
additive | minif2f | 35 | 2 | false | theorem algebra_sqineq_unitcircatbpamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + (a - b) β€ 1 | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | by
have hβ : (a - b) ^ 2 = 1 - 2 * (a * b) := by
have hββ : (a - b) ^ 2 = a ^ 2 + b ^ 2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b) ^ 2) / 2 + (a - b) := by
have hββ : (a - b) ^ 2 ... | 240 | 240 | 0 | 1 | false | 42 | 42 | 0 | false | null |
additive | minif2f | 35 | 3 | false | theorem algebra_sqineq_unitcircatbpamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + (a - b) β€ 1 | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | 240 | 240 | 0 | 1 | false | 42 | 42 | 0 | false | null |
additive | minif2f | 35 | 4 | false | theorem algebra_sqineq_unitcircatbpamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + (a - b) β€ 1 | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | 240 | 240 | 0 | 1 | false | 42 | 42 | 0 | false | null |
additive | minif2f | 35 | 5 | false | theorem algebra_sqineq_unitcircatbpamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + (a - b) β€ 1 | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a *... | 240 | 240 | 0 | 1 | false | 42 | 42 | 0 | false | null |
additive | minif2f | 35 | 6 | false | theorem algebra_sqineq_unitcircatbpamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + (a - b) β€ 1 | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | 240 | 240 | 0 | 1 | false | 42 | 42 | 0 | false | null |
additive | minif2f | 35 | 7 | false | theorem algebra_sqineq_unitcircatbpamblt1
(a b: β)
(hβ : a^2 + b^2 = 1) :
a * b + (a - b) β€ 1 | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | by
have hβ : (a - b)^2 = 1 - 2 * (a * b) := by
have hββ : (a - b)^2 = a^2 + b^2 - 2 * (a * b) := by
ring
rw [hββ]
rw [hβ]
<;> ring
<;> linarith
have hβ : a * b + (a - b) β€ 1 := by
have hβ : a * b + (a - b) = (1 - (a - b)^2) / 2 + (a - b) := by
have hββ : (a - b)^2 = 1 - 2 * (a... | 240 | 240 | 0 | 1 | false | 42 | 42 | 0 | false | null |
additive | minif2f | 36 | 0 | true | theorem algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3 (x y z : β€)
(hβ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z) :
x + y + z + 6 β£ x ^ 3 + y ^ 3 + z ^ 3 | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z... | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z + 6) * (x ^ 2 + y ^ 2 + z ^... | 688 | 577 | 111 | 1.1924 | true | 59 | 43 | 16 | false | null |
additive | minif2f | 36 | 1 | false | theorem algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3 (x y z : β€)
(hβ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z) :
x + y + z + 6 β£ x ^ 3 + y ^ 3 + z ^ 3 | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z... | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z + ... | 688 | 583 | 105 | 1.1801 | true | 59 | 53 | 6 | false | null |
additive | minif2f | 36 | 2 | false | theorem algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3 (x y z : β€)
(hβ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z) :
x + y + z + 6 β£ x ^ 3 + y ^ 3 + z ^ 3 | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z... | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z... | 688 | 688 | 0 | 1 | false | 59 | 59 | 0 | false | null |
additive | minif2f | 36 | 3 | false | theorem algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3 (x y z : β€)
(hβ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z) :
x + y + z + 6 β£ x ^ 3 + y ^ 3 + z ^ 3 | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z... | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z + 6) * (x ^ 2 + y ^ 2 + z ... | 688 | 619 | 69 | 1.1115 | true | 59 | 47 | 12 | false | null |
additive | minif2f | 36 | 4 | false | theorem algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3 (x y z : β€)
(hβ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z) :
x + y + z + 6 β£ x ^ 3 + y ^ 3 + z ^ 3 | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z... | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z... | 688 | 688 | 0 | 1 | false | 59 | 59 | 0 | false | null |
additive | minif2f | 36 | 5 | false | theorem algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3 (x y z : β€)
(hβ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z) :
x + y + z + 6 β£ x ^ 3 + y ^ 3 + z ^ 3 | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z... | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z +... | 688 | 688 | 0 | 1 | false | 59 | 59 | 0 | false | null |
additive | minif2f | 36 | 6 | false | theorem algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3 (x y z : β€)
(hβ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z) :
x + y + z + 6 β£ x ^ 3 + y ^ 3 + z ^ 3 | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z... | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z... | 688 | 688 | 0 | 1 | false | 59 | 59 | 0 | false | null |
additive | minif2f | 36 | 7 | false | theorem algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3 (x y z : β€)
(hβ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z) :
x + y + z + 6 β£ x ^ 3 + y ^ 3 + z ^ 3 | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z... | by
have hβ : 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) = x * y * z := by
have hββ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2 * (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) := by
ring_nf
<;>
linarith
rw [hββ] at hβ
linarith
have hβ : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z... | 688 | 688 | 0 | 1 | false | 59 | 59 | 0 | false | null |
additive | minif2f | 37 | 0 | false | theorem amc12_2000_p1
(i m o : β)
(hβ : i β m β§ m β o β§ o β i)
(hβ : i*m*o = 2001) :
i+m+o β€ 671 | by
have hβ : i > 0 := by
by_contra h
-- If i were 0, then the product i * m * o would be 0, but it's 2001.
have hβ : i = 0 := by
omega
rw [hβ] at hβ
norm_num at hβ
<;> omega
have hβ : m > 0 := by
by_contra h
-- If m were 0, then the product i * m * o would be 0, but it's 200... | by
have hβ : i > 0 := by
by_contra h
have hβ : i = 0 := by
omega
rw [hβ] at hβ
norm_num at hβ
<;> omega
have hβ : m > 0 := by
by_contra h
have hβ : m = 0 := by
omega
rw [hβ] at hβ
norm_num at hβ
<;> omega
have hβ : o > 0 := by
by_contra h
have hβ
:... | 1,473 | 899 | 574 | 1.6385 | true | 359 | 295 | 64 | false | null |
additive | minif2f | 37 | 1 | false | theorem amc12_2000_p1
(i m o : β)
(hβ : i β m β§ m β o β§ o β i)
(hβ : i*m*o = 2001) :
i+m+o β€ 671 | by
have hβ : i > 0 := by
by_contra h
-- If i were 0, then the product i * m * o would be 0, but it's 2001.
have hβ : i = 0 := by
omega
rw [hβ] at hβ
norm_num at hβ
<;> omega
have hβ : m > 0 := by
by_contra h
-- If m were 0, then the product i * m * o would be 0, but it's 200... | by
have hβ : i > 0 := by
by_contra h
have hβ : i = 0 := by
omega
rw [hβ] at hβ
norm_num at hβ
<;> omega
have hβ : m > 0 := by
by_contra h
have hβ : m = 0 := by
omega
rw [hβ] at hβ
norm_num at hβ
<;> omega
have hβ : o > 0 := by
by_contra h
have hβ
:... | 1,473 | 940 | 533 | 1.567 | true | 359 | 303.5 | 55.5 | false | null |
additive | minif2f | 37 | 2 | true | theorem amc12_2000_p1
(i m o : β)
(hβ : i β m β§ m β o β§ o β i)
(hβ : i*m*o = 2001) :
i+m+o β€ 671 | by
have hβ : i > 0 := by
by_contra h
-- If i were 0, then the product i * m * o would be 0, but it's 2001.
have hβ : i = 0 := by
omega
rw [hβ] at hβ
norm_num at hβ
<;> omega
have hβ : m > 0 := by
by_contra h
-- If m were 0, then the product i * m * o would be 0, but it's 200... | by
have hβ : i > 0 := by
by_contra h
have hβ : i = 0 := by
omega
rw [hβ] at hβ
norm_num at hβ
<;> omega
have hβ : m > 0 := by
by_contra h
have hβ : m = 0 := by
omega
rw [hβ] at hβ
norm_num at hβ
<;> omega
have hβ : o > 0 := by
by_contra h
have hβ
: o = 0... | 1,473 | 861 | 612 | 1.7108 | true | 359 | 285.5 | 73.5 | false | null |
additive | minif2f | 37 | 3 | false | theorem amc12_2000_p1
(i m o : β)
(hβ : i β m β§ m β o β§ o β i)
(hβ : i*m*o = 2001) :
i+m+o β€ 671 | by
have hβ : i > 0 := by
by_contra h
-- If i were 0, then the product i * m * o would be 0, but it's 2001.
have hβ : i = 0 := by
omega
rw [hβ] at hβ
norm_num at hβ
<;> omega
have hβ : m > 0 := by
by_contra h
-- If m were 0, then the product i * m * o would be 0, but it's 200... | by
have hβ : i > 0 := by
by_contra h
have hβ : i = 0 := by
omega
rw [hβ] at hβ
norm_num at hβ
<;> omega
have hβ : m > 0 := by
by_contra h
have hβ : m = 0 := by
omega
rw [hβ] at hβ
norm_num at hβ
<;> omega
have hβ : o > 0 := by
by_contra h
have hβ
:... | 1,473 | 875 | 598 | 1.6834 | true | 359 | 289 | 70 | false | null |
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