url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have hbn : 0 < (1 - schnirelmannDensity B) := by
rw [hbeta] at hbo
rw [lt_sub_iff_add_lt, zero_add, lt_iff_le_and_ne]
exact β¨schnirelmannDensity_le_one, hboβ© | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
β’ schnirelmannDensity A * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ schnirelmannDensity A * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
β’ schnirelmannDensity A * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | simp only [add_le_add_iff_right, sub_pos, sub_neg] | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ schnirelmannDensity A * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ schnirelmannDensity A * βn * (1 - Ξ²) β€ β(countelements A n) * (1 - Ξ²) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ schnirelmannDensity A * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [β le_div_iff (hbn)] | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ schnirelmannDensity A * βn * (1 - Ξ²) β€ β(countelements A n) * (1 - Ξ²) | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ schnirelmannDensity A * βn β€ β(countelements A n) * (1 - Ξ²) / (1 - schnirelmannDensity B) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ schnirelmannDensity A * βn * (1 - Ξ²) β€ β(countelements A n) * (1 - Ξ²)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [mul_div_assoc] | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ schnirelmannDensity A * βn β€ β(countelements A n) * (1 - Ξ²) / (1 - schnirelmannDensity B) | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ schnirelmannDensity A * βn β€ β(countelements A n) * ((1 - Ξ²) / (1 - schnirelmannDensity B)) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ schnirelmannDensity A * βn β€ β(countelements A n) * (1 - Ξ²) / (1 - schnirelmannDensity B)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have hun : (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1 := by
rw [div_self]
positivity | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ schnirelmannDensity A * βn β€ β(countelements A n) * ((1 - Ξ²) / (1 - schnirelmannDensity B)) | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
hun : (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1
β’ schnirelmannDensity A * βn β€ β(countelements A n) * ((1 - Ξ²) / (1 - schnirelmannDensity B)) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ schnirelmannDensity A * βn β€ β(countelements A n) * ((1 - Ξ²) / (1 - schnirelmannDensity B))
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [hun, mul_one] | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
hun : (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1
β’ schnirelmannDensity A * βn β€ β(countelements A n) * ((1 - Ξ²) / (1 - schnirelmannDensity B)) | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
hun : (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1
β’ schnirelmannDensity A * βn β€ β(countelements A n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
hun : (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1
β’ schnirelmannDensity A * βn β€ β(countelements A n) * ((1 - Ξ²) / (1 - schnirelmannDensity B))
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | exact schnirelmannDensity_mul_le_card_filter | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
hun : (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1
β’ schnirelmannDensity A * βn β€ β(countelements A n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
hun : (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1
β’ schnirelmannDensity A * βn β€ β(countelements A n)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [hbeta] at hbo | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
β’ 0 < 1 - schnirelmannDensity B | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : Β¬schnirelmannDensity B = 1
β’ 0 < 1 - schnirelmannDensity B | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
β’ 0 < 1 - schnirelmannDensity B
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [lt_sub_iff_add_lt, zero_add, lt_iff_le_and_ne] | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : Β¬schnirelmannDensity B = 1
β’ 0 < 1 - schnirelmannDensity B | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : Β¬schnirelmannDensity B = 1
β’ schnirelmannDensity B β€ 1 β§ schnirelmannDensity B β 1 | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : Β¬schnirelmannDensity B = 1
β’ 0 < 1 - schnirelmannDensity B
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | exact β¨schnirelmannDensity_le_one, hboβ© | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : Β¬schnirelmannDensity B = 1
β’ schnirelmannDensity B β€ 1 β§ schnirelmannDensity B β 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : Β¬schnirelmannDensity B = 1
β’ schnirelmannDensity B β€ 1 β§ schnirelmannDensity B β 1
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [div_self] | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1 | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ 1 - schnirelmannDensity B β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | positivity | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ 1 - schnirelmannDensity B β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
β’ 1 - schnirelmannDensity B β 0
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | ring_nf | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hc2 : Ξ± * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn
β’ Ξ± * βn * (1 - Ξ²) + Ξ² * βn = (Ξ± * (1 - Ξ²) + Ξ²) * βn | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hc2 : Ξ± * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn
β’ Ξ± * βn * (1 - Ξ²) + Ξ² * βn = (Ξ± * (1 - Ξ²) + Ξ²) * βn
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | let Ξ± := schnirelmannDensity A | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | have halpha : Ξ± = schnirelmannDensity A := rfl | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | let Ξ² := schnirelmannDensity B | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | have hbeta : Ξ² = schnirelmannDensity B := rfl | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | let Ξ³ := schnirelmannDensity (A + B) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | have hgamma : Ξ³ = schnirelmannDensity (A + B) := rfl | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | rw [β halpha, β hbeta, β hgamma] | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 1 - Ξ³ β€ (1 - Ξ±) * (1 - Ξ²) | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 1 - schnirelmannDensity (A + B) β€ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | linarith | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
h : 1 - Ξ³ β€ 1 - (Ξ± + Ξ² - Ξ± * Ξ²)
β’ 1 - Ξ³ β€ (1 - Ξ±) * (1 - Ξ²) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
h : 1 - Ξ³ β€ 1 - (Ξ± + Ξ² - Ξ± * Ξ²)
β’ 1 - Ξ³ β€ (1 - Ξ±) * (1 - Ξ²)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | rw [sub_le_iff_le_add, add_comm_sub] | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 1 - Ξ³ β€ 1 - (Ξ± + Ξ² - Ξ± * Ξ²) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 1 β€ 1 + (Ξ³ - (Ξ± + Ξ² - Ξ± * Ξ²)) | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 1 - Ξ³ β€ 1 - (Ξ± + Ξ² - Ξ± * Ξ²)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | nth_rewrite 1 [β add_zero 1] | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 1 β€ 1 + (Ξ³ - (Ξ± + Ξ² - Ξ± * Ξ²)) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 1 + 0 β€ 1 + (Ξ³ - (Ξ± + Ξ² - Ξ± * Ξ²)) | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 1 β€ 1 + (Ξ³ - (Ξ± + Ξ² - Ξ± * Ξ²))
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | rw [add_le_add_iff_left, le_sub_comm, sub_zero] | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 1 + 0 β€ 1 + (Ξ³ - (Ξ± + Ξ² - Ξ± * Ξ²)) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ Ξ± + Ξ² - Ξ± * Ξ² β€ Ξ³ | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 1 + 0 β€ 1 + (Ξ³ - (Ξ± + Ξ² - Ξ± * Ξ²))
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | rw [sub_eq_add_neg] | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ Ξ± + Ξ² - Ξ± * Ξ² β€ Ξ³ | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ Ξ± + Ξ² + -(Ξ± * Ξ²) β€ Ξ³ | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ Ξ± + Ξ² - Ξ± * Ξ² β€ Ξ³
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | exact h0 | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
h0 : Ξ± + Ξ² - Ξ± * Ξ² β€ Ξ³
β’ Ξ± + Ξ² + -(Ξ± * Ξ²) β€ Ξ³ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
h0 : Ξ± + Ξ² - Ξ± * Ξ² β€ Ξ³
β’ Ξ± + Ξ² + -(Ξ± * Ξ²) β€ Ξ³
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | rw [halpha, hbeta, hgamma] | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ Ξ± + Ξ² - Ξ± * Ξ² β€ Ξ³ | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ schnirelmannDensity A + schnirelmannDensity B - schnirelmannDensity A * schnirelmannDensity B β€
schnirelmannDensity (A + B) | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ Ξ± + Ξ² - Ξ± * Ξ² β€ Ξ³
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | apply le_schnirelmannDensity_add A B | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ schnirelmannDensity A + schnirelmannDensity B - schnirelmannDensity A * schnirelmannDensity B β€
schnirelmannDensity (A + B) | case hA
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 0 β A
case hB
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 0 β B | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ schnirelmannDensity A + schnirelmannDensity B - schnirelmannDensity A * schnirelmannDensity B β€
schnirelmannDensity (A + B)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | exact hA | case hA
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 0 β A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hA
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 0 β A
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | exact hB | case hB
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 0 β B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hB
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
Ξ³ : β := schnirelmannDensity (A + B)
hgamma : Ξ³ = schnirelmannDensity (A + B)
β’ 0 β B
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | mannTheorem | [243, 1] | [245, 8] | sorry | Aβ Bβ : Set β
n : β
A B : Set β
β’ min 1 (schnirelmannDensity A + schnirelmannDensity B) β€ schnirelmannDensity (A + B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Aβ Bβ : Set β
n : β
A B : Set β
β’ min 1 (schnirelmannDensity A + schnirelmannDensity B) β€ schnirelmannDensity (A + B)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.prod_top | [46, 1] | [47, 53] | simp [mem_prod, LatticeHom.coe_fst] | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ² : Lattice Ξ±
instβΒΉ : Lattice Ξ²
instβ : Lattice Ξ³
Lβ M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
aβ : Ξ±
L : Sublattice Ξ±
a : Ξ± Γ Ξ²
β’ a β L.prod β€ β a β comap (LatticeHom.fst Ξ± Ξ²) L | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ² : Lattice Ξ±
instβΒΉ : Lattice Ξ²
instβ : Lattice Ξ³
Lβ M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
aβ : Ξ±
L : Sublattice Ξ±
a : Ξ± Γ Ξ²
β’ a β L.prod β€ β a β comap (LatticeHom.fst Ξ± Ξ²) L
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.top_prod | [49, 1] | [50, 53] | simp [mem_prod, LatticeHom.coe_snd] | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ² : Lattice Ξ±
instβΒΉ : Lattice Ξ²
instβ : Lattice Ξ³
Lβ M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
aβ : Ξ±
L : Sublattice Ξ²
a : Ξ± Γ Ξ²
β’ a β β€.prod L β a β comap (LatticeHom.snd Ξ± Ξ²) L | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ² : Lattice Ξ±
instβΒΉ : Lattice Ξ²
instβ : Lattice Ξ³
Lβ M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
aβ : Ξ±
L : Sublattice Ξ²
a : Ξ± Γ Ξ²
β’ a β β€.prod L β a β comap (LatticeHom.snd Ξ± Ξ²) L
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.le_prod_iff | [61, 1] | [63, 36] | simp [SetLike.le_def, forall_and] | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ² : Lattice Ξ±
instβΒΉ : Lattice Ξ²
instβ : Lattice Ξ³
L Mβ : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
a : Ξ±
M : Sublattice Ξ²
N : Sublattice (Ξ± Γ Ξ²)
β’ N β€ L.prod M β N β€ comap (LatticeHom.fst Ξ± Ξ²) L β§ N β€ comap (LatticeHom.snd Ξ± Ξ²) M | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ² : Lattice Ξ±
instβΒΉ : Lattice Ξ²
instβ : Lattice Ξ³
L Mβ : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
a : Ξ±
M : Sublattice Ξ²
N : Sublattice (Ξ± Γ Ξ²)
β’ N β€ L.prod M β N β€ comap (LatticeHom.fst Ξ± Ξ²) L β§ N β€ comap (LatticeHom.snd Ξ± Ξ²) M
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.prod_eq_bot | [65, 1] | [66, 53] | simpa only [β coe_inj] using Set.prod_eq_empty_iff | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ² : Lattice Ξ±
instβΒΉ : Lattice Ξ²
instβ : Lattice Ξ³
L Mβ : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
a : Ξ±
M : Sublattice Ξ²
β’ L.prod M = β₯ β L = β₯ β¨ M = β₯ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ² : Lattice Ξ±
instβΒΉ : Lattice Ξ²
instβ : Lattice Ξ³
L Mβ : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
a : Ξ±
M : Sublattice Ξ²
β’ L.prod M = β₯ β L = β₯ β¨ M = β₯
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.prod_eq_top | [68, 1] | [69, 85] | simpa only [β coe_inj] using Set.prod_eq_univ | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instββ΄ : Lattice Ξ±
instβΒ³ : Lattice Ξ²
instβΒ² : Lattice Ξ³
L Mβ : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
a : Ξ±
instβΒΉ : Nonempty Ξ±
instβ : Nonempty Ξ²
M : Sublattice Ξ²
β’ L.prod M = β€ β L = β€ β§ M = β€ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instββ΄ : Lattice Ξ±
instβΒ³ : Lattice Ξ²
instβΒ² : Lattice Ξ³
L Mβ : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
a : Ξ±
instβΒΉ : Nonempty Ξ±
instβ : Nonempty Ξ²
M : Sublattice Ξ²
β’ L.prod M = β€ β L = β€ β§ M = β€
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.pi_empty | [93, 1] | [93, 96] | simp [mem_pi] | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
Lβ M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
aβ : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
L : (i : ΞΊ) β Sublattice (Ο i)
a : (i : ΞΊ) β Ο i
β’ a β pi β
L β a β β€ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
Lβ M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
aβ : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
L : (i : ΞΊ) β Sublattice (Ο i)
a : (i : ΞΊ) β Ο i
β’ a β pi β
L β a β β€
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.pi_top | [95, 1] | [96, 31] | simp [mem_pi] | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
L M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
sβ t : Set Ξ±
aβ : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
s : Set ΞΊ
a : (i : ΞΊ) β Ο i
β’ (a β pi s fun i => β€) β a β β€ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
L M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
sβ t : Set Ξ±
aβ : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
s : Set ΞΊ
a : (i : ΞΊ) β Ο i
β’ (a β pi s fun i => β€) β a β β€
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.pi_bot | [98, 1] | [99, 31] | simp [mem_pi] | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instββ΄ : Lattice Ξ±
instβΒ³ : Lattice Ξ²
instβΒ² : Lattice Ξ³
L M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
aβ : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβΒΉ : (i : ΞΊ) β Lattice (Ο i)
instβ : Nonempty ΞΊ
a : (i : ΞΊ) β Ο i
β’ (a β pi univ fun i => β₯) β a β β₯ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instββ΄ : Lattice Ξ±
instβΒ³ : Lattice Ξ²
instβΒ² : Lattice Ξ³
L M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
aβ : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβΒΉ : (i : ΞΊ) β Lattice (Ο i)
instβ : Nonempty ΞΊ
a : (i : ΞΊ) β Ο i
β’ (a β pi univ fun i => β₯) β a β β₯
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.le_pi | [101, 1] | [102, 101] | simp [SetLike.le_def] | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
Lβ Mβ : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
sβ t : Set Ξ±
a : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
s : Set ΞΊ
L : (i : ΞΊ) β Sublattice (Ο i)
M : Sublattice ((i : ΞΊ) β Ο i)
β’ M β€ pi s L β β i β s, M β€ comap (Pi.evalLatticeHom Ο i) (L i) | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
Lβ Mβ : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
sβ t : Set Ξ±
a : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
s : Set ΞΊ
L : (i : ΞΊ) β Sublattice (Ο i)
M : Sublattice ((i : ΞΊ) β Ο i)
β’ (β β¦x : (i : ΞΊ) β Ο iβ¦, x β M β β i β s, x i β L i) β β i β s, β β¦x : (i : ΞΊ) β Ο iβ¦, x β M β x i β L i | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
Lβ Mβ : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
sβ t : Set Ξ±
a : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
s : Set ΞΊ
L : (i : ΞΊ) β Sublattice (Ο i)
M : Sublattice ((i : ΞΊ) β Ο i)
β’ M β€ pi s L β β i β s, M β€ comap (Pi.evalLatticeHom Ο i) (L i)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.le_pi | [101, 1] | [102, 101] | aesop | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
Lβ Mβ : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
sβ t : Set Ξ±
a : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
s : Set ΞΊ
L : (i : ΞΊ) β Sublattice (Ο i)
M : Sublattice ((i : ΞΊ) β Ο i)
β’ (β β¦x : (i : ΞΊ) β Ο iβ¦, x β M β β i β s, x i β L i) β β i β s, β β¦x : (i : ΞΊ) β Ο iβ¦, x β M β x i β L i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
Lβ Mβ : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
sβ t : Set Ξ±
a : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
s : Set ΞΊ
L : (i : ΞΊ) β Sublattice (Ο i)
M : Sublattice ((i : ΞΊ) β Ο i)
β’ (β β¦x : (i : ΞΊ) β Ο iβ¦, x β M β β i β s, x i β L i) β β i β s, β β¦x : (i : ΞΊ) β Ο iβ¦, x β M β x i β L i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.pi_univ_eq_bot | [104, 1] | [105, 28] | simp_rw [β coe_inj] | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
Lβ M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
a : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
L : (i : ΞΊ) β Sublattice (Ο i)
β’ pi univ L = β₯ β β i, L i = β₯ | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
Lβ M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
a : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
L : (i : ΞΊ) β Sublattice (Ο i)
β’ β(pi univ L) = ββ₯ β β i, β(L i) = ββ₯ | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
Lβ M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
a : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
L : (i : ΞΊ) β Sublattice (Ο i)
β’ pi univ L = β₯ β β i, L i = β₯
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.pi_univ_eq_bot | [104, 1] | [105, 28] | simp | ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
Lβ M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
a : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
L : (i : ΞΊ) β Sublattice (Ο i)
β’ β(pi univ L) = ββ₯ β β i, β(L i) = ββ₯ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Sort u_1
Ξ± : Type u_2
Ξ² : Type u_3
Ξ³ : Type u_4
instβΒ³ : Lattice Ξ±
instβΒ² : Lattice Ξ²
instβΒΉ : Lattice Ξ³
Lβ M : Sublattice Ξ±
f : LatticeHom Ξ± Ξ²
s t : Set Ξ±
a : Ξ±
ΞΊ : Type u_5
Ο : ΞΊ β Type u_6
instβ : (i : ΞΊ) β Lattice (Ο i)
L : (i : ΞΊ) β Sublattice (Ο i)
β’ β(pi univ L) = ββ₯ β β i, β(L i) = ββ₯
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | mem_combiFrontier_iff | [36, 1] | [37, 97] | simp [combiFrontier] | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ x β combiFrontier π s β β t β s, x β (convexHull π) βt | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ x β combiFrontier π s β β t β s, x β (convexHull π) βt
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_empty | [44, 1] | [47, 42] | apply Set.eq_empty_of_subset_empty | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiFrontier π β
= β
| case a
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiFrontier π β
β β
| Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiFrontier π β
= β
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_empty | [44, 1] | [47, 42] | convert combiFrontier_subset_convexHull | case a
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiFrontier π β
β β
| case h.e'_4
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ β
= (convexHull π) ββ
| Please generate a tactic in lean4 to solve the state.
STATE:
case a
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiFrontier π β
β β
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_empty | [44, 1] | [47, 42] | rw [Finset.coe_empty, convexHull_empty] | case h.e'_4
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ β
= (convexHull π) ββ
| no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ β
= (convexHull π) ββ
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiInterior_empty | [49, 1] | [52, 42] | apply Set.eq_empty_of_subset_empty | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiInterior π β
= β
| case a
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiInterior π β
β β
| Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiInterior π β
= β
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiInterior_empty | [49, 1] | [52, 42] | convert combiInterior_subset_convexHull | case a
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiInterior π β
β β
| case h.e'_4
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ β
= (convexHull π) ββ
| Please generate a tactic in lean4 to solve the state.
STATE:
case a
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiInterior π β
β β
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiInterior_empty | [49, 1] | [52, 42] | rw [Finset.coe_empty, convexHull_empty] | case h.e'_4
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ β
= (convexHull π) ββ
| no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ β
= (convexHull π) ββ
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_singleton | [54, 1] | [59, 14] | refine eq_empty_of_subset_empty fun y hy β¦ ?_ | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiFrontier π {x} = β
| π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x y : E
hy : y β combiFrontier π {x}
β’ y β β
| Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiFrontier π {x} = β
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_singleton | [54, 1] | [59, 14] | rw [mem_combiFrontier_iff] at hy | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x y : E
hy : y β combiFrontier π {x}
β’ y β β
| π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x y : E
hy : β t β {x}, y β (convexHull π) βt
β’ y β β
| Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x y : E
hy : y β combiFrontier π {x}
β’ y β β
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_singleton | [54, 1] | [59, 14] | obtain β¨s, hs, hysβ© := hy | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x y : E
hy : β t β {x}, y β (convexHull π) βt
β’ y β β
| case intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
x y : E
s : Finset E
hs : s β {x}
hys : y β (convexHull π) βs
β’ y β β
| Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x y : E
hy : β t β {x}, y β (convexHull π) βt
β’ y β β
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_singleton | [54, 1] | [59, 14] | rw [Finset.eq_empty_of_ssubset_singleton hs] at hys | case intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
x y : E
s : Finset E
hs : s β {x}
hys : y β (convexHull π) βs
β’ y β β
| case intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
x y : E
s : Finset E
hs : s β {x}
hys : y β (convexHull π) ββ
β’ y β β
| Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
x y : E
s : Finset E
hs : s β {x}
hys : y β (convexHull π) βs
β’ y β β
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_singleton | [54, 1] | [59, 14] | simp at hys | case intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
x y : E
s : Finset E
hs : s β {x}
hys : y β (convexHull π) ββ
β’ y β β
| no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
x y : E
s : Finset E
hs : s β {x}
hys : y β (convexHull π) ββ
β’ y β β
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiInterior_singleton | [61, 1] | [64, 7] | unfold combiInterior | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiInterior π {x} = {x} | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ (convexHull π) β{x} \ combiFrontier π {x} = {x} | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiInterior π {x} = {x}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiInterior_singleton | [61, 1] | [64, 7] | rw [combiFrontier_singleton] | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ (convexHull π) β{x} \ combiFrontier π {x} = {x} | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ (convexHull π) β{x} \ β
= {x} | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ (convexHull π) β{x} \ combiFrontier π {x} = {x}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiInterior_singleton | [61, 1] | [64, 7] | simp | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ (convexHull π) β{x} \ β
= {x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ (convexHull π) β{x} \ β
= {x}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | apply Subset.antisymm _ _ | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ (convexHull π) βs = β t, β (_ : t β s), combiInterior π t | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ (convexHull π) βs β β t, β (_ : t β s), combiInterior π t
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ β t, β (_ : t β s), combiInterior π t β (convexHull π) βs | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ (convexHull π) βs = β t, β (_ : t β s), combiInterior π t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | refine s.strongInductionOn ?_ | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ (convexHull π) βs β β t, β (_ : t β s), combiInterior π t | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ β (s : Finset E),
(β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1) β
(convexHull π) βs β β t, β (_ : t β s), combiInterior π t | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ (convexHull π) βs β β t, β (_ : t β s), combiInterior π t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | rintro s ih x hx | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ β (s : Finset E),
(β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1) β
(convexHull π) βs β β t, β (_ : t β s), combiInterior π t | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
β’ x β β t, β (_ : t β s), combiInterior π t | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ β (s : Finset E),
(β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1) β
(convexHull π) βs β β t, β (_ : t β s), combiInterior π t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | by_cases h : x β combiFrontier π s | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
β’ x β β t, β (_ : t β s), combiInterior π t | case pos
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
h : x β combiFrontier π s
β’ x β β t, β (_ : t β s), combiInterior π t
case neg
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
h : x β combiFrontier π s
β’ x β β t, β (_ : t β s), combiInterior π t | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
β’ x β β t, β (_ : t β s), combiInterior π t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | rw [mem_combiFrontier_iff] at h | case pos
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
h : x β combiFrontier π s
β’ x β β t, β (_ : t β s), combiInterior π t | case pos
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
h : β t β s, x β (convexHull π) βt
β’ x β β t, β (_ : t β s), combiInterior π t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
h : x β combiFrontier π s
β’ x β β t, β (_ : t β s), combiInterior π t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | obtain β¨t, st, htβ© := h | case pos
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
h : β t β s, x β (convexHull π) βt
β’ x β β t, β (_ : t β s), combiInterior π t | case pos.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ tβ : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
t : Finset E
st : t β s
ht : x β (convexHull π) βt
β’ x β β t, β (_ : t β s), combiInterior π t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
h : β t β s, x β (convexHull π) βt
β’ x β β t, β (_ : t β s), combiInterior π t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | specialize ih _ st ht | case pos.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ tβ : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
t : Finset E
st : t β s
ht : x β (convexHull π) βt
β’ x β β t, β (_ : t β s), combiInterior π t | case pos.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ tβ : Finset E
xβ : E
s : Finset E
x : E
hx : x β (convexHull π) βs
t : Finset E
st : t β s
ht : x β (convexHull π) βt
ih : x β β t_1, β (_ : t_1 β t), combiInterior π t_1
β’ x β β t, β (_ : t β s), combiInterior π t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ tβ : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
t : Finset E
st : t β s
ht : x β (convexHull π) βt
β’ x β β t, β (_ : t β s), combiInterior π t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | simp only [exists_prop, Set.mem_iUnion] at ih β’ | case pos.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ tβ : Finset E
xβ : E
s : Finset E
x : E
hx : x β (convexHull π) βs
t : Finset E
st : t β s
ht : x β (convexHull π) βt
ih : x β β t_1, β (_ : t_1 β t), combiInterior π t_1
β’ x β β t, β (_ : t β s), combiInterior π t | case pos.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ tβ : Finset E
xβ : E
s : Finset E
x : E
hx : x β (convexHull π) βs
t : Finset E
st : t β s
ht : x β (convexHull π) βt
ih : β i β t, x β combiInterior π i
β’ β i β s, x β combiInterior π i | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ tβ : Finset E
xβ : E
s : Finset E
x : E
hx : x β (convexHull π) βs
t : Finset E
st : t β s
ht : x β (convexHull π) βt
ih : x β β t_1, β (_ : t_1 β t), combiInterior π t_1
β’ x β β t, β (_ : t β s), combiInterior π t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | obtain β¨Z, Zt, hZβ© := ih | case pos.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ tβ : Finset E
xβ : E
s : Finset E
x : E
hx : x β (convexHull π) βs
t : Finset E
st : t β s
ht : x β (convexHull π) βt
ih : β i β t, x β combiInterior π i
β’ β i β s, x β combiInterior π i | case pos.intro.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ tβ : Finset E
xβ : E
s : Finset E
x : E
hx : x β (convexHull π) βs
t : Finset E
st : t β s
ht : x β (convexHull π) βt
Z : Finset E
Zt : Z β t
hZ : x β combiInterior π Z
β’ β i β s, x β combiInterior π i | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ tβ : Finset E
xβ : E
s : Finset E
x : E
hx : x β (convexHull π) βs
t : Finset E
st : t β s
ht : x β (convexHull π) βt
ih : β i β t, x β combiInterior π i
β’ β i β s, x β combiInterior π i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | exact β¨_, Zt.trans st.1, hZβ© | case pos.intro.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ tβ : Finset E
xβ : E
s : Finset E
x : E
hx : x β (convexHull π) βs
t : Finset E
st : t β s
ht : x β (convexHull π) βt
Z : Finset E
Zt : Z β t
hZ : x β combiInterior π Z
β’ β i β s, x β combiInterior π i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ tβ : Finset E
xβ : E
s : Finset E
x : E
hx : x β (convexHull π) βs
t : Finset E
st : t β s
ht : x β (convexHull π) βt
Z : Finset E
Zt : Z β t
hZ : x β combiInterior π Z
β’ β i β s, x β combiInterior π i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | exact subset_iUnionβ s Subset.rfl β¨hx, hβ© | case neg
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
h : x β combiFrontier π s
β’ x β β t, β (_ : t β s), combiInterior π t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
sβ t : Finset E
xβ : E
s : Finset E
ih : β t β s, (convexHull π) βt β β t_1, β (_ : t_1 β t), combiInterior π t_1
x : E
hx : x β (convexHull π) βs
h : x β combiFrontier π s
β’ x β β t, β (_ : t β s), combiInterior π t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | exact iUnionβ_subset fun t ht β¦ diff_subset.trans $ convexHull_mono ht | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ β t, β (_ : t β s), combiInterior π t β (convexHull π) βs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : OrderedRing π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ β t, β (_ : t β s), combiInterior π t β (convexHull π) βs
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | ext x | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiFrontier π s =
{x | β w, β (_ : β y β s, 0 β€ w y) (_ : β y β s, w y = 1) (_ : β y β s, w y = 0), s.centerMass w id = x} | case h
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
xβ x : E
β’ x β combiFrontier π s β
x β {x | β w, β (_ : β y β s, 0 β€ w y) (_ : β y β s, w y = 1) (_ : β y β s, w y = 0), s.centerMass w id = x} | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
x : E
β’ combiFrontier π s =
{x | β w, β (_ : β y β s, 0 β€ w y) (_ : β y β s, w y = 1) (_ : β y β s, w y = 0), s.centerMass w id = x}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | simp_rw [combiFrontier, Set.mem_iUnion, Set.mem_setOf_eq] | case h
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
xβ x : E
β’ x β combiFrontier π s β
x β {x | β w, β (_ : β y β s, 0 β€ w y) (_ : β y β s, w y = 1) (_ : β y β s, w y = 0), s.centerMass w id = x} | case h
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
xβ x : E
β’ (β i, β (_ : i β s), x β (convexHull π) βi) β
β w, β (_ : β y β s, 0 β€ w y) (_ : β y β s, w y = 1) (_ : β y β s, w y = 0), s.centerMass w id = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
xβ x : E
β’ x β combiFrontier π s β
x β {x | β w, β (_ : β y β s, 0 β€ w y) (_ : β y β s, w y = 1) (_ : β y β s, w y = 0), s.centerMass w id = x}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | constructor | case h
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
xβ x : E
β’ (β i, β (_ : i β s), x β (convexHull π) βi) β
β w, β (_ : β y β s, 0 β€ w y) (_ : β y β s, w y = 1) (_ : β y β s, w y = 0), s.centerMass w id = x | case h.mp
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
xβ x : E
β’ (β i, β (_ : i β s), x β (convexHull π) βi) β
β w, β (_ : β y β s, 0 β€ w y) (_ : β y β s, w y = 1) (_ : β y β s, w y = 0), s.centerMass w id = x
case h.mpr
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
xβ x : E
β’ (β w, β (_ : β y β s, 0 β€ w y) (_ : β y β s, w y = 1) (_ : β y β s, w y = 0), s.centerMass w id = x) β
β i, β (_ : i β s), x β (convexHull π) βi | Please generate a tactic in lean4 to solve the state.
STATE:
case h
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
xβ x : E
β’ (β i, β (_ : i β s), x β (convexHull π) βi) β
β w, β (_ : β y β s, 0 β€ w y) (_ : β y β s, w y = 1) (_ : β y β s, w y = 0), s.centerMass w id = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | simp only [and_imp, exists_prop, exists_imp] | case h.mp
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
xβ x : E
β’ (β i, β (_ : i β s), x β (convexHull π) βi) β
β w, β (_ : β y β s, 0 β€ w y) (_ : β y β s, w y = 1) (_ : β y β s, w y = 0), s.centerMass w id = x | case h.mp
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
xβ x : E
β’ β x_1 β s,
x β (convexHull π) βx_1 β β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
xβ x : E
β’ (β i, β (_ : i β s), x β (convexHull π) βi) β
β w, β (_ : β y β s, 0 β€ w y) (_ : β y β s, w y = 1) (_ : β y β s, w y = 0), s.centerMass w id = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | intro t ts hx | case h.mp
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
xβ x : E
β’ β x_1 β s,
x β (convexHull π) βx_1 β β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | case h.mp
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
hx : x β (convexHull π) βt
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s t : Finset E
xβ x : E
β’ β x_1 β s,
x β (convexHull π) βx_1 β β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rw [Finset.convexHull_eq, Set.mem_setOf_eq] at hx | case h.mp
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
hx : x β (convexHull π) βt
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | case h.mp
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
hx : β w, (β y β t, 0 β€ w y) β§ β y β t, w y = 1 β§ t.centerMass w id = x
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
hx : x β (convexHull π) βt
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rcases hx with β¨w, hwβ, hwβ, hxβ© | case h.mp
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
hx : β w, (β y β t, 0 β€ w y) β§ β y β t, w y = 1 β§ t.centerMass w id = x
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | case h.mp.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
hx : β w, (β y β t, 0 β€ w y) β§ β y β t, w y = 1 β§ t.centerMass w id = x
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rcases Finset.exists_of_ssubset ts with β¨y, hys, hytβ© | case h.mp.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | case h.mp.intro.intro.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | let w' z := if z β t then w z else 0 | case h.mp.intro.intro.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | case h.mp.intro.intro.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | have hw'β : s.sum w' = 1 := by
rwa [β Finset.sum_subset ts.1, Finset.sum_extend_by_zero]
simp only [ite_eq_right_iff]
tauto | case h.mp.intro.intro.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | case h.mp.intro.intro.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | refine' β¨w', _, hw'β, β¨_, βΉy β sβΊ, _β©, _β© | case h.mp.intro.intro.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x | case h.mp.intro.intro.intro.intro.intro.refine'_1
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β y β s, 0 β€ w' y
case h.mp.intro.intro.intro.intro.intro.refine'_2
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ w' y = 0
case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ s.centerMass w' id = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β w, (β y β s, 0 β€ w y) β§ β y β s, w y = 1 β§ (β y β s, w y = 0) β§ s.centerMass w id = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rw [β Finset.centerMass_subset id ts.1] | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ s.centerMass w' id = x | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ t.centerMass w' id = x
case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β s, i β t β w' i = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ s.centerMass w' id = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | exact fun i _ hi => if_neg hi | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β s, i β t β w' i = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β s, i β t β w' i = 0
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rwa [β Finset.sum_subset ts.1, Finset.sum_extend_by_zero] | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
β’ s.sum w' = 1 | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
β’ β x β s, x β t β (if x β t then w x else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
β’ s.sum w' = 1
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | simp only [ite_eq_right_iff] | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
β’ β x β s, x β t β (if x β t then w x else 0) = 0 | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
β’ β x β s, x β t β x β t β w x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
β’ β x β s, x β t β (if x β t then w x else 0) = 0
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | tauto | π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
β’ β x β s, x β t β x β t β w x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
β’ β x β s, x β t β x β t β w x = 0
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rintro y - | case h.mp.intro.intro.intro.intro.intro.refine'_1
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β y β s, 0 β€ w' y | case h.mp.intro.intro.intro.intro.intro.refine'_1
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
yβ : E
hys : yβ β s
hyt : yβ β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
y : E
β’ 0 β€ w' y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_1
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β y β s, 0 β€ w' y
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | change 0 β€ ite (y β t) (w y) 0 | case h.mp.intro.intro.intro.intro.intro.refine'_1
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
yβ : E
hys : yβ β s
hyt : yβ β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
y : E
β’ 0 β€ w' y | case h.mp.intro.intro.intro.intro.intro.refine'_1
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
yβ : E
hys : yβ β s
hyt : yβ β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
y : E
β’ 0 β€ if y β t then w y else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_1
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
yβ : E
hys : yβ β s
hyt : yβ β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
y : E
β’ 0 β€ w' y
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | split_ifs | case h.mp.intro.intro.intro.intro.intro.refine'_1
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
yβ : E
hys : yβ β s
hyt : yβ β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
y : E
β’ 0 β€ if y β t then w y else 0 | case pos
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
yβ : E
hys : yβ β s
hyt : yβ β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
y : E
hβ : y β t
β’ 0 β€ w y
case neg
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
yβ : E
hys : yβ β s
hyt : yβ β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
y : E
hβ : y β t
β’ 0 β€ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_1
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
yβ : E
hys : yβ β s
hyt : yβ β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
y : E
β’ 0 β€ if y β t then w y else 0
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | apply hwβ y βΉ_βΊ | case pos
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
yβ : E
hys : yβ β s
hyt : yβ β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
y : E
hβ : y β t
β’ 0 β€ w y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
yβ : E
hys : yβ β s
hyt : yβ β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
y : E
hβ : y β t
β’ 0 β€ w y
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rfl | case neg
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
yβ : E
hys : yβ β s
hyt : yβ β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
y : E
hβ : y β t
β’ 0 β€ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
yβ : E
hys : yβ β s
hyt : yβ β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
y : E
hβ : y β t
β’ 0 β€ 0
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | apply if_neg βΉy β tβΊ | case h.mp.intro.intro.intro.intro.intro.refine'_2
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ w' y = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_2
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ w' y = 0
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rw [Finset.centerMass_eq_of_sum_1] | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ t.centerMass w' id = x | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β t, w' i β’ id i = x
case h.mp.intro.intro.intro.intro.intro.refine'_3.hw
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β t, w' i = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ t.centerMass w' id = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rwa [Finset.sum_extend_by_zero] | case h.mp.intro.intro.intro.intro.intro.refine'_3.hw
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β t, w' i = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_3.hw
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β t, w' i = 1
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rw [Finset.centerMass_eq_of_sum_1 _ _ hwβ] at hx | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β t, w' i β’ id i = x | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : β i β t, w i β’ id i = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β t, w' i β’ id i = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β t, w' i β’ id i = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rw [β hx] | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : β i β t, w i β’ id i = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β t, w' i β’ id i = x | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : β i β t, w i β’ id i = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β t, w' i β’ id i = β i β t, w i β’ id i | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : β i β t, w i β’ id i = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β t, w' i β’ id i = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | apply Finset.sum_congr rfl | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : β i β t, w i β’ id i = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β t, w' i β’ id i = β i β t, w i β’ id i | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : β i β t, w i β’ id i = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β x β t, w' x β’ id x = w x β’ id x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : β i β t, w i β’ id i = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β i β t, w' i β’ id i = β i β t, w i β’ id i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | intro x hx | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : β i β t, w i β’ id i = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β x β t, w' x β’ id x = w x β’ id x | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβΒΉ xβ : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hxβ : β i β t, w i β’ id i = xβ
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
x : E
hx : x β t
β’ w' x β’ id x = w x β’ id x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβ x : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hx : β i β t, w i β’ id i = x
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
β’ β x β t, w' x β’ id x = w x β’ id x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | change ite _ _ _ β’ _ = _ | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβΒΉ xβ : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hxβ : β i β t, w i β’ id i = xβ
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
x : E
hx : x β t
β’ w' x β’ id x = w x β’ id x | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβΒΉ xβ : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hxβ : β i β t, w i β’ id i = xβ
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
x : E
hx : x β t
β’ (if x β t then w x else 0) β’ id x = w x β’ id x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβΒΉ xβ : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hxβ : β i β t, w i β’ id i = xβ
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
x : E
hx : x β t
β’ w' x β’ id x = w x β’ id x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rw [if_pos hx] | case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβΒΉ xβ : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hxβ : β i β t, w i β’ id i = xβ
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
x : E
hx : x β t
β’ (if x β t then w x else 0) β’ id x = w x β’ id x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro.intro.intro.refine'_3
π : Type u_1
E : Type u_2
ΞΉ : Type u_3
instβΒ² : LinearOrderedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
s tβ : Finset E
xβΒΉ xβ : E
t : Finset E
ts : t β s
w : E β π
hwβ : β y β t, 0 β€ w y
hwβ : β y β t, w y = 1
hxβ : β i β t, w i β’ id i = xβ
y : E
hys : y β s
hyt : y β t
w' : E β π := fun z => if z β t then w z else 0
hw'β : s.sum w' = 1
x : E
hx : x β t
β’ (if x β t then w x else 0) β’ id x = w x β’ id x
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.