Datasets:
pkuAI4M
/
lean_github

Dataset card Data Studio Files
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https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.smallParts_card
[270, 1]
[273, 48]
rw [← parts_card_add h,add_tsub_cancel_right]
t : β„• P : β„• β†’ β„• h : Balanced t P ⊒ card (smallParts h) = t + 1 - card (largeParts h)
no goals
Please generate a tactic in lean4 to solve the state. STATE: t : β„• P : β„• β†’ β„• h : Balanced t P ⊒ card (smallParts h) = t + 1 - card (largeParts h) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
rw [parts_union h, sum_union (parts_disjoint h)]
t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ i in range (t + 1), f (P i) = card (smallParts h) * f (minP t P) + card (largeParts h) * f (minP t P + 1)
t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in smallParts h, f (P x) + βˆ‘ x in largeParts h, f (P x) = card (smallParts h) * f (minP t P) + card (largeParts h) * f (minP t P + 1)
Please generate a tactic in lean4 to solve the state. STATE: t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ i in range (t + 1), f (P i) = card (smallParts h) * f (minP t P) + card (largeParts h) * f (minP t P + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
congr
t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in smallParts h, f (P x) + βˆ‘ x in largeParts h, f (P x) = card (smallParts h) * f (minP t P) + card (largeParts h) * f (minP t P + 1)
case e_a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in smallParts h, f (P x) = card (smallParts h) * f (minP t P) case e_a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in largeParts h, f (P x) = card (largeParts h) * f (minP t P + 1)
Please generate a tactic in lean4 to solve the state. STATE: t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in smallParts h, f (P x) + βˆ‘ x in largeParts h, f (P x) = card (smallParts h) * f (minP t P) + card (largeParts h) * f (minP t P + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
rw [card_eq_sum_ones, sum_mul, one_mul]
case e_a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in smallParts h, f (P x) = card (smallParts h) * f (minP t P)
case e_a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in smallParts h, f (P x) = βˆ‘ x in smallParts h, f (minP t P)
Please generate a tactic in lean4 to solve the state. STATE: case e_a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in smallParts h, f (P x) = card (smallParts h) * f (minP t P) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
apply sum_congr
case e_a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in smallParts h, f (P x) = βˆ‘ x in smallParts h, f (minP t P)
case e_a.h t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ smallParts h = smallParts h case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ smallParts h β†’ f (P x) = f (minP t P)
Please generate a tactic in lean4 to solve the state. STATE: case e_a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in smallParts h, f (P x) = βˆ‘ x in smallParts h, f (minP t P) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
rfl
case e_a.h t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ smallParts h = smallParts h case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ smallParts h β†’ f (P x) = f (minP t P)
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ smallParts h β†’ f (P x) = f (minP t P)
Please generate a tactic in lean4 to solve the state. STATE: case e_a.h t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ smallParts h = smallParts h case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ smallParts h β†’ f (P x) = f (minP t P) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
rw [smallParts]
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ smallParts h β†’ f (P x) = f (minP t P)
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ filter (fun i => P i = minP t P) (range (t + 1)) β†’ f (P x) = f (minP t P)
Please generate a tactic in lean4 to solve the state. STATE: case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ smallParts h β†’ f (P x) = f (minP t P) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
intro x
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ filter (fun i => P i = minP t P) (range (t + 1)) β†’ f (P x) = f (minP t P)
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• ⊒ x ∈ filter (fun i => P i = minP t P) (range (t + 1)) β†’ f (P x) = f (minP t P)
Please generate a tactic in lean4 to solve the state. STATE: case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ filter (fun i => P i = minP t P) (range (t + 1)) β†’ f (P x) = f (minP t P) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
rw [mem_filter]
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• ⊒ x ∈ filter (fun i => P i = minP t P) (range (t + 1)) β†’ f (P x) = f (minP t P)
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• ⊒ x ∈ range (t + 1) ∧ P x = minP t P β†’ f (P x) = f (minP t P)
Please generate a tactic in lean4 to solve the state. STATE: case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• ⊒ x ∈ filter (fun i => P i = minP t P) (range (t + 1)) β†’ f (P x) = f (minP t P) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
intro hx
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• ⊒ x ∈ range (t + 1) ∧ P x = minP t P β†’ f (P x) = f (minP t P)
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• hx : x ∈ range (t + 1) ∧ P x = minP t P ⊒ f (P x) = f (minP t P)
Please generate a tactic in lean4 to solve the state. STATE: case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• ⊒ x ∈ range (t + 1) ∧ P x = minP t P β†’ f (P x) = f (minP t P) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
rw [hx.2]
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• hx : x ∈ range (t + 1) ∧ P x = minP t P ⊒ f (P x) = f (minP t P)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• hx : x ∈ range (t + 1) ∧ P x = minP t P ⊒ f (P x) = f (minP t P) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
rw [card_eq_sum_ones, sum_mul, one_mul]
case e_a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in largeParts h, f (P x) = card (largeParts h) * f (minP t P + 1)
case e_a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in largeParts h, f (P x) = βˆ‘ x in largeParts h, f (minP t P + 1)
Please generate a tactic in lean4 to solve the state. STATE: case e_a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in largeParts h, f (P x) = card (largeParts h) * f (minP t P + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
apply sum_congr
case e_a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in largeParts h, f (P x) = βˆ‘ x in largeParts h, f (minP t P + 1)
case e_a.h t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ largeParts h = largeParts h case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ largeParts h β†’ f (P x) = f (minP t P + 1)
Please generate a tactic in lean4 to solve the state. STATE: case e_a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ‘ x in largeParts h, f (P x) = βˆ‘ x in largeParts h, f (minP t P + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
rfl
case e_a.h t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ largeParts h = largeParts h case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ largeParts h β†’ f (P x) = f (minP t P + 1)
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ largeParts h β†’ f (P x) = f (minP t P + 1)
Please generate a tactic in lean4 to solve the state. STATE: case e_a.h t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ largeParts h = largeParts h case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ largeParts h β†’ f (P x) = f (minP t P + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
rw [largeParts]
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ largeParts h β†’ f (P x) = f (minP t P + 1)
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ filter (fun i => P i = minP t P + 1) (range (t + 1)) β†’ f (P x) = f (minP t P + 1)
Please generate a tactic in lean4 to solve the state. STATE: case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ largeParts h β†’ f (P x) = f (minP t P + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
intro x
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ filter (fun i => P i = minP t P + 1) (range (t + 1)) β†’ f (P x) = f (minP t P + 1)
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• ⊒ x ∈ filter (fun i => P i = minP t P + 1) (range (t + 1)) β†’ f (P x) = f (minP t P + 1)
Please generate a tactic in lean4 to solve the state. STATE: case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• ⊒ βˆ€ (x : β„•), x ∈ filter (fun i => P i = minP t P + 1) (range (t + 1)) β†’ f (P x) = f (minP t P + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
rw [mem_filter]
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• ⊒ x ∈ filter (fun i => P i = minP t P + 1) (range (t + 1)) β†’ f (P x) = f (minP t P + 1)
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• ⊒ x ∈ range (t + 1) ∧ P x = minP t P + 1 β†’ f (P x) = f (minP t P + 1)
Please generate a tactic in lean4 to solve the state. STATE: case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• ⊒ x ∈ filter (fun i => P i = minP t P + 1) (range (t + 1)) β†’ f (P x) = f (minP t P + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
intro hx
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• ⊒ x ∈ range (t + 1) ∧ P x = minP t P + 1 β†’ f (P x) = f (minP t P + 1)
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• hx : x ∈ range (t + 1) ∧ P x = minP t P + 1 ⊒ f (P x) = f (minP t P + 1)
Please generate a tactic in lean4 to solve the state. STATE: case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• ⊒ x ∈ range (t + 1) ∧ P x = minP t P + 1 β†’ f (P x) = f (minP t P + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_f
[276, 1]
[284, 41]
rw [hx.2]
case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• hx : x ∈ range (t + 1) ∧ P x = minP t P + 1 ⊒ f (P x) = f (minP t P + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case e_a.a t : β„• P : β„• β†’ β„• h : Balanced t P f : β„• β†’ β„• x : β„• hx : x ∈ range (t + 1) ∧ P x = minP t P + 1 ⊒ f (P x) = f (minP t P + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum'
[292, 1]
[294, 77]
rw [bal_sum h, mul_add, mul_one, ← add_assoc, ← add_mul, parts_card_add h]
t : β„• P : β„• β†’ β„• h : Balanced t P ⊒ psum t P = (t + 1) * minP t P + card (largeParts h)
no goals
Please generate a tactic in lean4 to solve the state. STATE: t : β„• P : β„• β†’ β„• h : Balanced t P ⊒ psum t P = (t + 1) * minP t P + card (largeParts h) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_n
[302, 1]
[307, 85]
unfold Bal at hb
t n : β„• P : β„• β†’ β„• hb : Bal t n P ⊒ minP t P = n / (t + 1) ∧ card (largeParts (_ : Balanced t P)) = n % (t + 1)
t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n ⊒ minP t P = n / (t + 1) ∧ card (largeParts (_ : Balanced t P)) = n % (t + 1)
Please generate a tactic in lean4 to solve the state. STATE: t n : β„• P : β„• β†’ β„• hb : Bal t n P ⊒ minP t P = n / (t + 1) ∧ card (largeParts (_ : Balanced t P)) = n % (t + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_n
[302, 1]
[307, 85]
cases' hb with hb1 hb2
t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n ⊒ minP t P = n / (t + 1) ∧ card (largeParts (_ : Balanced t P)) = n % (t + 1)
case intro t n : β„• P : β„• β†’ β„• hb1 : Balanced t P hb2 : psum t P = n ⊒ minP t P = n / (t + 1) ∧ card (largeParts (_ : Balanced t P)) = n % (t + 1)
Please generate a tactic in lean4 to solve the state. STATE: t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n ⊒ minP t P = n / (t + 1) ∧ card (largeParts (_ : Balanced t P)) = n % (t + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_n
[302, 1]
[307, 85]
rw [bal_sum' hb1, ← div_add_mod n (t + 1)] at hb2
case intro t n : β„• P : β„• β†’ β„• hb1 : Balanced t P hb2 : psum t P = n ⊒ minP t P = n / (t + 1) ∧ card (largeParts (_ : Balanced t P)) = n % (t + 1)
case intro t n : β„• P : β„• β†’ β„• hb1 : Balanced t P hb2✝ : psum t P = n hb2 : (t + 1) * minP t P + card (largeParts hb1) = (t + 1) * (n / (t + 1)) + n % (t + 1) ⊒ minP t P = n / (t + 1) ∧ card (largeParts (_ : Balanced t P)) = n % (t + 1)
Please generate a tactic in lean4 to solve the state. STATE: case intro t n : β„• P : β„• β†’ β„• hb1 : Balanced t P hb2 : psum t P = n ⊒ minP t P = n / (t + 1) ∧ card (largeParts (_ : Balanced t P)) = n % (t + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_n
[302, 1]
[307, 85]
exact mod_tplus1 (largeParts_card hb1) (le_of_lt_succ (mod_lt n (succ_pos t))) hb2
case intro t n : β„• P : β„• β†’ β„• hb1 : Balanced t P hb2✝ : psum t P = n hb2 : (t + 1) * minP t P + card (largeParts hb1) = (t + 1) * (n / (t + 1)) + n % (t + 1) ⊒ minP t P = n / (t + 1) ∧ card (largeParts (_ : Balanced t P)) = n % (t + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro t n : β„• P : β„• β†’ β„• hb1 : Balanced t P hb2✝ : psum t P = n hb2 : (t + 1) * minP t P + card (largeParts hb1) = (t + 1) * (n / (t + 1)) + n % (t + 1) ⊒ minP t P = n / (t + 1) ∧ card (largeParts (_ : Balanced t P)) = n % (t + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_n_f
[310, 1]
[316, 55]
unfold Bal at hb
t n : β„• P : β„• β†’ β„• hb : Bal t n P f : β„• β†’ β„• ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1)
t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n f : β„• β†’ β„• ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1)
Please generate a tactic in lean4 to solve the state. STATE: t n : β„• P : β„• β†’ β„• hb : Bal t n P f : β„• β†’ β„• ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_n_f
[310, 1]
[316, 55]
obtain hf := bal_sum_f hb.1
t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n f : β„• β†’ β„• ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1)
t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n f : β„• β†’ β„• hf : βˆ€ (f : β„• β†’ β„•), βˆ‘ i in range (t + 1), f (P i) = card (smallParts (_ : Balanced t P)) * f (minP t P) + card (largeParts (_ : Balanced t P)) * f (minP t P + 1) ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1)
Please generate a tactic in lean4 to solve the state. STATE: t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n f : β„• β†’ β„• ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_n_f
[310, 1]
[316, 55]
obtain ⟨mn, ln⟩ := bal_sum_n hb
t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n f : β„• β†’ β„• hf : βˆ€ (f : β„• β†’ β„•), βˆ‘ i in range (t + 1), f (P i) = card (smallParts (_ : Balanced t P)) * f (minP t P) + card (largeParts (_ : Balanced t P)) * f (minP t P + 1) ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1)
case intro t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n f : β„• β†’ β„• hf : βˆ€ (f : β„• β†’ β„•), βˆ‘ i in range (t + 1), f (P i) = card (smallParts (_ : Balanced t P)) * f (minP t P) + card (largeParts (_ : Balanced t P)) * f (minP t P + 1) mn : minP t P = n / (t + 1) ln : card (largeParts (_ : Balanced t P)) = n % (t + 1) ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1)
Please generate a tactic in lean4 to solve the state. STATE: t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n f : β„• β†’ β„• hf : βˆ€ (f : β„• β†’ β„•), βˆ‘ i in range (t + 1), f (P i) = card (smallParts (_ : Balanced t P)) * f (minP t P) + card (largeParts (_ : Balanced t P)) * f (minP t P + 1) ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_n_f
[310, 1]
[316, 55]
specialize hf f
case intro t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n f : β„• β†’ β„• hf : βˆ€ (f : β„• β†’ β„•), βˆ‘ i in range (t + 1), f (P i) = card (smallParts (_ : Balanced t P)) * f (minP t P) + card (largeParts (_ : Balanced t P)) * f (minP t P + 1) mn : minP t P = n / (t + 1) ln : card (largeParts (_ : Balanced t P)) = n % (t + 1) ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1)
case intro t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n f : β„• β†’ β„• mn : minP t P = n / (t + 1) ln : card (largeParts (_ : Balanced t P)) = n % (t + 1) hf : βˆ‘ i in range (t + 1), f (P i) = card (smallParts (_ : Balanced t P)) * f (minP t P) + card (largeParts (_ : Balanced t P)) * f (minP t P + 1) ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1)
Please generate a tactic in lean4 to solve the state. STATE: case intro t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n f : β„• β†’ β„• hf : βˆ€ (f : β„• β†’ β„•), βˆ‘ i in range (t + 1), f (P i) = card (smallParts (_ : Balanced t P)) * f (minP t P) + card (largeParts (_ : Balanced t P)) * f (minP t P + 1) mn : minP t P = n / (t + 1) ln : card (largeParts (_ : Balanced t P)) = n % (t + 1) ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_sum_n_f
[310, 1]
[316, 55]
rwa [mn, smallParts_card, ln] at hf
case intro t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n f : β„• β†’ β„• mn : minP t P = n / (t + 1) ln : card (largeParts (_ : Balanced t P)) = n % (t + 1) hf : βˆ‘ i in range (t + 1), f (P i) = card (smallParts (_ : Balanced t P)) * f (minP t P) + card (largeParts (_ : Balanced t P)) * f (minP t P + 1) ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro t n : β„• P : β„• β†’ β„• hb : Balanced t P ∧ psum t P = n f : β„• β†’ β„• mn : minP t P = n / (t + 1) ln : card (largeParts (_ : Balanced t P)) = n % (t + 1) hf : βˆ‘ i in range (t + 1), f (P i) = card (smallParts (_ : Balanced t P)) * f (minP t P) + card (largeParts (_ : Balanced t P)) * f (minP t P + 1) ⊒ βˆ‘ i in range (t + 1), f (P i) = (t + 1 - n % (t + 1)) * f (n / (t + 1)) + n % (t + 1) * f (n / (t + 1) + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_turan_help
[319, 1]
[320, 47]
rw [tN',sumSq,bal_sum_n_f hb fun i => i ^ 2]
t n : β„• P : β„• β†’ β„• hb : Bal t n P ⊒ sumSq t P = turanNumb' t n
no goals
Please generate a tactic in lean4 to solve the state. STATE: t n : β„• P : β„• β†’ β„• hb : Bal t n P ⊒ sumSq t P = turanNumb' t n TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_turan_help'
[323, 1]
[324, 79]
rw [bal_turan_help hp]
t n : β„• P Q : β„• β†’ β„• hp : Bal t n P hq : Bal t n Q ⊒ sumSq t P = sumSq t Q
t n : β„• P Q : β„• β†’ β„• hp : Bal t n P hq : Bal t n Q ⊒ turanNumb' t n = sumSq t Q
Please generate a tactic in lean4 to solve the state. STATE: t n : β„• P Q : β„• β†’ β„• hp : Bal t n P hq : Bal t n Q ⊒ sumSq t P = sumSq t Q TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_turan_help'
[323, 1]
[324, 79]
rw [bal_turan_help hq]
t n : β„• P Q : β„• β†’ β„• hp : Bal t n P hq : Bal t n Q ⊒ turanNumb' t n = sumSq t Q
no goals
Please generate a tactic in lean4 to solve the state. STATE: t n : β„• P Q : β„• β†’ β„• hp : Bal t n P hq : Bal t n Q ⊒ turanNumb' t n = sumSq t Q TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_turan_bd
[327, 1]
[328, 96]
rw [bal_turan_help hp]
t n : β„• P : β„• β†’ β„• hp : Bal t n P ⊒ sumSq t P + 2 * turanNumb t n = n ^ 2
t n : β„• P : β„• β†’ β„• hp : Bal t n P ⊒ turanNumb' t n + 2 * turanNumb t n = n ^ 2
Please generate a tactic in lean4 to solve the state. STATE: t n : β„• P : β„• β†’ β„• hp : Bal t n P ⊒ sumSq t P + 2 * turanNumb t n = n ^ 2 TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.bal_turan_bd
[327, 1]
[328, 96]
exact tn_turanNumb' t n
t n : β„• P : β„• β†’ β„• hp : Bal t n P ⊒ turanNumb' t n + 2 * turanNumb t n = n ^ 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: t n : β„• P : β„• β†’ β„• hp : Bal t n P ⊒ turanNumb' t n + 2 * turanNumb t n = n ^ 2 TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.disjoint_insert_erase
[345, 1]
[350, 37]
rw [disjoint_insert_right, mem_erase]
α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α B : Finset α v : α A : Finset α hd : Disjoint A B ⊒ Disjoint (erase A v) (insert v B)
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± A : Finset Ξ± hd : Disjoint A B ⊒ Β¬(v β‰  v ∧ v ∈ A) ∧ Disjoint (erase A v) B
Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α B : Finset α v : α A : Finset α hd : Disjoint A B ⊒ Disjoint (erase A v) (insert v B) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.disjoint_insert_erase
[345, 1]
[350, 37]
refine ⟨?_, disjoint_of_subset_left (erase_subset v A) hd⟩
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± A : Finset Ξ± hd : Disjoint A B ⊒ Β¬(v β‰  v ∧ v ∈ A) ∧ Disjoint (erase A v) B
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± A : Finset Ξ± hd : Disjoint A B ⊒ Β¬(v β‰  v ∧ v ∈ A)
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± A : Finset Ξ± hd : Disjoint A B ⊒ Β¬(v β‰  v ∧ v ∈ A) ∧ Disjoint (erase A v) B TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.disjoint_insert_erase
[345, 1]
[350, 37]
push_neg
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± A : Finset Ξ± hd : Disjoint A B ⊒ Β¬(v β‰  v ∧ v ∈ A)
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± A : Finset Ξ± hd : Disjoint A B ⊒ v β‰  v β†’ Β¬v ∈ A
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± A : Finset Ξ± hd : Disjoint A B ⊒ Β¬(v β‰  v ∧ v ∈ A) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.disjoint_insert_erase
[345, 1]
[350, 37]
intro h
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± A : Finset Ξ± hd : Disjoint A B ⊒ v β‰  v β†’ Β¬v ∈ A
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± A : Finset Ξ± hd : Disjoint A B h : v β‰  v ⊒ Β¬v ∈ A
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± A : Finset Ξ± hd : Disjoint A B ⊒ v β‰  v β†’ Β¬v ∈ A TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.disjoint_insert_erase
[345, 1]
[350, 37]
contradiction
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± A : Finset Ξ± hd : Disjoint A B h : v β‰  v ⊒ Β¬v ∈ A
no goals
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± A : Finset Ξ± hd : Disjoint A B h : v β‰  v ⊒ Β¬v ∈ A TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.turanPartition_iff_not_moveable
[373, 1]
[375, 31]
rw [Moveable, not_not]
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± ⊒ TuranPartition M ↔ Β¬Moveable M
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± ⊒ TuranPartition M ↔ Balanced M.t fun i => card (MultiPart.P M i)
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± ⊒ TuranPartition M ↔ Β¬Moveable M TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.turanPartition_iff_not_moveable
[373, 1]
[375, 31]
rfl
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± ⊒ TuranPartition M ↔ Balanced M.t fun i => card (MultiPart.P M i)
no goals
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± ⊒ TuranPartition M ↔ Balanced M.t fun i => card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.not_turanPartition_imp
[378, 1]
[389, 32]
rw [turanPartition_iff_not_moveable, Moveable, Balanced] at h
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± h : Β¬TuranPartition M ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± h : Β¬Β¬Β¬βˆ€ (i : β„•), i ∈ range (M.t + 1) β†’ βˆ€ (j : β„•), j ∈ range (M.t + 1) β†’ card (MultiPart.P M i) ≀ card (MultiPart.P M j) + 1 ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± h : Β¬TuranPartition M ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.not_turanPartition_imp
[378, 1]
[389, 32]
push_neg at h
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± h : Β¬Β¬Β¬βˆ€ (i : β„•), i ∈ range (M.t + 1) β†’ βˆ€ (j : β„•), j ∈ range (M.t + 1) β†’ card (MultiPart.P M i) ≀ card (MultiPart.P M j) + 1 ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± h : βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i) ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± h : Β¬Β¬Β¬βˆ€ (i : β„•), i ∈ range (M.t + 1) β†’ βˆ€ (j : β„•), j ∈ range (M.t + 1) β†’ card (MultiPart.P M i) ≀ card (MultiPart.P M j) + 1 ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.not_turanPartition_imp
[378, 1]
[389, 32]
obtain ⟨i, hi, j, hj, hc⟩ := h
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± h : βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i) ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
case intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± h : βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i) ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.not_turanPartition_imp
[378, 1]
[389, 32]
refine ⟨i,hi,j,hj,?_⟩
case intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
case intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) ⊒ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ βˆƒ j, j ∈ range (M.t + 1) ∧ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.not_turanPartition_imp
[378, 1]
[389, 32]
obtain ⟨v,hv⟩:= card_pos.1 <| pos_of_gt hc
case intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) ⊒ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
case intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) v : Ξ± hv : v ∈ MultiPart.P M i ⊒ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) ⊒ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.not_turanPartition_imp
[378, 1]
[389, 32]
have cne :=((lt_succ_self _).trans hc).ne.symm
case intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) v : Ξ± hv : v ∈ MultiPart.P M i ⊒ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
case intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) v : Ξ± hv : v ∈ MultiPart.P M i cne : card (MultiPart.P M i) β‰  card (MultiPart.P M j) ⊒ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) v : Ξ± hv : v ∈ MultiPart.P M i ⊒ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.not_turanPartition_imp
[378, 1]
[389, 32]
refine ⟨v, hv, λ eq => ?_, hc⟩
case intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) v : Ξ± hv : v ∈ MultiPart.P M i cne : card (MultiPart.P M i) β‰  card (MultiPart.P M j) ⊒ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
case intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) v : Ξ± hv : v ∈ MultiPart.P M i cne : card (MultiPart.P M i) β‰  card (MultiPart.P M j) eq : j = i ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) v : Ξ± hv : v ∈ MultiPart.P M i cne : card (MultiPart.P M i) β‰  card (MultiPart.P M j) ⊒ βˆƒ v, v ∈ MultiPart.P M i ∧ j β‰  i ∧ card (MultiPart.P M j) + 1 < card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.not_turanPartition_imp
[378, 1]
[389, 32]
rw [eq] at cne
case intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) v : Ξ± hv : v ∈ MultiPart.P M i cne : card (MultiPart.P M i) β‰  card (MultiPart.P M j) eq : j = i ⊒ False
case intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) v : Ξ± hv : v ∈ MultiPart.P M i cne : card (MultiPart.P M i) β‰  card (MultiPart.P M i) eq : j = i ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) v : Ξ± hv : v ∈ MultiPart.P M i cne : card (MultiPart.P M i) β‰  card (MultiPart.P M j) eq : j = i ⊒ False TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.not_turanPartition_imp
[378, 1]
[389, 32]
contradiction
case intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) v : Ξ± hv : v ∈ MultiPart.P M i cne : card (MultiPart.P M i) β‰  card (MultiPart.P M i) eq : j = i ⊒ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± i : β„• hi : i ∈ range (M.t + 1) j : β„• hj : j ∈ range (M.t + 1) hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i) v : Ξ± hv : v ∈ MultiPart.P M i cne : card (MultiPart.P M i) β‰  card (MultiPart.P M i) eq : j = i ⊒ False TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.insert_P'
[485, 1]
[490, 13]
intro v hv
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± M : MultiPart Ξ± h : Disjoint M.A B ⊒ βˆ€ (v : Ξ±), v ∈ B β†’ v ∈ MultiPart.P (minsert M h) (M.t + 1)
α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α B : Finset α M : MultiPart α h : Disjoint M.A B v : α hv : v ∈ B ⊒ v ∈ MultiPart.P (minsert M h) (M.t + 1)
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± M : MultiPart Ξ± h : Disjoint M.A B ⊒ βˆ€ (v : Ξ±), v ∈ B β†’ v ∈ MultiPart.P (minsert M h) (M.t + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.insert_P'
[485, 1]
[490, 13]
rw [insert_P]
α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α B : Finset α M : MultiPart α h : Disjoint M.A B v : α hv : v ∈ B ⊒ v ∈ MultiPart.P (minsert M h) (M.t + 1)
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± M : MultiPart Ξ± h : Disjoint M.A B v : Ξ± hv : v ∈ B ⊒ v ∈ if M.t + 1 β‰  M.t + 1 then MultiPart.P M (M.t + 1) else B
Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α B : Finset α M : MultiPart α h : Disjoint M.A B v : α hv : v ∈ B ⊒ v ∈ MultiPart.P (minsert M h) (M.t + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.insert_P'
[485, 1]
[490, 13]
split_ifs with h_1
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± M : MultiPart Ξ± h : Disjoint M.A B v : Ξ± hv : v ∈ B ⊒ v ∈ if M.t + 1 β‰  M.t + 1 then MultiPart.P M (M.t + 1) else B
case pos Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± M : MultiPart Ξ± h : Disjoint M.A B v : Ξ± hv : v ∈ B h_1 : M.t + 1 β‰  M.t + 1 ⊒ v ∈ MultiPart.P M (M.t + 1) case neg Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± M : MultiPart Ξ± h : Disjoint M.A B v : Ξ± hv : v ∈ B h_1 : Β¬M.t + 1 β‰  M.t + 1 ⊒ v ∈ B
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± M : MultiPart Ξ± h : Disjoint M.A B v : Ξ± hv : v ∈ B ⊒ v ∈ if M.t + 1 β‰  M.t + 1 then MultiPart.P M (M.t + 1) else B TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.insert_P'
[485, 1]
[490, 13]
contradiction
case pos Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± M : MultiPart Ξ± h : Disjoint M.A B v : Ξ± hv : v ∈ B h_1 : M.t + 1 β‰  M.t + 1 ⊒ v ∈ MultiPart.P M (M.t + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± M : MultiPart Ξ± h : Disjoint M.A B v : Ξ± hv : v ∈ B h_1 : M.t + 1 β‰  M.t + 1 ⊒ v ∈ MultiPart.P M (M.t + 1) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.insert_P'
[485, 1]
[490, 13]
exact hv
case neg Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± M : MultiPart Ξ± h : Disjoint M.A B v : Ξ± hv : v ∈ B h_1 : Β¬M.t + 1 β‰  M.t + 1 ⊒ v ∈ B
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± M : MultiPart Ξ± h : Disjoint M.A B v : Ξ± hv : v ∈ B h_1 : Β¬M.t + 1 β‰  M.t + 1 ⊒ v ∈ B TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.insert_C
[493, 1]
[499, 13]
intro h1
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± M : MultiPart Ξ± h : Disjoint M.A B ⊒ v ∈ MultiPart.P (minsert M h) (M.t + 1) β†’ v ∈ B
α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α B : Finset α v : α M : MultiPart α h : Disjoint M.A B h1 : v ∈ MultiPart.P (minsert M h) (M.t + 1) ⊒ v ∈ B
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± M : MultiPart Ξ± h : Disjoint M.A B ⊒ v ∈ MultiPart.P (minsert M h) (M.t + 1) β†’ v ∈ B TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.insert_C
[493, 1]
[499, 13]
rw [insert_P] at h1
α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α B : Finset α v : α M : MultiPart α h : Disjoint M.A B h1 : v ∈ MultiPart.P (minsert M h) (M.t + 1) ⊒ v ∈ B
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± M : MultiPart Ξ± h : Disjoint M.A B h1 : v ∈ if M.t + 1 β‰  M.t + 1 then MultiPart.P M (M.t + 1) else B ⊒ v ∈ B
Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α B : Finset α v : α M : MultiPart α h : Disjoint M.A B h1 : v ∈ MultiPart.P (minsert M h) (M.t + 1) ⊒ v ∈ B TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.insert_C
[493, 1]
[499, 13]
split_ifs at h1 with h2
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± M : MultiPart Ξ± h : Disjoint M.A B h1 : v ∈ if M.t + 1 β‰  M.t + 1 then MultiPart.P M (M.t + 1) else B ⊒ v ∈ B
case pos Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± M : MultiPart Ξ± h : Disjoint M.A B h2 : M.t + 1 β‰  M.t + 1 h1 : v ∈ MultiPart.P M (M.t + 1) ⊒ v ∈ B case neg Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± M : MultiPart Ξ± h : Disjoint M.A B h2 : Β¬M.t + 1 β‰  M.t + 1 h1 : v ∈ B ⊒ v ∈ B
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± M : MultiPart Ξ± h : Disjoint M.A B h1 : v ∈ if M.t + 1 β‰  M.t + 1 then MultiPart.P M (M.t + 1) else B ⊒ v ∈ B TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.insert_C
[493, 1]
[499, 13]
contradiction
case pos Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± M : MultiPart Ξ± h : Disjoint M.A B h2 : M.t + 1 β‰  M.t + 1 h1 : v ∈ MultiPart.P M (M.t + 1) ⊒ v ∈ B
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± M : MultiPart Ξ± h : Disjoint M.A B h2 : M.t + 1 β‰  M.t + 1 h1 : v ∈ MultiPart.P M (M.t + 1) ⊒ v ∈ B TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.insert_C
[493, 1]
[499, 13]
exact h1
case neg Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± M : MultiPart Ξ± h : Disjoint M.A B h2 : Β¬M.t + 1 β‰  M.t + 1 h1 : v ∈ B ⊒ v ∈ B
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± B : Finset Ξ± v : Ξ± M : MultiPart Ξ± h : Disjoint M.A B h2 : Β¬M.t + 1 β‰  M.t + 1 h1 : v ∈ B ⊒ v ∈ B TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.mem_part
[510, 1]
[511, 63]
intro hi hv
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• v : Ξ± M : MultiPart Ξ± ⊒ i ∈ range (M.t + 1) β†’ v ∈ MultiPart.P M i β†’ v ∈ M.A
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hv : v ∈ MultiPart.P M i ⊒ v ∈ M.A
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• v : Ξ± M : MultiPart Ξ± ⊒ i ∈ range (M.t + 1) β†’ v ∈ MultiPart.P M i β†’ v ∈ M.A TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.mem_part
[510, 1]
[511, 63]
rw [M.uni]
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hv : v ∈ MultiPart.P M i ⊒ v ∈ M.A
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hv : v ∈ MultiPart.P M i ⊒ v ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hv : v ∈ MultiPart.P M i ⊒ v ∈ M.A TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.mem_part
[510, 1]
[511, 63]
rw [mem_biUnion]
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hv : v ∈ MultiPart.P M i ⊒ v ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hv : v ∈ MultiPart.P M i ⊒ βˆƒ a, a ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M a
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hv : v ∈ MultiPart.P M i ⊒ v ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.mem_part
[510, 1]
[511, 63]
exact ⟨i, hi, hv⟩
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hv : v ∈ MultiPart.P M i ⊒ βˆƒ a, a ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M a
no goals
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hv : v ∈ MultiPart.P M i ⊒ βˆƒ a, a ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M a TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.inv_part
[514, 1]
[515, 43]
rw [M.uni, mem_biUnion] at hA
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± v : Ξ± M : MultiPart Ξ± hA : v ∈ M.A ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± v : Ξ± M : MultiPart Ξ± hA : βˆƒ a, a ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M a ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± v : Ξ± M : MultiPart Ξ± hA : v ∈ M.A ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.inv_part
[514, 1]
[515, 43]
exact hA
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± v : Ξ± M : MultiPart Ξ± hA : βˆƒ a, a ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M a ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
no goals
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± v : Ξ± M : MultiPart Ξ± hA : βˆƒ a, a ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M a ⊒ βˆƒ i, i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.biUnion_parts
[518, 1]
[524, 63]
ext x
α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α M : MultiPart α ⊒ M.A = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
case a Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± ⊒ x ∈ M.A ↔ x ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α M : MultiPart α ⊒ M.A = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.biUnion_parts
[518, 1]
[524, 63]
rw [mem_biUnion]
case a Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± ⊒ x ∈ M.A ↔ x ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
case a Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± ⊒ x ∈ M.A ↔ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a
Please generate a tactic in lean4 to solve the state. STATE: case a Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± ⊒ x ∈ M.A ↔ x ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.biUnion_parts
[518, 1]
[524, 63]
constructor
case a Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± ⊒ x ∈ M.A ↔ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a
case a.mp Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± ⊒ x ∈ M.A β†’ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a case a.mpr Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± ⊒ (βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a) β†’ x ∈ M.A
Please generate a tactic in lean4 to solve the state. STATE: case a Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± ⊒ x ∈ M.A ↔ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.biUnion_parts
[518, 1]
[524, 63]
intro hA
case a.mp Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± ⊒ x ∈ M.A β†’ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a
case a.mp Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± hA : x ∈ M.A ⊒ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a
Please generate a tactic in lean4 to solve the state. STATE: case a.mp Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± ⊒ x ∈ M.A β†’ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.biUnion_parts
[518, 1]
[524, 63]
obtain ⟨i,hi⟩:= (inv_part hA)
case a.mp Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± hA : x ∈ M.A ⊒ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a
case a.mp.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± hA : x ∈ M.A i : β„• hi : i ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M i ⊒ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a
Please generate a tactic in lean4 to solve the state. STATE: case a.mp Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± hA : x ∈ M.A ⊒ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.biUnion_parts
[518, 1]
[524, 63]
exact ⟨i,hi⟩
case a.mp.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± hA : x ∈ M.A i : β„• hi : i ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M i ⊒ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a.mp.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± hA : x ∈ M.A i : β„• hi : i ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M i ⊒ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.biUnion_parts
[518, 1]
[524, 63]
intro hA
case a.mpr Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± ⊒ (βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a) β†’ x ∈ M.A
case a.mpr Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± hA : βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a ⊒ x ∈ M.A
Please generate a tactic in lean4 to solve the state. STATE: case a.mpr Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± ⊒ (βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a) β†’ x ∈ M.A TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.biUnion_parts
[518, 1]
[524, 63]
obtain ⟨i, hi, hi2⟩ := hA
case a.mpr Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± hA : βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a ⊒ x ∈ M.A
case a.mpr.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± i : β„• hi : i ∈ range (M.t + 1) hi2 : x ∈ MultiPart.P M i ⊒ x ∈ M.A
Please generate a tactic in lean4 to solve the state. STATE: case a.mpr Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± hA : βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a ⊒ x ∈ M.A TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.biUnion_parts
[518, 1]
[524, 63]
exact mem_part hi hi2
case a.mpr.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± i : β„• hi : i ∈ range (M.t + 1) hi2 : x ∈ MultiPart.P M i ⊒ x ∈ M.A
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a.mpr.intro.intro Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : Ξ± i : β„• hi : i ∈ range (M.t + 1) hi2 : x ∈ MultiPart.P M i ⊒ x ∈ M.A TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.card_uni
[527, 1]
[529, 89]
rw [biUnion_parts M]
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± ⊒ card M.A = βˆ‘ i in range (M.t + 1), card (MultiPart.P M i)
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± ⊒ card (Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i) = βˆ‘ i in range (M.t + 1), card (MultiPart.P M i)
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± ⊒ card M.A = βˆ‘ i in range (M.t + 1), card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.card_uni
[527, 1]
[529, 89]
rw [card_biUnion]
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± ⊒ card (Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i) = βˆ‘ i in range (M.t + 1), card (MultiPart.P M i)
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± ⊒ βˆ€ (x : β„•), x ∈ range (M.t + 1) β†’ βˆ€ (y : β„•), y ∈ range (M.t + 1) β†’ x β‰  y β†’ Disjoint (MultiPart.P M x) (MultiPart.P M y)
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± ⊒ card (Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i) = βˆ‘ i in range (M.t + 1), card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.card_uni
[527, 1]
[529, 89]
intro x hx y hy ne
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± ⊒ βˆ€ (x : β„•), x ∈ range (M.t + 1) β†’ βˆ€ (y : β„•), y ∈ range (M.t + 1) β†’ x β‰  y β†’ Disjoint (MultiPart.P M x) (MultiPart.P M y)
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : β„• hx : x ∈ range (M.t + 1) y : β„• hy : y ∈ range (M.t + 1) ne : x β‰  y ⊒ Disjoint (MultiPart.P M x) (MultiPart.P M y)
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± ⊒ βˆ€ (x : β„•), x ∈ range (M.t + 1) β†’ βˆ€ (y : β„•), y ∈ range (M.t + 1) β†’ x β‰  y β†’ Disjoint (MultiPart.P M x) (MultiPart.P M y) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.card_uni
[527, 1]
[529, 89]
exact M.disj x hx y hy ne
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : β„• hx : x ∈ range (M.t + 1) y : β„• hy : y ∈ range (M.t + 1) ne : x β‰  y ⊒ Disjoint (MultiPart.P M x) (MultiPart.P M y)
no goals
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± x : β„• hx : x ∈ range (M.t + 1) y : β„• hy : y ∈ range (M.t + 1) ne : x β‰  y ⊒ Disjoint (MultiPart.P M x) (MultiPart.P M y) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.turan_bal
[532, 1]
[535, 54]
rw [turanPartition_iff_not_moveable] at hM
α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α M : MultiPart α hM : TuranPartition M ⊒ Bal M.t (card M.A) fun i => card (MultiPart.P M i)
α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α M : MultiPart α hM : ¬Moveable M ⊒ Bal M.t (card M.A) fun i => card (MultiPart.P M i)
Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α M : MultiPart α hM : TuranPartition M ⊒ Bal M.t (card M.A) fun i => card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.turan_bal
[532, 1]
[535, 54]
unfold Moveable at hM
α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α M : MultiPart α hM : ¬Moveable M ⊒ Bal M.t (card M.A) fun i => card (MultiPart.P M i)
α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α M : MultiPart α hM : ¬¬Balanced M.t fun i => card (MultiPart.P M i) ⊒ Bal M.t (card M.A) fun i => card (MultiPart.P M i)
Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α M : MultiPart α hM : ¬Moveable M ⊒ Bal M.t (card M.A) fun i => card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.turan_bal
[532, 1]
[535, 54]
rw [not_not] at hM
α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α M : MultiPart α hM : ¬¬Balanced M.t fun i => card (MultiPart.P M i) ⊒ Bal M.t (card M.A) fun i => card (MultiPart.P M i)
α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α M : MultiPart α hM : Balanced M.t fun i => card (MultiPart.P M i) ⊒ Bal M.t (card M.A) fun i => card (MultiPart.P M i)
Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α M : MultiPart α hM : ¬¬Balanced M.t fun i => card (MultiPart.P M i) ⊒ Bal M.t (card M.A) fun i => card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.turan_bal
[532, 1]
[535, 54]
rw [card_uni]
α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α M : MultiPart α hM : Balanced M.t fun i => card (MultiPart.P M i) ⊒ Bal M.t (card M.A) fun i => card (MultiPart.P M i)
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± hM : Balanced M.t fun i => card (MultiPart.P M i) ⊒ Bal M.t (βˆ‘ i in range (M.t + 1), card (MultiPart.P M i)) fun i => card (MultiPart.P M i)
Please generate a tactic in lean4 to solve the state. STATE: α : Type u_1 inst✝¹ : Fintype α inst✝ : DecidableEq α M : MultiPart α hM : Balanced M.t fun i => card (MultiPart.P M i) ⊒ Bal M.t (card M.A) fun i => card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.turan_bal
[532, 1]
[535, 54]
exact ⟨hM, rfl⟩
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± hM : Balanced M.t fun i => card (MultiPart.P M i) ⊒ Bal M.t (βˆ‘ i in range (M.t + 1), card (MultiPart.P M i)) fun i => card (MultiPart.P M i)
no goals
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± M : MultiPart Ξ± hM : Balanced M.t fun i => card (MultiPart.P M i) ⊒ Bal M.t (βˆ‘ i in range (M.t + 1), card (MultiPart.P M i)) fun i => card (MultiPart.P M i) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.uniq_part
[538, 1]
[542, 62]
intro hi hj hiv hjv
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± ⊒ i ∈ range (M.t + 1) β†’ j ∈ range (M.t + 1) β†’ v ∈ MultiPart.P M i β†’ v ∈ MultiPart.P M j β†’ i = j
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hiv : v ∈ MultiPart.P M i hjv : v ∈ MultiPart.P M j ⊒ i = j
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± ⊒ i ∈ range (M.t + 1) β†’ j ∈ range (M.t + 1) β†’ v ∈ MultiPart.P M i β†’ v ∈ MultiPart.P M j β†’ i = j TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.uniq_part
[538, 1]
[542, 62]
by_contra h
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hiv : v ∈ MultiPart.P M i hjv : v ∈ MultiPart.P M j ⊒ i = j
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hiv : v ∈ MultiPart.P M i hjv : v ∈ MultiPart.P M j h : Β¬i = j ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hiv : v ∈ MultiPart.P M i hjv : v ∈ MultiPart.P M j ⊒ i = j TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.uniq_part
[538, 1]
[542, 62]
apply not_disjoint_iff.2 ⟨v, hiv, hjv⟩ (M.disj i hi j hj h)
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hiv : v ∈ MultiPart.P M i hjv : v ∈ MultiPart.P M j h : Β¬i = j ⊒ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hiv : v ∈ MultiPart.P M i hjv : v ∈ MultiPart.P M j h : Β¬i = j ⊒ False TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.uniq_part'
[545, 1]
[549, 48]
intro hi hj hiv ne
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± ⊒ i ∈ range (M.t + 1) β†’ j ∈ range (M.t + 1) β†’ i β‰  j β†’ v ∈ MultiPart.P M i β†’ Β¬v ∈ MultiPart.P M j
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hiv : i β‰  j ne : v ∈ MultiPart.P M i ⊒ Β¬v ∈ MultiPart.P M j
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± ⊒ i ∈ range (M.t + 1) β†’ j ∈ range (M.t + 1) β†’ i β‰  j β†’ v ∈ MultiPart.P M i β†’ Β¬v ∈ MultiPart.P M j TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.uniq_part'
[545, 1]
[549, 48]
contrapose hiv
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hiv : i β‰  j ne : v ∈ MultiPart.P M i ⊒ Β¬v ∈ MultiPart.P M j
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) ne : v ∈ MultiPart.P M i hiv : ¬¬v ∈ MultiPart.P M j ⊒ Β¬i β‰  j
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hiv : i β‰  j ne : v ∈ MultiPart.P M i ⊒ Β¬v ∈ MultiPart.P M j TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.uniq_part'
[545, 1]
[549, 48]
push_neg at hiv
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) ne : v ∈ MultiPart.P M i hiv : ¬¬v ∈ MultiPart.P M j ⊒ Β¬i β‰  j
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) ne : v ∈ MultiPart.P M i hiv : v ∈ MultiPart.P M j ⊒ Β¬i β‰  j
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) ne : v ∈ MultiPart.P M i hiv : ¬¬v ∈ MultiPart.P M j ⊒ Β¬i β‰  j TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.uniq_part'
[545, 1]
[549, 48]
rw [not_ne_iff]
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) ne : v ∈ MultiPart.P M i hiv : v ∈ MultiPart.P M j ⊒ Β¬i β‰  j
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) ne : v ∈ MultiPart.P M i hiv : v ∈ MultiPart.P M j ⊒ i = j
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) ne : v ∈ MultiPart.P M i hiv : v ∈ MultiPart.P M j ⊒ Β¬i β‰  j TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.uniq_part'
[545, 1]
[549, 48]
exact uniq_part hi hj ne hiv
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) ne : v ∈ MultiPart.P M i hiv : v ∈ MultiPart.P M j ⊒ i = j
no goals
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• v : Ξ± M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) ne : v ∈ MultiPart.P M i hiv : v ∈ MultiPart.P M j ⊒ i = j TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.sub_part
[552, 1]
[554, 62]
rw [M.uni]
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) ⊒ MultiPart.P M i βŠ† M.A
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) ⊒ MultiPart.P M i βŠ† Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) ⊒ MultiPart.P M i βŠ† M.A TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.sub_part
[552, 1]
[554, 62]
intro x hx
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) ⊒ MultiPart.P M i βŠ† Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) x : Ξ± hx : x ∈ MultiPart.P M i ⊒ x ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) ⊒ MultiPart.P M i βŠ† Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.sub_part
[552, 1]
[554, 62]
rw [mem_biUnion]
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) x : Ξ± hx : x ∈ MultiPart.P M i ⊒ x ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) x : Ξ± hx : x ∈ MultiPart.P M i ⊒ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) x : Ξ± hx : x ∈ MultiPart.P M i ⊒ x ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.sub_part
[552, 1]
[554, 62]
exact ⟨i, hi, hx⟩
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) x : Ξ± hx : x ∈ MultiPart.P M i ⊒ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a
no goals
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) x : Ξ± hx : x ∈ MultiPart.P M i ⊒ βˆƒ a, a ∈ range (M.t + 1) ∧ x ∈ MultiPart.P M a TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.two_parts
[558, 1]
[564, 24]
rw [card_uni,← sum_erase_add (range (M.t + 1)) _ hj]
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hne : i β‰  j ⊒ card (MultiPart.P M i) + card (MultiPart.P M j) ≀ card M.A
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hne : i β‰  j ⊒ card (MultiPart.P M i) + card (MultiPart.P M j) ≀ βˆ‘ x in erase (range (M.t + 1)) j, card (MultiPart.P M x) + card (MultiPart.P M j)
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hne : i β‰  j ⊒ card (MultiPart.P M i) + card (MultiPart.P M j) ≀ card M.A TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.two_parts
[558, 1]
[564, 24]
apply Nat.add_le_add_right
Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hne : i β‰  j ⊒ card (MultiPart.P M i) + card (MultiPart.P M j) ≀ βˆ‘ x in erase (range (M.t + 1)) j, card (MultiPart.P M x) + card (MultiPart.P M j)
case h Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hne : i β‰  j ⊒ card (MultiPart.P M i) ≀ βˆ‘ x in erase (range (M.t + 1)) j, card (MultiPart.P M x)
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hne : i β‰  j ⊒ card (MultiPart.P M i) + card (MultiPart.P M j) ≀ βˆ‘ x in erase (range (M.t + 1)) j, card (MultiPart.P M x) + card (MultiPart.P M j) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.two_parts
[558, 1]
[564, 24]
rw [← sum_erase_add ((range (M.t + 1)).erase j) _ (mem_erase_of_ne_of_mem hne hi)]
case h Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hne : i β‰  j ⊒ card (MultiPart.P M i) ≀ βˆ‘ x in erase (range (M.t + 1)) j, card (MultiPart.P M x)
case h Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hne : i β‰  j ⊒ card (MultiPart.P M i) ≀ βˆ‘ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) + card (MultiPart.P M i)
Please generate a tactic in lean4 to solve the state. STATE: case h Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hne : i β‰  j ⊒ card (MultiPart.P M i) ≀ βˆ‘ x in erase (range (M.t + 1)) j, card (MultiPart.P M x) TACTIC:
https://github.com/jt496/Turan_4.git
329b6acff8f9b8f41609e3e5758ed80c61047eb5
Turan4/Turanpartition.lean
Turanpartition.two_parts
[558, 1]
[564, 24]
apply Nat.le_add_left
case h Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hne : i β‰  j ⊒ card (MultiPart.P M i) ≀ βˆ‘ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) + card (MultiPart.P M i)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h Ξ± : Type u_1 inst✝¹ : Fintype Ξ± inst✝ : DecidableEq Ξ± i j : β„• M : MultiPart Ξ± hi : i ∈ range (M.t + 1) hj : j ∈ range (M.t + 1) hne : i β‰  j ⊒ card (MultiPart.P M i) ≀ βˆ‘ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) + card (MultiPart.P M i) TACTIC: