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/MetricBetween.lean | SBtw.sbtw.trans_right' | [53, 1] | [55, 77] | linarith [h.dist, h'.dist, dist_triangle u v w] | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w✝ u v w x : V
h : sbtw u v x
h' : sbtw v w x
this : u ≠ w
⊢ Dist.dist u w + Dist.dist w x ≤ Dist.dist u x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w✝ u v w x : V
h : sbtw u v x
h' : sbtw v w x
this : u ≠ w
⊢ Dist.dist u w + Dist.dist w x ≤ Dist.dist u x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/SupClosed.lean | supClosure_prod | [16, 1] | [23, 32] | rintro ⟨_, _⟩ ⟨⟨u, hu, hus, rfl⟩, v, hv, hvt, rfl⟩ | α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
⊢ supClosure s ×ˢ supClosure t ≤ supClosure (s ×ˢ t) | case mk.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ (u.sup' hu id, v.sup' hv id) ∈ supClosure (s ×ˢ t) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
⊢ supClosure s ×ˢ supClosure t ≤ supClosure (s ×ˢ t)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/SupClosed.lean | supClosure_prod | [16, 1] | [23, 32] | refine ⟨u ×ˢ v, hu.product hv, ?_, ?_⟩ | case mk.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ (u.sup' hu id, v.sup' hv id) ∈ supClosure (s ×ˢ t) | case mk.intro.intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ ↑(u ×ˢ v) ⊆ s ×ˢ t
case mk.intro.intro.intro.intro.intro.intro.intro.refine_2
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ (u ×ˢ v).sup' ⋯ id = (u.sup' hu id, v.sup' hv id) | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ (u.sup' hu id, v.sup' hv id) ∈ supClosure (s ×ˢ t)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/SupClosed.lean | supClosure_prod | [16, 1] | [23, 32] | simpa only [coe_product] using Set.prod_mono hus hvt | case mk.intro.intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ ↑(u ×ˢ v) ⊆ s ×ˢ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ ↑(u ×ˢ v) ⊆ s ×ˢ t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/SupClosed.lean | supClosure_prod | [16, 1] | [23, 32] | simp [prodMk_sup'_sup'] | case mk.intro.intro.intro.intro.intro.intro.intro.refine_2
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ (u ×ˢ v).sup' ⋯ id = (u.sup' hu id, v.sup' hv id) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro.intro.intro.intro.refine_2
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ (u ×ˢ v).sup' ⋯ id = (u.sup' hu id, v.sup' hv id)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/SupClosed.lean | infClosure_prod | [32, 1] | [39, 32] | rintro ⟨_, _⟩ ⟨⟨u, hu, hus, rfl⟩, v, hv, hvt, rfl⟩ | α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
⊢ infClosure s ×ˢ infClosure t ≤ infClosure (s ×ˢ t) | case mk.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ (u.inf' hu id, v.inf' hv id) ∈ infClosure (s ×ˢ t) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
⊢ infClosure s ×ˢ infClosure t ≤ infClosure (s ×ˢ t)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/SupClosed.lean | infClosure_prod | [32, 1] | [39, 32] | refine ⟨u ×ˢ v, hu.product hv, ?_, ?_⟩ | case mk.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ (u.inf' hu id, v.inf' hv id) ∈ infClosure (s ×ˢ t) | case mk.intro.intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ ↑(u ×ˢ v) ⊆ s ×ˢ t
case mk.intro.intro.intro.intro.intro.intro.intro.refine_2
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ (u ×ˢ v).inf' ⋯ id = (u.inf' hu id, v.inf' hv id) | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ (u.inf' hu id, v.inf' hv id) ∈ infClosure (s ×ˢ t)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/SupClosed.lean | infClosure_prod | [32, 1] | [39, 32] | simpa only [coe_product] using Set.prod_mono hus hvt | case mk.intro.intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ ↑(u ×ˢ v) ⊆ s ×ˢ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ ↑(u ×ˢ v) ⊆ s ×ˢ t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/SupClosed.lean | infClosure_prod | [32, 1] | [39, 32] | simp [prodMk_inf'_inf'] | case mk.intro.intro.intro.intro.intro.intro.intro.refine_2
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ (u ×ˢ v).inf' ⋯ id = (u.inf' hu id, v.inf' hv id) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro.intro.intro.intro.refine_2
α : Type u_1
β : Type u_2
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
s✝ t✝ : Set α
a b : α
s : Set α
t : Set β
u : Finset α
hu : u.Nonempty
hus : ↑u ⊆ s
v : Finset β
hv : v.Nonempty
hvt : ↑v ⊆ t
⊢ (u ×ˢ v).inf' ⋯ id = (u.inf' hu id, v.inf' hv id)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/SupClosed.lean | latticeClosure_prod | [48, 1] | [50, 42] | simp_rw [← supClosure_infClosure] | α : Type u_1
β : Type u_2
inst✝¹ : DistribLattice α
inst✝ : DistribLattice β
s✝ s : Set α
t : Set β
⊢ latticeClosure (s ×ˢ t) = latticeClosure s ×ˢ latticeClosure t | α : Type u_1
β : Type u_2
inst✝¹ : DistribLattice α
inst✝ : DistribLattice β
s✝ s : Set α
t : Set β
⊢ supClosure (infClosure (s ×ˢ t)) = supClosure (infClosure s) ×ˢ supClosure (infClosure t) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝¹ : DistribLattice α
inst✝ : DistribLattice β
s✝ s : Set α
t : Set β
⊢ latticeClosure (s ×ˢ t) = latticeClosure s ×ˢ latticeClosure t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/SupClosed.lean | latticeClosure_prod | [48, 1] | [50, 42] | simp | α : Type u_1
β : Type u_2
inst✝¹ : DistribLattice α
inst✝ : DistribLattice β
s✝ s : Set α
t : Set β
⊢ supClosure (infClosure (s ×ˢ t)) = supClosure (infClosure s) ×ˢ supClosure (infClosure t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝¹ : DistribLattice α
inst✝ : DistribLattice β
s✝ s : Set α
t : Set β
⊢ supClosure (infClosure (s ×ˢ t)) = supClosure (infClosure s) ×ˢ supClosure (infClosure t)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rw [modPartitions] | α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
⊢ (modPartitions s d hd h).parts = image (fun i => image (fun x => i + d * x) (range ((s - i - 1) / d + 1))) (range d) | α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
⊢ { parts := image (fun i => filter (fun j => j % d = i) (range s)) (range d), supIndep := ⋯, sup_parts := ⋯,
not_bot_mem := ⋯ }.parts =
image (fun i => image (fun x => i + d * x) (range ((s - i - 1) / d + 1))) (range d) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
⊢ (modPartitions s d hd h).parts = image (fun i => image (fun x => i + d * x) (range ((s - i - 1) / d + 1))) (range d)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | ext x | α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
⊢ { parts := image (fun i => filter (fun j => j % d = i) (range s)) (range d), supIndep := ⋯, sup_parts := ⋯,
not_bot_mem := ⋯ }.parts =
image (fun i => image (fun x => i + d * x) (range ((s - i - 1) / d + 1))) (range d) | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
⊢ x ∈
{ parts := image (fun i => filter (fun j => j % d = i) (range s)) (range d), supIndep := ⋯, sup_parts := ⋯,
not_bot_mem := ⋯ }.parts ↔
x ∈ image (fun i => image (fun x => i + d * x) (range ((s - i - 1) / d + 1))) (range d) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
⊢ { parts := image (fun i => filter (fun j => j % d = i) (range s)) (range d), supIndep := ⋯, sup_parts := ⋯,
not_bot_mem := ⋯ }.parts =
image (fun i => image (fun x => i + d * x) (range ((s - i - 1) / d + 1))) (range d)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | simp only [mem_image, mem_range] | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
⊢ x ∈
{ parts := image (fun i => filter (fun j => j % d = i) (range s)) (range d), supIndep := ⋯, sup_parts := ⋯,
not_bot_mem := ⋯ }.parts ↔
x ∈ image (fun i => image (fun x => i + d * x) (range ((s - i - 1) / d + 1))) (range d) | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
⊢ (∃ a < d, filter (fun j => j % d = a) (range s) = x) ↔
∃ a < d, image (fun x => a + d * x) (range ((s - a - 1) / d + 1)) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
⊢ x ∈
{ parts := image (fun i => filter (fun j => j % d = i) (range s)) (range d), supIndep := ⋯, sup_parts := ⋯,
not_bot_mem := ⋯ }.parts ↔
x ∈ image (fun i => image (fun x => i + d * x) (range ((s - i - 1) / d + 1))) (range d)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | refine' exists_congr fun i ↦ and_congr_right fun hi ↦ _ | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
⊢ (∃ a < d, filter (fun j => j % d = a) (range s) = x) ↔
∃ a < d, image (fun x => a + d * x) (range ((s - a - 1) / d + 1)) = x | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
i : ℕ
hi : i < d
⊢ filter (fun j => j % d = i) (range s) = x ↔ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
⊢ (∃ a < d, filter (fun j => j % d = a) (range s) = x) ↔
∃ a < d, image (fun x => a + d * x) (range ((s - a - 1) / d + 1)) = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | suffices
((range ((s - i - 1) / d + 1)).image fun x ↦ i + d * x) = (range s).filter fun j ↦ j % d = i
by rw [this] | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
i : ℕ
hi : i < d
⊢ filter (fun j => j % d = i) (range s) = x ↔ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = x | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
i : ℕ
hi : i < d
⊢ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = filter (fun j => j % d = i) (range s) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
i : ℕ
hi : i < d
⊢ filter (fun j => j % d = i) (range s) = x ↔ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | clear x | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
i : ℕ
hi : i < d
⊢ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = filter (fun j => j % d = i) (range s) | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
⊢ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = filter (fun j => j % d = i) (range s) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
i : ℕ
hi : i < d
⊢ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = filter (fun j => j % d = i) (range s)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | ext j | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
⊢ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = filter (fun j => j % d = i) (range s) | case a.a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ j ∈ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) ↔ j ∈ filter (fun j => j % d = i) (range s) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
⊢ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = filter (fun j => j % d = i) (range s)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | simp only [mem_image, mem_filter, mem_range, Nat.lt_add_one_iff] | case a.a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ j ∈ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) ↔ j ∈ filter (fun j => j % d = i) (range s) | case a.a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ (∃ a ≤ (s - i - 1) / d, i + d * a = j) ↔ j < s ∧ j % d = i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ j ∈ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) ↔ j ∈ filter (fun j => j % d = i) (range s)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | constructor | case a.a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ (∃ a ≤ (s - i - 1) / d, i + d * a = j) ↔ j < s ∧ j % d = i | case a.a.mp
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ (∃ a ≤ (s - i - 1) / d, i + d * a = j) → j < s ∧ j % d = i
case a.a.mpr
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ j < s ∧ j % d = i → ∃ a ≤ (s - i - 1) / d, i + d * a = j | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ (∃ a ≤ (s - i - 1) / d, i + d * a = j) ↔ j < s ∧ j % d = i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rw [this] | α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
i : ℕ
hi : i < d
this : image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = filter (fun j => j % d = i) (range s)
⊢ filter (fun j => j % d = i) (range s) = x ↔ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
i : ℕ
hi : i < d
this : image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = filter (fun j => j % d = i) (range s)
⊢ filter (fun j => j % d = i) (range s) = x ↔ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rintro ⟨j, hj, rfl⟩ | case a.a.mp
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ (∃ a ≤ (s - i - 1) / d, i + d * a = j) → j < s ∧ j % d = i | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j ≤ (s - i - 1) / d
⊢ i + d * j < s ∧ (i + d * j) % d = i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.mp
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ (∃ a ≤ (s - i - 1) / d, i + d * a = j) → j < s ∧ j % d = i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rw [Nat.add_mul_mod_self_left, Nat.mod_eq_of_lt hi, eq_self_iff_true, and_true_iff, ←
Nat.lt_sub_iff_add_lt', mul_comm] | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j ≤ (s - i - 1) / d
⊢ i + d * j < s ∧ (i + d * j) % d = i | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j ≤ (s - i - 1) / d
⊢ j * d < s - i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j ≤ (s - i - 1) / d
⊢ i + d * j < s ∧ (i + d * j) % d = i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rwa [Nat.le_div_iff_mul_le hd.bot_lt, Nat.le_sub_iff_add_le, Nat.succ_le_iff] at hj | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j ≤ (s - i - 1) / d
⊢ j * d < s - i | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j * d ≤ s - i - 1
⊢ 1 ≤ s - i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j ≤ (s - i - 1) / d
⊢ j * d < s - i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rw [Nat.succ_le_iff] | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j * d ≤ s - i - 1
⊢ 1 ≤ s - i | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j * d ≤ s - i - 1
⊢ 0 < s - i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j * d ≤ s - i - 1
⊢ 1 ≤ s - i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | exact Nat.sub_pos_of_lt (hi.trans_le h) | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j * d ≤ s - i - 1
⊢ 0 < s - i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j * d ≤ s - i - 1
⊢ 0 < s - i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rintro ⟨hj, rfl⟩ | case a.a.mpr
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ j < s ∧ j % d = i → ∃ a ≤ (s - i - 1) / d, i + d * a = j | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ ∃ a ≤ (s - j % d - 1) / d, j % d + d * a = j | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.mpr
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ j < s ∧ j % d = i → ∃ a ≤ (s - i - 1) / d, i + d * a = j
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | refine' ⟨j / d, _, Nat.mod_add_div _ _⟩ | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ ∃ a ≤ (s - j % d - 1) / d, j % d + d * a = j | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ j / d ≤ (s - j % d - 1) / d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ ∃ a ≤ (s - j % d - 1) / d, j % d + d * a = j
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rwa [Nat.le_div_iff_mul_le' hd.bot_lt, Nat.le_sub_iff_add_le, Nat.le_sub_iff_add_le',
← add_assoc, mul_comm, Nat.mod_add_div, Nat.add_one_le_iff] | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ j / d ≤ (s - j % d - 1) / d | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ j % d ≤ s
case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ 1 ≤ s - j % d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ j / d ≤ (s - j % d - 1) / d
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rw [Nat.succ_le_iff] | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ 1 ≤ s - j % d | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ 0 < s - j % d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ 1 ≤ s - j % d
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | exact Nat.sub_pos_of_lt (hi.trans_le h) | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ 0 < s - j % d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ 0 < s - j % d
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | exact hi.le.trans h | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ j % d ≤ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ j % d ≤ s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.not_mem_bot | [34, 1] | [35, 81] | simp [← mem_faces_iff] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ s ∉ ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ s ∉ ⊥
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.eq_bot_of_forall_not_mem | [40, 1] | [41, 37] | ext s | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K✝ K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
K : SimplicialComplex 𝕜 E
h : ∀ (s : Finset E), s ∉ K
⊢ K = ⊥ | case faces.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K✝ K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s✝ t : Finset E
A : Set (Finset E)
m n : ℕ
K : SimplicialComplex 𝕜 E
h : ∀ (s : Finset E), s ∉ K
s : Finset E
⊢ s ∈ K.faces ↔ s ∈ ⊥.faces | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K✝ K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
K : SimplicialComplex 𝕜 E
h : ∀ (s : Finset E), s ∉ K
⊢ K = ⊥
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.eq_bot_of_forall_not_mem | [40, 1] | [41, 37] | exact iff_of_false (h s) id | case faces.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K✝ K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s✝ t : Finset E
A : Set (Finset E)
m n : ℕ
K : SimplicialComplex 𝕜 E
h : ∀ (s : Finset E), s ∉ K
s : Finset E
⊢ s ∈ K.faces ↔ s ∈ ⊥.faces | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case faces.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K✝ K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s✝ t : Finset E
A : Set (Finset E)
m n : ℕ
K : SimplicialComplex 𝕜 E
h : ∀ (s : Finset E), s ∉ K
s : Finset E
⊢ s ∈ K.faces ↔ s ∈ ⊥.faces
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.space_eq_empty | [43, 1] | [48, 74] | simp only [Set.eq_empty_iff_forall_not_mem, mem_space_iff, SimplicialComplex.ext_iff,
@forall_swap E, mem_faces_iff, exists_prop, not_exists, not_and, faces_bot] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ K.space = ∅ ↔ K = ⊥ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ (∀ y ∈ K, ∀ (x : E), x ∉ (convexHull 𝕜) ↑y) ↔ ∀ (x : Finset E), x ∉ K | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ K.space = ∅ ↔ K = ⊥
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.space_eq_empty | [43, 1] | [48, 74] | simp only [← Set.eq_empty_iff_forall_not_mem, convexHull_empty_iff, coe_eq_empty] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ (∀ y ∈ K, ∀ (x : E), x ∉ (convexHull 𝕜) ↑y) ↔ ∀ (x : Finset E), x ∉ K | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ (∀ y ∈ K, y = ∅) ↔ ∀ (x : Finset E), x ∉ K | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ (∀ y ∈ K, ∀ (x : E), x ∉ (convexHull 𝕜) ↑y) ↔ ∀ (x : Finset E), x ∉ K
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.space_eq_empty | [43, 1] | [48, 74] | exact forall₂_congr fun s hs ↦ iff_false_intro (K.nonempty hs).ne_empty | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ (∀ y ∈ K, y = ∅) ↔ ∀ (x : Finset E), x ∉ K | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ (∀ y ∈ K, y = ∅) ↔ ∀ (x : Finset E), x ∉ K
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.coe_eq_empty | [54, 1] | [56, 68] | simp [Set.eq_empty_iff_forall_not_mem, SimplicialComplex.ext_iff] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ ↑K = ∅ ↔ K = ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ ↑K = ∅ ↔ K = ⊥
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.facets_singleton | [64, 1] | [66, 74] | rw [Set.eq_singleton_iff_unique_mem] at hK ⊢ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hK : K.faces = {s}
⊢ K.facets = {s} | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hK : s ∈ K.faces ∧ ∀ x ∈ K.faces, x = s
⊢ s ∈ K.facets ∧ ∀ x ∈ K.facets, x = s | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hK : K.faces = {s}
⊢ K.facets = {s}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.facets_singleton | [64, 1] | [66, 74] | exact ⟨⟨hK.1, fun t ht _ => (hK.2 _ ht).symm⟩, fun t ht => hK.2 _ ht.1⟩ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hK : s ∈ K.faces ∧ ∀ x ∈ K.faces, x = s
⊢ s ∈ K.facets ∧ ∀ x ∈ K.facets, x = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hK : s ∈ K.faces ∧ ∀ x ∈ K.faces, x = s
⊢ s ∈ K.facets ∧ ∀ x ∈ K.facets, x = s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.mem_of_mem_convexHull | [86, 1] | [91, 38] | have h := K.inter_subset_convexHull hx hs ⟨by simp, hxs⟩ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : x ∈ K.vertices
hs : s ∈ K
hxs : x ∈ (convexHull 𝕜) ↑s
⊢ x ∈ s | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : x ∈ K.vertices
hs : s ∈ K
hxs : x ∈ (convexHull 𝕜) ↑s
h : x ∈ (convexHull 𝕜) (↑{x} ∩ ↑s)
⊢ x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : x ∈ K.vertices
hs : s ∈ K
hxs : x ∈ (convexHull 𝕜) ↑s
⊢ x ∈ s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.mem_of_mem_convexHull | [86, 1] | [91, 38] | by_contra H | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : x ∈ K.vertices
hs : s ∈ K
hxs : x ∈ (convexHull 𝕜) ↑s
h : x ∈ (convexHull 𝕜) (↑{x} ∩ ↑s)
⊢ x ∈ s | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : x ∈ K.vertices
hs : s ∈ K
hxs : x ∈ (convexHull 𝕜) ↑s
h : x ∈ (convexHull 𝕜) (↑{x} ∩ ↑s)
H : ¬x ∈ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : x ∈ K.vertices
hs : s ∈ K
hxs : x ∈ (convexHull 𝕜) ↑s
h : x ∈ (convexHull 𝕜) (↑{x} ∩ ↑s)
⊢ x ∈ s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.mem_of_mem_convexHull | [86, 1] | [91, 38] | rwa [← coe_inter, inter_comm, disjoint_iff_inter_eq_empty.1 (disjoint_singleton_right.2 H),
coe_empty, convexHull_empty] at h | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : x ∈ K.vertices
hs : s ∈ K
hxs : x ∈ (convexHull 𝕜) ↑s
h : x ∈ (convexHull 𝕜) (↑{x} ∩ ↑s)
H : ¬x ∈ s
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : x ∈ K.vertices
hs : s ∈ K
hxs : x ∈ (convexHull 𝕜) ↑s
h : x ∈ (convexHull 𝕜) (↑{x} ∩ ↑s)
H : ¬x ∈ s
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.mem_of_mem_convexHull | [86, 1] | [91, 38] | simp | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : x ∈ K.vertices
hs : s ∈ K
hxs : x ∈ (convexHull 𝕜) ↑s
⊢ x ∈ (convexHull 𝕜) ↑{x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : x ∈ K.vertices
hs : s ∈ K
hxs : x ∈ (convexHull 𝕜) ↑s
⊢ x ∈ (convexHull 𝕜) ↑{x}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.mem_ofSimplex | [145, 1] | [149, 18] | refine' ⟨_, fun h => ⟨h.1, s, rfl, h.2⟩⟩ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : AffineIndependent 𝕜 Subtype.val
⊢ t ∈ ofSimplex hs ↔ t.Nonempty ∧ t ⊆ s | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : AffineIndependent 𝕜 Subtype.val
⊢ t ∈ ofSimplex hs → t.Nonempty ∧ t ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : AffineIndependent 𝕜 Subtype.val
⊢ t ∈ ofSimplex hs ↔ t.Nonempty ∧ t ⊆ s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.mem_ofSimplex | [145, 1] | [149, 18] | rintro ⟨ht, u, rfl : u = s, hts⟩ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : AffineIndependent 𝕜 Subtype.val
⊢ t ∈ ofSimplex hs → t.Nonempty ∧ t ⊆ s | case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
t : Finset E
A : Set (Finset E)
m n : ℕ
ht : t.Nonempty
u : Finset E
hts : t ⊆ u
hs : AffineIndependent 𝕜 Subtype.val
⊢ t.Nonempty ∧ t ⊆ u | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : AffineIndependent 𝕜 Subtype.val
⊢ t ∈ ofSimplex hs → t.Nonempty ∧ t ⊆ s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.mem_ofSimplex | [145, 1] | [149, 18] | exact ⟨ht, hts⟩ | case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
t : Finset E
A : Set (Finset E)
m n : ℕ
ht : t.Nonempty
u : Finset E
hts : t ⊆ u
hs : AffineIndependent 𝕜 Subtype.val
⊢ t.Nonempty ∧ t ⊆ u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
t : Finset E
A : Set (Finset E)
m n : ℕ
ht : t.Nonempty
u : Finset E
hts : t ⊆ u
hs : AffineIndependent 𝕜 Subtype.val
⊢ t.Nonempty ∧ t ⊆ u
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.face_dimension_le_space_dimension | [152, 1] | [154, 100] | simpa using (K.indep hs).card_le_finrank_succ.trans (add_le_add_right (Submodule.finrank_le _) _) | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
⊢ s.card ≤ FiniteDimensional.finrank 𝕜 E + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
⊢ s.card ≤ FiniteDimensional.finrank 𝕜 E + 1
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.subfacet | [156, 1] | [165, 36] | have := id hs | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
⊢ ∃ t ∈ K.facets, s ⊆ t | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs this : s ∈ K
⊢ ∃ t ∈ K.facets, s ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
⊢ ∃ t ∈ K.facets, s ⊆ t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.subfacet | [156, 1] | [165, 36] | revert this | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs this : s ∈ K
⊢ ∃ t ∈ K.facets, s ⊆ t | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
⊢ s ∈ K → ∃ t ∈ K.facets, s ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs this : s ∈ K
⊢ ∃ t ∈ K.facets, s ⊆ t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.subfacet | [156, 1] | [165, 36] | refine strongDownwardInductionOn s ?_ (face_dimension_le_space_dimension hs) | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
⊢ s ∈ K → ∃ t ∈ K.facets, s ⊆ t | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
⊢ ∀ (t₁ : Finset E),
(∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t₁ ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t) →
t₁.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t₁ ∈ K → ∃ t ∈ K.facets, t₁ ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
⊢ s ∈ K → ∃ t ∈ K.facets, s ⊆ t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.subfacet | [156, 1] | [165, 36] | rintro t h - ht | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
⊢ ∀ (t₁ : Finset E),
(∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t₁ ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t) →
t₁.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t₁ ∈ K → ∃ t ∈ K.facets, t₁ ⊆ t | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
⊢ ∀ (t₁ : Finset E),
(∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t₁ ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t) →
t₁.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t₁ ∈ K → ∃ t ∈ K.facets, t₁ ⊆ t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.subfacet | [156, 1] | [165, 36] | by_cases htfacet : t ∈ K.facets | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1 | case pos
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
htfacet : t ∈ K.facets
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1
case neg
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
htfacet : t ∉ K.facets
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.subfacet | [156, 1] | [165, 36] | obtain ⟨u, hu, htu⟩ := (not_facet_iff_subface ht).mp htfacet | case neg
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
htfacet : t ∉ K.facets
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1 | case neg.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
htfacet : t ∉ K.facets
u : Finset E
hu : u ∈ K.faces
htu : t ⊂ u
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
htfacet : t ∉ K.facets
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.subfacet | [156, 1] | [165, 36] | obtain ⟨v, hv⟩ := h (face_dimension_le_space_dimension hu) htu hu | case neg.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
htfacet : t ∉ K.facets
u : Finset E
hu : u ∈ K.faces
htu : t ⊂ u
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1 | case neg.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
htfacet : t ∉ K.facets
u : Finset E
hu : u ∈ K.faces
htu : t ⊂ u
v : Finset E
hv : v ∈ K.facets ∧ u ⊆ v
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
htfacet : t ∉ K.facets
u : Finset E
hu : u ∈ K.faces
htu : t ⊂ u
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.subfacet | [156, 1] | [165, 36] | exact ⟨v, hv.1, htu.1.trans hv.2⟩ | case neg.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
htfacet : t ∉ K.facets
u : Finset E
hu : u ∈ K.faces
htu : t ⊂ u
v : Finset E
hv : v ∈ K.facets ∧ u ⊆ v
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
htfacet : t ∉ K.facets
u : Finset E
hu : u ∈ K.faces
htu : t ⊂ u
v : Finset E
hv : v ∈ K.facets ∧ u ⊆ v
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.subfacet | [156, 1] | [165, 36] | exact ⟨t, htfacet, Subset.rfl⟩ | case pos
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
htfacet : t ∈ K.facets
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
hs : s ∈ K
t : Finset E
h : ∀ {t₂ : Finset E}, t₂.card ≤ FiniteDimensional.finrank 𝕜 E + 1 → t ⊂ t₂ → t₂ ∈ K → ∃ t ∈ K.facets, t₂ ⊆ t
ht : t ∈ K
htfacet : t ∈ K.facets
⊢ ∃ t_1 ∈ K.facets, t ⊆ t_1
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.facets_eq_empty_iff | [167, 1] | [174, 21] | refine' ⟨fun h => _, _⟩ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
⊢ K.facets = ∅ ↔ K = ⊥ | case refine'_1
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
h : K.facets = ∅
⊢ K = ⊥
case refine'_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
⊢ K = ⊥ → K.facets = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
⊢ K.facets = ∅ ↔ K = ⊥
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.facets_eq_empty_iff | [167, 1] | [174, 21] | ext s | case refine'_1
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
h : K.facets = ∅
⊢ K = ⊥ | case refine'_1.faces.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s✝ t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
h : K.facets = ∅
s : Finset E
⊢ s ∈ K.faces ↔ s ∈ ⊥.faces | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
h : K.facets = ∅
⊢ K = ⊥
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.facets_eq_empty_iff | [167, 1] | [174, 21] | refine' iff_of_false (fun hs => _) (Set.not_mem_empty _) | case refine'_1.faces.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s✝ t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
h : K.facets = ∅
s : Finset E
⊢ s ∈ K.faces ↔ s ∈ ⊥.faces | case refine'_1.faces.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s✝ t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
h : K.facets = ∅
s : Finset E
hs : s ∈ K.faces
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1.faces.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s✝ t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
h : K.facets = ∅
s : Finset E
⊢ s ∈ K.faces ↔ s ∈ ⊥.faces
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.facets_eq_empty_iff | [167, 1] | [174, 21] | obtain ⟨t, ht, -⟩ := subfacet hs | case refine'_1.faces.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s✝ t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
h : K.facets = ∅
s : Finset E
hs : s ∈ K.faces
⊢ False | case refine'_1.faces.h.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s✝ t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
h : K.facets = ∅
s : Finset E
hs : s ∈ K.faces
t : Finset E
ht : t ∈ K.facets
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1.faces.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s✝ t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
h : K.facets = ∅
s : Finset E
hs : s ∈ K.faces
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.facets_eq_empty_iff | [167, 1] | [174, 21] | exact h.subset ht | case refine'_1.faces.h.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s✝ t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
h : K.facets = ∅
s : Finset E
hs : s ∈ K.faces
t : Finset E
ht : t ∈ K.facets
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1.faces.h.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s✝ t✝ : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
h : K.facets = ∅
s : Finset E
hs : s ∈ K.faces
t : Finset E
ht : t ∈ K.facets
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.facets_eq_empty_iff | [167, 1] | [174, 21] | rintro rfl | case refine'_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
⊢ K = ⊥ → K.facets = ∅ | case refine'_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
⊢ ⊥.facets = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
⊢ K = ⊥ → K.facets = ∅
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.facets_eq_empty_iff | [167, 1] | [174, 21] | exact facets_bot | case refine'_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
⊢ ⊥.facets = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : FiniteDimensional 𝕜 E
⊢ ⊥.facets = ∅
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.isExtreme_iff_openSegment_subset_diff | [22, 1] | [31, 38] | refine' ⟨fun h => ⟨h.1, fun x y hx hy z hz =>
⟨hAconv.openSegment_subset hx hy.1 hz, fun hzB => hy.2 (h.2 hx hy.1 hzB hz).2⟩⟩,
fun h => ⟨h.1, fun x hx y hy z hzB hz => ⟨_, _⟩⟩⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
hAconv : Convex 𝕜 s
⊢ IsExtreme 𝕜 s t ↔ t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
⊢ x ∈ t
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
⊢ y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
hAconv : Convex 𝕜 s
⊢ IsExtreme 𝕜 s t ↔ t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.isExtreme_iff_openSegment_subset_diff | [22, 1] | [31, 38] | by_contra hxB | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
⊢ x ∈ t | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
hxB : x ∉ t
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
⊢ x ∈ t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.isExtreme_iff_openSegment_subset_diff | [22, 1] | [31, 38] | rw [openSegment_symm] at hz | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
hxB : x ∉ t
⊢ False | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 y x
hxB : x ∉ t
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
hxB : x ∉ t
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.isExtreme_iff_openSegment_subset_diff | [22, 1] | [31, 38] | exact (h.2 hy ⟨hx, hxB⟩ hz).2 hzB | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 y x
hxB : x ∉ t
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 y x
hxB : x ∉ t
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.isExtreme_iff_openSegment_subset_diff | [22, 1] | [31, 38] | by_contra hyB | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
⊢ y ∈ t | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
hyB : y ∉ t
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
⊢ y ∈ t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.isExtreme_iff_openSegment_subset_diff | [22, 1] | [31, 38] | exact (h.2 hx ⟨hy, hyB⟩ hz).2 hzB | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
hyB : y ∉ t
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
hyB : y ∉ t
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | refine' ⟨fun h => _, fun hs => _⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
⊢ extremePoints 𝕜 ((convexHull 𝕜) s) = s ↔ ConvexIndependent 𝕜 fun p => ↑p | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
h : extremePoints 𝕜 ((convexHull 𝕜) s) = s
⊢ ConvexIndependent 𝕜 fun p => ↑p
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
hs : ConvexIndependent 𝕜 fun p => ↑p
⊢ extremePoints 𝕜 ((convexHull 𝕜) s) = s | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
⊢ extremePoints 𝕜 ((convexHull 𝕜) s) = s ↔ ConvexIndependent 𝕜 fun p => ↑p
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | rw [convexIndependent_set_iff_not_mem_convexHull_diff] at hs | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
hs : ConvexIndependent 𝕜 fun p => ↑p
⊢ extremePoints 𝕜 ((convexHull 𝕜) s) = s | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
⊢ extremePoints 𝕜 ((convexHull 𝕜) s) = s | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
hs : ConvexIndependent 𝕜 fun p => ↑p
⊢ extremePoints 𝕜 ((convexHull 𝕜) s) = s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | refine' extremePoints_convexHull_subset.antisymm fun x hxs => ⟨subset_convexHull 𝕜 _ hxs, _⟩ | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
⊢ extremePoints 𝕜 ((convexHull 𝕜) s) = s | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
⊢ ∀ ⦃x₁ : E⦄, x₁ ∈ (convexHull 𝕜) s → ∀ ⦃x₂ : E⦄, x₂ ∈ (convexHull 𝕜) s → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
⊢ extremePoints 𝕜 ((convexHull 𝕜) s) = s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | by_contra! h | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
⊢ ∀ ⦃x₁ : E⦄, x₁ ∈ (convexHull 𝕜) s → ∀ ⦃x₂ : E⦄, x₂ ∈ (convexHull 𝕜) s → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
h : ∃ x₁ ∈ (convexHull 𝕜) s, ∃ x₂ ∈ (convexHull 𝕜) s, x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
⊢ ∀ ⦃x₁ : E⦄, x₁ ∈ (convexHull 𝕜) s → ∀ ⦃x₂ : E⦄, x₂ ∈ (convexHull 𝕜) s → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | obtain ⟨x₁, hx₁, x₂, hx₂, hx⟩ := h | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
h : ∃ x₁ ∈ (convexHull 𝕜) s, ∃ x₂ ∈ (convexHull 𝕜) s, x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ False | case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
h : ∃ x₁ ∈ (convexHull 𝕜) s, ∃ x₂ ∈ (convexHull 𝕜) s, x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | suffices h : x₁ ∈ convexHull 𝕜 (s \ {x}) ∧ x₂ ∈ convexHull 𝕜 (s \ {x}) by
exact hs _ hxs (convex_iff_openSegment_subset.1 (convex_convexHull 𝕜 _) h.1 h.2 hx.1) | case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ False | case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ x₁ ∈ (convexHull 𝕜) (s \ {x}) ∧ x₂ ∈ (convexHull 𝕜) (s \ {x}) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | have hx₁₂ : segment 𝕜 x₁ x₂ ⊆ convexHull 𝕜 s := (convex_convexHull 𝕜 _).segment_subset hx₁ hx₂ | case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ x₁ ∈ (convexHull 𝕜) (s \ {x}) ∧ x₂ ∈ (convexHull 𝕜) (s \ {x}) | case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
hx₁₂ : segment 𝕜 x₁ x₂ ⊆ (convexHull 𝕜) s
⊢ x₁ ∈ (convexHull 𝕜) (s \ {x}) ∧ x₂ ∈ (convexHull 𝕜) (s \ {x}) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ x₁ ∈ (convexHull 𝕜) (s \ {x}) ∧ x₂ ∈ (convexHull 𝕜) (s \ {x})
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | sorry | case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
hx₁₂ : segment 𝕜 x₁ x₂ ⊆ (convexHull 𝕜) s
⊢ x₁ ∈ (convexHull 𝕜) (s \ {x}) ∧ x₂ ∈ (convexHull 𝕜) (s \ {x}) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
hx₁₂ : segment 𝕜 x₁ x₂ ⊆ (convexHull 𝕜) s
⊢ x₁ ∈ (convexHull 𝕜) (s \ {x}) ∧ x₂ ∈ (convexHull 𝕜) (s \ {x})
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | exact (convex_convexHull 𝕜 _).convexIndependent_extremePoints | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
h : extremePoints 𝕜 ((convexHull 𝕜) s) = s
⊢ ConvexIndependent 𝕜 fun p => ↑p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
h : extremePoints 𝕜 ((convexHull 𝕜) s) = s
⊢ ConvexIndependent 𝕜 fun p => ↑p
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | exact hs _ hxs (convex_iff_openSegment_subset.1 (convex_convexHull 𝕜 _) h.1 h.2 hx.1) | 𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
h : x₁ ∈ (convexHull 𝕜) (s \ {x}) ∧ x₂ ∈ (convexHull 𝕜) (s \ {x})
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
h : x₁ ∈ (convexHull 𝕜) (s \ {x}) ∧ x₂ ∈ (convexHull 𝕜) (s \ {x})
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | inter_frontier_self_inter_convexHull_extreme | [69, 1] | [73, 8] | refine' ⟨inter_subset_left, fun x₁ hx₁A x₂ hx₂A x hxs hx => ⟨⟨hx₁A, _⟩, hx₂A, _⟩⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
⊢ IsExtreme 𝕜 (closure s) (closure s ∩ frontier ((convexHull 𝕜) s)) | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₁ ∈ frontier ((convexHull 𝕜) s)
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₂ ∈ frontier ((convexHull 𝕜) s) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
⊢ IsExtreme 𝕜 (closure s) (closure s ∩ frontier ((convexHull 𝕜) s))
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | inter_frontier_self_inter_convexHull_extreme | [69, 1] | [73, 8] | sorry | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₁ ∈ frontier ((convexHull 𝕜) s)
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₂ ∈ frontier ((convexHull 𝕜) s) | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₂ ∈ frontier ((convexHull 𝕜) s) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₁ ∈ frontier ((convexHull 𝕜) s)
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₂ ∈ frontier ((convexHull 𝕜) s)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | inter_frontier_self_inter_convexHull_extreme | [69, 1] | [73, 8] | sorry | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₂ ∈ frontier ((convexHull 𝕜) s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₂ ∈ frontier ((convexHull 𝕜) s)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | frontier_extreme | [76, 1] | [81, 87] | convert
(inter_frontier_self_inter_convexHull_extreme :
IsExtreme 𝕜 (closure s) (closure s ∩ frontier (convexHull 𝕜 s))) using 1 | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hA₁ : Convex 𝕜 s
hA₂ : IsClosed s
⊢ IsExtreme 𝕜 s (frontier s) | case h.e'_6
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hA₁ : Convex 𝕜 s
hA₂ : IsClosed s
⊢ s = closure s
case h.e'_7
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hA₁ : Convex 𝕜 s
hA₂ : IsClosed s
⊢ frontier s = closure s ∩ frontier ((convexHull 𝕜) s) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hA₁ : Convex 𝕜 s
hA₂ : IsClosed s
⊢ IsExtreme 𝕜 s (frontier s)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | frontier_extreme | [76, 1] | [81, 87] | rw [Convex.convexHull_eq hA₁, inter_eq_self_of_subset_right frontier_subset_closure] | case h.e'_7
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hA₁ : Convex 𝕜 s
hA₂ : IsClosed s
⊢ frontier s = closure s ∩ frontier ((convexHull 𝕜) s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_7
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hA₁ : Convex 𝕜 s
hA₂ : IsClosed s
⊢ frontier s = closure s ∩ frontier ((convexHull 𝕜) s)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | frontier_extreme | [76, 1] | [81, 87] | exact (IsClosed.closure_eq hA₂).symm | case h.e'_6
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hA₁ : Convex 𝕜 s
hA₂ : IsClosed s
⊢ s = closure s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_6
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hA₁ : Convex 𝕜 s
hA₂ : IsClosed s
⊢ s = closure s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.frontier_extreme_to_closure | [84, 1] | [87, 8] | use frontier_subset_closure | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hAconv : Convex 𝕜 s
⊢ IsExtreme 𝕜 (closure s) (frontier s) | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hAconv : Convex 𝕜 s
⊢ ∀ ⦃x₁ : E⦄,
x₁ ∈ closure s →
∀ ⦃x₂ : E⦄,
x₂ ∈ closure s → ∀ ⦃x : E⦄, x ∈ frontier s → x ∈ openSegment 𝕜 x₁ x₂ → x₁ ∈ frontier s ∧ x₂ ∈ frontier s | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hAconv : Convex 𝕜 s
⊢ IsExtreme 𝕜 (closure s) (frontier s)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.frontier_extreme_to_closure | [84, 1] | [87, 8] | sorry | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hAconv : Convex 𝕜 s
⊢ ∀ ⦃x₁ : E⦄,
x₁ ∈ closure s →
∀ ⦃x₂ : E⦄,
x₂ ∈ closure s → ∀ ⦃x : E⦄, x ∈ frontier s → x ∈ openSegment 𝕜 x₁ x₂ → x₁ ∈ frontier s ∧ x₂ ∈ frontier s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hAconv : Convex 𝕜 s
⊢ ∀ ⦃x₁ : E⦄,
x₁ ∈ closure s →
∀ ⦃x₂ : E⦄,
x₂ ∈ closure s → ∀ ⦃x : E⦄, x ∈ frontier s → x ∈ openSegment 𝕜 x₁ x₂ → x₁ ∈ frontier s ∧ x₂ ∈ frontier s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | rintro x hxB | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
⊢ t ⊆ frontier s | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
⊢ x ∈ frontier s | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
⊢ t ⊆ frontier s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | obtain ⟨y, hyA, hyB⟩ := nonempty_of_ssubset ⟨hAB.1, hBA⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
⊢ x ∈ frontier s | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ frontier s | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
⊢ x ∈ frontier s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | rw [frontier_eq_closure_inter_closure] | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ frontier s | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ closure s ∩ closure sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ frontier s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | use subset_closure (hAB.1 hxB) | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ closure s ∩ closure sᶜ | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ closure sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ closure s ∩ closure sᶜ
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | rw [mem_closure_iff_seq_limit] | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ closure sᶜ | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ sᶜ) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ closure sᶜ
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | let z : ℕ → E := fun n => (1 + 1 / n.succ : 𝕜) • x - (1 / n.succ : 𝕜) • y | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ sᶜ) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
z : ℕ → E := fun n => (1 + 1 / ↑n.succ) • x - (1 / ↑n.succ) • y
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ sᶜ) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ sᶜ) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | use z | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
z : ℕ → E := fun n => (1 + 1 / ↑n.succ) • x - (1 / ↑n.succ) • y
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ sᶜ) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) | case h
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
z : ℕ → E := fun n => (1 + 1 / ↑n.succ) • x - (1 / ↑n.succ) • y
⊢ (∀ (n : ℕ), z n ∈ sᶜ) ∧ Filter.Tendsto z Filter.atTop (nhds x) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
z : ℕ → E := fun n => (1 + 1 / ↑n.succ) • x - (1 / ↑n.succ) • y
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ sᶜ) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | sorry | case h
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
z : ℕ → E := fun n => (1 + 1 / ↑n.succ) • x - (1 / ↑n.succ) • y
⊢ (∀ (n : ℕ), z n ∈ sᶜ) ∧ Filter.Tendsto z Filter.atTop (nhds x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
z : ℕ → E := fun n => (1 + 1 / ↑n.succ) • x - (1 / ↑n.succ) • y
⊢ (∀ (n : ℕ), z n ∈ sᶜ) ∧ Filter.Tendsto z Filter.atTop (nhds x)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | refine' Subset.antisymm (fun x hx => _) (closure_mono interior_subset) | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x : E
s : Set E
hAconv : Convex 𝕜 s
hAnemp : (interior s).Nonempty
⊢ closure s = closure (interior s) | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
hAnemp : (interior s).Nonempty
x : E
hx : x ∈ closure s
⊢ x ∈ closure (interior s) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x : E
s : Set E
hAconv : Convex 𝕜 s
hAnemp : (interior s).Nonempty
⊢ closure s = closure (interior s)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | obtain ⟨y, hy⟩ := hAnemp | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
hAnemp : (interior s).Nonempty
x : E
hx : x ∈ closure s
⊢ x ∈ closure (interior s) | case intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x : E
hx : x ∈ closure s
y : E
hy : y ∈ interior s
⊢ x ∈ closure (interior s) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
hAnemp : (interior s).Nonempty
x : E
hx : x ∈ closure s
⊢ x ∈ closure (interior s)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | rw [mem_closure_iff_seq_limit] at hx ⊢ | case intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x : E
hx : x ∈ closure s
y : E
hy : y ∈ interior s
⊢ x ∈ closure (interior s) | case intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x : E
hx : ∃ x_1, (∀ (n : ℕ), x_1 n ∈ s) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x)
y : E
hy : y ∈ interior s
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ interior s) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x : E
hx : x ∈ closure s
y : E
hy : y ∈ interior s
⊢ x ∈ closure (interior s)
TACTIC:
|
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