awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
⊢ ∃ (y : (↑(↑t : Set E) : Type u)) (w : E → 𝕜),
    (∀ y_1 ∈ t.erase (↑y : E), 0 ≤ w y_1) ∧
      Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t.erase (↑y : E), w y) 1 ∧ (t.erase (↑y : E)).centerMass w id = t.centerMass f id
have refine_1 : ∀ y ∈ t.erase (↑(Subtype.mk (p := fun (x : E) ↦ x ∈ (↑t : Set E)) i₀ hi₀) : E), 0 ≤ k y
Tactic: refine ⟨⟨i₀, hi₀⟩, k, ?_, by convert! ksum, ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
refine_1 : ∀ y ∈ t.erase (↑(Subtype.mk (p := fun (x : E) ↦ x ∈ (↑t : Set E)) i₀ hi₀) : E), 0 ≤ k y
⊢ ∃ (y : (↑(↑t : Set E) : Type u)) (w : E → 𝕜),
    (∀ y_1 ∈ t.erase (↑y : E), 0 ≤ w y_1) ∧
      Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t.erase (↑y : E), w y) 1 ∧ (t.erase (↑y : E)).centerMass w id = t.centerMass f id
have refine_2 : (t.erase (↑(Subtype.mk (p := fun (x : E) ↦ x ∈ (↑t : Set E)) i₀ hi₀) : E)).centerMass k id = t.centerMass f id
Tactic: refine ⟨⟨i₀, hi₀⟩, k, ?_, by convert! ksum, ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
⊢ ∀ y ∈ t.erase (↑(Subtype.mk (p := fun (x : E) ↦ x ∈ (↑t : Set E)) i₀ hi₀) : E), 0 ≤ k y
have refine_1 : ∀ (y : E), ¬y = i₀ → y ∈ t → f i₀ / g i₀ * g y ≤ f y
Tactic: simp only [k, and_imp, sub_nonneg, mem_erase, Ne]
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
e : E
h : ¬e = i₀
het : e ∈ t
⊢ f i₀ / g i₀ * g e ≤ f e
have pos : ∀ (hes : e ∈ s), f i₀ / g i₀ * g e ≤ f e
Tactic: by_cases hes : e ∈ s
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
e : E
h : ¬e = i₀
het : e ∈ t
pos : ∀ (hes : e ∈ s), f i₀ / g i₀ * g e ≤ f e
⊢ f i₀ / g i₀ * g e ≤ f e
have neg : ∀ (hes : e ∉ s), f i₀ / g i₀ * g e ≤ f e
Tactic: by_cases hes : e ∈ s
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
e : E
h : ¬e = i₀
het : e ∈ t
hes : e ∈ s
⊢ 0 < g e
have hes₁ : e ∈ t ∧ 0 < g e
Tactic: rw [mem_filter] at hes
Cleared hypotheses: hes
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
e : E
h : ¬e = i₀
het : e ∈ t
hes : e ∈ s
⊢ f i₀ / g i₀ * g e ≤ f e
have hge : 0 < g e
Tactic: have hge : 0 < g e := by
        rw [mem_filter] at hes
        exact hes.2
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
e : E
h : ¬e = i₀
het : e ∈ t
hes : e ∈ s
hge : 0 < g e
⊢ f i₀ / g i₀ * g e ≤ f e
have pos : f i₀ / g i₀ ≤ f e / g e
Tactic: rw [← le_div_iff₀ hge]
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
e : E
h : ¬e = i₀
het : e ∈ t
hes : e ∉ s
⊢ f i₀ / g i₀ * g e ≤ 0
have ha : 0 ≤ f i₀ / g i₀
Tactic: apply mul_nonpos_of_nonneg_of_nonpos
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
e : E
h : ¬e = i₀
het : e ∈ t
hes : e ∉ s
ha : 0 ≤ f i₀ / g i₀
⊢ f i₀ / g i₀ * g e ≤ 0
have hb : g e ≤ 0
Tactic: apply mul_nonpos_of_nonneg_of_nonpos
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
e : E
h : ¬e = i₀
het : e ∈ t
hes : e ∉ s
⊢ f i₀ / g i₀ * g e ≤ f e
have _ : f i₀ / g i₀ * g e ≤ 0
Tactic: calc
        _ ≤ 0 := by
          apply mul_nonpos_of_nonneg_of_nonpos
          · apply div_nonneg (fpos i₀ (mem_of_subset (filter_subset _ t) mem)) (le_of_lt hg)
          · simpa only [s, mem_filter, het, true_and, not_lt] using hes
        _ ≤ f e := fpos e het
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
⊢ (t.erase (↑(Subtype.mk (p := fun (x : E) ↦ x ∈ (↑t : Set E)) i₀ hi₀) : E)).centerMass k id = t.centerMass f id
have refine_2 : ∑ i ∈ t.erase i₀, k i • id i = t.centerMass f id
Tactic: rw [Subtype.coe_mk, centerMass_eq_of_sum_1 _ id ksum]
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
⊢ k i₀ • i₀ = 0
have rw : HSMul.hSMul.{u_1, u, u} (α := 𝕜) 0 i₀ = 0
Tactic: rw [hk, zero_smul]
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
rw : HSMul.hSMul.{u_1, u, u} (α := 𝕜) 0 i₀ = 0
⊢ k i₀ • i₀ = 0
have rw₁ : Eq.{u + 1} (α := E) 0 0
Tactic: rw [hk, zero_smul]
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
inst_5 : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, f y) 1
g : E → 𝕜
gcombo : Eq (α := E) (∑ e ∈ t, g e • e) 0
gsum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t, g e) 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun (z : E) ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : Eq.{u_1 + 1} (α := 𝕜) (∑ e ∈ t.erase i₀, k e) 1
⊢ ∑ i ∈ t.erase i₀, k i • id i = t.centerMass f id
have e : ∑ e ∈ t, (f e - f i₀ / g i₀ * g e) • e = t.centerMass f id
Tactic: calc
      ∑ e ∈ t.erase i₀, k e • e = ∑ e ∈ t, k e • e := sum_erase _ (by rw [hk, zero_smul])
      _ = ∑ e ∈ t, (f e - f i₀ / g i₀ * g e) • e := rfl
      _ = t.centerMass f id := by
        simp only [sub_smul, mul_smul, sum_sub_distrib, ← smul_sum, gcombo, smul_zero, sub_zero,
          centerMass, fsum, inv_one, one_smul, id]
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
⊢ (minCardFinsetOfMemConvexHull (s := s) hx).Nonempty
have rw : Set.Nonempty.{u} (α := E) ((convexHull 𝕜 : (a : Set E) → Set E) (↑(minCardFinsetOfMemConvexHull (s := s) hx) : Set E))
Tactic: rw [← Finset.coe_nonempty, ← @convexHull_nonempty_iff 𝕜]
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
k : ℕ := #(minCardFinsetOfMemConvexHull (s := s) hx) - 1
⊢ AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
    (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))
have hk : #(minCardFinsetOfMemConvexHull (s := s) hx) = k + 1
Tactic: have hk : #(minCardFinsetOfMemConvexHull hx) = k + 1 :=
    (Nat.succ_pred_eq_of_pos (Finset.card_pos.mpr (minCardFinsetOfMemConvexHull_nonempty hx))).symm
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
k : ℕ := #(minCardFinsetOfMemConvexHull (s := s) hx) - 1
hk : #(minCardFinsetOfMemConvexHull (s := s) hx) = k + 1
⊢ AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
    (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))
have by_contra : ∀
  (h :
    ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
        (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))),
  False
Tactic: by_contra h
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
k : ℕ := #(minCardFinsetOfMemConvexHull (s := s) hx) - 1
hk : #(minCardFinsetOfMemConvexHull (s := s) hx) = k + 1
h :
  ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
      (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))
⊢ False
have obtain : ∀ (p : (↑(↑(minCardFinsetOfMemConvexHull (s := s) hx) : Set E) : Type u))
  (hp :
    Membership.mem (γ := Set E)
      ((convexHull 𝕜 : (a : Set E) → Set E) (↑((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) : Set E) :
        Set E)
      x),
  False
Tactic: obtain ⟨p, hp⟩ := mem_convexHull_erase h (mem_minCardFinsetOfMemConvexHull hx)
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
k : ℕ := #(minCardFinsetOfMemConvexHull (s := s) hx) - 1
hk : #(minCardFinsetOfMemConvexHull (s := s) hx) = k + 1
h :
  ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
      (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))
p : (↑(↑(minCardFinsetOfMemConvexHull (s := s) hx) : Set E) : Type u)
hp :
  Membership.mem (γ := Set E)
    ((convexHull 𝕜 : (a : Set E) → Set E) (↑((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) : Set E) :
      Set E)
    x
⊢ False
have contra : #(minCardFinsetOfMemConvexHull (s := s) hx) ≤ #((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E))
Tactic: have contra := minCardFinsetOfMemConvexHull_card_le_card hx (Set.Subset.trans
    (Finset.erase_subset (p : E) (minCardFinsetOfMemConvexHull hx))
    (minCardFinsetOfMemConvexHull_subseteq hx)) hp
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
k : ℕ := #(minCardFinsetOfMemConvexHull (s := s) hx) - 1
hk : #(minCardFinsetOfMemConvexHull (s := s) hx) = k + 1
h :
  ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
      (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))
p : (↑(↑(minCardFinsetOfMemConvexHull (s := s) hx) : Set E) : Type u)
hp :
  Membership.mem (γ := Set E)
    ((convexHull 𝕜 : (a : Set E) → Set E) (↑((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) : Set E) :
      Set E)
    x
contra : #(minCardFinsetOfMemConvexHull (s := s) hx) ≤ #((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E))
⊢ False
have contra₁ : ¬#((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) < #(minCardFinsetOfMemConvexHull (s := s) hx)
Tactic: rw [← not_lt] at contra
Cleared hypotheses: contra
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
k : ℕ := #(minCardFinsetOfMemConvexHull (s := s) hx) - 1
hk : #(minCardFinsetOfMemConvexHull (s := s) hx) = k + 1
h :
  ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
      (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))
p : (↑(↑(minCardFinsetOfMemConvexHull (s := s) hx) : Set E) : Type u)
hp :
  Membership.mem (γ := Set E)
    ((convexHull 𝕜 : (a : Set E) → Set E) (↑((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) : Set E) :
      Set E)
    x
contra : ¬#((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) < #(minCardFinsetOfMemConvexHull (s := s) hx)
⊢ False
have apply : #((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) < #(minCardFinsetOfMemConvexHull (s := s) hx)
Tactic: apply contra
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
k : ℕ := #(minCardFinsetOfMemConvexHull (s := s) hx) - 1
hk : #(minCardFinsetOfMemConvexHull (s := s) hx) = k + 1
h :
  ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
      (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))
p : (↑(↑(minCardFinsetOfMemConvexHull (s := s) hx) : Set E) : Type u)
hp :
  Membership.mem (γ := Set E)
    ((convexHull 𝕜 : (a : Set E) → Set E) (↑((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) : Set E) :
      Set E)
    x
contra : ¬#((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) < #(minCardFinsetOfMemConvexHull (s := s) hx)
⊢ #((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) < #(minCardFinsetOfMemConvexHull (s := s) hx)
have rw : k + 1 - 1 < k + 1
Tactic: rw [card_erase_of_mem p.2, hk]
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
k : ℕ := #(minCardFinsetOfMemConvexHull (s := s) hx) - 1
hk : #(minCardFinsetOfMemConvexHull (s := s) hx) = k + 1
⊢ AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
    (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))
have by_contra : ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
      (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx)) →
  False
Tactic: classical
  by_contra h
  obtain ⟨p, hp⟩ := mem_convexHull_erase h (mem_minCardFinsetOfMemConvexHull hx)
  have contra := minCardFinsetOfMemConvexHull_card_le_card hx (Set.Subset.trans
    (Finset.erase_subset (p : E) (minCardFinsetOfMemConvexHull hx))
    (minCardFinsetOfMemConvexHull_subseteq hx)) hp
  rw [← not_lt] at contra
  apply contra
  rw [card_erase_of_mem p.2, hk]
  exact lt_add_one _
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
k : ℕ := #(minCardFinsetOfMemConvexHull (s := s) hx) - 1
hk : #(minCardFinsetOfMemConvexHull (s := s) hx) = k + 1
by_contra by_contra₁ by_contra₂ :
  ∀
    (h :
      ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
          (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))),
    False
⊢ AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
    (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))
have by_contra₃ : ∀
  (h :
    ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
        (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx)))
  (p : (↑(↑(minCardFinsetOfMemConvexHull (s := s) hx) : Set E) : Type u))
  (hp :
    Membership.mem (γ := Set E)
      ((convexHull 𝕜 : (a : Set E) → Set E) (↑((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) : Set E) :
        Set E)
      x),
  False
Tactic: classical
  by_contra h
  obtain ⟨p, hp⟩ := mem_convexHull_erase h (mem_minCardFinsetOfMemConvexHull hx)
  have contra := minCardFinsetOfMemConvexHull_card_le_card hx (Set.Subset.trans
    (Finset.erase_subset (p : E) (minCardFinsetOfMemConvexHull hx))
    (minCardFinsetOfMemConvexHull_subseteq hx)) hp
  rw [← not_lt] at contra
  apply contra
  rw [card_erase_of_mem p.2, hk]
  exact lt_add_one _