l3lab/ntp-mathlib huggingface dataset:
/-- A version of `Convex.sum_mem` for `finsum`s. If `s` is a convex set, `w : ι → R` is a family of
nonnegative weights with sum one and `z : ι → E` is a family of elements of a module over `R` such
that `z i ∈ s` whenever `w i ≠ 0`, then the sum `∑ᶠ i, w i • z i` belongs to `s`. See also
`PartitionOfUnity.finsum_smul_mem_convex`. -/
theorem Convex.finsum_mem {ι : Sort*} {w : ι → R} {z : ι → E} {s : Set E} (hs : Convex R s)
    (h₀ : ∀ i, 0 ≤ w i) (h₁ : ∑ᶠ i, w i = 1) (hz : ∀ i, w i ≠ 0 → z i ∈ s) :
    (∑ᶠ i, w i • z i) ∈ s := by
  have hfin_w : (support (w ∘ PLift.down)).Finite := by
    by_contra H
    rw [finsum, dif_neg H] at h₁
    exact zero_ne_one h₁
  

Proof state:
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι✝ : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s✝ : Set E
i j : ι✝
c : R
t : Finset ι✝
w✝ : ι✝ → R
z✝ : ι✝ → E
ι : Sort u_8
w : ι → R
z : ι → E
s : Set E
hs : Convex R s
h₀ : ∀ (i : ι), 0 ≤ w i
h₁ : ∑ᶠ (i : ι), w i = 1
hz : ∀ (i : ι), w i ≠ 0 → z i ∈ s
hfin_w : Set.Finite (support (w ∘ PLift.down))
⊢ ∑ᶠ (i : ι), w i • z i ∈ s

Next tactic:
have hsub : support ((fun i => w i • z i) ∘ PLift.down) ⊆ hfin_w.toFinset :=
    (support_smul_subset_left _ _).trans hfin_w.coe_toFinset.ge
/-- A version of `Convex.sum_mem` for `finsum`s. If `s` is a convex set, `w : ι → R` is a family of
nonnegative weights with sum one and `z : ι → E` is a family of elements of a module over `R` such
that `z i ∈ s` whenever `w i ≠ 0`, then the sum `∑ᶠ i, w i • z i` belongs to `s`. See also
`PartitionOfUnity.finsum_smul_mem_convex`. -/
theorem Convex.finsum_mem {ι : Sort*} {w : ι → R} {z : ι → E} {s : Set E} (hs : Convex R s)
    (h₀ : ∀ i, 0 ≤ w i) (h₁ : ∑ᶠ i, w i = 1) (hz : ∀ i, w i ≠ 0 → z i ∈ s) :
    (∑ᶠ i, w i • z i) ∈ s := by
  have hfin_w : (support (w ∘ PLift.down)).Finite := by
    by_contra H
    rw [finsum, dif_neg H] at h₁
    exact zero_ne_one h₁
  have hsub : support ((fun i => w i • z i) ∘ PLift.down) ⊆ hfin_w.toFinset :=
    (support_smul_subset_left _ _).trans hfin_w.coe_toFinset.ge
  

Proof state:
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι✝ : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s✝ : Set E
i j : ι✝
c : R
t : Finset ι✝
w✝ : ι✝ → R
z✝ : ι✝ → E
ι : Sort u_8
w : ι → R
z : ι → E
s : Set E
hs : Convex R s
h₀ : ∀ (i : ι), 0 ≤ w i
h₁ : ∑ᶠ (i : ι), w i = 1
hz : ∀ (i : ι), w i ≠ 0 → z i ∈ s
hfin_w : Set.Finite (support (w ∘ PLift.down))
hsub : support ((fun i => w i • z i) ∘ PLift.down) ⊆ ↑(Finite.toFinset hfin_w)
⊢ ∑ᶠ (i : ι), w i • z i ∈ s

Next tactic:
rw [finsum_eq_sum_plift_of_support_subset hsub]
/-- A version of `Convex.sum_mem` for `finsum`s. If `s` is a convex set, `w : ι → R` is a family of
nonnegative weights with sum one and `z : ι → E` is a family of elements of a module over `R` such
that `z i ∈ s` whenever `w i ≠ 0`, then the sum `∑ᶠ i, w i • z i` belongs to `s`. See also
`PartitionOfUnity.finsum_smul_mem_convex`. -/
theorem Convex.finsum_mem {ι : Sort*} {w : ι → R} {z : ι → E} {s : Set E} (hs : Convex R s)
    (h₀ : ∀ i, 0 ≤ w i) (h₁ : ∑ᶠ i, w i = 1) (hz : ∀ i, w i ≠ 0 → z i ∈ s) :
    (∑ᶠ i, w i • z i) ∈ s := by
  have hfin_w : (support (w ∘ PLift.down)).Finite := by
    by_contra H
    rw [finsum, dif_neg H] at h₁
    exact zero_ne_one h₁
  have hsub : support ((fun i => w i • z i) ∘ PLift.down) ⊆ hfin_w.toFinset :=
    (support_smul_subset_left _ _).trans hfin_w.coe_toFinset.ge
  rw [finsum_eq_sum_plift_of_support_subset hsub]
  

Proof state:
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι✝ : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s✝ : Set E
i j : ι✝
c : R
t : Finset ι✝
w✝ : ι✝ → R
z✝ : ι✝ → E
ι : Sort u_8
w : ι → R
z : ι → E
s : Set E
hs : Convex R s
h₀ : ∀ (i : ι), 0 ≤ w i
h₁ : ∑ᶠ (i : ι), w i = 1
hz : ∀ (i : ι), w i ≠ 0 → z i ∈ s
hfin_w : Set.Finite (support (w ∘ PLift.down))
hsub : support ((fun i => w i • z i) ∘ PLift.down) ⊆ ↑(Finite.toFinset hfin_w)
⊢ ∑ i in Finite.toFinset hfin_w, w i.down • z i.down ∈ s

Next tactic:
refine' hs.sum_mem (fun _ _ => h₀ _) _ fun i hi => hz _ _
/-- A version of `Convex.sum_mem` for `finsum`s. If `s` is a convex set, `w : ι → R` is a family of
nonnegative weights with sum one and `z : ι → E` is a family of elements of a module over `R` such
that `z i ∈ s` whenever `w i ≠ 0`, then the sum `∑ᶠ i, w i • z i` belongs to `s`. See also
`PartitionOfUnity.finsum_smul_mem_convex`. -/
theorem Convex.finsum_mem {ι : Sort*} {w : ι → R} {z : ι → E} {s : Set E} (hs : Convex R s)
    (h₀ : ∀ i, 0 ≤ w i) (h₁ : ∑ᶠ i, w i = 1) (hz : ∀ i, w i ≠ 0 → z i ∈ s) :
    (∑ᶠ i, w i • z i) ∈ s := by
  have hfin_w : (support (w ∘ PLift.down)).Finite := by
    by_contra H
    rw [finsum, dif_neg H] at h₁
    exact zero_ne_one h₁
  have hsub : support ((fun i => w i • z i) ∘ PLift.down) ⊆ hfin_w.toFinset :=
    (support_smul_subset_left _ _).trans hfin_w.coe_toFinset.ge
  rw [finsum_eq_sum_plift_of_support_subset hsub]
  refine' hs.sum_mem (fun _ _ => h₀ _) _ fun i hi => hz _ _
  · 

Proof state:
case refine'_1
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι✝ : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s✝ : Set E
i j : ι✝
c : R
t : Finset ι✝
w✝ : ι✝ → R
z✝ : ι✝ → E
ι : Sort u_8
w : ι → R
z : ι → E
s : Set E
hs : Convex R s
h₀ : ∀ (i : ι), 0 ≤ w i
h₁ : ∑ᶠ (i : ι), w i = 1
hz : ∀ (i : ι), w i ≠ 0 → z i ∈ s
hfin_w : Set.Finite (support (w ∘ PLift.down))
hsub : support ((fun i => w i • z i) ∘ PLift.down) ⊆ ↑(Finite.toFinset hfin_w)
⊢ ∑ i in Finite.toFinset hfin_w, w i.down = 1

Next tactic:
rwa [finsum, dif_pos hfin_w] at h₁
/-- A version of `Convex.sum_mem` for `finsum`s. If `s` is a convex set, `w : ι → R` is a family of
nonnegative weights with sum one and `z : ι → E` is a family of elements of a module over `R` such
that `z i ∈ s` whenever `w i ≠ 0`, then the sum `∑ᶠ i, w i • z i` belongs to `s`. See also
`PartitionOfUnity.finsum_smul_mem_convex`. -/
theorem Convex.finsum_mem {ι : Sort*} {w : ι → R} {z : ι → E} {s : Set E} (hs : Convex R s)
    (h₀ : ∀ i, 0 ≤ w i) (h₁ : ∑ᶠ i, w i = 1) (hz : ∀ i, w i ≠ 0 → z i ∈ s) :
    (∑ᶠ i, w i • z i) ∈ s := by
  have hfin_w : (support (w ∘ PLift.down)).Finite := by
    by_contra H
    rw [finsum, dif_neg H] at h₁
    exact zero_ne_one h₁
  have hsub : support ((fun i => w i • z i) ∘ PLift.down) ⊆ hfin_w.toFinset :=
    (support_smul_subset_left _ _).trans hfin_w.coe_toFinset.ge
  rw [finsum_eq_sum_plift_of_support_subset hsub]
  refine' hs.sum_mem (fun _ _ => h₀ _) _ fun i hi => hz _ _
  · rwa [finsum, dif_pos hfin_w] at h₁
  · 

Proof state:
case refine'_2
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι✝ : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s✝ : Set E
i✝ j : ι✝
c : R
t : Finset ι✝
w✝ : ι✝ → R
z✝ : ι✝ → E
ι : Sort u_8
w : ι → R
z : ι → E
s : Set E
hs : Convex R s
h₀ : ∀ (i : ι), 0 ≤ w i
h₁ : ∑ᶠ (i : ι), w i = 1
hz : ∀ (i : ι), w i ≠ 0 → z i ∈ s
hfin_w : Set.Finite (support (w ∘ PLift.down))
hsub : support ((fun i => w i • z i) ∘ PLift.down) ⊆ ↑(Finite.toFinset hfin_w)
i : PLift ι
hi : i ∈ Finite.toFinset hfin_w
⊢ w i.down ≠ 0

Next tactic:
rwa [hfin_w.mem_toFinset] at hi
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  

Proof state:
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
⊢ Convex R s ↔
    ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s

Next tactic:
refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
  

Proof state:
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
⊢ (∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s) →
    Convex R s

Next tactic:
intro h x hx y hy a b ha hb hab
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
  intro h x hx y hy a b ha hb hab
  

Proof state:
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • x + b • y ∈ s

Next tactic:
by_cases h_cases : x = y
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
  intro h x hx y hy a b ha hb hab
  by_cases h_cases : x = y
  · 

Proof state:
case pos
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : x = y
⊢ a • x + b • y ∈ s

Next tactic:
rw [h_cases, ← add_smul, hab, one_smul]
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
  intro h x hx y hy a b ha hb hab
  by_cases h_cases : x = y
  · rw [h_cases, ← add_smul, hab, one_smul]
    

Proof state:
case pos
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : x = y
⊢ y ∈ s

Next tactic:
exact hy
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
  intro h x hx y hy a b ha hb hab
  by_cases h_cases : x = y
  · rw [h_cases, ← add_smul, hab, one_smul]
    exact hy
  · 

Proof state:
case neg
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : ¬x = y
⊢ a • x + b • y ∈ s

Next tactic:
convert h {x, y} (fun z => if z = y then b else a) _ _ _
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
  intro h x hx y hy a b ha hb hab
  by_cases h_cases : x = y
  · rw [h_cases, ← add_smul, hab, one_smul]
    exact hy
  · convert h {x, y} (fun z => if z = y then b else a) _ _ _
    -- Porting note: Original proof had 2 `simp_intro i hi`
    · 

Proof state:
case h.e'_4
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : ¬x = y
⊢ a • x + b • y = ∑ x in {x, y}, (if x = y then b else a) • x

Next tactic:
simp only [sum_pair h_cases, if_neg h_cases, if_pos trivial]
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
  intro h x hx y hy a b ha hb hab
  by_cases h_cases : x = y
  · rw [h_cases, ← add_smul, hab, one_smul]
    exact hy
  · convert h {x, y} (fun z => if z = y then b else a) _ _ _
    -- Porting note: Original proof had 2 `simp_intro i hi`
    · simp only [sum_pair h_cases, if_neg h_cases, if_pos trivial]
    · 

Proof state:
case neg.convert_1
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : ¬x = y
⊢ ∀ i ∈ {x, y}, 0 ≤ (fun z => if z = y then b else a) i

Next tactic:
intro i _
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
  intro h x hx y hy a b ha hb hab
  by_cases h_cases : x = y
  · rw [h_cases, ← add_smul, hab, one_smul]
    exact hy
  · convert h {x, y} (fun z => if z = y then b else a) _ _ _
    -- Porting note: Original proof had 2 `simp_intro i hi`
    · simp only [sum_pair h_cases, if_neg h_cases, if_pos trivial]
    · intro i _
      

Proof state:
case neg.convert_1
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i✝ j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : ¬x = y
i : E
a✝ : i ∈ {x, y}
⊢ 0 ≤ (fun z => if z = y then b else a) i

Next tactic:
simp only
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
  intro h x hx y hy a b ha hb hab
  by_cases h_cases : x = y
  · rw [h_cases, ← add_smul, hab, one_smul]
    exact hy
  · convert h {x, y} (fun z => if z = y then b else a) _ _ _
    -- Porting note: Original proof had 2 `simp_intro i hi`
    · simp only [sum_pair h_cases, if_neg h_cases, if_pos trivial]
    · intro i _
      simp only
      

Proof state:
case neg.convert_1
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i✝ j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : ¬x = y
i : E
a✝ : i ∈ {x, y}
⊢ 0 ≤ if i = y then b else a

Next tactic:
split_ifs
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
  intro h x hx y hy a b ha hb hab
  by_cases h_cases : x = y
  · rw [h_cases, ← add_smul, hab, one_smul]
    exact hy
  · convert h {x, y} (fun z => if z = y then b else a) _ _ _
    -- Porting note: Original proof had 2 `simp_intro i hi`
    · simp only [sum_pair h_cases, if_neg h_cases, if_pos trivial]
    · intro i _
      simp only
      split_ifs <;> 

Proof state:
case pos
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i✝ j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : ¬x = y
i : E
a✝ : i ∈ {x, y}
h✝ : i = y
⊢ 0 ≤ b

Next tactic:
assumption
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
  intro h x hx y hy a b ha hb hab
  by_cases h_cases : x = y
  · rw [h_cases, ← add_smul, hab, one_smul]
    exact hy
  · convert h {x, y} (fun z => if z = y then b else a) _ _ _
    -- Porting note: Original proof had 2 `simp_intro i hi`
    · simp only [sum_pair h_cases, if_neg h_cases, if_pos trivial]
    · intro i _
      simp only
      split_ifs <;> 

Proof state:
case neg
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i✝ j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : ¬x = y
i : E
a✝ : i ∈ {x, y}
h✝ : ¬i = y
⊢ 0 ≤ a

Next tactic:
assumption
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
  intro h x hx y hy a b ha hb hab
  by_cases h_cases : x = y
  · rw [h_cases, ← add_smul, hab, one_smul]
    exact hy
  · convert h {x, y} (fun z => if z = y then b else a) _ _ _
    -- Porting note: Original proof had 2 `simp_intro i hi`
    · simp only [sum_pair h_cases, if_neg h_cases, if_pos trivial]
    · intro i _
      simp only
      split_ifs <;> assumption
    · 

Proof state:
case neg.convert_2
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : ¬x = y
⊢ ∑ i in {x, y}, (fun z => if z = y then b else a) i = 1

Next tactic:
simp only [sum_pair h_cases, if_neg h_cases, if_pos trivial, hab]
theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
    (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x in t, w x • x) ∈ s := by
  refine' ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, _⟩
  intro h x hx y hy a b ha hb hab
  by_cases h_cases : x = y
  · rw [h_cases, ← add_smul, hab, one_smul]
    exact hy
  · convert h {x, y} (fun z => if z = y then b else a) _ _ _
    -- Porting note: Original proof had 2 `simp_intro i hi`
    · simp only [sum_pair h_cases, if_neg h_cases, if_pos trivial]
    · intro i _
      simp only
      split_ifs <;> assumption
    · simp only [sum_pair h_cases, if_neg h_cases, if_pos trivial, hab]
    · 

Proof state:
case neg.convert_3
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : ¬x = y
⊢ ∀ x_1 ∈ {x, y}, x_1 ∈ s

Next tactic:
intro i hi