Theorem: padicValRat.lt_sum_of_lt
File: Mathlib/NumberTheory/Padics/PadicVal.lean

tactic: induction' hS using Finset.Nonempty.cons_induction with k s S' Hnot Hne Hind
state_before: p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hS : S.Nonempty
hF : ∀ i ∈ S, padicValRat p (F j) < padicValRat p (F i)
hn1 : ∀ (i : ℕ), 0 < F i
⊢ padicValRat p (F j) < padicValRat p (∑ i ∈ S, F i)
state_after: case singleton
p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
k : ℕ
hF : ∀ i ∈ {k}, padicValRat p (F j) < padicValRat p (F i)
⊢ padicValRat p (F j) < padicValRat p (∑ i ∈ {k}, F i)

case cons
p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
s : ℕ
S' : Finset ℕ
Hnot : s ∉ S'
Hne : S'.Nonempty
Hind : (∀ i ∈ S', padicValRat p (F j) < padicValRat p (F i)) → padicValRat p (F j) < padicValRat p (∑ i ∈ S', F i)
hF : ∀ i ∈ Finset.cons s S' Hnot, padicValRat p (F j) < padicValRat p (F i)
⊢ padicValRat p (F j) < padicValRat p (∑ i ∈ Finset.cons s S' Hnot, F i)

tactic: rw [Finset.sum_singleton]
state_before: case singleton
p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
k : ℕ
hF : ∀ i ∈ {k}, padicValRat p (F j) < padicValRat p (F i)
⊢ padicValRat p (F j) < padicValRat p (∑ i ∈ {k}, F i)
state_after: case singleton
p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
k : ℕ
hF : ∀ i ∈ {k}, padicValRat p (F j) < padicValRat p (F i)
⊢ padicValRat p (F j) < padicValRat p (F k)

tactic: exact hF k (by simp)
state_before: case singleton
p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
k : ℕ
hF : ∀ i ∈ {k}, padicValRat p (F j) < padicValRat p (F i)
⊢ padicValRat p (F j) < padicValRat p (F k)
state_after: no goals

tactic: simp
state_before: p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
k : ℕ
hF : ∀ i ∈ {k}, padicValRat p (F j) < padicValRat p (F i)
⊢ k ∈ {k}
state_after: no goals

tactic: rw [Finset.cons_eq_insert, Finset.sum_insert Hnot]
state_before: case cons
p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
s : ℕ
S' : Finset ℕ
Hnot : s ∉ S'
Hne : S'.Nonempty
Hind : (∀ i ∈ S', padicValRat p (F j) < padicValRat p (F i)) → padicValRat p (F j) < padicValRat p (∑ i ∈ S', F i)
hF : ∀ i ∈ Finset.cons s S' Hnot, padicValRat p (F j) < padicValRat p (F i)
⊢ padicValRat p (F j) < padicValRat p (∑ i ∈ Finset.cons s S' Hnot, F i)
state_after: case cons
p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
s : ℕ
S' : Finset ℕ
Hnot : s ∉ S'
Hne : S'.Nonempty
Hind : (∀ i ∈ S', padicValRat p (F j) < padicValRat p (F i)) → padicValRat p (F j) < padicValRat p (∑ i ∈ S', F i)
hF : ∀ i ∈ Finset.cons s S' Hnot, padicValRat p (F j) < padicValRat p (F i)
⊢ padicValRat p (F j) < padicValRat p (F s + ∑ x ∈ S', F x)

tactic: exact padicValRat.lt_add_of_lt
  (ne_of_gt (add_pos (hn1 s) (Finset.sum_pos (fun i _ => hn1 i) Hne)))
  (hF _ (by simp [Finset.mem_insert, true_or]))
  (Hind (fun i hi => hF _ (by rw [Finset.cons_eq_insert,Finset.mem_insert]; exact Or.inr hi)))
state_before: case cons
p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
s : ℕ
S' : Finset ℕ
Hnot : s ∉ S'
Hne : S'.Nonempty
Hind : (∀ i ∈ S', padicValRat p (F j) < padicValRat p (F i)) → padicValRat p (F j) < padicValRat p (∑ i ∈ S', F i)
hF : ∀ i ∈ Finset.cons s S' Hnot, padicValRat p (F j) < padicValRat p (F i)
⊢ padicValRat p (F j) < padicValRat p (F s + ∑ x ∈ S', F x)
state_after: no goals

tactic: simp [Finset.mem_insert, true_or]
state_before: p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
s : ℕ
S' : Finset ℕ
Hnot : s ∉ S'
Hne : S'.Nonempty
Hind : (∀ i ∈ S', padicValRat p (F j) < padicValRat p (F i)) → padicValRat p (F j) < padicValRat p (∑ i ∈ S', F i)
hF : ∀ i ∈ Finset.cons s S' Hnot, padicValRat p (F j) < padicValRat p (F i)
⊢ s ∈ Finset.cons s S' Hnot
state_after: no goals

tactic: rw [Finset.cons_eq_insert,Finset.mem_insert]
state_before: p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
s : ℕ
S' : Finset ℕ
Hnot : s ∉ S'
Hne : S'.Nonempty
Hind : (∀ i ∈ S', padicValRat p (F j) < padicValRat p (F i)) → padicValRat p (F j) < padicValRat p (∑ i ∈ S', F i)
hF : ∀ i ∈ Finset.cons s S' Hnot, padicValRat p (F j) < padicValRat p (F i)
i : ℕ
hi : i ∈ S'
⊢ i ∈ Finset.cons s S' Hnot
state_after: p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
s : ℕ
S' : Finset ℕ
Hnot : s ∉ S'
Hne : S'.Nonempty
Hind : (∀ i ∈ S', padicValRat p (F j) < padicValRat p (F i)) → padicValRat p (F j) < padicValRat p (∑ i ∈ S', F i)
hF : ∀ i ∈ Finset.cons s S' Hnot, padicValRat p (F j) < padicValRat p (F i)
i : ℕ
hi : i ∈ S'
⊢ i = s ∨ i ∈ S'

tactic: exact Or.inr hi
state_before: p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
s : ℕ
S' : Finset ℕ
Hnot : s ∉ S'
Hne : S'.Nonempty
Hind : (∀ i ∈ S', padicValRat p (F j) < padicValRat p (F i)) → padicValRat p (F j) < padicValRat p (∑ i ∈ S', F i)
hF : ∀ i ∈ Finset.cons s S' Hnot, padicValRat p (F j) < padicValRat p (F i)
i : ℕ
hi : i ∈ S'
⊢ i = s ∨ i ∈ S'
state_after: no goals