Theorem: MeasureTheory.SignedMeasure.of_symmDiff_compl_positive_negative
File: Mathlib/MeasureTheory/Decomposition/SignedHahn.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ ↑s (i ∆ j) = 0 ∧ ↑s (iᶜ ∆ jᶜ) = 0
state_after: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
  (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
    ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ ↑s (i ∆ j) = 0 ∧ ↑s (iᶜ ∆ jᶜ) = 0

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet j

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet i

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet jᶜ

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet iᶜ
tactic: rw [restrict_le_restrict_iff s 0, restrict_le_restrict_iff 0 s] at hi' hj'
state_before: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
  (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
    ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ ↑s (i ∆ j) = 0 ∧ ↑s (iᶜ ∆ jᶜ) = 0

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet j

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet i

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet jᶜ

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet iᶜ
state_after: case left
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
  (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
    ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ ↑s (i ∆ j) = 0

case right
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
  (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
    ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ ↑s (iᶜ ∆ jᶜ) = 0

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet j

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet i

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet jᶜ

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet iᶜ
tactic: constructor
state_before: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet j

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet i

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet jᶜ

α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s✝ : SignedMeasure α
i✝ j✝ : Set α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ
hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ
⊢ MeasurableSet iᶜ
state_after: no goals
tactic: all_goals measurability