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namespace OAI
section
namespace PartitionPilot.Forcing
universe u
def assignmentCut (levels alpha A : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (fun z => ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels) A
def assignmentUpper (levels alpha A : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (fun z => z ∉ assignmentCut levels alpha A) A
def assignmentSplice (levels alpha Aq Ap : ZFSet.{u}) : ZFSet.{u} :=
assignmentCut levels alpha Aq ∪ assignmentUpper levels alpha Ap
theorem mem_assignmentCut (levels alpha A z : ZFSet.{u}) :
z ∈ assignmentCut levels alpha A ↔
z ∈ A ∧ ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels :=
ZFSet.mem_sep
theorem mem_assignmentUpper (levels alpha A z : ZFSet.{u}) :
z ∈ assignmentUpper levels alpha A ↔
z ∈ A ∧ ¬ ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels := by
rw [assignmentUpper, ZFSet.mem_sep, mem_assignmentCut]
constructor
· rintro ⟨hz, outside⟩
exact ⟨hz, fun lower => outside ⟨hz, lower⟩⟩
· rintro ⟨hz, outside⟩
exact ⟨hz, fun lower => outside lower.2⟩
theorem mem_assignmentSplice (levels alpha Aq Ap z : ZFSet.{u}) :
z ∈ assignmentSplice levels alpha Aq Ap ↔
(z ∈ Aq ∧ ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels) ∨
(z ∈ Ap ∧ ¬ ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels) := by
rw [assignmentSplice, ZFSet.mem_union, mem_assignmentCut, mem_assignmentUpper]
theorem assignmentCut_subset (levels alpha A : ZFSet.{u}) :
assignmentCut levels alpha A ⊆ A :=
fun _ hz => ((mem_assignmentCut levels alpha A _).1 hz).1
theorem assignmentUpper_subset (levels alpha A : ZFSet.{u}) :
assignmentUpper levels alpha A ⊆ A :=
fun _ hz => ((mem_assignmentUpper levels alpha A _).1 hz).1
theorem assignmentSplice_subset (levels alpha Aq Ap : ZFSet.{u}) :
assignmentSplice levels alpha Aq Ap ⊆ Aq ∪ Ap := by
intro z hz
rcases (mem_assignmentSplice levels alpha Aq Ap z).1 hz with lower | upper
· exact ZFSet.mem_union.2 (Or.inl lower.1)
· exact ZFSet.mem_union.2 (Or.inr upper.1)
theorem assignmentCut_mono {levels alpha A B : ZFSet.{u}} (subset : A ⊆ B) :
assignmentCut levels alpha A ⊆ assignmentCut levels alpha B := by
intro z hz
obtain ⟨hz, lower⟩ := (mem_assignmentCut levels alpha A z).1 hz
exact (mem_assignmentCut levels alpha B z).2 ⟨subset hz, lower⟩
theorem assignmentCut_nested {levels alpha beta A : ZFSet.{u}}
(below : alpha ⊆ beta) :
assignmentCut levels alpha (assignmentCut levels beta A) =
assignmentCut levels alpha A := by
apply ZFSet.ext
intro z
rw [mem_assignmentCut, mem_assignmentCut, mem_assignmentCut]
constructor
· rintro ⟨⟨hz, _⟩, lower⟩
exact ⟨hz, lower⟩
· rintro ⟨hz, xi, hxi, entry⟩
exact ⟨⟨hz, xi, below hxi, entry⟩, xi, hxi, entry⟩
theorem assignmentCut_eq_of_lower {levels alpha A : ZFSet.{u}}
(lower : ∀ z ∈ A, ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels) :
assignmentCut levels alpha A = A := by
apply ZFSet.ext
intro z
rw [mem_assignmentCut]
exact ⟨fun hz => hz.1, fun hz => ⟨hz, lower z hz⟩⟩
theorem assignmentCut_full {J eta levels A : ZFSet.{u}}
(function : ZFSet.IsFunc J eta levels) (subset : A ⊆ J) :
assignmentCut levels eta A = A := by
apply assignmentCut_eq_of_lower
intro z hz
obtain ⟨xi, entry, _⟩ := function.2 z (subset hz)
exact ⟨xi, (ZFSet.pair_mem_prod.1 (function.1 entry)).2, entry⟩
theorem assignmentCut_splice (levels alpha Aq Ap : ZFSet.{u}) :
assignmentCut levels alpha (assignmentSplice levels alpha Aq Ap) =
assignmentCut levels alpha Aq := by
apply ZFSet.ext
intro z
rw [mem_assignmentCut, mem_assignmentSplice, mem_assignmentCut]
constructor
· rintro ⟨lower | upper, level⟩
· exact lower
· exact False.elim (upper.2 level)
· intro lower
exact ⟨Or.inl lower, lower.2⟩
theorem assignmentUpper_splice (levels alpha Aq Ap : ZFSet.{u}) :
assignmentUpper levels alpha (assignmentSplice levels alpha Aq Ap) =
assignmentUpper levels alpha Ap := by
apply ZFSet.ext
intro z
rw [mem_assignmentUpper, mem_assignmentSplice, mem_assignmentUpper]
constructor
· rintro ⟨lower | upper, outside⟩
· exact False.elim (outside lower.2)
· exact upper
· intro upper
exact ⟨Or.inr upper, upper.2⟩
theorem assignmentSplice_self (levels alpha A : ZFSet.{u}) :
assignmentSplice levels alpha A A = A := by
classical
apply ZFSet.ext
intro z
rw [mem_assignmentSplice]
constructor
· exact fun h => h.elim And.left And.left
· intro hz
by_cases lower : ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels
· exact Or.inl ⟨hz, lower⟩
· exact Or.inr ⟨hz, lower⟩
theorem assignmentSplice_of_lower {levels alpha Aq Ap : ZFSet.{u}}
(lower : ∀ z ∈ Aq, ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels) :
assignmentSplice levels alpha Aq Ap = Aq ∪ assignmentUpper levels alpha Ap ∧
assignmentCut levels alpha (assignmentSplice levels alpha Aq Ap) = Aq := by
have equal := assignmentCut_eq_of_lower lower
exact ⟨by rw [assignmentSplice, equal], by rw [assignmentCut_splice, equal]⟩
theorem assignmentSplice_extends {levels alpha Aq Ap : ZFSet.{u}}
(contains : assignmentCut levels alpha Ap ⊆ assignmentCut levels alpha Aq) :
Ap ⊆ assignmentSplice levels alpha Aq Ap := by
classical
intro z hz
by_cases lower : ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels
· exact ZFSet.mem_union.2 (Or.inl (contains
((mem_assignmentCut levels alpha Ap z).2 ⟨hz, lower⟩)))
· exact (mem_assignmentSplice levels alpha Aq Ap z).2 (Or.inr ⟨hz, lower⟩)
end PartitionPilot.Forcing
namespace PartitionPilot.Forcing
universe u
variable {levels alpha beta : ZFSet.{u}} (below : alpha ⊆ beta)
include below
theorem assignmentCut_splice_below (Aq Ap : ZFSet.{u}) :
assignmentCut levels alpha (assignmentSplice levels beta Aq Ap) =
assignmentCut levels alpha Aq := by
calc
_ = assignmentCut levels alpha
(assignmentCut levels beta (assignmentSplice levels beta Aq Ap)) :=
(assignmentCut_nested below).symm
_ = assignmentCut levels alpha (assignmentCut levels beta Aq) := by rw [assignmentCut_splice]
_ = _ := assignmentCut_nested below
theorem assignmentUpper_splice_above (Aq Ap : ZFSet.{u}) :
assignmentUpper levels beta (assignmentSplice levels alpha Aq Ap) =
assignmentUpper levels beta Ap := by
apply ZFSet.ext
intro z
rw [mem_assignmentUpper, mem_assignmentSplice, mem_assignmentUpper]
constructor
· rintro ⟨lower | upper, outside⟩
· obtain ⟨xi, hxi, entry⟩ := lower.2
exact False.elim (outside ⟨xi, below hxi, entry⟩)
· exact ⟨upper.1, outside⟩
· rintro ⟨member, outside⟩
exact ⟨Or.inr ⟨member, fun ⟨xi, hxi, entry⟩ => outside ⟨xi, below hxi, entry⟩⟩, outside⟩
theorem assignmentSplice_nested (Ar Aq Ap : ZFSet.{u}) :
assignmentSplice levels alpha Ar (assignmentSplice levels beta Aq Ap) =
assignmentSplice levels beta (assignmentSplice levels alpha Ar Aq) Ap := by
classical
apply ZFSet.ext
intro z
simp only [mem_assignmentSplice]
by_cases lower : ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels
· obtain ⟨xi, hxi, entry⟩ := lower
have lower : ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels := ⟨xi, hxi, entry⟩
have higher : ∃ xi ∈ beta, ZFSet.pair z xi ∈ levels := ⟨xi, below hxi, entry⟩
simp only [lower, higher, and_true, not_true_eq_false, and_false, or_false]
· by_cases higher : ∃ xi ∈ beta, ZFSet.pair z xi ∈ levels
· simp only [lower, higher, and_true, not_true_eq_false, and_false, or_false,
not_false_eq_true, false_or]
· simp only [lower, higher, and_true, and_false, not_false_eq_true, false_or]
end PartitionPilot.Forcing
end
end OAI
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