Nuprl Lemma : fset-add-union
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s1,s2:fset(T)]. ∀[x:T]. (fset-add(eq;x;s1) ⋃ s2 = [x / s1 ⋃ s2] ∈ fset(T))
Proof
Definitions occuring in Statement :
fset-add: fset-add(eq;x;s),
fset-union: x ⋃ y,
fset: fset(T),
cons: [a / b],
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False
Lemmas referenced :
fset-add-as-cons,
equal_wf,
squash_wf,
true_wf,
fset-union_wf,
fset-add_wf,
iff_weakening_equal,
fset-extensionality,
decidable__fset-member,
fset-member_wf,
or_wf,
member-fset-add,
uiff_wf,
fset-member_witness,
member-fset-union,
fset_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
cumulativity,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
inrFormation,
inlFormation,
voidElimination,
addLevel,
orFunctionality,
rename,
independent_pairEquality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s1,s2:fset(T)]. \mforall{}[x:T]. (fset-add(eq;x;s1) \mcup{} s2 = [x / s1 \mcup{} s2])
Date html generated:
2017_04_17-AM-09_19_50
Last ObjectModification:
2017_02_27-PM-05_23_02
Theory : finite!sets
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