Nuprl Lemma : fset-constrained-image-union
∀[T,A:Type]. ∀[eqt:EqDecider(T)]. ∀[eqa:EqDecider(A)]. ∀[f:T ⟶ A]. ∀[P:A ⟶ 𝔹]. ∀[x,y:fset(T)].
(f"(x ⋃ y) s.t. P = f"(x) s.t. P ⋃ f"(y) s.t. P ∈ fset(A))
Proof
Definitions occuring in Statement :
fset-constrained-image: f"(s) s.t. P,
fset-union: x ⋃ y,
fset: fset(T),
deq: EqDecider(T),
bool: 𝔹,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
guard: {T},
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
exists: ∃x:A. B[x],
cand: A c∧ B,
false: False
Lemmas referenced :
fset-extensionality,
fset-constrained-image_wf,
fset-union_wf,
fset-member_witness,
fset-member_wf,
or_wf,
member-fset-union,
uiff_wf,
fset_wf,
bool_wf,
deq_wf,
member-fset-constrained-image-iff,
decidable__fset-member,
squash_wf,
exists_wf,
equal_wf,
assert_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
cumulativity,
functionExtensionality,
applyEquality,
hypothesis,
productElimination,
independent_isectElimination,
independent_pairFormation,
because_Cache,
independent_functionElimination,
rename,
addLevel,
dependent_functionElimination,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
functionEquality,
universeEquality,
unionElimination,
inlFormation,
inrFormation,
lambdaFormation,
lambdaEquality,
productEquality,
promote_hyp,
imageElimination,
dependent_pairFormation,
imageMemberEquality,
baseClosed,
voidElimination
Latex:
\mforall{}[T,A:Type]. \mforall{}[eqt:EqDecider(T)]. \mforall{}[eqa:EqDecider(A)]. \mforall{}[f:T {}\mrightarrow{} A]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[x,y:fset(T)].
(f"(x \mcup{} y) s.t. P = f"(x) s.t. P \mcup{} f"(y) s.t. P)
Date html generated:
2017_04_17-AM-09_21_20
Last ObjectModification:
2017_02_27-PM-05_24_07
Theory : finite!sets
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