Nuprl Lemma : fpf-domain-join
∀[A:Type]
∀f,g:a:A fp-> Top. ∀eq:EqDecider(A). ∀x:A. ((x ∈ fpf-domain(f ⊕ g)) ⇐⇒ (x ∈ fpf-domain(f)) ∨ (x ∈ fpf-domain(g)))
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-domain: fpf-domain(f),
fpf: a:A fp-> B[a],
l_member: (x ∈ l),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
prop: ℙ
Lemmas referenced :
member-fpf-domain,
fpf-join_wf,
top_wf,
deq_wf,
fpf_wf,
istype-universe,
fpf-join-dom,
istype-assert,
fpf-dom_wf,
l_member_wf,
fpf-domain_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
sqequalRule,
lambdaEquality_alt,
hypothesis,
inhabitedIsType,
universeIsType,
instantiate,
universeEquality,
productElimination,
independent_functionElimination,
unionIsType,
independent_pairFormation,
promote_hyp,
unionElimination,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}[A:Type]
\mforall{}f,g:a:A fp-> Top. \mforall{}eq:EqDecider(A). \mforall{}x:A.
((x \mmember{} fpf-domain(f \moplus{} g)) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} fpf-domain(f)) \mvee{} (x \mmember{} fpf-domain(g)))
Date html generated:
2020_05_20-AM-09_02_35
Last ObjectModification:
2019_11_27-PM-02_36_29
Theory : finite!partial!functions
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