Nuprl Lemma : fpf-dom-type
∀[X,Y:Type]. ∀[eq:EqDecider(Y)]. ∀[f:x:X fp-> Top]. ∀[x:Y]. (x ∈ X) supposing ((↑x ∈ dom(f)) and strong-subtype(X;Y))
Proof
Definitions occuring in Statement :
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
strong-subtype: strong-subtype(A;B),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
strong-subtype-l_member-type,
fpf-domain_wf,
member-fpf-domain,
assert_wf,
fpf-dom_wf,
subtype-fpf3,
top_wf,
strong-subtype_wf,
fpf_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
applyEquality,
sqequalRule,
productElimination,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
lambdaFormation,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[X,Y:Type]. \mforall{}[eq:EqDecider(Y)]. \mforall{}[f:x:X fp-> Top]. \mforall{}[x:Y].
(x \mmember{} X) supposing ((\muparrow{}x \mmember{} dom(f)) and strong-subtype(X;Y))
Date html generated:
2018_05_21-PM-09_17_33
Last ObjectModification:
2018_02_09-AM-10_16_34
Theory : finite!partial!functions
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