Nuprl Lemma : subtype-fpf-cap-top2
∀[X,T:Type]. ∀[eq:EqDecider(X)]. ∀[g:x:X fp-> Type]. ∀[x:X]. T ⊆r g(x)?Top supposing (↑x ∈ dom(g)) ⇒ (T ⊆r g(x))
Proof
Definitions occuring in Statement :
fpf-cap: f(x)?z,
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
top: Top,
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fpf-cap: f(x)?z,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
assert_wf,
fpf-dom_wf,
subtype_rel_wf,
fpf-ap_wf,
fpf_wf,
deq_wf,
bool_wf,
equal-wf-T-base,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
axiomEquality,
hypothesis,
functionEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
applyEquality,
because_Cache,
instantiate,
lambdaEquality,
universeEquality,
independent_isectElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
baseClosed,
voidElimination,
voidEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
dependent_functionElimination
Latex:
\mforall{}[X,T:Type]. \mforall{}[eq:EqDecider(X)]. \mforall{}[g:x:X fp-> Type]. \mforall{}[x:X].
T \msubseteq{}r g(x)?Top supposing (\muparrow{}x \mmember{} dom(g)) {}\mRightarrow{} (T \msubseteq{}r g(x))
Date html generated:
2018_05_21-PM-09_19_23
Last ObjectModification:
2018_02_09-AM-10_17_33
Theory : finite!partial!functions
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