Nuprl Lemma : subtype-fpf-cap
∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[f,g:x:X fp-> Type]. {∀[x:X]. (f(x)?Top ⊆r g(x)?Top)} supposing g ⊆ f
Proof
Definitions occuring in Statement :
fpf-sub: f ⊆ g,
fpf-cap: f(x)?z,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
top: Top,
guard: {T},
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
fpf-sub: f ⊆ g,
implies: P ⇒ Q,
cand: A c∧ B,
not: ¬A,
false: False,
guard: {T},
fpf-cap: f(x)?z,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
prop: ℙ
Lemmas referenced :
fpf-dom_wf,
subtype-fpf2,
top_wf,
bool_wf,
fpf-ap_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
fpf-cap_wf,
fpf-sub_wf,
fpf_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
applyEquality,
instantiate,
because_Cache,
sqequalRule,
lambdaEquality,
universeEquality,
hypothesis,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
independent_functionElimination,
productElimination,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
baseClosed,
isect_memberFormation,
unionElimination,
equalityElimination,
axiomEquality
Latex:
\mforall{}[X:Type]. \mforall{}[eq:EqDecider(X)]. \mforall{}[f,g:x:X fp-> Type].
\{\mforall{}[x:X]. (f(x)?Top \msubseteq{}r g(x)?Top)\} supposing g \msubseteq{} f
Date html generated:
2018_05_21-PM-09_19_17
Last ObjectModification:
2018_02_09-AM-10_17_30
Theory : finite!partial!functions
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