Nuprl Lemma : fpf-cap-join-subtype
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[f,g:a:A fp-> Type]. ∀[a:A]. (f ⊕ g(a)?Top ⊆r f(a)?Top)
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-cap: f(x)?z,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
top: Top,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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-cap: f(x)?z,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False
Lemmas referenced :
fpf_wf,
deq_wf,
fpf-join-cap,
subtype-fpf2,
top_wf,
fpf-dom_wf,
bool_wf,
eqtt_to_assert,
subtype_rel_self,
fpf-ap_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
axiomEquality,
hypothesis,
hypothesisEquality,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
because_Cache,
instantiate,
extract_by_obid,
cumulativity,
lambdaEquality,
universeEquality,
applyEquality,
independent_isectElimination,
lambdaFormation,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f,g:a:A fp-> Type]. \mforall{}[a:A]. (f \moplus{} g(a)?Top \msubseteq{}r f(a)?Top)
Date html generated:
2018_05_21-PM-09_29_54
Last ObjectModification:
2018_02_09-AM-10_24_33
Theory : finite!partial!functions
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