Nuprl Lemma : fpf-compatible-triple
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[f,g,h:x:T fp-> Type].
({(g ⊆ h ⊕ f ⊕ g ∧ f ⊆ h ⊕ f ⊕ g) ∧ h ⊕ g ⊆ h ⊕ f ⊕ g ∧ h ⊕ f ⊆ h ⊕ f ⊕ g}) supposing (h || g and h || f and f || g)
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-compatible: f || g,
fpf-sub: f ⊆ g,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fpf-sub: f ⊆ g,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
fpf-compatible: f || g,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Lemmas referenced :
fpf-join-dom,
fpf-join_wf,
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
or_wf,
fpf-sub_witness,
fpf-compatible_wf,
fpf_wf,
deq_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
fpf-ap_wf,
fpf-join-ap-sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
universeEquality,
dependent_functionElimination,
hypothesis,
productElimination,
independent_functionElimination,
inrFormation,
applyEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
addLevel,
orFunctionality,
because_Cache,
independent_pairFormation,
inlFormation,
unionElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
dependent_pairFormation,
promote_hyp
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[f,g,h:x:T fp-> Type].
(\{(g \msubseteq{} h \moplus{} f \moplus{} g \mwedge{} f \msubseteq{} h \moplus{} f \moplus{} g) \mwedge{} h \moplus{} g \msubseteq{} h \moplus{} f \moplus{} g \mwedge{} h \moplus{} f \msubseteq{} h \moplus{} f \moplus{} g\}) supposing
(h || g and
h || f and
f || g)
Date html generated:
2018_05_21-PM-09_30_41
Last ObjectModification:
2018_02_09-AM-10_25_12
Theory : finite!partial!functions
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