Nuprl Lemma : fpf-sub-join
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f,g:a:A fp-> B[a]]. {f ⊆ f ⊕ g ∧ g ⊆ f ⊕ g} supposing 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},
so_apply: x[s],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
top: Top,
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
fpf-sub-join-left,
subtype-fpf2,
top_wf,
fpf-sub-join-right,
fpf-sub_witness,
fpf-join_wf,
fpf-compatible_wf,
fpf_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
because_Cache,
hypothesis,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
productElimination,
independent_pairEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f,g:a:A fp-> B[a]].
\{f \msubseteq{} f \moplus{} g \mwedge{} g \msubseteq{} f \moplus{} g\} supposing f || g
Date html generated:
2018_05_21-PM-09_22_24
Last ObjectModification:
2018_02_09-AM-10_18_42
Theory : finite!partial!functions
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