Nuprl Lemma : fpf-sub_witness
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f,g:a:A fp-> B[a]]. (f ⊆ g ⇒ (λx,y. <Ax, Ax> ∈ f ⊆ g))
Proof
Definitions occuring in Statement :
fpf-sub: f ⊆ g,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q,
member: t ∈ T,
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
universe: Type,
axiom: Ax
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
fpf-sub: f ⊆ g,
all: ∀x:A. B[x],
cand: A c∧ B,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top,
prop: ℙ,
guard: {T}
Lemmas referenced :
assert_witness,
fpf-dom_wf,
subtype-fpf2,
top_wf,
assert_wf,
fpf-sub_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
lambdaEquality,
independent_pairEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
independent_functionElimination,
axiomEquality,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
productElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f,g:a:A fp-> B[a]]. (f \msubseteq{} g {}\mRightarrow{} (\mlambda{}x,y. <Ax, Ax> \mmember{} f \000C\msubseteq{} g))
Date html generated:
2018_05_21-PM-09_18_41
Last ObjectModification:
2018_02_09-AM-10_17_15
Theory : finite!partial!functions
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