Nuprl Lemma : fpf-compatible_monotonic
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f1,g1,f2,g2:a:A fp-> B[a]].
(f1 || g1) supposing (f2 || g2 and g1 ⊆ g2 and f1 ⊆ f2)
Proof
Definitions occuring in Statement :
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],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fpf-compatible: f || g,
fpf-sub: f ⊆ g,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
guard: {T}
Lemmas referenced :
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
all_wf,
equal_wf,
fpf-ap_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
independent_pairFormation,
equalityElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
productEquality,
extract_by_obid,
isectElimination,
cumulativity,
applyEquality,
lambdaEquality,
functionExtensionality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
functionEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f1,g1,f2,g2:a:A fp-> B[a]].
(f1 || g1) supposing (f2 || g2 and g1 \msubseteq{} g2 and f1 \msubseteq{} f2)
Date html generated:
2018_05_21-PM-09_19_59
Last ObjectModification:
2018_02_09-AM-10_17_48
Theory : finite!partial!functions
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