Nuprl Lemma : fpf-join-sub2
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f1,g,f2:a:A fp-> B[a]]. (f1 ⊕ f2 ⊆ g) supposing (f2 ⊆ g and f1 ⊆ g)
Proof
Definitions occuring in Statement :
fpf-join: 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 :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
fpf-sub: f ⊆ g,
fpf-compatible: f || g,
top: Top,
cand: A c∧ B
Lemmas referenced :
fpf-sub_witness,
fpf-join_wf,
fpf-sub_wf,
fpf_wf,
deq_wf,
fpf-join-sub,
equal_wf,
squash_wf,
true_wf,
fpf-join-idempotent,
iff_weakening_equal,
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
because_Cache,
functionEquality,
universeEquality,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
hyp_replacement,
applyLambdaEquality,
productEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f1,g,f2:a:A fp-> B[a]].
(f1 \moplus{} f2 \msubseteq{} g) supposing (f2 \msubseteq{} g and f1 \msubseteq{} g)
Date html generated:
2018_05_21-PM-09_22_32
Last ObjectModification:
2018_02_09-AM-10_18_45
Theory : finite!partial!functions
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