Nuprl Lemma : fpf-sub-join-left
∀[A:Type]. ∀[B1,B2:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f:a:A fp-> B1[a]]. ∀[g:a:A fp-> Top]. f ⊆ f ⊕ g
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-sub: f ⊆ g,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fpf-sub: f ⊆ g,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
prop: ℙ,
squash: ↓T,
true: True,
guard: {T}
Lemmas referenced :
fpf-join-dom,
top_wf,
subtype-fpf2,
assert_wf,
fpf-dom_wf,
equal_wf,
squash_wf,
true_wf,
fpf-ap_wf,
fpf-join-ap-left,
iff_weakening_equal,
assert_witness,
fpf-join_wf,
fpf_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
hypothesis,
cumulativity,
dependent_functionElimination,
applyEquality,
functionExtensionality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
productElimination,
independent_functionElimination,
inlFormation,
independent_pairFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairEquality,
axiomEquality,
functionEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B1,B2:A {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f:a:A fp-> B1[a]]. \mforall{}[g:a:A fp-> Top].
f \msubseteq{} f \moplus{} g
Date html generated:
2018_05_21-PM-09_22_11
Last ObjectModification:
2018_02_09-AM-10_18_35
Theory : finite!partial!functions
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