Nuprl Lemma : fpf-join-sub
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f1,g1,f2,g2:a:A fp-> B[a]].
(f1 ⊕ f2 ⊆ g1 ⊕ g2) supposing (f2 ⊆ g2 and f1 ⊆ g1 and f2 || g1)
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],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fpf-sub: f ⊆ g,
all: ∀x:A. B[x],
implies: P ⇒ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
or: P ∨ Q,
guard: {T},
not: ¬A,
false: False,
fpf-compatible: f || g,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
fpf-join-dom,
assert_wf,
fpf-dom_wf,
fpf-join_wf,
top_wf,
subtype-fpf2,
fpf-sub_witness,
fpf-sub_wf,
fpf-compatible_wf,
fpf_wf,
deq_wf,
equal-wf-T-base,
bool_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
fpf-join-ap-sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
dependent_functionElimination,
hypothesis,
productElimination,
independent_functionElimination,
independent_pairFormation,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
unionElimination,
inlFormation,
inrFormation,
baseClosed,
equalityElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f1,g1,f2,g2:a:A fp-> B[a]].
(f1 \moplus{} f2 \msubseteq{} g1 \moplus{} g2) supposing (f2 \msubseteq{} g2 and f1 \msubseteq{} g1 and f2 || g1)
Date html generated:
2018_05_21-PM-09_22_28
Last ObjectModification:
2018_02_09-AM-10_18_43
Theory : finite!partial!functions
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