Nuprl Lemma : fpf-join-ap-left
∀[A:Type]. ∀[B,C:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f:a:A fp-> B[a]]. ∀[g:a:A fp-> C[a]]. ∀[x:A].
f ⊕ g(x) = f(x) ∈ B[x] supposing ↑x ∈ dom(f)
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fpf-ap: f(x),
fpf-join: f ⊕ g,
pi2: snd(t),
fpf-cap: f(x)?z,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
fpf_wf,
deq_wf,
bool_wf,
fpf-ap_wf,
equal-wf-T-base,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
applyEquality,
lambdaEquality,
functionExtensionality,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
independent_functionElimination,
unionElimination,
equalityElimination,
productElimination,
dependent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B,C:A {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[g:a:A fp-> C[a]]. \mforall{}[x:A].
f \moplus{} g(x) = f(x) supposing \muparrow{}x \mmember{} dom(f)
Date html generated:
2018_05_21-PM-09_21_48
Last ObjectModification:
2018_02_09-AM-10_18_27
Theory : finite!partial!functions
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