Nuprl Lemma : fpf-restrict-compatible
∀[A:Type]. ∀[P:A ⟶ 𝔹]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f,g:x:A fp-> B[x]].
fpf-restrict(f;P) || g supposing f || g
Proof
Definitions occuring in Statement :
fpf-restrict: fpf-restrict(f;P),
fpf-compatible: f || g,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
bool: 𝔹,
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,
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
guard: {T},
prop: ℙ,
subtype_rel: A ⊆r B
Lemmas referenced :
ap_fpf_restrict_lemma,
fpf-restrict-dom,
assert_wf,
fpf-dom_wf,
fpf-restrict_wf2,
top_wf,
subtype-fpf2,
all_wf,
equal_wf,
fpf-ap_wf,
fpf_wf,
deq_wf,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isect_memberFormation,
lambdaFormation,
productElimination,
hypothesisEquality,
independent_functionElimination,
isectElimination,
because_Cache,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
independent_isectElimination,
independent_pairFormation,
productEquality,
axiomEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f,g:x:A fp-> B[x]].
fpf-restrict(f;P) || g supposing f || g
Date html generated:
2018_05_21-PM-09_31_30
Last ObjectModification:
2018_02_09-AM-10_25_52
Theory : finite!partial!functions
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