Nuprl Lemma : fpf-compatible-single2
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[x:A]. ∀[v:B[x]]. x : v || f supposing ¬↑x ∈ dom(f)
Proof
Definitions occuring in Statement :
fpf-single: x : v,
fpf-compatible: f || g,
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],
not: ¬A,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
not: ¬A,
implies: P ⇒ Q,
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
fpf-compatible: f || g
Lemmas referenced :
fpf-compatible-symmetry,
fpf-single_wf,
iff_weakening_uiff,
fpf-compatible_wf,
isect_wf,
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
equal_wf,
fpf-ap_wf,
fpf-compatible-single-iff,
not_wf,
fpf_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
instantiate,
hypothesis,
independent_isectElimination,
cumulativity,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
independent_functionElimination,
productElimination,
dependent_functionElimination,
axiomEquality,
productEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[x:A]. \mforall{}[v:B[x]].
x : v || f supposing \mneg{}\muparrow{}x \mmember{} dom(f)
Date html generated:
2018_05_21-PM-09_29_20
Last ObjectModification:
2018_02_09-AM-10_24_18
Theory : finite!partial!functions
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