Nuprl Lemma : fpf-compatible-singles-iff
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[x,y:A]. ∀[v:B[x]]. ∀[u:B[y]].
uiff(x : v || y : u;v = u ∈ B[x] supposing x = y ∈ A)
Proof
Definitions occuring in Statement :
fpf-single: x : v,
fpf-compatible: f || g,
deq: EqDecider(T),
uiff: uiff(P;Q),
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,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
fpf-compatible: f || g,
all: ∀x:A. B[x],
top: Top,
implies: P ⇒ Q,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B
Lemmas referenced :
fpf_ap_single_lemma,
fpf-single-dom,
equal_wf,
fpf-compatible_wf,
fpf-single_wf,
fpf-compatible-singles,
assert_wf,
fpf-dom_wf,
top_wf,
isect_wf,
subtype_rel-equal,
and_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
hypothesisEquality,
independent_functionElimination,
isectElimination,
because_Cache,
productElimination,
independent_isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
applyEquality,
instantiate,
lambdaFormation,
productEquality,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
independent_pairEquality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[x,y:A]. \mforall{}[v:B[x]]. \mforall{}[u:B[y]].
uiff(x : v || y : u;v = u supposing x = y)
Date html generated:
2018_05_21-PM-09_29_23
Last ObjectModification:
2018_05_19-PM-04_38_34
Theory : finite!partial!functions
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