Nuprl Lemma : fpf-join-single-property
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[a:A]. ∀[v:B[a]]. ∀[eq:EqDecider(A)]. ∀[b:A].
({(↑b ∈ dom(f)) ∧ (f ⊕ a : v(b) = f(b) ∈ B[b])}) supposing ((↑b ∈ dom(f ⊕ a : v)) and (¬(b = a ∈ A)))
Proof
Definitions occuring in Statement :
fpf-single: x : v,
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],
guard: {T},
so_apply: x[s],
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
or: P ∨ Q,
cand: A c∧ B,
top: Top,
uiff: uiff(P;Q),
uimplies: b supposing a,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
fpf-join-dom,
fpf-single_wf,
fpf-single-dom,
fpf-dom_wf,
subtype-fpf2,
top_wf,
assert_wf,
fpf-join_wf,
not_wf,
equal_wf,
deq_wf,
fpf_wf,
assert_witness,
fpf-join-ap-sq,
bool_wf,
eqtt_to_assert,
fpf-ap_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
dependent_functionElimination,
instantiate,
hypothesis,
productElimination,
independent_functionElimination,
unionElimination,
independent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
lambdaFormation,
functionEquality,
universeEquality,
isect_memberFormation,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
dependent_pairFormation,
promote_hyp
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[a:A]. \mforall{}[v:B[a]]. \mforall{}[eq:EqDecider(A)]. \mforall{}[b:A].
(\{(\muparrow{}b \mmember{} dom(f)) \mwedge{} (f \moplus{} a : v(b) = f(b))\}) supposing ((\muparrow{}b \mmember{} dom(f \moplus{} a : v)) and (\mneg{}(b = a)))
Date html generated:
2018_05_21-PM-09_29_08
Last ObjectModification:
2018_02_09-AM-10_24_11
Theory : finite!partial!functions
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