Nuprl Lemma : fpf-vals-nil
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[P:A ⟶ 𝔹]. ∀[f:x:A fp-> B[x]]. ∀[a:A].
(fpf-vals(eq;P;f) ~ []) supposing ((∀b:A. (↑(P b) ⇐⇒ b = a ∈ A)) and (¬↑a ∈ dom(f)))
Proof
Definitions occuring in Statement :
fpf-vals: fpf-vals(eq;P;f),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
nil: [],
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
fpf-vals: fpf-vals(eq;P;f),
let: let,
fpf: a:A fp-> B[a],
pi1: fst(t),
pi2: snd(t),
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
not: ¬A,
false: False,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
fpf-dom: x ∈ dom(f),
cons: [a / b]
Lemmas referenced :
all_wf,
iff_wf,
assert_wf,
equal_wf,
not_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
fpf_wf,
bool_wf,
deq_wf,
deq-member_wf,
equal-wf-T-base,
l_member_wf,
bnot_wf,
cons_wf,
nil_wf,
iff_transitivity,
iff_weakening_uiff,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
assert_of_bnot,
list_wf,
nil_member,
false_wf,
filter_nil_lemma,
no_repeats_wf,
cons_member,
filter_cons_lemma,
no_repeats_cons,
uiff_transitivity,
or_wf,
list_induction,
filter_wf5,
subtype_rel_dep_function,
subtype_rel_self,
set_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
and_wf,
remove-repeats_wf,
remove-repeats-no_repeats,
remove-repeats_property,
zip_nil_lemma,
list-cases,
equal-wf-base,
product_subtype_list,
null_nil_lemma,
btrue_wf,
null_wf3,
subtype_rel_list,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
because_Cache,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
universeEquality,
isect_memberFormation,
sqequalAxiom,
equalityTransitivity,
equalitySymmetry,
baseClosed,
unionElimination,
equalityElimination,
independent_functionElimination,
independent_pairFormation,
dependent_functionElimination,
impliesFunctionality,
hyp_replacement,
applyLambdaEquality,
promote_hyp,
setEquality,
setElimination,
rename,
dependent_pairFormation,
instantiate,
dependent_set_memberEquality,
inrFormation,
inlFormation,
hypothesis_subsumption
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:x:A fp-> B[x]]. \mforall{}[a:A].
(fpf-vals(eq;P;f) \msim{} []) supposing ((\mforall{}b:A. (\muparrow{}(P b) \mLeftarrow{}{}\mRightarrow{} b = a)) and (\mneg{}\muparrow{}a \mmember{} dom(f)))
Date html generated:
2018_05_21-PM-09_26_18
Last ObjectModification:
2018_02_09-AM-10_21_46
Theory : finite!partial!functions
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