Nuprl Lemma : pairs-fpf_property
∀[A,B:Type].
∀eq1:EqDecider(A). ∀eq2:EqDecider(B). ∀L:(A × B) List.
(no_repeats(A;fpf-domain(fpf(L)))
∧ (∀a:A. ((a ∈ fpf-domain(fpf(L))) ⇐⇒ ∃b:B. (<a, b> ∈ L)))
∧ ∀a∈dom(fpf(L)). l=fpf(L)(a) ⇒ no_repeats(B;l) ∧ (∀b:B. ((b ∈ l) ⇐⇒ (<a, b> ∈ L))))
Proof
Definitions occuring in Statement :
pairs-fpf: fpf(L),
fpf-all: ∀x∈dom(f). v=f(x) ⇒ P[x; v],
fpf-domain: fpf-domain(f),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
pair: <a, b>,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
pairs-fpf: fpf(L),
fpf-domain: fpf-domain(f),
pi1: fst(t),
top: Top,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
squash: ↓T,
pi2: snd(t),
true: True,
fpf-all: ∀x∈dom(f). v=f(x) ⇒ P[x; v],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
fpf-ap: f(x),
eqof: eqof(d),
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
not: ¬A,
false: False,
or: P ∨ Q,
guard: {T},
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
list_wf,
deq_wf,
remove-repeats_property,
map_wf,
pi1_wf_top,
member_map,
l_member_wf,
remove-repeats_wf,
exists_wf,
equal_wf,
pi2_wf,
squash_wf,
true_wf,
assert_wf,
fpf-dom_wf,
pairs-fpf_wf,
subtype-fpf2,
top_wf,
list_induction,
no_repeats_wf,
reduce_wf,
ifthenelse_wf,
insert_wf,
nil_wf,
reduce_nil_lemma,
no_repeats_nil,
reduce_cons_lemma,
bool_wf,
equal-wf-T-base,
no_repeats-insert,
bnot_wf,
not_wf,
eqof_wf,
uiff_transitivity,
eqtt_to_assert,
safe-assert-deq,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
all_wf,
iff_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
or_wf,
and_wf,
member-insert,
subtype_rel_product,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
cons_member,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
cumulativity,
hypothesisEquality,
universeEquality,
sqequalRule,
dependent_functionElimination,
lambdaEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
allFunctionality,
independent_functionElimination,
promote_hyp,
dependent_pairFormation,
hyp_replacement,
equalitySymmetry,
applyEquality,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
because_Cache,
independent_isectElimination,
setElimination,
rename,
unionElimination,
equalityElimination,
addLevel,
impliesFunctionality,
levelHypothesis,
inlFormation,
inrFormation,
dependent_set_memberEquality,
applyLambdaEquality,
orFunctionality,
instantiate
Latex:
\mforall{}[A,B:Type].
\mforall{}eq1:EqDecider(A). \mforall{}eq2:EqDecider(B). \mforall{}L:(A \mtimes{} B) List.
(no\_repeats(A;fpf-domain(fpf(L)))
\mwedge{} (\mforall{}a:A. ((a \mmember{} fpf-domain(fpf(L))) \mLeftarrow{}{}\mRightarrow{} \mexists{}b:B. (<a, b> \mmember{} L)))
\mwedge{} \mforall{}a\mmember{}dom(fpf(L)). l=fpf(L)(a) {}\mRightarrow{} no\_repeats(B;l) \mwedge{} (\mforall{}b:B. ((b \mmember{} l) \mLeftarrow{}{}\mRightarrow{} (<a, b> \mmember{} L))))
Date html generated:
2018_05_21-PM-09_32_00
Last ObjectModification:
2018_02_09-AM-10_26_47
Theory : finite!partial!functions
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