Nuprl Lemma : fpf-cap-single-join
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[x:A]. ∀[v,z,f:Top]. (x : v ⊕ f(x)?z ~ v)
Proof
Definitions occuring in Statement :
fpf-single: x : v,
fpf-join: f ⊕ g,
fpf-cap: f(x)?z,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
top: Top,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
fpf-single: x : v,
fpf-join: f ⊕ g,
fpf-cap: f(x)?z,
pi1: fst(t),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
deq: EqDecider(T),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
eqof: eqof(d),
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fpf_ap_pair_lemma,
list_ind_cons_lemma,
list_ind_nil_lemma,
deq_member_cons_lemma,
deq_member_nil_lemma,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
deq-member_wf,
filter_wf5,
pi1_wf_top,
list_wf,
l_member_wf,
bnot_wf,
bor_wf,
bfalse_wf,
assert-deq-member,
eqof_wf,
top_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
applyEquality,
setElimination,
rename,
hypothesisEquality,
lambdaFormation,
unionElimination,
equalityElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
lambdaEquality,
setEquality,
universeEquality,
isect_memberFormation,
sqequalAxiom
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[x:A]. \mforall{}[v,z,f:Top]. (x : v \moplus{} f(x)?z \msim{} v)
Date html generated:
2018_05_21-PM-09_25_01
Last ObjectModification:
2018_02_09-AM-10_20_48
Theory : finite!partial!functions
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