Nuprl Lemma : fpf-join-is-empty
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[f,g:x:A fp-> Top]. (fpf-is-empty(f ⊕ g) ~ fpf-is-empty(f) ∧b fpf-is-empty(g))
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-is-empty: fpf-is-empty(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
band: p ∧b q,
uall: ∀[x:A]. B[x],
top: Top,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
guard: {T},
implies: P ⇒ Q,
all: ∀x:A. B[x],
sq_type: SQType(T),
fpf-dom: x ∈ dom(f),
pi1: fst(t),
fpf-join: f ⊕ g,
fpf-is-empty: fpf-is-empty(f),
fpf: a:A fp-> B[a],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
cons: [a / b],
btrue: tt,
band: p ∧b q,
eq_int: (i =z j),
bfalse: ff,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
so_apply: x[s1;s2;s3],
top: Top,
so_lambda: so_lambda3,
append: as @ bs,
or: P ∨ Q,
true: True,
assert: ↑b,
l_all: (∀x∈L.P[x]),
prop: ℙ,
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
eqof: eqof(d),
iff: P ⇐⇒ Q,
squash: ↓T,
ge: i ≥ j ,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
nat: ℕ,
nequal: a ≠ b ∈ T
Lemmas referenced :
deq_wf,
top_wf,
fpf_wf,
bool_subtype_base,
bool_wf,
subtype_base_sq,
length_of_cons_lemma,
deq_member_cons_lemma,
list_ind_cons_lemma,
product_subtype_list,
length_of_nil_lemma,
deq_member_nil_lemma,
list_ind_nil_lemma,
list-cases,
length_wf,
eq_int_wf,
btrue_wf,
filter_trivial,
int_seg_wf,
length-append,
append_wf,
filter_wf5,
l_member_wf,
bnot_wf,
bor_wf,
deq-member_wf,
eqtt_to_assert,
assert_of_eq_int,
iff_imp_equal_bool,
zero-add-sq,
equal_wf,
add_functionality_wrt_eq,
length_of_null_list,
non_neg_length,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
length_append,
iff_weakening_equal,
eqof_wf,
int_subtype_base,
length_wf_nat,
set_subtype_base,
le_wf,
iff_weakening_uiff,
assert_wf,
equal-wf-base,
iff_transitivity,
bool_cases,
band_wf,
bfalse_wf,
assert_of_band,
istype-assert,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
add-associates,
add-zero,
add-commutes
Rules used in proof :
universeEquality,
because_Cache,
isect_memberEquality,
lambdaEquality,
hypothesisEquality,
axiomSqEquality,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
sqequalRule,
productElimination,
independent_isectElimination,
hypothesis,
cumulativity,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
instantiate,
thin,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
hypothesis_subsumption,
promote_hyp,
voidEquality,
voidElimination,
unionElimination,
natural_numberEquality,
lambdaFormation,
Error :memTop,
addEquality,
lambdaEquality_alt,
lambdaFormation_alt,
universeIsType,
setElimination,
rename,
applyEquality,
setIsType,
closedConclusion,
inhabitedIsType,
equalityElimination,
independent_pairFormation,
imageElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
imageMemberEquality,
baseClosed,
equalityIstype,
sqequalBase,
productIsType,
intEquality,
productEquality,
isect_memberEquality_alt
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f,g:x:A fp-> Top].
(fpf-is-empty(f \moplus{} g) \msim{} fpf-is-empty(f) \mwedge{}\msubb{} fpf-is-empty(g))
Date html generated:
2020_05_20-AM-09_02_39
Last ObjectModification:
2020_01_25-AM-09_13_00
Theory : finite!partial!functions
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