Nuprl Lemma : priority-select-tt
∀[T:Type]
∀as:T List. ∀f,g:T ⟶ 𝔹.
(priority-select(f;g;as) = (inl tt) ∈ (𝔹?)
⇐⇒ (∃a∈as. (↑(f a)) ∧ (∀b:T. ((b ∈ as) ⇒ ¬↑(g b) supposing b < a)))) supposing
(no_repeats(T;as) and
sorted(as) and
(T ⊆r ℤ))
Proof
Definitions occuring in Statement :
priority-select: priority-select(f;g;as),
l_exists: (∃x∈L. P[x]),
no_repeats: no_repeats(T;l),
sorted: sorted(L),
l_member: (x ∈ l),
list: T List,
assert: ↑b,
btrue: tt,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
sorted: sorted(L),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
so_apply: x[s],
l_exists: (∃x∈L. P[x]),
cand: A c∧ B,
l_member: (x ∈ l),
nat: ℕ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
label: ...$L... t,
sq_type: SQType(T)
Lemmas referenced :
less_than'_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
decidable__lt,
length_wf,
int_seg_wf,
no_repeats_witness,
priority-select-property,
exists_wf,
assert_wf,
all_wf,
not_wf,
l_exists_wf,
l_member_wf,
less_than_wf,
equal-wf-T-base,
bool_wf,
unit_wf2,
priority-select_wf,
iff_wf,
no_repeats_wf,
sorted_wf,
subtype_rel_wf,
list_wf,
lelt_wf,
strict-sorted,
less_than_transitivity2,
nat_properties,
le_weakening2,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
select_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
axiomEquality,
hypothesis,
thin,
rename,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
productElimination,
independent_pairEquality,
voidElimination,
extract_by_obid,
isectElimination,
because_Cache,
independent_isectElimination,
setElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
cumulativity,
imageElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productEquality,
functionExtensionality,
functionEquality,
isectEquality,
setEquality,
addLevel,
impliesFunctionality,
unionEquality,
baseClosed,
universeEquality,
hyp_replacement,
Error :applyLambdaEquality,
dependent_set_memberEquality,
instantiate
Latex:
\mforall{}[T:Type]
\mforall{}as:T List. \mforall{}f,g:T {}\mrightarrow{} \mBbbB{}.
(priority-select(f;g;as) = (inl tt)
\mLeftarrow{}{}\mRightarrow{} (\mexists{}a\mmember{}as. (\muparrow{}(f a)) \mwedge{} (\mforall{}b:T. ((b \mmember{} as) {}\mRightarrow{} \mneg{}\muparrow{}(g b) supposing b < a)))) supposing
(no\_repeats(T;as) and
sorted(as) and
(T \msubseteq{}r \mBbbZ{}))
Date html generated:
2016_10_25-AM-10_49_42
Last ObjectModification:
2016_07_12-AM-06_58_40
Theory : general
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