Nuprl Lemma : priority-select-property
∀[T:Type]
∀as:T List. ∀f,g:T ⟶ 𝔹.
((priority-select(f;g;as) = (inl tt) ∈ (𝔹?) ⇐⇒ ∃i:ℕ||as||. ((↑(f as[i])) ∧ (∀j:ℕi. (¬↑(g as[j])))))
∧ (priority-select(f;g;as) = (inl ff) ∈ (𝔹?) ⇐⇒ ∃i:ℕ||as||. ((↑(g as[i])) ∧ (∀j:ℕi + 1. (¬↑(f as[j])))))
∧ (priority-select(f;g;as) = (inr ⋅ ) ∈ (𝔹?) ⇐⇒ ∀i:ℕ||as||. ((¬↑(f as[i])) ∧ (¬↑(g as[i])))))
Proof
Definitions occuring in Statement :
priority-select: priority-select(f;g;as),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
assert: ↑b,
bfalse: ff,
btrue: tt,
bool: 𝔹,
it: ⋅,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inr: inr x ,
inl: inl x,
union: left + right,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
select: L[n],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_apply: x[s],
squash: ↓T,
less_than: a < b,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
guard: {T},
uimplies: b supposing a,
int_seg: {i..j-},
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
isl: isl(x),
cand: A c∧ B,
priority-select: priority-select(f;g;as),
uiff: uiff(P;Q),
btrue: tt,
unit: Unit,
bool: 𝔹,
exposed-bfalse: exposed-bfalse,
bfalse: ff,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
colength: colength(L),
less_than': less_than'(a;b),
le: A ≤ B,
cons: [a / b],
ge: i ≥ j ,
nat: ℕ,
true: True,
nat_plus: ℕ+,
assert: ↑b,
outl: outl(x),
subtract: n - m
Lemmas referenced :
istype-universe,
list_wf,
istype-assert,
priority-select_wf,
unit_wf2,
length_of_cons_lemma,
istype-base,
stuck-spread,
length_of_nil_lemma,
int_term_value_add_lemma,
itermAdd_wf,
not_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
int_seg_properties,
select_wf,
assert_wf,
length_wf,
int_seg_wf,
exists_wf,
equal-wf-T-base,
iff_wf,
bool_wf,
all_wf,
list_induction,
it_wf,
btrue_neq_bfalse,
btrue_wf,
bfalse_wf,
list_accum_nil_lemma,
assert_of_bnot,
eqff_to_assert,
uiff_transitivity,
eqtt_to_assert,
list_accum_cons_lemma,
bnot_wf,
istype-nat,
le_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
decidable__equal_int,
spread_cons_lemma,
int_subtype_base,
set_subtype_base,
int_formula_prop_eq_lemma,
intformeq_wf,
subtype_base_sq,
subtract-1-ge-0,
istype-le,
istype-false,
colength_wf_list,
colength-cons-not-zero,
product_subtype_list,
list-cases,
istype-less_than,
ge_wf,
nat_properties,
non_neg_length,
cons_wf,
select-cons-hd,
false_wf,
add-is-int-iff,
nat_plus_properties,
length_wf_nat,
add_nat_plus,
equal_wf,
outl_wf,
list_accum_wf,
ifthenelse_wf,
isl_wf,
zero-add,
add-commutes,
add-swap,
add-associates,
select-cons-tl,
add-member-int_seg2,
select_cons_tl,
assert_functionality_wrt_uiff,
iff_weakening_uiff,
add-subtract-cancel,
subtract-add-cancel,
select-nthtl0,
int_seg_subtype_nat,
subtype_rel_list,
top_wf,
select_cons_tl_sq2
Rules used in proof :
universeEquality,
instantiate,
equalitySymmetry,
sqequalBase,
unionIsType,
equalityIstype,
productIsType,
functionIsType,
inhabitedIsType,
addEquality,
baseClosed,
imageElimination,
universeIsType,
independent_pairFormation,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
unionElimination,
dependent_functionElimination,
productElimination,
independent_isectElimination,
rename,
setElimination,
applyEquality,
natural_numberEquality,
closedConclusion,
productEquality,
because_Cache,
hypothesis,
functionEquality,
lambdaEquality_alt,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
inrEquality_alt,
applyLambdaEquality,
equalityTransitivity,
dependent_set_memberEquality_alt,
equalityElimination,
intEquality,
baseApply,
hypothesis_subsumption,
promote_hyp,
functionIsTypeImplies,
axiomSqEquality,
intWeakElimination,
inlEquality_alt,
pointwiseFunctionality,
imageMemberEquality,
unionEquality,
hyp_replacement,
cumulativity,
Error :memTop,
independent_pairEquality
Latex:
\mforall{}[T:Type]
\mforall{}as:T List. \mforall{}f,g:T {}\mrightarrow{} \mBbbB{}.
((priority-select(f;g;as) = (inl tt) \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}||as||. ((\muparrow{}(f as[i])) \mwedge{} (\mforall{}j:\mBbbN{}i. (\mneg{}\muparrow{}(g as[j])))))
\mwedge{} (priority-select(f;g;as) = (inl ff)
\mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}||as||. ((\muparrow{}(g as[i])) \mwedge{} (\mforall{}j:\mBbbN{}i + 1. (\mneg{}\muparrow{}(f as[j])))))
\mwedge{} (priority-select(f;g;as) = (inr \mcdot{} ) \mLeftarrow{}{}\mRightarrow{} \mforall{}i:\mBbbN{}||as||. ((\mneg{}\muparrow{}(f as[i])) \mwedge{} (\mneg{}\muparrow{}(g as[i])))))
Date html generated:
2020_05_20-AM-08_07_00
Last ObjectModification:
2020_01_31-PM-02_51_23
Theory : general
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