Nuprl Lemma : priority-select-inr
∀[T:Type]. ∀[as:T List]. ∀[f,g:T ⟶ 𝔹]. uiff(priority-select(f;g;as) = (inr ⋅ ) ∈ (𝔹?);(∀a∈as.(¬↑(f a)) ∧ (¬↑(g a))))
Proof
Definitions occuring in Statement :
priority-select: priority-select(f;g;as),
l_all: (∀x∈L.P[x]),
list: T List,
assert: ↑b,
bool: 𝔹,
it: ⋅,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inr: inr x ,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
l_all: (∀x∈L.P[x]),
not: ¬A,
false: False,
int_seg: {i..j-},
rev_implies: P ⇐ Q,
guard: {T},
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,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
priority-select-property,
assert_wf,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_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_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
equal-wf-T-base,
bool_wf,
unit_wf2,
priority-select_wf,
l_all_wf2,
not_wf,
l_member_wf,
uiff_wf,
all_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
productElimination,
addLevel,
independent_pairFormation,
independent_isectElimination,
independent_functionElimination,
sqequalRule,
lambdaEquality,
independent_pairEquality,
voidElimination,
applyEquality,
functionExtensionality,
cumulativity,
because_Cache,
setElimination,
rename,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
imageElimination,
universeEquality,
instantiate,
unionEquality,
baseClosed,
productEquality,
setEquality,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[as:T List]. \mforall{}[f,g:T {}\mrightarrow{} \mBbbB{}].
uiff(priority-select(f;g;as) = (inr \mcdot{} );(\mforall{}a\mmember{}as.(\mneg{}\muparrow{}(f a)) \mwedge{} (\mneg{}\muparrow{}(g a))))
Date html generated:
2018_05_21-PM-06_49_18
Last ObjectModification:
2017_07_26-PM-04_56_50
Theory : general
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