Nuprl Lemma : l_before_filter
∀[T:Type]. ∀l:T List. ∀P:T ⟶ 𝔹. ∀x,y:T. (x before y ∈ filter(P;l) ⇐⇒ (↑(P x)) ∧ (↑(P y)) ∧ x before y ∈ l)
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
l_before: x before y ∈ l,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
rev_implies: P ⇐ Q,
top: Top,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False
Lemmas referenced :
assert_witness,
sublist_wf,
cons_wf,
nil_wf,
assert_wf,
l_all_wf_nil,
l_all_nil,
select_wf,
length_of_nil_lemma,
int_seg_properties,
decidable__le,
full-omega-unsat,
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,
length_wf,
l_all_cons,
l_all_wf,
l_member_wf,
sublist_filter,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
subtype_rel_self,
set_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
independent_pairFormation,
lambdaFormation,
introduction,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
extract_by_obid,
isectElimination,
applyEquality,
hypothesisEquality,
independent_functionElimination,
productEquality,
lambdaEquality,
voidElimination,
voidEquality,
isect_memberEquality,
dependent_functionElimination,
functionExtensionality,
cumulativity,
because_Cache,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
promote_hyp,
setEquality,
Error :inhabitedIsType,
Error :universeIsType,
Error :functionIsType,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}l:T List. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}x,y:T. (x before y \mmember{} filter(P;l) \mLeftarrow{}{}\mRightarrow{} (\muparrow{}(P x)) \mwedge{} (\muparrow{}(P y)) \mwedge{} x before y \mmember{} l)
Date html generated:
2019_06_20-PM-01_25_34
Last ObjectModification:
2018_09_26-PM-05_30_39
Theory : list_1
Home
Index