Nuprl Lemma : l_before_filter_subtype
∀[T:Type]
∀l:T List. ∀P:T ⟶ 𝔹. ∀x,y:T. ((↑(P x)) ⇒ (↑(P y)) ⇒ (x before y ∈ filter(P;l) ⇐⇒ x before y ∈ filter(P;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,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
l_before: x before y ∈ l,
sublist: L1 ⊆ L2,
member: t ∈ T,
top: Top,
exists: ∃x:A. B[x],
cand: A c∧ B,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
ge: i ≥ j ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
istype: istype(T),
nat: ℕ,
less_than: a < b,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
true: True,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
sq_type: SQType(T),
select: L[n],
cons: [a / b],
label: ...$L... t,
subtract: n - m
Lemmas referenced :
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
equal_wf,
squash_wf,
true_wf,
select_wf,
filter_wf5,
non_neg_length,
int_seg_properties,
decidable__le,
le_wf,
less_than_wf,
length_wf_nat,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_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_not_lemma,
int_formula_prop_wf,
decidable__lt,
length_wf,
intformless_wf,
int_formula_prop_less_lemma,
iff_weakening_equal,
int_seg_wf,
increasing_wf,
istype-false,
cons_wf,
nil_wf,
itermAdd_wf,
int_term_value_add_lemma,
l_before_wf,
assert_wf,
filter_type,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
select-cons-hd,
int_seg_subtype_special,
int_seg_cases,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
Error :isect_memberEquality_alt,
voidElimination,
hypothesis,
productElimination,
Error :dependent_pairFormation_alt,
hypothesisEquality,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
Error :inhabitedIsType,
universeEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
natural_numberEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
unionElimination,
closedConclusion,
setEquality,
Error :setIsType,
applyLambdaEquality,
approximateComputation,
independent_functionElimination,
int_eqEquality,
imageMemberEquality,
baseClosed,
functionExtensionality,
Error :functionIsType,
Error :equalityIsType1,
addEquality,
instantiate,
cumulativity,
intEquality,
hypothesis_subsumption
Latex:
\mforall{}[T:Type]
\mforall{}l:T List. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}x,y:T.
((\muparrow{}(P x)) {}\mRightarrow{} (\muparrow{}(P y)) {}\mRightarrow{} (x before y \mmember{} filter(P;l) \mLeftarrow{}{}\mRightarrow{} x before y \mmember{} filter(P;l)))
Date html generated:
2019_06_20-PM-01_25_39
Last ObjectModification:
2018_10_15-PM-03_20_51
Theory : list_1
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