Nuprl Lemma : cons_filter2
∀[T:Type]. ∀[x:T]. ∀[L:T List]. ∀[P:ℕ||L|| + 1 ⟶ 𝔹].
(filter2(P;[x / L]) = if P 0 then [x / filter2(λi.(P (i + 1));L)] else filter2(λi.(P (i + 1));L) fi ∈ (T List))
Proof
Definitions occuring in Statement :
filter2: filter2(P;L),
length: ||as||,
cons: [a / b],
list: T List,
int_seg: {i..j-},
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
filter2: filter2(P;L),
all: ∀x:A. B[x],
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
nat: ℕ,
subtract: n - m
Lemmas referenced :
reduce2_cons_lemma,
int_seg_wf,
length_wf,
bool_wf,
list_wf,
false_wf,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
lelt_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
cons_wf,
squash_wf,
true_wf,
reduce2_shift,
nil_wf,
le_wf,
add-member-int_seg2,
decidable__le,
subtract_wf,
intformle_wf,
itermSubtract_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
reduce2_wf,
nat_wf,
decidable__equal_int,
add-associates,
add-swap,
add-commutes,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
functionEquality,
isectElimination,
natural_numberEquality,
addEquality,
cumulativity,
hypothesisEquality,
axiomEquality,
because_Cache,
universeEquality,
applyEquality,
dependent_set_memberEquality,
independent_pairFormation,
lambdaFormation,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
productElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
independent_functionElimination,
equalityElimination,
imageElimination,
functionExtensionality,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. \mforall{}[L:T List]. \mforall{}[P:\mBbbN{}||L|| + 1 {}\mrightarrow{} \mBbbB{}].
(filter2(P;[x / L])
= if P 0 then [x / filter2(\mlambda{}i.(P (i + 1));L)] else filter2(\mlambda{}i.(P (i + 1));L) fi )
Date html generated:
2017_10_01-AM-08_35_09
Last ObjectModification:
2017_07_26-PM-04_25_43
Theory : list!
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