Nuprl Lemma : filter-equal
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L1,L2:T List].
(filter(P;L1) = filter(P;L2) ∈ (T List)) supposing
((∀i:ℕ||L1||. ((L1[i] = L2[i] ∈ T) ∨ ((¬↑(P L1[i])) ∧ (¬↑(P L2[i]))))) and
(||L1|| = ||L2|| ∈ ℤ))
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
filter: filter(P;l),
list: T List,
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
true: True,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
unit: Unit,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
cons: [a / b],
colength: colength(L),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
le: A ≤ B,
subtype_rel: A ⊆r B,
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
nat_plus: ℕ+
Lemmas referenced :
select-cons,
squash_wf,
true_wf,
add-subtract-cancel,
bool_subtype_base,
iff_imp_equal_bool,
le_int_wf,
bfalse_wf,
iff_functionality_wrt_iff,
iff_weakening_uiff,
assert_of_le_int,
iff_weakening_equal,
assert_of_bnot,
eqff_to_assert,
uiff_transitivity,
eqtt_to_assert,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
list-cases,
length_of_nil_lemma,
stuck-spread,
istype-base,
filter_nil_lemma,
nil_wf,
int_seg_wf,
int_seg_properties,
product_subtype_list,
colength-cons-not-zero,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
length_of_cons_lemma,
filter_cons_lemma,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
length_wf,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_wf,
add-is-int-iff,
false_wf,
select_wf,
cons_wf,
decidable__lt,
istype-assert,
istype-nat,
list_wf,
bool_wf,
istype-universe,
istype-false,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
assert_elim,
not_assert_elim,
btrue_neq_bfalse
Rules used in proof :
imageMemberEquality,
cumulativity,
equalityElimination,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
unionElimination,
baseClosed,
because_Cache,
Error :functionIsType,
equalityTransitivity,
equalitySymmetry,
productElimination,
Error :equalityIstype,
sqequalBase,
promote_hyp,
hypothesis_subsumption,
instantiate,
applyLambdaEquality,
addEquality,
applyEquality,
Error :dependent_set_memberEquality_alt,
imageElimination,
baseApply,
closedConclusion,
intEquality,
pointwiseFunctionality,
Error :unionIsType,
Error :productIsType,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L1,L2:T List].
(filter(P;L1) = filter(P;L2)) supposing
((\mforall{}i:\mBbbN{}||L1||. ((L1[i] = L2[i]) \mvee{} ((\mneg{}\muparrow{}(P L1[i])) \mwedge{} (\mneg{}\muparrow{}(P L2[i]))))) and
(||L1|| = ||L2||))
Date html generated:
2019_06_20-PM-02_13_14
Last ObjectModification:
2019_06_20-PM-02_08_13
Theory : list_1
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