Nuprl Lemma : no_repeats_filter2
∀[T:Type]. ∀[l:T List]. ∀[P:{x:T| (x ∈ l)} ⟶ 𝔹]. no_repeats(T;filter(P;l)) supposing no_repeats(T;l)
Proof
Definitions occuring in Statement :
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
filter: filter(P;l),
list: T List,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
no_repeats: no_repeats(T;l),
not: ¬A,
false: False,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
squash: ↓T
Lemmas referenced :
no_repeats_witness,
no_repeats_filter,
l_member_wf,
list-subtype,
equal_wf,
select_wf,
nat_properties,
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,
not_wf,
nat_wf,
less_than_wf,
length_wf,
equal_functionality_wrt_subtype_rel2,
filter_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_functionElimination,
hypothesis,
sqequalRule,
isect_memberEquality,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
setEquality,
cumulativity,
independent_isectElimination,
lambdaFormation,
voidElimination,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidEquality,
independent_pairFormation,
computeAll,
functionExtensionality,
applyEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[l:T List]. \mforall{}[P:\{x:T| (x \mmember{} l)\} {}\mrightarrow{} \mBbbB{}].
no\_repeats(T;filter(P;l)) supposing no\_repeats(T;l)
Date html generated:
2017_04_17-AM-07_27_04
Last ObjectModification:
2017_02_27-PM-04_06_16
Theory : list_1
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