Nuprl Lemma : no_repeats_append_iff
∀[T:Type]. ∀[l1,l2:T List]. uiff(no_repeats(T;l1 @ l2);l_disjoint(T;l1;l2) ∧ no_repeats(T;l1) ∧ no_repeats(T;l2))
Proof
Definitions occuring in Statement :
l_disjoint: l_disjoint(T;l1;l2),
no_repeats: no_repeats(T;l),
append: as @ bs,
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
prop: ℙ,
false: False,
implies: P ⇒ Q,
not: ¬A,
all: ∀x:A. B[x],
l_disjoint: l_disjoint(T;l1;l2),
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
top: Top,
nat: ℕ,
or: P ∨ Q,
decidable: Dec(P),
no_repeats: no_repeats(T;l),
true: True,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
less_than: a < b,
cand: A c∧ B,
l_member: (x ∈ l),
le: A ≤ B,
sq_type: SQType(T)
Lemmas referenced :
istype-universe,
list_wf,
l_disjoint_wf,
append_wf,
no_repeats_wf,
no_repeats_witness,
no_repeats_append,
sublist_append1,
no_repeats-sublist,
sublist_append2,
istype-nat,
istype-less_than,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
istype-void,
length-append,
select_wf,
length_wf,
decidable__lt,
istype-le,
equal_wf,
squash_wf,
true_wf,
select_append_front,
subtype_rel_self,
iff_weakening_equal,
subtract_wf,
itermSubtract_wf,
intformless_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
select_append_back,
select_member,
top_wf,
subtype_rel_list,
length_append,
non_neg_length,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
int_formula_prop_eq_lemma,
intformeq_wf
Rules used in proof :
universeEquality,
instantiate,
productIsType,
universeIsType,
because_Cache,
independent_functionElimination,
inhabitedIsType,
functionIsTypeImplies,
voidElimination,
dependent_functionElimination,
lambdaEquality_alt,
independent_pairEquality,
productElimination,
sqequalRule,
independent_isectElimination,
independent_pairFormation,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
cumulativity,
isectIsTypeImplies,
functionIsType,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
natural_numberEquality,
isect_memberEquality_alt,
rename,
setElimination,
equalityIstype,
unionElimination,
lambdaFormation_alt,
dependent_set_memberEquality_alt,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
addEquality,
sqequalBase,
intEquality,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[l1,l2:T List].
uiff(no\_repeats(T;l1 @ l2);l\_disjoint(T;l1;l2) \mwedge{} no\_repeats(T;l1) \mwedge{} no\_repeats(T;l2))
Date html generated:
2019_10_15-AM-10_22_15
Last ObjectModification:
2019_08_05-PM-01_59_05
Theory : list_1
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