Nuprl Lemma : member_interleaving
∀[T:Type]. ∀L,L1,L2:T List. (interleaving(T;L1;L2;L) ⇒ {∀x:T. ((x ∈ L) ⇐⇒ (x ∈ L1) ∨ (x ∈ L2))})
Proof
Definitions occuring in Statement :
interleaving: interleaving(T;L1;L2;L),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
universe: Type
Definitions unfolded in proof :
guard: {T},
interleaving: interleaving(T;L1;L2;L),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
cand: A c∧ B,
finite': finite'(T),
ge: i ≥ j ,
decidable: Dec(P),
false: False,
uiff: uiff(P;Q),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
sq_type: SQType(T),
squash: ↓T,
true: True,
surject: Surj(A;B;f),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b
Lemmas referenced :
l_member_wf,
istype-universe,
nat_wf,
length_wf_nat,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
length_wf,
disjoint_sublists_wf,
list_wf,
disjoint_sublists_witness,
nsub_finite',
subtype_base_sq,
nat_properties,
decidable__equal_int,
add-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
false_wf,
subtype_rel_self,
int_seg_wf,
inject_wf,
squash_wf,
true_wf,
iff_weakening_equal,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
less_than_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
or_wf,
int_seg_subtype_nat,
istype-false,
select_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
disjoint_sublists_sublist,
member_sublist
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
unionIsType,
productIsType,
equalityIsType4,
applyEquality,
intEquality,
lambdaEquality_alt,
natural_numberEquality,
independent_isectElimination,
addEquality,
because_Cache,
inhabitedIsType,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
cumulativity,
setElimination,
rename,
applyLambdaEquality,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
functionEquality,
imageElimination,
functionIsType,
imageMemberEquality,
dependent_set_memberEquality_alt,
hyp_replacement,
inlFormation_alt,
equalityIsType1,
productEquality,
inrFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}L,L1,L2:T List. (interleaving(T;L1;L2;L) {}\mRightarrow{} \{\mforall{}x:T. ((x \mmember{} L) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L1) \mvee{} (x \mmember{} L2))\})
Date html generated:
2019_10_15-AM-10_55_21
Last ObjectModification:
2018_10_09-AM-10_18_20
Theory : list!
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