Nuprl Lemma : l_intersection-size
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[a,b,c:T List].
(no_repeats(T;a) ⇒ no_repeats(T;b) ⇒ a ⊆ c ⇒ b ⊆ c ⇒ ((||a|| + ||b||) ≤ (||c|| + ||(a ⋂ b)||)))
Proof
Definitions occuring in Statement :
l_intersection: (L1 ⋂ L2),
l_contains: A ⊆ B,
no_repeats: no_repeats(T;l),
length: ||as||,
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
le: A ≤ B,
implies: P ⇒ Q,
add: n + m,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
l_intersection: (L1 ⋂ L2),
uimplies: b supposing a,
l_contains: A ⊆ B,
l_member: (x ∈ l),
l_all: (∀x∈L.P[x]),
exists: ∃x:A. B[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
guard: {T},
all: ∀x:A. B[x],
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
false: False,
nat: ℕ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
cand: A c∧ B,
pi1: fst(t),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
inject: Inj(A;B;f),
no_repeats: no_repeats(T;l),
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
label: ...$L... t
Lemmas referenced :
no_repeats_filter,
non_neg_length,
int_seg_properties,
length_wf,
decidable__le,
lelt_wf,
length_wf_nat,
nat_properties,
full-omega-unsat,
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_not_lemma,
int_seg_wf,
select_wf,
decidable__lt,
int_formula_prop_less_lemma,
no_repeats_l_index-inj,
l_intersection_wf,
l_contains_wf,
no_repeats_wf,
le_witness_for_triv,
list_wf,
deq_wf,
less_than_wf,
nat_wf,
member-intersection,
select_member,
l_index_wf,
l_member_wf,
pigeon-hole,
add_nat_wf,
add-is-int-iff,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
le_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
subtract_wf,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_term_value_subtract_lemma,
int_formula_prop_wf,
itermAdd_wf,
deq-member_wf,
assert-deq-member,
add-member-int_seg1,
intformeq_wf,
decidable__equal_int_seg,
int_seg_subtype_nat,
set_subtype_base,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
not_wf,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_isectElimination,
hypothesis,
sqequalRule,
promote_hyp,
productElimination,
Error :dependent_pairFormation_alt,
hypothesisEquality,
Error :functionExtensionality_alt,
Error :dependent_set_memberEquality_alt,
applyEquality,
independent_pairFormation,
natural_numberEquality,
setElimination,
rename,
dependent_functionElimination,
Error :universeIsType,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
independent_functionElimination,
voidElimination,
approximateComputation,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
Error :functionIsType,
Error :equalityIsType1,
imageElimination,
functionExtensionality,
cumulativity,
Error :inhabitedIsType,
universeEquality,
Error :productIsType,
addEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
equalityElimination,
instantiate,
hyp_replacement,
Error :equalityIsType4,
intEquality,
imageMemberEquality,
productEquality,
Error :equalityIsType3,
Error :equalityIsType2
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[a,b,c:T List].
(no\_repeats(T;a)
{}\mRightarrow{} no\_repeats(T;b)
{}\mRightarrow{} a \msubseteq{} c
{}\mRightarrow{} b \msubseteq{} c
{}\mRightarrow{} ((||a|| + ||b||) \mleq{} (||c|| + ||(a \mcap{} b)||)))
Date html generated:
2019_06_20-PM-01_58_00
Last ObjectModification:
2018_10_03-PM-10_28_04
Theory : decidable!equality
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