Nuprl Lemma : mem_diff
∀s:DSet. ∀as:DisList{s}. ∀bs:|s| List. ∀c:|s|. c ∈b (as - bs) = (c ∈b as) ∧b (¬b(c ∈b bs))
Proof
Definitions occuring in Statement :
diff: as - bs,
dislist: DisList{s},
mem: a ∈b as,
list: T List,
band: p ∧b q,
bnot: ¬bb,
bool: 𝔹,
all: ∀x:A. B[x],
equal: s = t ∈ T,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
member: t ∈ T,
all: ∀x:A. B[x],
dislist: DisList{s},
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
prop: ℙ,
gt: i > j,
dset: DSet,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
ndiff: a -- b,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
ge: i ≥ j ,
decidable: Dec(P)
Lemmas referenced :
mem_wf,
diff_wf,
eqtt_to_assert,
bnot_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
bfalse_wf,
mem_iff_count_nzero,
assert_wf,
gt_wf,
count_wf,
not_wf,
set_car_wf,
list_wf,
dislist_wf,
dset_wf,
iff_imp_equal_bool,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_bnot,
dislist_properties,
squash_wf,
true_wf,
istype-int,
count_diff,
subtype_rel_self,
iff_weakening_equal,
le_int_wf,
subtract_wf,
assert_of_le_int,
subtract-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
itermSubtract_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
false_wf,
le_wf,
non_neg_length,
count_bounds,
decidable__lt,
imax_unfold,
imax_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
because_Cache,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_pairFormation_alt,
equalityIsType1,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
independent_pairFormation,
universeIsType,
natural_numberEquality,
productIsType,
productEquality,
isect_memberEquality_alt,
applyEquality,
lambdaEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
approximateComputation,
int_eqEquality
Latex:
\mforall{}s:DSet. \mforall{}as:DisList\{s\}. \mforall{}bs:|s| List. \mforall{}c:|s|. c \mmember{}\msubb{} (as - bs) = (c \mmember{}\msubb{} as) \mwedge{}\msubb{} (\mneg{}\msubb{}(c \mmember{}\msubb{} bs))
Date html generated:
2019_10_16-PM-01_04_20
Last ObjectModification:
2018_10_08-AM-11_17_19
Theory : list_2
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