Nuprl Lemma : mem_lmin
∀s:DSet. ∀as,bs:|s| List. ∀c:|s|. c ∈b lmin(s;as;bs) = (c ∈b as) ∧b (c ∈b bs)
Proof
Definitions occuring in Statement :
lmin: lmin(s;as;bs),
mem: a ∈b as,
list: T List,
band: p ∧b q,
bool: 𝔹,
all: ∀x:A. B[x],
equal: s = t ∈ T,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
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,
prop: ℙ,
dset: DSet,
gt: i > j,
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
true: True,
subtype_rel: A ⊆r B
Lemmas referenced :
iff_imp_equal_bool,
mem_wf,
lmin_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
bfalse_wf,
mem_iff_count_nzero,
assert_wf,
iff_weakening_uiff,
assert_of_band,
set_car_wf,
list_wf,
dset_wf,
le_int_wf,
count_wf,
equal-wf-T-base,
le_wf,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
istype-int,
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_formula_prop_wf,
gt_wf,
lt_int_wf,
less_than_wf,
bnot_wf,
uiff_transitivity,
assert_of_le_int,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
imin_unfold,
iff_weakening_equal,
imin_wf,
squash_wf,
true_wf,
count_lmin,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_pairFormation_alt,
equalityIsType1,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
because_Cache,
voidElimination,
independent_pairFormation,
universeIsType,
productIsType,
productEquality,
isect_memberEquality_alt,
setElimination,
rename,
baseClosed,
imageElimination,
natural_numberEquality,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
applyEquality,
imageMemberEquality,
universeEquality
Latex:
\mforall{}s:DSet. \mforall{}as,bs:|s| List. \mforall{}c:|s|. c \mmember{}\msubb{} lmin(s;as;bs) = (c \mmember{}\msubb{} as) \mwedge{}\msubb{} (c \mmember{}\msubb{} bs)
Date html generated:
2019_10_16-PM-01_04_29
Last ObjectModification:
2018_10_08-AM-10_22_39
Theory : list_2
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