Nuprl Lemma : ss-mem-basic-and
∀[X:SeparationSpace]. ∀u,v:ss-basic(X). ∀x:Point(X). uiff(x ∈ ss-basic-and(u;v);x ∈ u ∧ x ∈ v)
Proof
Definitions occuring in Statement :
ss-basic-and: ss-basic-and(u;v),
ss-mem-basic: x ∈ B,
ss-basic: ss-basic(X),
ss-point: Point(ss),
separation-space: SeparationSpace,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
ss-basic-and: ss-basic-and(u;v),
ss-mem-basic: x ∈ B,
l_all: (∀x∈L.P[x]),
ss-basic: ss-basic(X),
squash: ↓T,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
false: False,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
ss-point: Point(ss),
record-select: r.x,
real-ss: ℝ,
mk-ss: Point=P #=Sep cotrans=C,
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
real: ℝ,
guard: {T},
iff: P ⇐⇒ Q,
less_than: a < b,
subtract: n - m,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q
Lemmas referenced :
sq_stable__ss-mem-basic,
ss-mem-basic_wf,
ss-basic-and_wf,
ss-point_wf,
ss-basic_wf,
separation-space_wf,
length-append,
non_neg_length,
ss-fun_wf,
real-ss_wf,
real_wf,
decidable__lt,
length_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
append_wf,
select_append_front,
rless_wf,
ss-ap_wf,
subtype_rel_self,
iff_weakening_equal,
int_seg_wf,
add-member-int_seg1,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtract_wf,
select_append_back,
int_seg_properties,
add-associates,
minus-one-mul,
add-swap,
add-mul-special,
zero-mul,
add-zero,
select-append,
subtype_rel_list,
top_wf,
int_seg_subtype_nat,
istype-false,
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,
less_than_wf,
add-is-int-iff,
false_wf,
length_append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
universeIsType,
productIsType,
inhabitedIsType,
setElimination,
rename,
dependent_set_memberEquality_alt,
productElimination,
Error :memTop,
productEquality,
addEquality,
unionElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
voidElimination,
spreadEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
closedConclusion,
equalityElimination,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
pointwiseFunctionality,
baseApply
Latex:
\mforall{}[X:SeparationSpace]. \mforall{}u,v:ss-basic(X). \mforall{}x:Point(X). uiff(x \mmember{} ss-basic-and(u;v);x \mmember{} u \mwedge{} x \mmember{} v)
Date html generated:
2020_05_20-PM-01_22_21
Last ObjectModification:
2020_02_08-AM-11_38_55
Theory : intuitionistic!topology
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