Nuprl Lemma : path-end-mem-basic
∀X:SeparationSpace. ∀f:Point(Path(X)). ∀B:ss-basic(X).
(f@r1 ∈ B ⇒ (∃z:{z:ℝ| z ∈ [r0, r1)} . ∀t:{t:ℝ| t ∈ [z, r1]} . f@t ∈ B))
Proof
Definitions occuring in Statement :
ss-mem-basic: x ∈ B,
ss-basic: ss-basic(X),
path-at: p@t,
path-ss: Path(X),
ss-point: Point(ss),
separation-space: SeparationSpace,
rcoint: [l, u),
rccint: [l, u],
i-member: r ∈ I,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
uimplies: b supposing a,
prop: ℙ,
sq_stable: SqStable(P),
true: True,
squash: ↓T,
less_than: a < b,
real-cont: real-cont(f;a;b),
exists: ∃x:A. B[x],
top: Top,
uiff: uiff(P;Q),
real-fun: real-fun(f;a;b),
real: ℝ,
btrue: tt,
bfalse: ff,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
record-update: r[x := v],
mk-ss: Point=P #=Sep cotrans=C,
real-ss: ℝ,
record-select: r.x,
ss-point: Point(ss),
subtype_rel: A ⊆r B,
guard: {T},
rfun: I ⟶ℝ,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
sq_exists: ∃x:A [B[x]],
rless: x < y,
or: P ∨ Q,
rneq: x ≠ y,
nat_plus: ℕ+,
ss-mem-basic: x ∈ B,
l_all: (∀x∈L.P[x]),
ss-basic: ss-basic(X),
int_seg: {i..j-},
lelt: i ≤ j < k,
i-member: r ∈ I,
rcoint: [l, u),
rccint: [l, u],
pi1: fst(t),
rnonneg: rnonneg(x),
rleq: x ≤ y,
so_apply: x[s],
nat: ℕ,
so_lambda: λ2x.t[x],
subtract: n - m,
sq_type: SQType(T)
Lemmas referenced :
ss-mem-basic_wf,
path-at_wf,
member_rccint_lemma,
rleq-int,
istype-false,
rleq_weakening_equal,
int-to-real_wf,
rleq_wf,
ss-basic_wf,
ss-point_wf,
path-ss_wf,
separation-space_wf,
rleq_weakening_rless,
trivial-rleq-radd,
radd_wf,
req-iff-rsub-is-0,
itermConstant_wf,
itermVar_wf,
itermSubtract_wf,
ss-eq_weakening,
ss-fun_wf,
rleq_transitivity,
rccint-icompact,
rabs-difference-bound-rleq,
iff_weakening_uiff,
rabs_wf,
sq_stable__rleq,
rmax_lb,
rless-int,
rmax_strict_lb,
rleq-rmax,
rmax_wf,
member_rcoint_lemma,
rless_wf,
istype-void,
rsub_wf,
rless-implies-rless,
small-reciprocal-real,
req_wf,
real-ss_wf,
real-ss-eq,
real_wf,
subtype_rel_self,
ss-ap_wf,
rccint_wf,
i-member_wf,
ss-eq_functionality,
ss-ap_functionality,
path-at_functionality,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq_functionality_wrt_implies,
itermAdd_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
rdiv_wf,
rless_functionality_wrt_implies,
real_term_value_add_lemma,
select_wf,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
length_wf,
int_seg_wf,
rcoint_wf,
le_witness_for_triv,
exists_wf,
all_wf,
primrec-wf2,
istype-less_than,
subtract_wf,
length_wf_nat,
le-add-cancel2,
add-commutes,
add-zero,
zero-mul,
add-mul-special,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
add-associates,
condition-implies-le,
not-le-2,
int_seg_subtype,
subtype_rel_function,
istype-le,
int_term_value_subtract_lemma,
rmax_ub,
decidable__equal_int,
int_formula_prop_eq_lemma,
intformeq_wf,
int_subtype_base,
subtype_base_sq,
rleq_weakening,
sq_stable__subtype_rel,
subtype_rel_sets_simple
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
dependent_functionElimination,
Error :memTop,
hypothesis,
natural_numberEquality,
productElimination,
independent_functionElimination,
independent_pairFormation,
because_Cache,
independent_isectElimination,
dependent_set_memberEquality_alt,
productIsType,
functionIsType,
imageElimination,
promote_hyp,
productEquality,
baseClosed,
imageMemberEquality,
dependent_pairFormation_alt,
voidElimination,
isect_memberEquality_alt,
setIsType,
applyEquality,
rename,
setElimination,
inhabitedIsType,
lambdaEquality_alt,
approximateComputation,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
inrFormation_alt,
equalityIstype,
functionExtensionality,
spreadEquality,
equalityIsType1,
functionIsTypeImplies,
setEquality,
closedConclusion,
functionEquality,
multiplyEquality,
minusEquality,
addEquality,
inlFormation_alt,
intEquality,
cumulativity,
instantiate
Latex:
\mforall{}X:SeparationSpace. \mforall{}f:Point(Path(X)). \mforall{}B:ss-basic(X).
(f@r1 \mmember{} B {}\mRightarrow{} (\mexists{}z:\{z:\mBbbR{}| z \mmember{} [r0, r1)\} . \mforall{}t:\{t:\mBbbR{}| t \mmember{} [z, r1]\} . f@t \mmember{} B))
Date html generated:
2020_05_20-PM-01_22_52
Last ObjectModification:
2020_01_06-PM-07_34_21
Theory : intuitionistic!topology
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