Nuprl Lemma : path-in-union
∀[A,B:SeparationSpace].
∀f:Point(Path(A + B))
(((∀x:{x:ℝ| (r0 ≤ x) ∧ (x ≤ r1)} . (↑isl(f@x))) ∧ (λx.outl(f x) ∈ Point(Path(A))))
∨ ((∀x:{x:ℝ| (r0 ≤ x) ∧ (x ≤ r1)} . (↑isr(f@x))) ∧ (λx.outr(f x) ∈ Point(Path(B)))))
Proof
Definitions occuring in Statement :
path-at: p@t,
path-ss: Path(X),
union-ss: ss1 + ss2,
ss-point: Point(ss),
separation-space: SeparationSpace,
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
outr: outr(x),
outl: outl(x),
assert: ↑b,
isr: isr(x),
isl: isl(x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
union-ss: ss1 + ss2,
ss-point: Point(ss),
mk-ss: Point=P #=Sep cotrans=C,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
isl: isl(x),
and: P ∧ Q,
prop: ℙ,
isr: isr(x),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
guard: {T},
uimplies: b supposing a,
true: True,
ss-sep: x # y,
union-sep: union-sep(ss1;ss2;p;q),
ss-eq: x ≡ y,
top: Top,
assert: ↑b,
bnot: ¬bb,
exists: ∃x:A. B[x],
sq_type: SQType(T),
or: P ∨ Q,
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
cand: A c∧ B,
outl: outl(x),
path-at: p@t,
outr: outr(x)
Lemmas referenced :
path-at_wf,
union-ss_wf,
member_rccint_lemma,
rec_select_update_lemma,
btrue_wf,
bfalse_wf,
real_wf,
rleq_wf,
int-to-real_wf,
extensional-interval-to-bool-constant,
rleq-int,
istype-false,
i-member_wf,
rccint_wf,
path-at_functionality,
req_wf,
ss-point_wf,
path-ss_wf,
separation-space_wf,
ss-eq_wf,
istype-void,
assert-bnot,
bool_cases_sqequal,
eqff_to_assert,
istype-assert,
assert_witness,
bool_subtype_base,
bool_wf,
subtype_base_sq,
eqtt_to_assert,
rleq_weakening_equal,
istype-true,
path-ss-point,
unit_ss_point_lemma,
unit-ss_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
dependent_functionElimination,
Error :memTop,
inhabitedIsType,
unionElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setIsType,
universeIsType,
productIsType,
natural_numberEquality,
because_Cache,
productElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
independent_isectElimination,
setElimination,
rename,
voidElimination,
isect_memberEquality_alt,
inrFormation_alt,
promote_hyp,
dependent_pairFormation_alt,
functionIsType,
cumulativity,
instantiate,
inlFormation_alt,
equalityElimination,
applyEquality
Latex:
\mforall{}[A,B:SeparationSpace].
\mforall{}f:Point(Path(A + B))
(((\mforall{}x:\{x:\mBbbR{}| (r0 \mleq{} x) \mwedge{} (x \mleq{} r1)\} . (\muparrow{}isl(f@x))) \mwedge{} (\mlambda{}x.outl(f x) \mmember{} Point(Path(A))))
\mvee{} ((\mforall{}x:\{x:\mBbbR{}| (r0 \mleq{} x) \mwedge{} (x \mleq{} r1)\} . (\muparrow{}isr(f@x))) \mwedge{} (\mlambda{}x.outr(f x) \mmember{} Point(Path(B)))))
Date html generated:
2020_05_20-PM-01_21_15
Last ObjectModification:
2020_02_08-AM-11_40_36
Theory : intuitionistic!topology
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