Nuprl Lemma : ss-homotopic_inversion
∀X:SeparationSpace. ∀[x0,x1:Point(X)]. ∀a,b:Point(Path(X)). (ss-homotopic(X;x0;x1;a;b) ⇒ ss-homotopic(X;x0;x1;b;a))
Proof
Definitions occuring in Statement :
ss-homotopic: ss-homotopic(X;x0;x1;a;b),
path-ss: Path(X),
ss-point: Point(ss),
separation-space: SeparationSpace,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
ss-homotopic: ss-homotopic(X;x0;x1;a;b),
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
cand: A c∧ B,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
path-at: p@t,
ss-eq: x ≡ y,
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
assert: ↑b,
ifthenelse: if b then t else f fi ,
nonneg-poly: nonneg-poly(p),
bl-all: (∀x∈L.P[x])_b,
reduce: reduce(f;k;as),
list_ind: list_ind,
int_term_to_ipoly: int_term_to_ipoly(t),
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
add_ipoly: add_ipoly(p;q),
add-ipoly-prepend: add-ipoly-prepend(p;q;l),
itermConstant: "const",
cons: [a / b],
minus-poly: minus-poly(p),
map: map(f;as),
nil: [],
it: ⋅,
rev-append: rev(as) + bs,
list_accum: list_accum,
band: p ∧b q,
nonneg-monomial: nonneg-monomial(m),
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
bfalse: ff,
btrue: tt,
even-int-list: even-int-list(L),
bor: p ∨bq,
null: null(as),
true: True,
req_int_terms: t1 ≡ t2
Lemmas referenced :
path-ss-point,
rleq_wf,
int-to-real_wf,
real_wf,
path-at_wf,
path-ss_wf,
member_rccint_lemma,
rleq-implies-rleq,
rsub_wf,
trivial-rsub-rleq,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
path-at_functionality,
sq_stable__rleq,
unit-ss-eq,
req_functionality,
rsub_functionality,
req_weakening,
req-same,
ss-eq_wf,
unit-ss_wf,
unit_ss_point_lemma,
i-member_wf,
rccint_wf,
rleq-int,
istype-false,
ss-homotopic_wf,
ss-point_wf,
separation-space_wf,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
req-iff-rsub-is-0,
real-term-nonneg,
istype-int,
real_term_value_sub_lemma,
real_term_value_const_lemma,
rleq_weakening,
real_polynomial_null,
real_term_value_var_lemma,
ss-eq_functionality,
ss-eq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
Error :memTop,
hypothesis,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
productEquality,
natural_numberEquality,
hypothesisEquality,
universeIsType,
setElimination,
rename,
dependent_functionElimination,
independent_isectElimination,
independent_pairFormation,
because_Cache,
equalityTransitivity,
equalitySymmetry,
productIsType,
setIsType,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
functionIsType,
applyEquality,
dependent_pairFormation_alt,
independent_pairEquality,
voidElimination,
functionIsTypeImplies,
inhabitedIsType,
approximateComputation,
int_eqEquality
Latex:
\mforall{}X:SeparationSpace
\mforall{}[x0,x1:Point(X)]. \mforall{}a,b:Point(Path(X)). (ss-homotopic(X;x0;x1;a;b) {}\mRightarrow{} ss-homotopic(X;x0;x1;b;a))
Date html generated:
2020_05_20-PM-01_20_38
Last ObjectModification:
2020_01_06-AM-11_24_24
Theory : intuitionistic!topology
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