Nuprl Lemma : ss-free-homotopic_inversion
∀X:SeparationSpace. ∀a,b:Point(X). (ss-free-homotopic(X;a;b) ⇒ ss-free-homotopic(X;b;a))
Proof
Definitions occuring in Statement :
ss-free-homotopic: ss-free-homotopic(X;a;b),
ss-point: Point(ss),
separation-space: SeparationSpace,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
ss-free-homotopic: ss-free-homotopic(X;a;b),
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
cand: A c∧ B,
prop: ℙ,
uall: ∀[x:A]. B[x],
top: Top,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
uiff: uiff(P;Q),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
path-at: p@t
Lemmas referenced :
ss-eq_wf,
path-at_wf,
member_rccint_lemma,
rleq_weakening_equal,
int-to-real_wf,
rleq-int,
false_wf,
rleq_wf,
ss-free-homotopic_wf,
ss-point_wf,
separation-space_wf,
path-ss-point,
real_wf,
rleq-implies-rleq,
rsub_wf,
trivial-rsub-rleq,
unit-ss_wf,
unit_ss_point_lemma,
set_wf,
all_wf,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
path-at_functionality,
unit-ss-eq,
req_functionality,
rsub_functionality,
req_weakening,
ss-eq_functionality,
ss-eq_weakening,
rleq_functionality,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
cut,
independent_pairFormation,
hypothesis,
productEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
because_Cache,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
dependent_set_memberEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
functionEquality,
applyEquality,
approximateComputation,
int_eqEquality,
intEquality
Latex:
\mforall{}X:SeparationSpace. \mforall{}a,b:Point(X). (ss-free-homotopic(X;a;b) {}\mRightarrow{} ss-free-homotopic(X;b;a))
Date html generated:
2020_05_20-PM-01_20_29
Last ObjectModification:
2018_07_05-PM-03_20_55
Theory : intuitionistic!topology
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