Nuprl Lemma : path-comp-fun
∀[T:Type]. ∀[B:SeparationSpace]. (path-comp-property(B) ⇒ path-comp-property(T ⟶ B))
Proof
Definitions occuring in Statement :
path-comp-property: path-comp-property(X),
fun-ss: A ⟶ ss,
separation-space: SeparationSpace,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
path-comp-property: path-comp-property(X),
all: ∀x:A. B[x],
member: t ∈ T,
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,
subtype_rel: A ⊆r B,
ss-point: Point(ss),
record-select: r.x,
fun-ss: A ⟶ ss,
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,
guard: {T},
uiff: uiff(P;Q),
path-at: p@t,
exists: ∃x:A. B[x],
pi1: fst(t),
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
rneq: x ≠ y,
or: P ∨ Q,
less_than: a < b,
true: True,
nat_plus: ℕ+,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
rleq: x ≤ y,
rnonneg: rnonneg(x),
rge: x ≥ y,
path-comp-rel: path-comp-rel(X;f;g;h),
so_lambda: λ2x.t[x],
so_apply: x[s],
i-member: r ∈ I,
rccint: [l, u],
subinterval: I ⊆ J ,
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
ss-eq_wf,
fun-ss_wf,
path-at_wf,
member_rccint_lemma,
rleq-int,
istype-false,
rleq_weakening_equal,
int-to-real_wf,
rleq_wf,
ss-point_wf,
path-ss_wf,
path-comp-property_wf,
separation-space_wf,
istype-universe,
path-ss-point,
fun-ss-point,
real_wf,
iff_weakening_uiff,
subtype_rel_self,
fun-ss-eq,
unit-ss_wf,
unit_ss_point_lemma,
path-comp-rel_wf,
sq_stable__ss-eq,
rcc-subinterval,
rccint_wf,
rdiv_wf,
rless-int,
rless_wf,
rleq-int-fractions3,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
le_witness_for_triv,
rleq-int-fractions2,
rleq_functionality_wrt_implies,
subtype_rel_sets_simple,
i-member_wf,
rmul-nonneg-case1,
sq_stable__rleq,
rmul_wf,
rmul_preserves_rleq2,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
rinv_wf2,
rleq_functionality,
req_transitivity,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rsub_wf,
rleq-implies-rleq,
rmul-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
dependent_functionElimination,
Error :memTop,
natural_numberEquality,
productElimination,
independent_functionElimination,
independent_pairFormation,
because_Cache,
independent_isectElimination,
dependent_set_memberEquality_alt,
productIsType,
instantiate,
universeEquality,
setElimination,
rename,
lambdaEquality_alt,
applyEquality,
setIsType,
functionEquality,
inhabitedIsType,
functionIsType,
promote_hyp,
functionExtensionality,
dependent_pairFormation_alt,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
closedConclusion,
inrFormation_alt,
unionElimination,
approximateComputation,
voidElimination,
independent_pairEquality,
functionIsTypeImplies,
int_eqEquality
Latex:
\mforall{}[T:Type]. \mforall{}[B:SeparationSpace]. (path-comp-property(B) {}\mRightarrow{} path-comp-property(T {}\mrightarrow{} B))
Date html generated:
2020_05_20-PM-01_21_36
Last ObjectModification:
2020_01_06-AM-11_19_59
Theory : intuitionistic!topology
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