Nuprl Lemma : path-comp-for-reals
path-comp-property(ℝ)
Proof
Definitions occuring in Statement :
path-comp-property: path-comp-property(X),
real-ss: ℝ
Definitions unfolded in proof :
path-comp-property: path-comp-property(X),
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
rfun: I ⟶ℝ,
i-member: r ∈ I,
rccint: [l, u],
ss-point: Point(ss),
record-select: r.x,
real-ss: ℝ,
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,
real: ℝ,
ss-eq: x ≡ y,
ss-sep: x # y,
unit-ss: 𝕀,
set-ss: {x:ss | P[x]},
r-ap: f(x),
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
path-at: p@t,
path-comp-rel: path-comp-rel(X;f;g;h),
exists: ∃x:A. B[x],
squash: ↓T,
sq_stable: SqStable(P),
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
uiff: uiff(P;Q),
nat_plus: ℕ+,
true: True,
less_than: a < b,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
path-ss-point,
real-path-comp-exists,
subtype_rel_set,
real_wf,
rleq_wf,
int-to-real_wf,
ss-point_wf,
real-ss_wf,
rfun_wf,
rccint_wf,
ss-eq_wf,
unit-ss_wf,
unit_ss_point_lemma,
subtype_rel_self,
rec_select_update_lemma,
member_rccint_lemma,
not-rneq,
rneq_irrefl,
rneq_functionality,
req_weakening,
req_inversion,
rneq_wf,
req_wf,
i-member_wf,
rleq-int,
istype-false,
path-at_wf,
path-ss_wf,
sq_stable__rleq,
r-ap_wf,
istype-void,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
istype-int,
intformeq_wf,
full-omega-unsat,
rneq-int,
rleq_transitivity,
istype-less_than,
rleq-int-fractions3,
subtype_rel_sets_simple,
rinv_wf2,
itermConstant_wf,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
rless_wf,
rless-int,
rdiv_wf,
rmul_preserves_rleq2,
rmul-nonneg-case1,
rmul_wf,
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,
rleq-int-fractions2,
rleq-implies-rleq,
rsub_wf,
rmul-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
Error :memTop,
hypothesis,
sqequalRule,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
functionEquality,
setEquality,
productEquality,
natural_numberEquality,
lambdaEquality_alt,
functionIsType,
setIsType,
universeIsType,
productIsType,
because_Cache,
independent_isectElimination,
independent_functionElimination,
setElimination,
rename,
productElimination,
voidElimination,
inhabitedIsType,
independent_pairFormation,
dependent_set_memberEquality_alt,
dependent_pairFormation_alt,
imageElimination,
baseClosed,
imageMemberEquality,
closedConclusion,
isect_memberEquality_alt,
equalityIsType4,
approximateComputation,
promote_hyp,
inrFormation_alt,
int_eqEquality
Latex:
path-comp-property(\mBbbR{})
Date html generated:
2020_05_20-PM-01_20_57
Last ObjectModification:
2020_02_08-AM-11_41_35
Theory : intuitionistic!topology
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