Nuprl Lemma : cantor-interval-req
∀a,b:ℝ. ∀f:ℕ ⟶ 𝔹. ∀n:ℕ.
(((fst(cantor-interval(a;b;f;n))) = (fst(cantor_ivl(a;b;f;n))))
∧ ((snd(cantor-interval(a;b;f;n))) = (snd(cantor_ivl(a;b;f;n)))))
Proof
Definitions occuring in Statement :
cantor-interval: cantor-interval(a;b;f;n),
cantor_ivl: cantor_ivl(a;b;f;n),
req: x = y,
real: ℝ,
nat: ℕ,
bool: 𝔹,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
cand: A c∧ B,
has-value: (a)↓,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
nat_plus: ℕ+,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rneq: x ≠ y,
nequal: a ≠ b ∈ T ,
true: True,
int_nzero: ℤ-o,
pi2: snd(t),
guard: {T},
sq_type: SQType(T),
subtract: n - m,
pi1: fst(t),
efficient-exp-ext,
fastexp: i^n,
primrec: primrec(n;b;c),
unit-interval-fan: unit-interval-fan(f;n),
cantor-interval: cantor-interval(a;b;f;n),
cantor_ivl: cantor_ivl(a;b;f;n),
or: P ∨ Q,
decidable: Dec(P),
so_apply: x[s],
so_lambda: λ2x.t[x],
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
rmul_comm,
int-rinv-cancel,
real_term_value_minus_lemma,
radd-int,
rsub_functionality,
rsub-int,
req_inversion,
rmul-rinv3,
rinv-mul-as-rdiv,
exp-positive-stronger,
itermMinus_wf,
rsub_wf,
rminus_wf,
exp-positive,
rmul_preserves_req,
rmul-int,
exp-fastexp,
assert_of_lt_int,
primrec-unroll,
int_term_value_mul_lemma,
decidable__equal_int,
exp_step,
req-int,
req_wf,
exp-non-zero,
exp_wf3,
int_formula_prop_eq_lemma,
intformeq_wf,
rneq-int,
exp_wf2,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
eqtt_to_assert,
exp_wf_nat_plus,
int-value-type,
set-value-type,
nat_plus_wf,
value-type-has-value,
unit-interval-fan_wf,
lt_int_wf,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rmul-identity1,
rinv1,
rmul_functionality,
req_transitivity,
int-rmul-req,
radd_functionality,
rdiv_functionality,
int-rdiv-req,
req_functionality,
req_weakening,
req-iff-rsub-is-0,
itermAdd_wf,
itermMultiply_wf,
rinv_wf2,
rmul_wf,
rless_wf,
rless-int,
int-to-real_wf,
rdiv_wf,
int-rmul_wf,
radd_wf,
nequal_wf,
true_wf,
equal-wf-base,
int-rdiv_wf,
int_subtype_base,
subtype_base_sq,
primrec0_lemma,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
le_wf,
pi2_wf,
cantor_ivl_wf,
equal_wf,
pi1_wf_top,
real_wf,
subtype_rel_self,
false_wf,
int_seg_subtype_nat,
int_seg_wf,
bool_wf,
nat_wf,
subtype_rel_function,
cantor-interval_wf,
req_witness,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
efficient-exp-ext
Rules used in proof :
minusEquality,
promote_hyp,
equalityElimination,
multiplyEquality,
addEquality,
imageMemberEquality,
inrFormation,
baseClosed,
addLevel,
cumulativity,
instantiate,
sqleReflexivity,
callbyvalueReduce,
functionEquality,
unionElimination,
dependent_set_memberEquality,
functionExtensionality,
equalitySymmetry,
equalityTransitivity,
productEquality,
because_Cache,
applyEquality,
independent_pairEquality,
productElimination,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. \mforall{}n:\mBbbN{}.
(((fst(cantor-interval(a;b;f;n))) = (fst(cantor\_ivl(a;b;f;n))))
\mwedge{} ((snd(cantor-interval(a;b;f;n))) = (snd(cantor\_ivl(a;b;f;n)))))
Date html generated:
2018_05_22-PM-02_09_33
Last ObjectModification:
2018_05_21-AM-00_48_41
Theory : reals
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