Nuprl Lemma : uniform-partition-increasing
∀a,b:ℝ. ((a < b) ⇒ (∀k:ℕ+. frs-increasing(uniform-partition([a, b];k))))
Proof
Definitions occuring in Statement :
uniform-partition: uniform-partition(I;k),
frs-increasing: frs-increasing(p),
rccint: [l, u],
rless: x < y,
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
frs-increasing: frs-increasing(p),
uniform-partition: uniform-partition(I;k),
top: Top,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
partition: partition(I),
nat: ℕ,
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
rneq: x ≠ y,
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
le: A ≤ B,
true: True,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rdiv: (x/y)
Lemmas referenced :
rccint-icompact,
rleq_weakening_rless,
nat_plus_wf,
rless_wf,
real_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
less_than_wf,
int_seg_wf,
length_wf,
uniform-partition_wf,
rccint_wf,
partition_wf,
subtract_wf,
int_seg_properties,
mklist_wf,
nat_plus_properties,
sq_stable__less_than,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
rdiv_wf,
radd_wf,
rmul_wf,
rsub_wf,
int-to-real_wf,
rless-int,
decidable__lt,
mklist_length,
squash_wf,
true_wf,
mklist_select,
iff_weakening_equal,
rmul_preserves_rless,
rminus_wf,
rinv_wf2,
itermAdd_wf,
int_term_value_add_lemma,
rless-implies-rless,
real_term_polynomial,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rless_functionality,
itermMultiply_wf,
itermMinus_wf,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
req_transitivity,
radd_functionality,
rmul_functionality,
req_weakening,
rmul-rinv3,
rminus_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
hypothesis,
isectElimination,
independent_isectElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
natural_numberEquality,
applyEquality,
lambdaEquality,
dependent_set_memberEquality,
functionEquality,
intEquality,
because_Cache,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll,
inrFormation,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}a,b:\mBbbR{}. ((a < b) {}\mRightarrow{} (\mforall{}k:\mBbbN{}\msupplus{}. frs-increasing(uniform-partition([a, b];k))))
Date html generated:
2017_10_03-AM-09_46_57
Last ObjectModification:
2017_07_28-AM-07_59_59
Theory : reals
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