Nuprl Lemma : case-real3_wf
∀[f:ℕ+ ⟶ 𝔹]. ∀[b:ℝ]. ∀[a:ℝ supposing ∃n:ℕ+. (↑(f n))].
case-real3(a;b;f) ∈ ℝ supposing ∀n,m:ℕ+. ((↑(f n)) ⇒ (¬↑(f m)) ⇒ (|(a m) - b m| ≤ 4))
Proof
Definitions occuring in Statement :
case-real3: case-real3(a;b;f),
real: ℝ,
absval: |i|,
nat_plus: ℕ+,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
case-real3: case-real3(a;b;f),
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
nat: ℕ
Lemmas referenced :
accelerate_wf,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
case-real3-seq_wf,
nat_plus_wf,
istype-assert,
istype-le,
absval_wf,
subtract_wf,
uimplies_subtype,
real_wf,
assert_wf,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality_alt,
natural_numberEquality,
dependent_functionElimination,
hypothesis,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
universeIsType,
hypothesisEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsType,
because_Cache,
applyEquality,
functionEquality,
intEquality,
productEquality,
setElimination,
rename,
inhabitedIsType,
isectIsTypeImplies,
isectIsType,
productIsType
Latex:
\mforall{}[f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbB{}]. \mforall{}[b:\mBbbR{}]. \mforall{}[a:\mBbbR{} supposing \mexists{}n:\mBbbN{}\msupplus{}. (\muparrow{}(f n))].
case-real3(a;b;f) \mmember{} \mBbbR{} supposing \mforall{}n,m:\mBbbN{}\msupplus{}. ((\muparrow{}(f n)) {}\mRightarrow{} (\mneg{}\muparrow{}(f m)) {}\mRightarrow{} (|(a m) - b m| \mleq{} 4))
Date html generated:
2019_10_29-AM-09_37_43
Last ObjectModification:
2019_06_14-PM-03_11_14
Theory : reals
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