Nuprl Lemma : case-real3-req1
∀[f:ℕ+ ⟶ 𝔹]
∀[b,a:ℝ]. case-real3(a;b;f) = a supposing ∀n,m:ℕ+. ((↑(f n)) ⇒ (¬↑(f m)) ⇒ (|(a m) - b m| ≤ 4))
supposing ∃n:ℕ+. (↑(f n))
Proof
Definitions occuring in Statement :
case-real3: case-real3(a;b;f),
req: x = y,
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,
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,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
case-real3: case-real3(a;b;f),
all: ∀x:A. B[x],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
false: False,
real: ℝ,
subtype_rel: A ⊆r B,
nat: ℕ,
bdd-diff: bdd-diff(f;g),
le: A ≤ B,
less_than': less_than'(a;b),
case-real3-seq: case-real3-seq(a;b;f),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
absval: |i|,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m
Lemmas referenced :
req-iff-bdd-diff,
case-real3_wf,
istype-assert,
accelerate-bdd-diff,
nat_plus_properties,
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,
req_witness,
nat_plus_wf,
istype-le,
absval_wf,
subtract_wf,
real_wf,
bool_wf,
bdd-diff_inversion,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
istype-false,
bdd-diff_functionality,
bdd-diff_weakening,
minus-one-mul,
add-mul-special,
zero-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
isect_memberEquality_alt,
productIsType,
inhabitedIsType,
applyEquality,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
setElimination,
rename,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
voidElimination,
universeIsType,
functionIsType,
because_Cache,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies,
independent_pairFormation,
lambdaFormation_alt,
equalityElimination,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
minusEquality
Latex:
\mforall{}[f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbB{}]
\mforall{}[b,a:\mBbbR{}].
case-real3(a;b;f) = a supposing \mforall{}n,m:\mBbbN{}\msupplus{}. ((\muparrow{}(f n)) {}\mRightarrow{} (\mneg{}\muparrow{}(f m)) {}\mRightarrow{} (|(a m) - b m| \mleq{} 4))
supposing \mexists{}n:\mBbbN{}\msupplus{}. (\muparrow{}(f n))
Date html generated:
2019_10_29-AM-09_37_52
Last ObjectModification:
2019_06_14-PM-03_20_01
Theory : reals
Home
Index