Nuprl Lemma : eventually-equal-implies-bdd-diff
∀f,g:ℕ+ ⟶ ℤ. ((∃m:ℕ+. ∀n:{m...}. ((f n) = (g n) ∈ ℤ)) ⇒ bdd-diff(f;g))
Proof
Definitions occuring in Statement :
bdd-diff: bdd-diff(f;g),
int_upper: {i...},
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
int_upper: {i...},
nat: ℕ,
le: A ≤ B,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_apply: x[s],
less_than': less_than'(a;b),
false: False,
not: ¬A,
sq_type: SQType(T),
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
absval: |i|
Lemmas referenced :
bdd-diff-iff-eventual,
exists_wf,
nat_wf,
all_wf,
int_upper_wf,
le_wf,
absval_wf,
subtract_wf,
less_than_transitivity1,
less_than_wf,
nat_plus_wf,
equal_wf,
false_wf,
subtype_base_sq,
int_subtype_base,
int_upper_properties,
nat_plus_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
dependent_pairFormation,
isectElimination,
hypothesis,
sqequalRule,
lambdaEquality,
setElimination,
rename,
because_Cache,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
natural_numberEquality,
independent_isectElimination,
intEquality,
functionEquality,
independent_pairFormation,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}f,g:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}. ((\mexists{}m:\mBbbN{}\msupplus{}. \mforall{}n:\{m...\}. ((f n) = (g n))) {}\mRightarrow{} bdd-diff(f;g))
Date html generated:
2017_10_02-PM-07_13_07
Last ObjectModification:
2017_07_05-PM-04_24_39
Theory : reals
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