Nuprl Lemma : trivial-bdd-diff
∀[f,g:ℕ+ ⟶ ℤ]. bdd-diff(f;g) supposing ∀n:ℕ+. ((f n) = (g n) ∈ ℤ)
Proof
Definitions occuring in Statement :
bdd-diff: bdd-diff(f;g),
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
bdd-diff: bdd-diff(f;g),
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
absval: |i|,
subtract: n - m
Lemmas referenced :
zero-mul,
add-mul-special,
minus-one-mul,
iff_weakening_equal,
nat_wf,
true_wf,
squash_wf,
equal_wf,
all_wf,
subtract_wf,
absval_wf,
less_than'_wf,
le_wf,
false_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
lemma_by_obid,
rename,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
isectElimination,
productElimination,
independent_pairEquality,
voidElimination,
applyEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
setElimination,
intEquality,
functionEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
independent_functionElimination,
isect_memberEquality,
voidEquality
Latex:
\mforall{}[f,g:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. bdd-diff(f;g) supposing \mforall{}n:\mBbbN{}\msupplus{}. ((f n) = (g n))
Date html generated:
2016_05_18-AM-06_46_20
Last ObjectModification:
2016_01_17-AM-01_45_18
Theory : reals
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