Nuprl Lemma : increasing_lower_bound
∀[k:ℕ]. ∀[f:ℕk ⟶ ℤ]. ∀[x:ℕk]. ((f 0) + x) ≤ (f x) supposing increasing(f;k)
Proof
Definitions occuring in Statement :
increasing: increasing(f;k),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than': less_than'(a;b),
prop: ℙ,
nat: ℕ,
guard: {T},
all: ∀x:A. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
true: True,
increasing: increasing(f;k)
Lemmas referenced :
less_than'_wf,
false_wf,
less_than_transitivity2,
lelt_wf,
increasing_wf,
int_seg_wf,
nat_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
decidable__le,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
zero-add,
add-commutes,
le-add-cancel,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add_functionality_wrt_le,
decidable__lt,
not-lt-2,
le-add-cancel-alt,
le-add-cancel2,
le_weakening2,
minus-zero,
int_seg_subtype_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
lemma_by_obid,
isectElimination,
applyEquality,
addEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
hypothesis,
setElimination,
rename,
independent_isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionEquality,
intEquality,
voidElimination,
intWeakElimination,
independent_functionElimination,
unionElimination,
voidEquality,
minusEquality,
multiplyEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[f:\mBbbN{}k {}\mrightarrow{} \mBbbZ{}]. \mforall{}[x:\mBbbN{}k]. ((f 0) + x) \mleq{} (f x) supposing increasing(f;k)
Date html generated:
2016_05_13-PM-04_02_38
Last ObjectModification:
2015_12_26-AM-10_57_16
Theory : int_1
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