Nuprl Lemma : increasing_implies_le
∀[k:ℕ]. ∀[f:ℕk ⟶ ℤ]. {∀[x,y:ℕk]. (f x) ≤ (f y) supposing x ≤ y} 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],
guard: {T},
le: A ≤ B,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
guard: {T},
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,
prop: ℙ,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
less_than'_wf,
le_wf,
int_seg_wf,
increasing_wf,
nat_wf,
decidable__int_equal,
subtype_base_sq,
int_subtype_base,
le_weakening,
increasing_implies,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
add-associates,
le_weakening2
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
isect_memberEquality,
natural_numberEquality,
functionEquality,
intEquality,
voidElimination,
unionElimination,
promote_hyp,
instantiate,
cumulativity,
independent_isectElimination,
independent_functionElimination,
independent_pairFormation,
lambdaFormation,
addEquality,
voidEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[f:\mBbbN{}k {}\mrightarrow{} \mBbbZ{}]. \{\mforall{}[x,y:\mBbbN{}k]. (f x) \mleq{} (f y) supposing x \mleq{} y\} supposing increasing(f;k)
Date html generated:
2018_05_21-PM-00_04_00
Last ObjectModification:
2018_05_19-AM-07_10_38
Theory : int_1
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