Nuprl Lemma : increasing_le
∀[k,m:ℕ]. k ≤ m supposing ∃f:ℕk ⟶ ℕm. 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,
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
exists: ∃x:A. B[x],
true: True,
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
lelt: i ≤ j < k,
less_than': less_than'(a;b),
squash: ↓T,
sq_stable: SqStable(P),
so_apply: x[s],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
not: ¬A,
and: P ∧ Q,
le: A ≤ B,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
sq_type: SQType(T),
less_than: a < b,
nat_plus: ℕ+,
increasing: increasing(f;k)
Lemmas referenced :
le_weakening2,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-ge-2,
subtract_wf,
decidable__le,
le_wf,
false_wf,
sq_stable__le,
nat_wf,
increasing_wf,
int_seg_wf,
exists_wf,
less_than'_wf,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties,
le-add-cancel-alt,
not-le-2,
minus-zero,
not-equal-2,
int_subtype_base,
subtype_base_sq,
decidable__int_equal,
lelt_wf,
int_seg_properties,
equal_wf,
le_weakening,
base_wf,
subtype_rel-equal,
int_seg_subtype,
member_wf,
not-lt-2,
decidable__lt,
increasing_implies,
set_subtype_base,
zero-mul,
add-mul-special,
mul-swap,
mul-commutes,
mul-associates,
mul-distributes,
omega-shadow,
mul-distributes-right,
two-mul,
one-mul,
le_reflexive,
subtype_rel_self,
not-equal-implies-less,
le-add-cancel2,
add-member-int_seg2
Rules used in proof :
minusEquality,
intEquality,
voidEquality,
addEquality,
unionElimination,
independent_pairFormation,
dependent_set_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
applyEquality,
functionExtensionality,
because_Cache,
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_pairEquality,
productElimination,
isect_memberEquality,
dependent_functionElimination,
lambdaEquality,
sqequalRule,
voidElimination,
independent_functionElimination,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cumulativity,
instantiate,
promote_hyp,
sqequalIntensionalEquality,
dependent_pairFormation,
applyLambdaEquality,
hyp_replacement,
multiplyEquality
Latex:
\mforall{}[k,m:\mBbbN{}]. k \mleq{} m supposing \mexists{}f:\mBbbN{}k {}\mrightarrow{} \mBbbN{}m. increasing(f;k)
Date html generated:
2019_06_20-AM-11_33_27
Last ObjectModification:
2018_08_03-PM-05_11_14
Theory : int_1
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