Nuprl Lemma : implies-strictly-increasing-seq
∀[n:ℕ]. ∀[s:ℕn ⟶ ℤ]. ((∀i:ℕn - 1. s i < s (i + 1)) ⇒ strictly-increasing-seq(n;s))
Proof
Definitions occuring in Statement :
strictly-increasing-seq: strictly-increasing-seq(n;s),
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
strictly-increasing-seq: strictly-increasing-seq(n;s),
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
so_apply: x[s],
guard: {T},
ge: i ≥ j ,
less_than: a < b,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
squash: ↓T,
sq_type: SQType(T)
Lemmas referenced :
all_wf,
int_seg_wf,
subtract_wf,
less_than_wf,
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
add-associates,
nat_wf,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
le-add-cancel2,
and_wf,
le_wf,
add-member-int_seg2,
decidable__le,
not-le-2,
zero-add,
add-zero,
member-less_than,
less_than_transitivity2,
le_weakening2,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
int_subtype_base,
equal_wf,
not-ge-2,
minus-minus,
le-add-cancel,
add-is-int-iff,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
minus-zero,
omega-shadow,
mul-distributes,
mul-associates,
le-add-cancel-alt,
subtype_rel_sets,
subtype_base_sq,
decidable__int_equal,
not-equal-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
voidElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
minusEquality,
isect_memberEquality,
voidEquality,
intEquality,
functionEquality,
intWeakElimination,
dependent_pairFormation,
sqequalIntensionalEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
addLevel,
multiplyEquality,
levelHypothesis,
imageMemberEquality,
setEquality,
instantiate,
cumulativity
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[s:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. ((\mforall{}i:\mBbbN{}n - 1. s i < s (i + 1)) {}\mRightarrow{} strictly-increasing-seq(n;s))
Date html generated:
2017_04_14-AM-07_26_19
Last ObjectModification:
2017_02_27-PM-02_56_14
Theory : bar-induction
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