Nuprl Lemma : increasing_implies
∀[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,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
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,
increasing: increasing(f;k),
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
lelt: i ≤ j < k,
true: True,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
sq_type: SQType(T),
top: Top,
subtract: n - m,
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+
Lemmas referenced :
istype-less_than,
member-less_than,
int_seg_wf,
increasing_wf,
istype-int,
istype-nat,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
subtract-1-ge-0,
le-add-cancel-alt,
add_functionality_wrt_le,
less-iff-le,
not-lt-2,
false_wf,
decidable__lt,
int_subtype_base,
subtype_base_sq,
zero-mul,
add-mul-special,
add-swap,
add-associates,
zero-add,
add-commutes,
minus-one-mul,
lelt_wf,
subtract_wf,
equal-wf-T-base,
less_than_wf,
int_seg_properties,
sq_stable__le,
add-member-int_seg2,
decidable__le,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul-top,
add-zero,
le-add-cancel2,
istype-le,
decidable__int_equal,
not-equal-2,
le_antisymmetry_iff,
minus-minus,
le-add-cancel,
less_than_functionality,
istype-sqequal,
le_weakening2,
le_weakening,
set_subtype_base,
add-is-int-iff,
le_reflexive,
one-mul,
two-mul,
mul-distributes-right,
mul-associates,
omega-shadow,
mul-distributes,
mul-swap,
mul-commutes,
minus-zero,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
isect_memberEquality_alt,
applyEquality,
independent_isectElimination,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
universeIsType,
natural_numberEquality,
functionIsType,
lambdaFormation_alt,
intWeakElimination,
independent_pairFormation,
productElimination,
imageElimination,
independent_functionElimination,
voidElimination,
lambdaEquality_alt,
dependent_functionElimination,
functionIsTypeImplies,
lambdaEquality,
unionElimination,
cumulativity,
instantiate,
promote_hyp,
equalityTransitivity,
minusEquality,
multiplyEquality,
addEquality,
voidEquality,
isect_memberEquality,
dependent_set_memberEquality,
baseClosed,
intEquality,
functionExtensionality,
applyLambdaEquality,
equalitySymmetry,
hyp_replacement,
lambdaFormation,
imageMemberEquality,
closedConclusion,
dependent_set_memberEquality_alt,
Error :memTop,
productIsType,
dependent_pairFormation_alt,
equalityIstype,
baseApply,
sqequalBase
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[f:\mBbbN{}k {}\mrightarrow{} \mBbbZ{}]. \{\mforall{}[x,y:\mBbbN{}k]. f x < f y supposing x < y\} supposing increasing(f;k)
Date html generated:
2020_05_19-PM-09_36_05
Last ObjectModification:
2020_01_04-PM-07_56_48
Theory : int_1
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