Nuprl Lemma : increasing_is_id
∀[k:ℕ]. ∀[f:ℕk ⟶ ℕk]. ∀[i:ℕk]. ((f i) = i ∈ ℤ) 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],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
true: True,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
increasing: increasing(f;k),
sq_type: SQType(T)
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
int_seg_wf,
increasing_wf,
istype-false,
istype-le,
subtract-1-ge-0,
le_weakening2,
istype-nat,
subtract_wf,
decidable__le,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
istype-void,
istype-int,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__lt,
not-lt-2,
add-mul-special,
zero-mul,
le-add-cancel-alt,
istype-sqequal,
set_subtype_base,
lelt_wf,
int_subtype_base,
int_seg_properties,
increasing_implies,
le-add-cancel2,
false_wf,
decidable__int_equal,
subtype_base_sq,
minus-zero,
not-equal-2,
equal_wf,
le_wf,
one-mul,
mul-distributes-right,
two-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :universeIsType,
sqequalRule,
Error :lambdaEquality_alt,
dependent_functionElimination,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
productElimination,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
because_Cache,
functionExtensionality,
applyEquality,
closedConclusion,
Error :functionIsType,
equalityTransitivity,
equalitySymmetry,
unionElimination,
addEquality,
minusEquality,
Error :productIsType,
Error :dependent_pairFormation_alt,
baseApply,
baseClosed,
intEquality,
Error :equalityIsType1,
promote_hyp,
voidEquality,
isect_memberEquality,
lambdaEquality,
dependent_set_memberEquality,
lambdaFormation,
cumulativity,
instantiate,
sqequalIntensionalEquality,
dependent_pairFormation,
multiplyEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[f:\mBbbN{}k {}\mrightarrow{} \mBbbN{}k]. \mforall{}[i:\mBbbN{}k]. ((f i) = i) supposing increasing(f;k)
Date html generated:
2019_06_20-AM-11_33_32
Last ObjectModification:
2018_10_18-PM-04_02_07
Theory : int_1
Home
Index