Nuprl Lemma : fun_exp-increasing
∀[T:Type]. ((T ⊆r ℤ) ⇒ (∀f:T ⟶ T. ((∀n:T. n < f n) ⇒ (∀n:T. ∀i,j:ℕ. (i < j ⇒ f^i n < f^j n)))))
Proof
Definitions occuring in Statement :
fun_exp: f^n,
nat: ℕ,
less_than: a < b,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
nat: ℕ,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
subtract: n - m,
sq_type: SQType(T),
squash: ↓T,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
compose: f o g,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
less_than: a < b
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
member-less_than,
fun_exp_wf,
subtype_base_sq,
int_subtype_base,
add-associates,
nat_wf,
add-swap,
subtract_wf,
squash_wf,
true_wf,
fun_exp_add,
false_wf,
le_wf,
iff_weakening_equal,
fun_exp1_lemma,
decidable__le,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-zero,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
not-ge-2,
less-iff-le,
minus-minus,
add_nat_wf,
equal_wf,
less_than_transitivity2,
le_weakening2,
all_wf,
subtype_rel_wf,
add-mul-special,
zero-mul,
le_reflexive,
one-mul,
two-mul,
mul-distributes-right,
omega-shadow,
mul-distributes,
mul-associates,
mul-commutes,
le-add-cancel-alt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
applyEquality,
cumulativity,
functionExtensionality,
because_Cache,
instantiate,
intEquality,
isect_memberEquality,
voidEquality,
minusEquality,
addEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
unionElimination,
functionEquality,
multiplyEquality,
addLevel,
levelHypothesis
Latex:
\mforall{}[T:Type]
((T \msubseteq{}r \mBbbZ{}) {}\mRightarrow{} (\mforall{}f:T {}\mrightarrow{} T. ((\mforall{}n:T. n < f n) {}\mRightarrow{} (\mforall{}n:T. \mforall{}i,j:\mBbbN{}. (i < j {}\mRightarrow{} f\^{}i n < f\^{}j n)))))
Date html generated:
2017_04_14-AM-07_35_03
Last ObjectModification:
2017_02_27-PM-03_08_09
Theory : fun_1
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