Nuprl Lemma : iteration_terminates
∀[T:Type]
∀f:T ⟶ T. ∀m:T ⟶ ℕ.
∀x:T. ∃n:ℕ. ((f (f^n x)) = (f^n x) ∈ T)
supposing ∀x:T. (((m (f x)) ≤ (m x)) ∧ (f x) = x ∈ T supposing (m (f x)) = (m x) ∈ ℤ)
Proof
Definitions occuring in Statement :
fun_exp: f^n,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
and: P ∧ Q,
le: A ≤ B,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
true: True,
nat_plus: ℕ+,
prop: ℙ,
guard: {T},
squash: ↓T,
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
ge: i ≥ j ,
less_than: a < b
Lemmas referenced :
le_witness_for_triv,
istype-le,
istype-nat,
istype-int,
set_subtype_base,
le_wf,
int_subtype_base,
istype-universe,
fun_exp0_lemma,
istype-void,
decidable__le,
subtract_wf,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
minus-zero,
add-associates,
zero-add,
add-commutes,
le-add-cancel,
fun_exp_wf,
less-iff-le,
minus-minus,
add_functionality_wrt_le,
istype-less_than,
primrec-wf2,
or_wf,
equal_wf,
fun_exp_add1_sub,
equal-wf-base,
decidable__lt,
le_weakening2,
le-add-cancel2,
decidable__int_equal,
not-equal-2,
not-lt-2,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
fun_exp_add,
istype_wf,
add_nat_wf,
sq_stable__le,
istype-sqequal,
nat_properties,
le-add-cancel-alt,
le_reflexive,
one-mul,
two-mul,
mul-distributes-right,
omega-shadow
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
Error :lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
extract_by_obid,
isectElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_isectElimination,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
rename,
Error :universeIsType,
Error :functionIsType,
Error :productIsType,
applyEquality,
setElimination,
because_Cache,
Error :isectIsType,
Error :equalityIstype,
intEquality,
closedConclusion,
natural_numberEquality,
sqequalBase,
instantiate,
universeEquality,
voidElimination,
unionElimination,
Error :inlFormation_alt,
independent_pairFormation,
addEquality,
minusEquality,
multiplyEquality,
independent_functionElimination,
Error :inrFormation_alt,
Error :unionIsType,
Error :dependent_set_memberEquality_alt,
Error :setIsType,
hyp_replacement,
applyLambdaEquality,
productEquality,
isectEquality,
Error :equalityIsType1,
imageElimination,
imageMemberEquality,
baseClosed,
unionEquality,
Error :dependent_pairFormation_alt,
promote_hyp
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T. \mforall{}m:T {}\mrightarrow{} \mBbbN{}.
\mforall{}x:T. \mexists{}n:\mBbbN{}. ((f (f\^{}n x)) = (f\^{}n x))
supposing \mforall{}x:T. (((m (f x)) \mleq{} (m x)) \mwedge{} (f x) = x supposing (m (f x)) = (m x))
Date html generated:
2019_06_20-PM-00_27_05
Last ObjectModification:
2018_11_23-PM-00_41_17
Theory : fun_1
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