Nuprl Lemma : fun_exp-fixedpoint
∀[T:Type]. ∀[f:T ⟶ T]. ∀[x:T]. ∀[n:ℕ]. ((f^n x) = x ∈ T) supposing (f x) = x ∈ T
Proof
Definitions occuring in Statement :
fun_exp: f^n,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
prop: ℙ,
all: ∀x:A. B[x],
top: Top,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
compose: f o g
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
fun_exp0_lemma,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
nat_wf,
equal_wf,
fun_exp_unroll,
le_weakening2,
le_wf,
eq_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
le_weakening,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
voidEquality,
unionElimination,
independent_pairFormation,
productElimination,
addEquality,
applyEquality,
intEquality,
minusEquality,
because_Cache,
cumulativity,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
dependent_set_memberEquality,
baseApply,
closedConclusion,
baseClosed,
equalityElimination,
impliesFunctionality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T]. \mforall{}[x:T]. \mforall{}[n:\mBbbN{}]. ((f\^{}n x) = x) supposing (f x) = x
Date html generated:
2017_04_14-AM-07_35_00
Last ObjectModification:
2017_02_27-PM-03_07_54
Theory : fun_1
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