Nuprl Lemma : fun_exp_com
∀[T:Type]. ∀[n,m:ℕ]. ∀[f:T ⟶ T]. ∀[x:T]. ((f^m (f^n x)) = (f^n (f^m x)) ∈ T)
Proof
Definitions occuring in Statement :
fun_exp: f^n,
nat: ℕ,
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,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
nat: ℕ,
top: Top,
all: ∀x:A. B[x],
sq_stable: SqStable(P)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
fun_exp_add_apply,
fun_exp_wf,
iff_weakening_equal,
add-commutes,
add_nat_wf,
nat_wf,
sq_stable__le,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
because_Cache,
cumulativity,
functionExtensionality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
setElimination,
rename,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
addEquality,
lambdaFormation,
dependent_functionElimination,
axiomEquality,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[f:T {}\mrightarrow{} T]. \mforall{}[x:T]. ((f\^{}m (f\^{}n x)) = (f\^{}n (f\^{}m x)))
Date html generated:
2017_04_14-AM-07_34_45
Last ObjectModification:
2017_02_27-PM-03_07_38
Theory : fun_1
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