Nuprl Lemma : fun_exp_add_apply1
∀[T:Type]. ∀[n:ℕ]. ∀[f:T ⟶ T]. ∀[x:T].  ((f^n (f x)) = (f^n + 1 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]
, 
add: n + m
, 
natural_number: $n
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
nat: ℕ
, 
le: A ≤ B
, 
and: P ∧ Q
, 
less_than': less_than'(a;b)
, 
false: False
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
all: ∀x:A. B[x]
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
uiff: uiff(P;Q)
, 
uimplies: b supposing a
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
subtract: n - m
, 
subtype_rel: A ⊆r B
, 
top: Top
, 
true: True
, 
guard: {T}
, 
eq_int: (i =z j)
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
compose: f o g
, 
btrue: tt
Lemmas referenced : 
nat_wf, 
false_wf, 
le_wf, 
fun_exp_wf, 
decidable__le, 
not-le-2, 
sq_stable__le, 
condition-implies-le, 
minus-add, 
minus-one-mul, 
zero-add, 
minus-one-mul-top, 
add-associates, 
add-swap, 
add-commutes, 
add_functionality_wrt_le, 
add-zero, 
le-add-cancel, 
equal_wf, 
squash_wf, 
true_wf, 
fun_exp_add_apply, 
iff_weakening_equal, 
fun_exp_unroll
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
hypothesis, 
hypothesisEquality, 
sqequalRule, 
sqequalHypSubstitution, 
isect_memberEquality, 
isectElimination, 
thin, 
axiomEquality, 
because_Cache, 
functionEquality, 
cumulativity, 
extract_by_obid, 
universeEquality, 
dependent_set_memberEquality, 
natural_numberEquality, 
independent_pairFormation, 
lambdaFormation, 
applyEquality, 
addEquality, 
setElimination, 
rename, 
dependent_functionElimination, 
unionElimination, 
voidElimination, 
productElimination, 
independent_functionElimination, 
independent_isectElimination, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
lambdaEquality, 
voidEquality, 
intEquality, 
minusEquality, 
functionExtensionality, 
equalityTransitivity, 
equalitySymmetry
Latex:
\mforall{}[T:Type].  \mforall{}[n:\mBbbN{}].  \mforall{}[f:T  {}\mrightarrow{}  T].  \mforall{}[x:T].    ((f\^{}n  (f  x))  =  (f\^{}n  +  1  x))
Date html generated:
2017_04_14-AM-07_34_41
Last ObjectModification:
2017_02_27-PM-03_07_34
Theory : fun_1
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