Nuprl Lemma : fun_exp_add
∀[T:Type]. ∀[n,m:ℕ]. ∀[f:T ⟶ T]. (f^n + m = (f^n o f^m) ∈ (T ⟶ T))
Proof
Definitions occuring in Statement :
fun_exp: f^n,
compose: f o g,
nat: ℕ,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
add: n + m,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
sq_stable: SqStable(P),
true: True,
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
fun_exp: f^n
Lemmas referenced :
uall_wf,
squash_wf,
true_wf,
nat_wf,
equal_wf,
fun_exp_wf,
add_nat_wf,
sq_stable__le,
le_wf,
fun_exp_compose,
iff_weakening_equal,
primrec_add,
compose_wf,
int_seg_wf,
primrec_wf
Rules used in proof :
cut,
applyEquality,
instantiate,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
because_Cache,
sqequalRule,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
lambdaFormation,
natural_numberEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
functionExtensionality,
independent_isectElimination,
productElimination,
isect_memberFormation,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[T:Type]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[f:T {}\mrightarrow{} T]. (f\^{}n + m = (f\^{}n o f\^{}m))
Date html generated:
2017_04_14-AM-07_34_39
Last ObjectModification:
2017_02_27-PM-03_07_33
Theory : fun_1
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