Nuprl Lemma : fun_exp_compose2
∀[A,B:Type]. ∀[h:A ⟶ A]. ∀[n:ℕ]. (λg.(g o h)^n = (λg.(g o h^n)) ∈ ((A ⟶ B) ⟶ A ⟶ B))
Proof
Definitions occuring in Statement :
fun_exp: f^n,
compose: f o g,
nat: ℕ,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
top: Top,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
compose: f o g,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
fun_exp_unroll,
false_wf,
le_wf,
equal_wf,
squash_wf,
true_wf,
eta_conv,
iff_weakening_equal,
decidable__le,
subtract_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,
le_weakening2,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
le_weakening,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
compose_wf,
fun_exp_wf,
not-le-2,
not-equal-2,
nat_wf,
fun_exp_add_apply1,
fun_exp_apply_add1
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,
dependent_set_memberEquality,
independent_pairFormation,
isect_memberEquality,
voidEquality,
because_Cache,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
unionElimination,
addEquality,
intEquality,
minusEquality,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
hyp_replacement,
applyLambdaEquality,
functionExtensionality
Latex:
\mforall{}[A,B:Type]. \mforall{}[h:A {}\mrightarrow{} A]. \mforall{}[n:\mBbbN{}]. (\mlambda{}g.(g o h)\^{}n = (\mlambda{}g.(g o h\^{}n)))
Date html generated:
2017_04_14-AM-07_34_50
Last ObjectModification:
2017_02_27-PM-03_07_51
Theory : fun_1
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