Nuprl Lemma : fun_exp_compose
∀[T:Type]. ∀[n:ℕ]. ∀[h,f:T ⟶ T]. ((f^n o h) = primrec(n;h;λi,g. (f o g)) ∈ (T ⟶ T))
Proof
Definitions occuring in Statement :
fun_exp: f^n,
compose: f o g,
primrec: primrec(n;b;c),
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,
fun_exp: f^n,
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,
compose: f o g,
sq_type: SQType(T),
squash: ↓T
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
primrec0_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,
subtype_base_sq,
int_subtype_base,
primrec_add,
le_wf,
not-le-2,
compose_wf,
int_seg_wf,
primrec1_lemma,
equal_wf,
primrec_wf,
squash_wf,
true_wf,
fun_exp_wf,
comp_assoc,
iff_weakening_equal
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,
isect_memberEquality,
axiomEquality,
functionEquality,
cumulativity,
voidEquality,
unionElimination,
independent_pairFormation,
productElimination,
addEquality,
applyEquality,
intEquality,
minusEquality,
because_Cache,
universeEquality,
functionExtensionality,
instantiate,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[h,f:T {}\mrightarrow{} T]. ((f\^{}n o h) = primrec(n;h;\mlambda{}i,g. (f o g)))
Date html generated:
2017_04_14-AM-07_34_37
Last ObjectModification:
2017_02_27-PM-03_07_44
Theory : fun_1
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