Nuprl Lemma : primrec_add
∀[T:Type]. ∀[n,m:ℕ]. ∀[b:T]. ∀[c:ℕn + m ⟶ T ⟶ T]. (primrec(n + m;b;c) ~ primrec(n;primrec(m;b;c);λi,t. (c (i + m) t))\000C)
Proof
Definitions occuring in Statement :
primrec: primrec(n;b;c),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type,
sqequal: s ~ 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,
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
int_seg_wf,
primrec0_lemma,
zero-add,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
subtype_rel_dep_function,
subtype_rel_sets,
and_wf,
le_wf,
decidable__lt,
not-lt-2,
nat_wf,
le-add-cancel2,
subtype_rel_self,
primrec-unroll,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
uiff_transitivity,
assert_of_bnot,
not_functionality_wrt_uiff,
assert_wf,
bnot_wf,
not_wf,
general_arith_equation1,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
omega-shadow,
mul-associates,
mul-swap
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
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,
axiomSqEquality,
functionEquality,
addEquality,
because_Cache,
voidEquality,
Error :functionIsType,
Error :universeIsType,
Error :inhabitedIsType,
unionElimination,
independent_pairFormation,
productElimination,
applyEquality,
intEquality,
minusEquality,
setEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
universeEquality,
multiplyEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[b:T]. \mforall{}[c:\mBbbN{}n + m {}\mrightarrow{} T {}\mrightarrow{} T].
(primrec(n + m;b;c) \msim{} primrec(n;primrec(m;b;c);\mlambda{}i,t. (c (i + m) t)))
Date html generated:
2019_06_20-AM-11_27_47
Last ObjectModification:
2018_09_26-AM-10_58_09
Theory : call!by!value_2
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