Nuprl Lemma : primrec-wf-upper
∀[k:ℤ]. ∀[P:{k...} ⟶ ℙ]. ∀[b:P[k]]. ∀[s:∀n:{k...}. (P[n] ⇒ P[n + 1])]. ∀[n:{k...}].
(primrec(n - k;b;λi,x. (s (i + k) x)) ∈ P[n])
Proof
Definitions occuring in Statement :
primrec: primrec(n;b;c),
int_upper: {i...},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
nat: ℕ,
ge: i ≥ j ,
cand: A c∧ B,
less_than: a < b,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
prop: ℙ
Lemmas referenced :
subtract_nat_wf,
minus-minus,
decidable__lt,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
primrec-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
subtype_base_sq,
int_subtype_base,
add-zero,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
iff_weakening_uiff,
assert_of_bnot,
istype-assert,
int_upper_wf,
not-lt-2,
subtract-1-ge-0,
subtype_rel-equal,
less-iff-le,
le-add-cancel2,
subtract_wf,
subtype_rel_function,
le_weakening2,
zero-add,
subtype_rel_self,
add-is-int-iff,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
omega-shadow,
istype-nat,
decidable__le,
istype-false,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
istype-void,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
le-add-cancel,
istype-le,
le_reflexive,
istype-int_upper,
istype-int
Rules used in proof :
intWeakElimination,
Error :lambdaEquality_alt,
Error :functionIsTypeImplies,
equalityElimination,
instantiate,
cumulativity,
intEquality,
Error :dependent_pairFormation_alt,
Error :equalityIstype,
promote_hyp,
functionExtensionality,
baseApply,
closedConclusion,
multiplyEquality,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
axiomEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
Error :isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsType,
Error :universeIsType,
applyEquality,
Error :dependent_set_memberEquality_alt,
addEquality,
setElimination,
rename,
natural_numberEquality,
extract_by_obid,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
Error :lambdaFormation_alt,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality,
universeEquality
Latex:
\mforall{}[k:\mBbbZ{}]. \mforall{}[P:\{k...\} {}\mrightarrow{} \mBbbP{}]. \mforall{}[b:P[k]]. \mforall{}[s:\mforall{}n:\{k...\}. (P[n] {}\mRightarrow{} P[n + 1])]. \mforall{}[n:\{k...\}].
(primrec(n - k;b;\mlambda{}i,x. (s (i + k) x)) \mmember{} P[n])
Date html generated:
2019_06_20-PM-01_04_41
Last ObjectModification:
2019_06_20-PM-01_01_23
Theory : call!by!value_2
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