Nuprl Lemma : primrec-wf-int_seg
∀[a,b:ℤ].
∀[P:{a..b-} ⟶ ℙ]. ∀[init:P[a]]. ∀[s:∀n:ℕb - 1 - a. (P[a + n] ⇒ P[a + n + 1])]. ∀[n:{a..b-}].
(primrec(n - a;init;s) ∈ P[n])
supposing a < b
Proof
Definitions occuring in Statement :
primrec: primrec(n;b;c),
int_seg: {i..j-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
guard: {T},
prop: ℙ,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
sq_stable: SqStable(P),
nat_plus: ℕ+
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
primrec-unroll,
istype-void,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
subtype_base_sq,
int_subtype_base,
add-zero,
subtract_wf,
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_seg_wf,
not-lt-2,
subtract-1-ge-0,
subtype_rel-equal,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-commutes,
le-add-cancel2,
decidable__le,
istype-false,
not-le-2,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-swap,
le-add-cancel,
decidable__lt,
le-add-cancel-alt,
istype-le,
subtype_rel_function,
le_weakening2,
subtype_rel_self,
istype-nat,
subtract_nat_wf,
sq_stable__le,
add-mul-special,
one-mul,
zero-mul,
add-member-int_seg1,
le_reflexive,
istype-int,
sq_stable__and,
le_wf,
sq_stable__less_than,
member-less_than,
add-is-int-iff,
two-mul,
mul-distributes-right,
omega-shadow,
mul-associates,
mul-distributes,
mul-swap,
mul-commutes
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
independent_pairFormation,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :universeIsType,
sqequalRule,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
because_Cache,
Error :isect_memberEquality_alt,
unionElimination,
equalityElimination,
instantiate,
cumulativity,
intEquality,
Error :dependent_pairFormation_alt,
Error :equalityIstype,
promote_hyp,
Error :functionIsType,
applyEquality,
functionExtensionality,
closedConclusion,
addEquality,
Error :dependent_set_memberEquality_alt,
minusEquality,
Error :productIsType,
imageMemberEquality,
baseClosed,
multiplyEquality,
Error :isectIsTypeImplies,
universeEquality,
baseApply
Latex:
\mforall{}[a,b:\mBbbZ{}].
\mforall{}[P:\{a..b\msupminus{}\} {}\mrightarrow{} \mBbbP{}]. \mforall{}[init:P[a]]. \mforall{}[s:\mforall{}n:\mBbbN{}b - 1 - a. (P[a + n] {}\mRightarrow{} P[a + n + 1])]. \mforall{}[n:\{a..b\msupminus{}\}].
(primrec(n - a;init;s) \mmember{} P[n])
supposing a < b
Date html generated:
2019_06_20-AM-11_27_36
Last ObjectModification:
2019_01_28-PM-05_23_48
Theory : call!by!value_2
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