Nuprl Lemma : primtailrec-unroll
∀[n:ℤ]. ∀[b,c:Top]. (primtailrec(n;0;b;c) ~ if n <z 1 then b else c (n - 1) primtailrec(n - 1;0;b;c) fi )
Proof
Definitions occuring in Statement :
primtailrec: primtailrec(n;i;b;f),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
subtract: n - m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
primtailrec: primtailrec(n;i;b;f),
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
gt: i > j,
nat: ℕ,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
prop: ℙ,
subtract: n - m,
has-value: (a)↓,
decidable: Dec(P)
Lemmas referenced :
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-void,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
istype-less_than,
istype-assert,
not-gt-2,
istype-top,
istype-int,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
subtract-1-ge-0,
istype-nat,
value-type-has-value,
int-value-type,
zero-add,
subtract-add-cancel,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
add-subtract-cancel,
int_subtype_base,
subtract_wf,
decidable__le,
istype-false,
not-le-2,
minus-minus,
le-add-cancel,
istype-le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
because_Cache,
lessCases,
axiomSqEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
independent_pairFormation,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
callbyvalueReduce,
sqleReflexivity,
Error :functionIsType,
setElimination,
rename,
intWeakElimination,
Error :universeIsType,
Error :lambdaEquality_alt,
Error :functionIsTypeImplies,
intEquality,
addEquality,
minusEquality,
Error :dependent_set_memberEquality_alt
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[b,c:Top].
(primtailrec(n;0;b;c) \msim{} if n <z 1 then b else c (n - 1) primtailrec(n - 1;0;b;c) fi )
Date html generated:
2019_06_20-AM-11_27_28
Last ObjectModification:
2019_01_28-PM-05_02_19
Theory : call!by!value_2
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