Nuprl Lemma : rpolydiv-rec
∀[n:ℤ]. ∀[a,z:Top].
((rpolydiv(n;a;z) (n - 1) ~ a n) ∧ (∀i:ℕn - 1. (rpolydiv(n;a;z) i ~ (a (i + 1)) + (z * (rpolydiv(n;a;z) (i + 1))))))
Proof
Definitions occuring in Statement :
rpolydiv: rpolydiv(n;a;z),
rmul: a * b,
radd: a + b,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
and: P ∧ Q,
rpolydiv: rpolydiv(n;a;z),
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
int_seg: {i..j-},
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
lelt: i ≤ j < k,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
istype-top,
istype-int,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
primrec0_lemma,
int_seg_wf,
subtract_wf,
primrec-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
int_seg_properties,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
itermAdd_wf,
int_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
independent_pairFormation,
because_Cache,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
instantiate,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
hypothesisEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
equalityTransitivity,
equalitySymmetry,
lambdaFormation_alt,
setElimination,
rename,
inhabitedIsType,
equalityElimination,
productElimination,
equalityIstype,
promote_hyp,
axiomSqEquality
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[a,z:Top].
((rpolydiv(n;a;z) (n - 1) \msim{} a n)
\mwedge{} (\mforall{}i:\mBbbN{}n - 1. (rpolydiv(n;a;z) i \msim{} (a (i + 1)) + (z * (rpolydiv(n;a;z) (i + 1))))))
Date html generated:
2019_10_29-AM-10_15_26
Last ObjectModification:
2019_01_14-PM-07_12_14
Theory : reals
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