Nuprl Lemma : q-linear-unroll
∀[k:ℕ+]. ∀[X:ℕ ⟶ ℚ]. ∀[y:ℚ List].
q-linear(k;j.X[j];y) = (q-linear(k - 1;j.X[j];y) + (X[k] * y[k - 1])) ∈ ℚ supposing k ≤ ||y||
Proof
Definitions occuring in Statement :
q-linear: q-linear(k;i.X[i];y),
qmul: r * s,
qadd: r + s,
rationals: ℚ,
select: L[n],
length: ||as||,
list: T List,
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
q-linear: q-linear(k;i.X[i];y),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
so_apply: x[s],
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
rationals_wf,
qadd_wf,
nat_wf,
nat_plus_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
false_wf,
le_wf,
sum_unroll_hi_q,
decidable__lt,
intformand_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
qmul_wf,
int_seg_properties,
decidable__le,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
select_wf,
length_wf,
int_seg_wf,
qsum_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
nat_plus_subtype_nat,
iff_weakening_equal,
subtract-add-cancel,
qadd_assoc,
list_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
functionExtensionality,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
because_Cache,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality,
independent_pairFormation,
lambdaFormation,
int_eqEquality,
addEquality,
productElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination,
axiomEquality,
functionEquality
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[X:\mBbbN{} {}\mrightarrow{} \mBbbQ{}]. \mforall{}[y:\mBbbQ{} List].
q-linear(k;j.X[j];y) = (q-linear(k - 1;j.X[j];y) + (X[k] * y[k - 1])) supposing k \mleq{} ||y||
Date html generated:
2018_05_22-AM-00_17_30
Last ObjectModification:
2017_07_26-PM-06_53_27
Theory : rationals
Home
Index