Nuprl Lemma : rng_prod_unroll_hi
∀[r:CRng]. ∀[n:ℕ+]. ∀[F:ℕn ⟶ |r|]. ((Π(r) 0 ≤ i < n. F[i]) = ((Π(r) 0 ≤ i < n - 1. F[i]) * F[n - 1]) ∈ |r|)
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
equal: s = t ∈ T,
rng_prod: rng_prod,
crng: CRng,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
rng: Rng,
crng: CRng,
nat_plus: ℕ+,
guard: {T},
implies: P ⇒ Q,
sq_type: SQType(T),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
prop: ℙ,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_apply: x[s],
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
ycomb: Y,
itop: Π(op,id) lb ≤ i < ub. E[i],
grp_id: e,
pi1: fst(t),
pi2: snd(t),
grp_op: *,
mul_mon_of_rng: r↓xmn,
mon_itop: Π lb ≤ i < ub. E[i],
subtract: n - m,
rng_prod: rng_prod,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
grp_car: |g|,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
squash: ↓T
Lemmas referenced :
crng_wf,
nat_plus_wf,
rng_car_wf,
int_seg_wf,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
lelt_wf,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
false_wf,
rng_one_wf,
rng_times_wf,
infix_ap_wf,
iff_weakening_equal,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
rng_prod_wf,
mul_mon_of_rng_wf_c,
mon_itop_unroll_hi,
true_wf,
squash_wf,
equal_wf
Rules used in proof :
hypothesisEquality,
rename,
setElimination,
functionEquality,
independent_functionElimination,
independent_isectElimination,
intEquality,
cumulativity,
isectElimination,
instantiate,
unionElimination,
hypothesis,
natural_numberEquality,
because_Cache,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
approximateComputation,
lambdaFormation,
independent_pairFormation,
dependent_set_memberEquality,
functionExtensionality,
applyEquality,
sqequalRule,
baseClosed,
imageMemberEquality,
productElimination,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[F:\mBbbN{}n {}\mrightarrow{} |r|].
((\mPi{}(r) 0 \mleq{} i < n. F[i]) = ((\mPi{}(r) 0 \mleq{} i < n - 1. F[i]) * F[n - 1]))
Date html generated:
2018_05_21-PM-09_33_22
Last ObjectModification:
2017_12_14-PM-07_02_32
Theory : matrices
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