Nuprl Lemma : rprod-split-first
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. rprod(n;m;k.x[k]) = (x[n] * rprod(n + 1;m;k.x[k])) supposing n ≤ m
Proof
Definitions occuring in Statement :
rprod: rprod(n;m;k.x[k]),
req: x = y,
rmul: a * b,
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rprod-split,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
req_witness,
rprod_wf,
int_seg_wf,
rmul_wf,
decidable__lt,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
real_wf,
rprod-single,
req_weakening,
req_functionality,
rmul_functionality
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
addEquality,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
applyEquality,
dependent_set_memberEquality_alt,
productIsType,
closedConclusion,
setElimination,
rename,
productElimination,
functionIsType,
inhabitedIsType
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. rprod(n;m;k.x[k]) = (x[n] * rprod(n + 1;m;k.x[k])) supposing n \mleq{} m
Date html generated:
2019_10_29-AM-10_18_56
Last ObjectModification:
2019_01_19-AM-11_01_55
Theory : reals
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