Nuprl Lemma : dot-product-split-first
∀[n:ℕ+]. ∀[x,y:ℝ^n]. (x⋅y = (((x 0) * (y 0)) + λi.(x (i + 1))⋅λi.(y (i + 1))))
Proof
Definitions occuring in Statement :
dot-product: x⋅y,
real-vec: ℝ^n,
req: x = y,
rmul: a * b,
radd: a + b,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
nat: ℕ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
sq_type: SQType(T),
guard: {T},
dot-product: x⋅y,
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
req-vec: req-vec(n;x;y)
Lemmas referenced :
sq_stable__req,
real-vec_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
nat_plus_wf,
dot-product_wf,
radd_wf,
rmul_wf,
decidable__lt,
istype-less_than,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
int_seg_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
rsum_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
add-member-int_seg2,
rsum-empty,
int-to-real_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
req_functionality,
rsum-single,
req_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
dot-product-split,
nat_plus_subtype_nat,
real-vec-subtype,
radd_functionality,
dot-product_functionality,
add-commutes
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
baseClosed,
imageElimination,
because_Cache,
universeIsType,
dependent_set_memberEquality_alt,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
applyEquality,
productIsType,
addEquality,
productElimination,
instantiate,
cumulativity,
intEquality,
closedConclusion,
minusEquality,
setIsType,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
baseApply,
sqequalBase,
lambdaFormation_alt
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. (x\mcdot{}y = (((x 0) * (y 0)) + \mlambda{}i.(x (i + 1))\mcdot{}\mlambda{}i.(y (i + 1))))
Date html generated:
2019_10_30-AM-08_06_15
Last ObjectModification:
2019_07_01-AM-10_46_09
Theory : reals
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