Nuprl Lemma : dot-product-split
∀[n:ℕ]. ∀[k:ℕn]. ∀[x,y:ℝ^n]. (x⋅y = (x⋅y + λi.(x (k + i))⋅λi.(y (k + i))))
Proof
Definitions occuring in Statement :
dot-product: x⋅y,
real-vec: ℝ^n,
req: x = y,
radd: a + b,
int_seg: {i..j-},
nat: 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],
real-vec: ℝ^n,
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
sq_stable: SqStable(P),
guard: {T},
sq_type: SQType(T),
true: True,
so_apply: x[s],
top: Top,
so_lambda: λ2x.t[x],
subtract: n - m,
dot-product: x⋅y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
req-vec: req-vec(n;x;y),
pointwise-req: x[k] = y[k] for k ∈ [n,m]
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[k:\mBbbN{}n]. \mforall{}[x,y:\mBbbR{}\^{}n]. (x\mcdot{}y = (x\mcdot{}y + \mlambda{}i.(x (k + i))\mcdot{}\mlambda{}i.(y (k + i))))
Date html generated:
2020_05_20-PM-00_35_49
Last ObjectModification:
2020_01_02-PM-01_54_14
Theory : reals
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