### Nuprl Lemma : le-tuple-sum

`∀[P:Type]. ∀[as:P List]. ∀[G:P ⟶ Type]. ∀[x:tuple-type(map(G;as))]. ∀[f:i:P ⟶ (G i) ⟶ ℕ].`
`  ∀k:ℕ||as||. ((f as[k] x.k) ≤ tuple-sum(f;as;x))`

Proof

Definitions occuring in Statement :  select-tuple: `x.n` tuple-sum: `tuple-sum(f;L;x)` tuple-type: `tuple-type(L)` select: `L[n]` length: `||as||` map: `map(f;as)` list: `T List` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀x:A. B[x]` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` and: `P ∧ Q` prop: `ℙ` le: `A ≤ B` or: `P ∨ Q` tuple-sum: `tuple-sum(f;L;x)` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` ifthenelse: `if b then t else f fi ` btrue: `tt` int_seg: `{i..j-}` lelt: `i ≤ j < k` cons: `[a / b]` less_than': `less_than'(a;b)` colength: `colength(L)` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` decidable: `Dec(P)` subtype_rel: `A ⊆r B` bfalse: `ff` bool: `𝔹` unit: `Unit` subtract: `n - m` eq_int: `(i =z j)` select-tuple: `x.n` top: `Top` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` sq_stable: `SqStable(P)` true: `True` pi1: `fst(t)` pi2: `snd(t)` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` cand: `A c∧ B`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut thin lambdaFormation_alt extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination Error :memTop,  sqequalRule independent_pairFormation universeIsType voidElimination isect_memberEquality_alt productElimination equalityTransitivity equalitySymmetry functionIsTypeImplies inhabitedIsType isectIsTypeImplies unionElimination baseClosed functionIsType applyEquality because_Cache promote_hyp hypothesis_subsumption equalityIstype dependent_set_memberEquality_alt instantiate applyLambdaEquality imageElimination baseApply closedConclusion intEquality sqequalBase equalityElimination addEquality productIsType cumulativity universeEquality imageMemberEquality minusEquality

Latex:
\mforall{}[P:Type].  \mforall{}[as:P  List].  \mforall{}[G:P  {}\mrightarrow{}  Type].  \mforall{}[x:tuple-type(map(G;as))].  \mforall{}[f:i:P  {}\mrightarrow{}  (G  i)  {}\mrightarrow{}  \mBbbN{}].
\mforall{}k:\mBbbN{}||as||.  ((f  as[k]  x.k)  \mleq{}  tuple-sum(f;as;x))

Date html generated: 2020_05_19-PM-10_00_19
Last ObjectModification: 2020_01_01-AM-10_58_21

Theory : tuples

Home Index