### Nuprl Lemma : double-lsum-swap

`∀[T,S:Type]. ∀[K:T List]. ∀[L:S List]. ∀[f:T ⟶ S ⟶ ℤ].`
`  (Σ(Σ(f[t;s] | s ∈ L) | t ∈ K) = Σ(Σ(f[t;s] | t ∈ K) | s ∈ L) ∈ ℤ)`

Proof

Definitions occuring in Statement :  lsum: `Σ(f[x] | x ∈ L)` list: `T List` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` function: `x:A ⟶ B[x]` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
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]` top: `Top` and: `P ∧ Q` prop: `ℙ` or: `P ∨ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` nil: `[]` it: `⋅` sq_type: `SQType(T)` less_than: `a < b` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf istype-less_than list-cases lsum_nil_lemma list_wf equal_wf squash_wf true_wf lsum-0 subtype_rel_self iff_weakening_equal product_subtype_list colength-cons-not-zero colength_wf_list istype-le subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le le_wf lsum_cons_lemma istype-nat istype-universe lsum_wf l_member_wf add_functionality_wrt_eq lsum-add
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 isect_memberEquality_alt voidElimination sqequalRule independent_pairFormation universeIsType axiomEquality isectIsTypeImplies inhabitedIsType functionIsTypeImplies unionElimination functionIsType because_Cache equalityTransitivity equalitySymmetry applyEquality imageElimination intEquality imageMemberEquality baseClosed instantiate productElimination promote_hyp hypothesis_subsumption equalityIstype dependent_set_memberEquality_alt applyLambdaEquality baseApply closedConclusion sqequalBase universeEquality addEquality setIsType

Latex:
\mforall{}[T,S:Type].  \mforall{}[K:T  List].  \mforall{}[L:S  List].  \mforall{}[f:T  {}\mrightarrow{}  S  {}\mrightarrow{}  \mBbbZ{}].
(\mSigma{}(\mSigma{}(f[t;s]  |  s  \mmember{}  L)  |  t  \mmember{}  K)  =  \mSigma{}(\mSigma{}(f[t;s]  |  t  \mmember{}  K)  |  s  \mmember{}  L))

Date html generated: 2020_05_19-PM-09_48_02
Last ObjectModification: 2019_11_12-PM-11_50_13

Theory : list_1

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