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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