Nuprl Lemma : pair_support_double_sum
∀[n,m:ℕ]. ∀[f:ℕn ⟶ ℕm ⟶ ℤ]. ∀[x1,x2:ℕn]. ∀[y1,y2:ℕm].
(sum(f[x;y] | x < n; y < m) = (f[x1;y1] + f[x2;y2]) ∈ ℤ) supposing
((∀x:ℕn. ∀y:ℕm. ((¬((x = x1 ∈ ℤ) ∧ (y = y1 ∈ ℤ)))
⇒ (¬((x = x2 ∈ ℤ) ∧ (y = y2 ∈ ℤ)))
⇒ (f[x;y] = 0 ∈ ℤ))) and
((¬(x1 = x2 ∈ ℤ)) ∨ (¬(y1 = y2 ∈ ℤ))))
Proof
Definitions occuring in Statement :
double_sum: sum(f[x; y] | x < n; y < m)
,
int_seg: {i..j-}
,
nat: ℕ
,
uimplies: b supposing a
,
uall: ∀[x:A]. B[x]
,
so_apply: x[s1;s2]
,
all: ∀x:A. B[x]
,
not: ¬A
,
implies: P
⇒ Q
,
or: P ∨ Q
,
and: P ∧ Q
,
function: x:A ⟶ B[x]
,
add: n + m
,
natural_number: $n
,
int: ℤ
,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
uimplies: b supposing a
,
double_sum: sum(f[x; y] | x < n; y < m)
,
all: ∀x:A. B[x]
,
int_seg: {i..j-}
,
decidable: Dec(P)
,
or: P ∨ Q
,
implies: P
⇒ Q
,
prop: ℙ
,
and: P ∧ Q
,
so_apply: x[s1;s2]
,
nat: ℕ
,
subtype_rel: A ⊆r B
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
rev_implies: P
⇐ Q
,
iff: P
⇐⇒ Q
,
true: True
,
top: Top
,
exists: ∃x:A. B[x]
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
ge: i ≥ j
,
lelt: i ≤ j < k
,
false: False
,
guard: {T}
,
not: ¬A
,
squash: ↓T
,
sq_type: SQType(T)
Lemmas referenced :
decidable__equal_int,
not_wf,
equal_wf,
istype-int,
int_seg_wf,
int_subtype_base,
nat_wf,
singleton_support_sum,
sum_wf,
equal-wf-base,
set_subtype_base,
lelt_wf,
iff_weakening_equal,
subtype_rel_self,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
int_seg_properties,
true_wf,
squash_wf,
empty_support,
int_term_value_add_lemma,
itermAdd_wf,
member_wf,
int_formula_prop_or_lemma,
intformor_wf,
pair_support,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
unionElimination,
sqequalRule,
Error :functionIsType,
Error :inhabitedIsType,
hypothesisEquality,
Error :universeIsType,
isectElimination,
productEquality,
intEquality,
Error :equalityIsType4,
applyEquality,
functionExtensionality,
natural_numberEquality,
Error :isect_memberEquality_alt,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :unionIsType,
Error :lambdaEquality_alt,
independent_isectElimination,
Error :lambdaFormation_alt,
instantiate,
baseClosed,
imageMemberEquality,
independent_pairFormation,
voidEquality,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
voidElimination,
productElimination,
independent_functionElimination,
universeEquality,
imageElimination,
lambdaFormation,
lambdaEquality,
dependent_set_memberEquality,
addEquality,
promote_hyp,
cumulativity
Latex:
\mforall{}[n,m:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}m {}\mrightarrow{} \mBbbZ{}]. \mforall{}[x1,x2:\mBbbN{}n]. \mforall{}[y1,y2:\mBbbN{}m].
(sum(f[x;y] | x < n; y < m) = (f[x1;y1] + f[x2;y2])) supposing
((\mforall{}x:\mBbbN{}n. \mforall{}y:\mBbbN{}m. ((\mneg{}((x = x1) \mwedge{} (y = y1))) {}\mRightarrow{} (\mneg{}((x = x2) \mwedge{} (y = y2))) {}\mRightarrow{} (f[x;y] = 0))) and
((\mneg{}(x1 = x2)) \mvee{} (\mneg{}(y1 = y2))))
Date html generated:
2019_06_20-PM-02_29_38
Last ObjectModification:
2018_10_05-AM-11_03_56
Theory : num_thy_1
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