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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