Nuprl Lemma : double_sum_difference
∀[n,m:ℕ]. ∀[f,g:ℕn ⟶ ℕm ⟶ ℤ]. ∀[d:ℤ].
sum(f[x;y] | x < n; y < m) = (sum(g[x;y] | x < n; y < m) + d) ∈ ℤ
supposing sum(f[x;y] - g[x;y] | x < n; y < m) = d ∈ 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],
function: x:A ⟶ B[x],
subtract: n - m,
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,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
double_sum: sum(f[x; y] | x < n; y < m),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
false: False,
uiff: uiff(P;Q),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
equal-wf-T-base,
double_sum_wf,
subtract_wf,
int_seg_wf,
int_subtype_base,
nat_wf,
sum_difference,
sum_wf,
subtype_base_sq,
sum_functionality,
int_seg_properties,
nat_properties,
decidable__equal_int,
add-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermVar_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
hypothesis,
Error :universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
natural_numberEquality,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
Error :inhabitedIsType,
functionEquality,
Error :functionIsType,
independent_isectElimination,
instantiate,
cumulativity,
dependent_functionElimination,
independent_functionElimination,
lambdaFormation,
productElimination,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation
Latex:
\mforall{}[n,m:\mBbbN{}]. \mforall{}[f,g:\mBbbN{}n {}\mrightarrow{} \mBbbN{}m {}\mrightarrow{} \mBbbZ{}]. \mforall{}[d:\mBbbZ{}].
sum(f[x;y] | x < n; y < m) = (sum(g[x;y] | x < n; y < m) + d)
supposing sum(f[x;y] - g[x;y] | x < n; y < m) = d
Date html generated:
2019_06_20-PM-02_29_44
Last ObjectModification:
2018_09_26-PM-06_05_08
Theory : num_thy_1
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