Nuprl Lemma : sum-unroll
∀[n,f:Top]. (Σ(f[x] | x < n) ~ if (0) < (n) then Σ(f[x] | x < n - 1) + f[n - 1] else 0)
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
less: if (a) < (b) then c else d,
subtract: n - m,
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
nat: ℕ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x]),
has-value: (a)↓,
so_apply: x[s],
so_lambda: λ2x.t[x]
Lemmas referenced :
top_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-void,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
decidable__le,
subtract_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
int_subtype_base,
nat_properties,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
ge_wf,
subtract-1-ge-0,
decidable__lt,
value-type-has-value,
int-value-type,
less_than_anti-reflexive,
add-zero,
add-commutes,
exception-not-value,
has-value_wf_base,
is-exception_wf,
subtract-add-cancel,
base_wf,
subtype_rel_self,
zero-add,
sum-has-value,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
thin,
sqequalHypSubstitution,
hypothesis,
axiomSqEquality,
Error :inhabitedIsType,
hypothesisEquality,
Error :isect_memberEquality_alt,
isectElimination,
Error :isectIsTypeImplies,
Error :universeIsType,
extract_by_obid,
Error :lambdaFormation_alt,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
lessCases,
independent_pairFormation,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :equalityIsType4,
baseApply,
closedConclusion,
applyEquality,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
because_Cache,
Error :equalityIsType1,
intEquality,
Error :dependent_set_memberEquality_alt,
approximateComputation,
Error :lambdaEquality_alt,
int_eqEquality,
applyLambdaEquality,
setElimination,
rename,
intWeakElimination,
Error :functionIsTypeImplies,
addEquality,
callbyvalueReduce,
sqequalSqle,
divergentSqle,
callbyvalueCallbyvalue,
sqleReflexivity,
callbyvalueExceptionCases,
axiomSqleEquality,
addExceptionCases,
exceptionSqequal,
callbyvalueAdd,
lessExceptionCases,
dependent_pairFormation,
voidEquality,
isect_memberEquality,
isect_memberFormation,
lambdaFormation,
callbyvalueLess
Latex:
\mforall{}[n,f:Top]. (\mSigma{}(f[x] | x < n) \msim{} if (0) < (n) then \mSigma{}(f[x] | x < n - 1) + f[n - 1] else 0)
Date html generated:
2019_06_20-PM-01_18_01
Last ObjectModification:
2018_10_15-PM-01_51_02
Theory : int_2
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