Nuprl Lemma : rng_lsum_0
∀r:Rng. ∀A:Type. ∀as:A List. (Σ{r} x ∈ as. 0 = 0 ∈ |r|)
Proof
Definitions occuring in Statement :
rng_lsum: Σ{r} x ∈ as. f[x],
list: T List,
all: ∀x:A. B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
rng_zero: 0,
rng_car: |r|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
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],
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
rng: Rng,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
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,
rng_lsum_nil_lemma,
rng_zero_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-void,
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,
rng_lsum_cons_lemma,
equal_wf,
rng_car_wf,
rng_plus_comm,
iff_weakening_equal,
rng_plus_wf,
rng_plus_zero,
istype-nat,
list_wf,
istype-universe,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
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,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
imageMemberEquality,
universeEquality
Latex:
\mforall{}r:Rng. \mforall{}A:Type. \mforall{}as:A List. (\mSigma{}\{r\} x \mmember{} as. 0 = 0)
Date html generated:
2020_05_20-AM-09_03_37
Last ObjectModification:
2019_12_26-PM-04_07_33
Theory : matrices
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