Nuprl Lemma : length_zero
∀[T:Type]. ∀[l:T List]. uiff(||l|| = 0 ∈ ℤ;l = [] ∈ (T List))
Proof
Definitions occuring in Statement :
length: ||as||,
nil: [],
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
or: P ∨ Q,
cons: [a / b],
top: Top,
prop: ℙ,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
ge: i ≥ j ,
decidable: Dec(P),
false: False,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
list-cases,
length_of_nil_lemma,
nil_wf,
product_subtype_list,
length_of_cons_lemma,
equal-wf-T-base,
length_wf,
equal_wf,
squash_wf,
true_wf,
length_of_null_list,
subtype_rel_self,
iff_weakening_equal,
list_wf,
le_weakening2,
non_neg_length,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
Error :universeIsType,
intEquality,
baseClosed,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
instantiate,
independent_functionElimination,
independent_pairEquality,
axiomEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality
Latex:
\mforall{}[T:Type]. \mforall{}[l:T List]. uiff(||l|| = 0;l = [])
Date html generated:
2019_06_20-PM-01_20_09
Last ObjectModification:
2018_09_26-PM-05_20_46
Theory : list_1
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