Nuprl Lemma : colength-positive
∀[T:Type]. ∀[L:T List].
(0 < colength(L)
⇒ {(fst(L) ∈ T) ∧ (snd(L) ∈ T List) ∧ (colength(L) = (1 + colength(snd(L))) ∈ ℤ) ∧ (L ~ [fst(L) / (snd(L))])})
Proof
Definitions occuring in Statement :
cons: [a / b],
list: T List,
colength: colength(L),
less_than: a < b,
uall: ∀[x:A]. B[x],
guard: {T},
pi1: fst(t),
pi2: snd(t),
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
guard: {T},
list: T List,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uimplies: b supposing a,
nil: [],
colength: colength(L),
has-value: (a)↓,
cons: [a / b],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
bfalse: ff,
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
colist-ext,
isaxiom_wf_listunion,
colist_wf,
bool_wf,
subtype_rel_b-union-left,
unit_wf2,
axiom-listunion,
subtype_rel_b-union-right,
non-axiom-listunion,
colength_wf_list,
nat_wf,
equal_wf,
less_than_wf,
list_wf,
colength_wf,
subtype_partial_sqtype_base,
set_subtype_base,
le_wf,
int_subtype_base,
value-type-has-value,
int-value-type,
has-value_wf-partial,
set-value-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
isectElimination,
hypothesisEquality,
promote_hyp,
productElimination,
hypothesis_subsumption,
hypothesis,
applyEquality,
sqequalRule,
unionElimination,
equalityElimination,
productEquality,
independent_isectElimination,
imageElimination,
voidElimination,
independent_pairFormation,
addEquality,
natural_numberEquality,
cumulativity,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
independent_pairEquality,
axiomEquality,
sqequalAxiom,
because_Cache,
isect_memberEquality,
universeEquality,
intEquality,
callbyvalueAdd,
baseClosed,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List].
(0 < colength(L)
{}\mRightarrow{} \{(fst(L) \mmember{} T)
\mwedge{} (snd(L) \mmember{} T List)
\mwedge{} (colength(L) = (1 + colength(snd(L))))
\mwedge{} (L \msim{} [fst(L) / (snd(L))])\})
Date html generated:
2017_04_14-AM-07_54_20
Last ObjectModification:
2017_02_27-PM-03_21_16
Theory : list_0
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