Nuprl Lemma : colength-positive2
∀[T:Type]. ∀[L:T List].
∀n:ℕ
(0 < n
⇒ (colength(L) = n ∈ ℤ)
⇒ {(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),
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
guard: {T},
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ
Lemmas referenced :
colength-positive,
less_than_transitivity1,
colength_wf_list,
le_weakening,
equal_wf,
nat_wf,
less_than_wf,
list_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
introduction,
independent_functionElimination,
equalitySymmetry,
natural_numberEquality,
setElimination,
rename,
applyEquality,
because_Cache,
sqequalRule,
independent_isectElimination,
dependent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
sqequalAxiom,
intEquality,
lambdaEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List].
\mforall{}n:\mBbbN{}
(0 < n
{}\mRightarrow{} (colength(L) = n)
{}\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:
2016_05_14-AM-06_26_17
Last ObjectModification:
2015_12_26-PM-00_42_03
Theory : list_0
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