Nuprl Lemma : list_append_ind
Alternative Induction Principle for Lists
Used for multiset induction.
∀[T:Type]. ∀[Q:(T List) ⟶ ℙ].
(Q[[]] ⇒ (∀x:T. Q[[x]]) ⇒ (∀ys,ys':T List. (Q[ys] ⇒ Q[ys'] ⇒ Q[ys @ ys'])) ⇒ {∀zs:T List. Q[zs]})
Proof
Definitions occuring in Statement :
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3]
Lemmas referenced :
list_induction,
list_wf,
all_wf,
append_wf,
cons_wf,
nil_wf,
list_ind_cons_lemma,
list_ind_nil_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
lambdaEquality,
applyEquality,
hypothesis,
independent_functionElimination,
rename,
because_Cache,
dependent_functionElimination,
functionEquality,
cumulativity,
universeEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[T:Type]. \mforall{}[Q:(T List) {}\mrightarrow{} \mBbbP{}].
(Q[[]]
{}\mRightarrow{} (\mforall{}x:T. Q[[x]])
{}\mRightarrow{} (\mforall{}ys,ys':T List. (Q[ys] {}\mRightarrow{} Q[ys'] {}\mRightarrow{} Q[ys @ ys']))
{}\mRightarrow{} \{\mforall{}zs:T List. Q[zs]\})
Date html generated:
2016_07_08-PM-04_49_05
Last ObjectModification:
2015_12_26-PM-02_13_23
Theory : list_1
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