Nuprl Lemma : permutation-invariant
∀[T:Type]. ∀[P:(T List) ⟶ ℙ].
((∀as:T List. ∀a:T. (P[[a / as]] ⇒ P[as @ [a]]))
⇒ (∀as:T List. ∀a1,a2:T. (P[[a1; [a2 / as]]] ⇒ P[[a2; [a1 / as]]]))
⇒ (∀as,bs:T List. (permutation(T;as;bs) ⇒ (P[as] ⇐⇒ P[bs]))))
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
prop: ℙ,
so_apply: x[s],
so_apply: x[s1;s2],
trans: Trans(T;x,y.E[x; y]),
all: ∀x:A. B[x],
refl: Refl(T;x,y.E[x; y]),
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
guard: {T}
Lemmas referenced :
permutation-invariant2,
list_wf,
permutation_inversion,
permutation_wf,
all_wf,
cons_wf,
append_wf,
nil_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
applyEquality,
hypothesis,
independent_functionElimination,
because_Cache,
independent_pairFormation,
dependent_functionElimination,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:(T List) {}\mrightarrow{} \mBbbP{}].
((\mforall{}as:T List. \mforall{}a:T. (P[[a / as]] {}\mRightarrow{} P[as @ [a]]))
{}\mRightarrow{} (\mforall{}as:T List. \mforall{}a1,a2:T. (P[[a1; [a2 / as]]] {}\mRightarrow{} P[[a2; [a1 / as]]]))
{}\mRightarrow{} (\mforall{}as,bs:T List. (permutation(T;as;bs) {}\mRightarrow{} (P[as] \mLeftarrow{}{}\mRightarrow{} P[bs]))))
Date html generated:
2016_05_14-PM-02_30_59
Last ObjectModification:
2015_12_26-PM-04_23_11
Theory : list_1
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