Nuprl Lemma : permutation-last
∀[A:Type]
∀x:A. ∀L1,L2:A List.
(permutation(A;L1 @ [x];L2) ⇐⇒ ∃as,bs:A List. ((L2 = (as @ [x / bs]) ∈ (A List)) ∧ permutation(A;L1;as @ 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],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3]
Lemmas referenced :
permutation_weakening,
permutation-rotate,
permutation_functionality_wrt_permutation,
permutation-cons2,
permutation-cons,
list_ind_nil_lemma,
list_ind_cons_lemma,
equal_wf,
and_wf,
exists_wf,
iff_weakening_equal,
list_wf,
true_wf,
squash_wf,
nil_wf,
cons_wf,
append_wf,
permutation_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
applyEquality,
lambdaEquality,
imageElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[A:Type]
\mforall{}x:A. \mforall{}L1,L2:A List.
(permutation(A;L1 @ [x];L2)
\mLeftarrow{}{}\mRightarrow{} \mexists{}as,bs:A List. ((L2 = (as @ [x / bs])) \mwedge{} permutation(A;L1;as @ bs)))
Date html generated:
2016_05_14-PM-02_32_11
Last ObjectModification:
2016_01_15-AM-07_45_19
Theory : list_1
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