Nuprl Lemma : permutation-cons
∀[A:Type]
∀x:A. ∀L1,L2:A List.
(permutation(A;[x / L1];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],
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],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
l_contains: A ⊆ B,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
select: L[n],
cons: [a / b],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cand: A c∧ B
Lemmas referenced :
permutation_wf,
cons_wf,
exists_wf,
list_wf,
equal_wf,
append_wf,
length_wf,
length-append,
permutation_inversion,
permutation-contains,
length_of_cons_lemma,
false_wf,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
lelt_wf,
l_member_decomp,
list_ind_cons_lemma,
list_ind_nil_lemma,
permutation_transitivity,
permutation-rotate,
nil_wf,
permutation_weakening,
append_functionality_wrt_permutation,
cons_cancel_wrt_permutation
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
productElimination,
sqequalRule,
lambdaEquality,
productEquality,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
addEquality,
hyp_replacement
Latex:
\mforall{}[A:Type]
\mforall{}x:A. \mforall{}L1,L2:A List.
(permutation(A;[x / L1];L2)
\mLeftarrow{}{}\mRightarrow{} \mexists{}as,bs:A List. ((L2 = (as @ [x / bs])) \mwedge{} permutation(A;L1;as @ bs)))
Date html generated:
2017_04_17-AM-08_23_42
Last ObjectModification:
2017_02_27-PM-04_45_25
Theory : list_1
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