Nuprl Lemma : permutation-invariant2
∀[T:Type]. ∀[R:(T List) ⟶ (T List) ⟶ ℙ].
(Trans(T List;as,bs.R[as;bs])
⇒ Refl(T List;as,bs.R[as;bs])
⇒ (∀as:T List. ∀a:T. R[[a / as];as @ [a]])
⇒ (∀as:T List. ∀a1,a2:T. R[[a1; [a2 / as]];[a2; [a1 / as]]])
⇒ (∀as,bs:T List. (permutation(T;as;bs) ⇒ R[as;bs])))
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
trans: Trans(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
so_lambda: λ2x y.t[x; y],
prop: ℙ,
so_apply: x[s1;s2],
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
subtype_rel: A ⊆r B,
member: t ∈ T,
and: P ∧ Q,
exists: ∃x:A. B[x],
permutation: permutation(T;L1;L2),
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
refl: Refl(T;x,y.E[x; y]),
lelt: i ≤ j < k,
int_seg: {i..j-},
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
squash: ↓T,
less_than: a < b,
or: P ∨ Q,
decidable: Dec(P),
trans: Trans(T;x,y.E[x; y]),
cons: [a / b],
it: ⋅,
nil: [],
list_ind: list_ind,
length: ||as||,
less_than': less_than'(a;b),
ge: i ≥ j ,
le: A ≤ B,
top: Top,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
subtract: n - m,
select: L[n],
sq_type: SQType(T),
flip: (i, j),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
mklist: mklist(n;f),
permute_list: (L o f),
rotate: rot(n)
Lemmas referenced :
istype-universe,
trans_wf,
refl_wf,
nil_wf,
append_wf,
subtype_rel_self,
cons_wf,
list_wf,
permutation_wf,
istype-less_than,
member-less_than,
length_wf,
int_seg_wf,
inject_wf,
permute_list_wf,
permutation-generators2,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
istype-nat,
length_wf_nat,
permute_list-identity,
permute_list-compose,
decidable__lt,
istype-le,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
flip_wf,
compose_wf,
permute_list_length,
product_subtype_list,
list-cases,
nat_wf,
less_than_wf,
lelt_wf,
int_term_value_add_lemma,
itermAdd_wf,
satisfiable-full-omega-tt,
non_neg_length,
false_wf,
length_of_cons_lemma,
list_extensionality,
iff_weakening_equal,
select_wf,
nat_properties,
permute_list_select,
true_wf,
squash_wf,
equal_wf,
not_wf,
bnot_wf,
subtype_base_sq,
assert_wf,
equal-wf-T-base,
bool_wf,
eq_int_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
select_cons_tl,
rotate_wf,
primrec0_lemma,
istype-base,
stuck-spread,
length_of_nil_lemma,
length_cons,
length-append,
select_append_front,
select_cons_tl_sq2,
istype-void,
istype-assert,
equal-wf-base,
select-cons-hd,
length-singleton,
add-is-int-iff,
select_append_back
Rules used in proof :
universeEquality,
instantiate,
functionIsType,
applyLambdaEquality,
hyp_replacement,
independent_functionElimination,
because_Cache,
universeIsType,
inhabitedIsType,
setIsType,
rename,
setElimination,
dependent_functionElimination,
equalitySymmetry,
sqequalBase,
independent_isectElimination,
natural_numberEquality,
lambdaEquality_alt,
intEquality,
sqequalRule,
applyEquality,
equalityIstype,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
hypothesis,
dependent_set_memberEquality_alt,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productIsType,
voidElimination,
independent_pairFormation,
Error :memTop,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
imageElimination,
unionElimination,
equalityTransitivity,
hypothesis_subsumption,
promote_hyp,
computeAll,
lambdaEquality,
dependent_pairFormation,
addEquality,
lambdaFormation,
dependent_set_memberEquality,
voidEquality,
isect_memberEquality,
cumulativity,
baseClosed,
imageMemberEquality,
equalityElimination,
impliesFunctionality,
closedConclusion,
baseApply,
pointwiseFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[R:(T List) {}\mrightarrow{} (T List) {}\mrightarrow{} \mBbbP{}].
(Trans(T List;as,bs.R[as;bs])
{}\mRightarrow{} Refl(T List;as,bs.R[as;bs])
{}\mRightarrow{} (\mforall{}as:T List. \mforall{}a:T. R[[a / as];as @ [a]])
{}\mRightarrow{} (\mforall{}as:T List. \mforall{}a1,a2:T. R[[a1; [a2 / as]];[a2; [a1 / as]]])
{}\mRightarrow{} (\mforall{}as,bs:T List. (permutation(T;as;bs) {}\mRightarrow{} R[as;bs])))
Date html generated:
2020_05_19-PM-09_44_58
Last ObjectModification:
2019_12_26-AM-11_46_41
Theory : list_1
Home
Index