Nuprl Lemma : swapped_select
∀[T:Type]. ∀[L1,L2:T List]. ∀[i,j:ℕ||L1||].
{(((L2[i] = L1[j] ∈ T) ∧ (L2[j] = L1[i] ∈ T)) ∧ (||L2|| = ||L1|| ∈ ℤ) ∧ (L1 = swap(L2;i;j) ∈ (T List)))
∧ (∀[x:ℕ||L2||]. (L2[x] = L1[x] ∈ T) supposing ((¬(x = j ∈ ℤ)) and (¬(x = i ∈ ℤ))))}
supposing L2 = swap(L1;i;j) ∈ (T List)
Proof
Definitions occuring in Statement :
swap: swap(L;i;j),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
not: ¬A,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
guard: {T},
and: P ∧ Q,
prop: ℙ,
int_seg: {i..j-},
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cand: A c∧ B,
lelt: i ≤ j < k,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
le: A ≤ B,
less_than: a < b,
flip: (i, j),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
not_wf,
equal_wf,
int_seg_wf,
length_wf,
list_wf,
swap_wf,
squash_wf,
true_wf,
swap_length,
iff_weakening_equal,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
intformeq_wf,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
swap_swap,
and_wf,
less_than_wf,
lelt_wf,
swap_select,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
hypothesis,
isect_memberEquality,
isectElimination,
hypothesisEquality,
extract_by_obid,
intEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
cumulativity,
because_Cache,
universeEquality,
applyEquality,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
lambdaFormation,
equalityElimination,
promote_hyp,
instantiate,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[L1,L2:T List]. \mforall{}[i,j:\mBbbN{}||L1||].
\{(((L2[i] = L1[j]) \mwedge{} (L2[j] = L1[i])) \mwedge{} (||L2|| = ||L1||) \mwedge{} (L1 = swap(L2;i;j)))
\mwedge{} (\mforall{}[x:\mBbbN{}||L2||]. (L2[x] = L1[x]) supposing ((\mneg{}(x = j)) and (\mneg{}(x = i))))\}
supposing L2 = swap(L1;i;j)
Date html generated:
2017_10_01-AM-08_38_08
Last ObjectModification:
2017_07_26-PM-04_26_53
Theory : list!
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