Nuprl Lemma : swap_cons
∀[T:Type]. ∀[L:T List]. ∀[x:T]. ∀[i,j:ℕ+||L|| + 1]. (swap([x / L];i;j) = [x / swap(L;i - 1;j - 1)] ∈ (T List))
Proof
Definitions occuring in Statement :
swap: swap(L;i;j),
length: ||as||,
cons: [a / b],
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
uiff: uiff(P;Q),
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
sq_type: SQType(T),
cons: [a / b],
select: L[n],
less_than': less_than'(a;b),
top: Top,
ge: i ≥ j ,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
so_apply: x[s],
so_lambda: λ2x.t[x],
flip: (i, j),
nequal: a ≠ b ∈ T
Lemmas referenced :
list_extensionality,
swap_wf,
cons_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
length_of_cons_lemma,
istype-le,
istype-less_than,
length_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
add-is-int-iff,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
false_wf,
equal_wf,
add_functionality_wrt_eq,
swap_length,
iff_weakening_equal,
length_cons,
istype-nat,
int_seg_wf,
list_wf,
istype-universe,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
subtype_rel_self,
istype-false,
less_than_wf,
le_wf,
istype-void,
nat_properties,
swap_select,
true_wf,
squash_wf,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
uiff_transitivity,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
not_wf,
bnot_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
assert_wf,
lelt_wf,
set_subtype_base,
bool_wf,
equal-wf-base,
eq_int_wf,
non_neg_length,
select_cons_tl,
select_wf,
select-cons-tl,
subtract-add-cancel,
btrue_neq_bfalse,
assert_elim,
bfalse_wf,
btrue_wf,
eq_int_eq_true
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
hypothesis,
setElimination,
rename,
dependent_set_memberEquality_alt,
productElimination,
independent_pairFormation,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
sqequalRule,
universeIsType,
voidElimination,
productIsType,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
imageMemberEquality,
lambdaFormation_alt,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
addEquality,
instantiate,
universeEquality,
cumulativity,
equalityIsType1,
equalityIsType2,
equalityElimination,
equalityIsType4
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[x:T]. \mforall{}[i,j:\mBbbN{}\msupplus{}||L|| + 1]. (swap([x / L];i;j) = [x / swap(L;i - 1;j - 1)])
Date html generated:
2020_05_20-AM-07_49_00
Last ObjectModification:
2019_12_26-PM-04_46_39
Theory : list!
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