Nuprl Lemma : l_before_swap
∀[T:Type]
∀L:T List. ∀i:ℕ||L|| - 1. ∀a,b:T.
(a before b ∈ swap(L;i;i + 1) ⇒ (a before b ∈ L ∨ ((a = L[i + 1] ∈ T) ∧ (b = L[i] ∈ T))))
Proof
Definitions occuring in Statement :
swap: swap(L;i;j),
l_before: x before y ∈ l,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
l_before: x before y ∈ l,
sublist: L1 ⊆ L2,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
squash: ↓T,
so_lambda: λ2x.t[x],
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
so_apply: x[s],
iff: P ⇐⇒ Q,
less_than: a < b,
select: L[n],
cons: [a / b],
subtract: n - m,
flip: (i, j),
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
cand: A c∧ B,
nequal: a ≠ b ∈ T
Lemmas referenced :
length_of_cons_lemma,
length_of_nil_lemma,
swap_length,
lelt_wf,
length_wf,
add-member-int_seg2,
subtract_wf,
all_wf,
squash_wf,
true_wf,
int_seg_wf,
equal_wf,
select_wf,
cons_wf,
nil_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
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,
non_neg_length,
swap_wf,
add-member-int_seg1,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
swap_select,
int_seg_subtype,
false_wf,
le_weakening,
subtype_rel_self,
iff_weakening_equal,
flip_wf,
length_wf_nat,
or_wf,
sublist_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
l_before_wf,
subtract-is-int-iff,
list_wf,
sublist_pair,
increasing_implies,
le_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not_wf,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
because_Cache,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
natural_numberEquality,
cumulativity,
independent_isectElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
addEquality,
imageMemberEquality,
baseClosed,
instantiate,
functionExtensionality,
hyp_replacement,
applyLambdaEquality,
productEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
inlFormation,
inrFormation,
equalityElimination,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}i:\mBbbN{}||L|| - 1. \mforall{}a,b:T.
(a before b \mmember{} swap(L;i;i + 1) {}\mRightarrow{} (a before b \mmember{} L \mvee{} ((a = L[i + 1]) \mwedge{} (b = L[i]))))
Date html generated:
2018_05_21-PM-06_21_15
Last ObjectModification:
2018_05_19-PM-05_35_18
Theory : list!
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