Nuprl Lemma : sublist_pair
∀[T:Type]. ∀L:T List. ∀i,j:ℕ||L||. [L[i]; L[j]] ⊆ L supposing i < j
Proof
Definitions occuring in Statement :
sublist: L1 ⊆ L2,
select: L[n],
length: ||as||,
cons: [a / b],
nil: [],
list: T List,
int_seg: {i..j-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
int_seg: {i..j-},
sublist: L1 ⊆ L2,
exists: ∃x:A. B[x],
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
nat: ℕ,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
guard: {T},
increasing: increasing(f;k),
subtract: n - m,
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 ,
select: L[n],
cons: [a / b],
eq_int: (i =z j)
Lemmas referenced :
member-less_than,
length_of_cons_lemma,
length_of_nil_lemma,
ifthenelse_wf,
eq_int_wf,
int_seg_wf,
increasing_wf,
istype-void,
istype-le,
select_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,
decidable__lt,
length_wf,
intformless_wf,
int_formula_prop_less_lemma,
nil_wf,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
istype-less_than,
length_wf_nat,
nat_properties,
list_wf,
istype-universe,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__equal_int,
int_subtype_base,
int_seg_subtype_special,
int_seg_cases
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
Error :memTop,
dependent_pairFormation_alt,
lambdaEquality_alt,
productElimination,
imageElimination,
universeIsType,
natural_numberEquality,
productIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
voidElimination,
functionExtensionality,
applyEquality,
because_Cache,
functionIsType,
equalityIstype,
unionElimination,
approximateComputation,
independent_functionElimination,
int_eqEquality,
addEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
inhabitedIsType,
instantiate,
universeEquality,
equalityElimination,
promote_hyp,
cumulativity,
intEquality,
hypothesis_subsumption
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}i,j:\mBbbN{}||L||. [L[i]; L[j]] \msubseteq{} L supposing i < j
Date html generated:
2020_05_19-PM-09_42_10
Last ObjectModification:
2020_01_04-PM-08_26_15
Theory : list_1
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