Nuprl Lemma : l_before_no_repeats
∀[T:Type]. ∀L:T List. ∀i,j:ℕ||L||. uiff(j < i;L[j] before L[i] ∈ L) supposing no_repeats(T;L)
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
no_repeats: no_repeats(T;l),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
less_than: a < b,
uiff: uiff(P;Q),
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,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
int_seg: {i..j-},
prop: ℙ,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
sq_type: SQType(T),
l_before: x before y ∈ l,
sublist: L1 ⊆ L2,
no_repeats: no_repeats(T;l),
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
ge: i ≥ j ,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
select: L[n],
cons: [a / b],
subtract: n - m
Lemmas referenced :
l_before_antisymmetry,
l_before_transitivity,
int_seg_subtype_nat,
nat_wf,
subtype_rel_dep_function,
increasing_implies,
le_wf,
nat_properties,
length_wf_nat,
non_neg_length,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
int_subtype_base,
subtype_base_sq,
list_wf,
int_seg_wf,
no_repeats_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
intformle_wf,
intformand_wf,
decidable__le,
length_wf,
select_wf,
l_before_wf,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_or_lemma,
int_formula_prop_not_lemma,
intformeq_wf,
itermVar_wf,
intformless_wf,
intformor_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
decidable__lt,
equal_wf,
or_wf,
decidable__or,
int_seg_properties,
less_than_wf,
l_before_select,
member-less_than,
no_repeats_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
independent_pairFormation,
setElimination,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
productElimination,
intEquality,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
cumulativity,
imageElimination,
universeEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
setEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}i,j:\mBbbN{}||L||. uiff(j < i;L[j] before L[i] \mmember{} L) supposing no\_repeats(T;L)
Date html generated:
2016_05_14-AM-07_46_46
Last ObjectModification:
2016_01_15-AM-08_34_23
Theory : list_1
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