Nuprl Lemma : strict-sorted
∀[T:Type]. ∀[as:T List]. uiff(sorted(as) ∧ no_repeats(T;as);∀[i:ℕ||as||]. ∀[j:ℕi]. as[j] < as[i]) supposing T ⊆r ℤ
Proof
Definitions occuring in Statement :
no_repeats: no_repeats(T;l),
sorted: sorted(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,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n,
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
sorted: sorted(L),
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
no_repeats: no_repeats(T;l),
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
sq_type: SQType(T),
label: ...$L... t
Lemmas referenced :
int_seg_wf,
member-less_than,
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,
length_wf,
intformless_wf,
int_formula_prop_less_lemma,
sorted_wf,
no_repeats_wf,
less_than'_wf,
no_repeats_witness,
uall_wf,
less_than_wf,
list_wf,
subtype_rel_wf,
int_seg_subtype_nat,
false_wf,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
nat_wf,
subtype_base_sq,
int_subtype_base,
not_wf,
lelt_wf,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
decidable__equal_int,
le_wf,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
extract_by_obid,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
cumulativity,
imageElimination,
applyEquality,
productEquality,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
universeEquality,
lambdaFormation,
applyLambdaEquality,
instantiate,
dependent_set_memberEquality,
addLevel,
levelHypothesis
Latex:
\mforall{}[T:Type]
\mforall{}[as:T List]. uiff(sorted(as) \mwedge{} no\_repeats(T;as);\mforall{}[i:\mBbbN{}||as||]. \mforall{}[j:\mBbbN{}i]. as[j] < as[i])
supposing T \msubseteq{}r \mBbbZ{}
Date html generated:
2018_05_21-PM-06_48_07
Last ObjectModification:
2017_07_26-PM-04_56_38
Theory : general
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