Nuprl Lemma : no_repeats_iff
∀[T:Type]. ∀[l:T List]. uiff(no_repeats(T;l);∀[x,y:T]. ¬(x = y ∈ T) supposing x before y ∈ l)
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
no_repeats: no_repeats(T;l),
list: T List,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
l_before: x before y ∈ l,
no_repeats: no_repeats(T;l),
sublist: L1 ⊆ L2,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
subtype_rel: A ⊆r B,
prop: ℙ,
all: ∀x:A. B[x],
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
nat: ℕ,
guard: {T},
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
less_than': less_than'(a;b),
top: Top,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
increasing: increasing(f;k),
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
length: ||as||,
list_ind: list_ind,
nil: [],
it: ⋅,
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
unit: Unit,
bool: 𝔹,
eq_int: (i =z j)
Lemmas referenced :
int_seg_wf,
length_wf,
cons_wf,
nil_wf,
increasing_wf,
length_wf_nat,
select_wf,
length_of_cons_lemma,
length_of_nil_lemma,
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,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
non_neg_length,
istype-le,
istype-less_than,
nat_properties,
istype-nat,
istype-void,
list_wf,
istype-universe,
lelt_wf,
false_wf,
iff_weakening_equal,
equal_wf,
true_wf,
squash_wf,
not_wf,
satisfiable-full-omega-tt,
int_seg_subtype_nat,
nat_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
ifthenelse_wf,
eq_int_wf,
subtype_rel_dep_function,
int_seg_subtype,
istype-false,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
decidable__equal_int,
int_subtype_base,
int_seg_subtype_special,
int_seg_cases,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
select-cons-tl,
select-cons-hd,
equal-wf-base,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaFormation_alt,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
lambdaEquality_alt,
dependent_functionElimination,
because_Cache,
functionIsTypeImplies,
productIsType,
functionIsType,
universeIsType,
extract_by_obid,
isectElimination,
natural_numberEquality,
setElimination,
rename,
productElimination,
functionExtensionality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
Error :memTop,
imageElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
addEquality,
dependent_set_memberEquality_alt,
applyLambdaEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
isectIsType,
independent_pairEquality,
instantiate,
universeEquality,
baseClosed,
imageMemberEquality,
lambdaFormation,
dependent_set_memberEquality,
voidEquality,
isect_memberEquality,
computeAll,
intEquality,
lambdaEquality,
dependent_pairFormation,
cumulativity,
closedConclusion,
promote_hyp,
equalityElimination,
hypothesis_subsumption
Latex:
\mforall{}[T:Type]. \mforall{}[l:T List]. uiff(no\_repeats(T;l);\mforall{}[x,y:T]. \mneg{}(x = y) supposing x before y \mmember{} l)
Date html generated:
2020_05_19-PM-09_42_24
Last ObjectModification:
2020_01_04-PM-08_14_09
Theory : list_1
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