Nuprl Lemma : no_repeats-remove-first
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ 𝔹]. no_repeats(T;remove-first(P;L)) supposing no_repeats(T;L)
Proof
Definitions occuring in Statement :
remove-first: remove-first(P;L),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
no_repeats: no_repeats(T;l),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
prop: ℙ,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_apply: x[s],
gt: i > j,
subtype_rel: A ⊆r B,
l_all: (∀x∈L.P[x]),
less_than: a < b
Lemmas referenced :
select-remove-first,
lelt_wf,
length_wf,
remove-first_wf,
l_member_wf,
length-remove-first-le,
decidable__all_int_seg,
not_wf,
assert_wf,
select_wf,
list-subtype,
int_seg_properties,
nat_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,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_seg_wf,
decidable__not,
decidable__assert,
equal_wf,
nat_wf,
less_than_wf,
no_repeats_wf,
bool_wf,
list_wf,
no_repeats_witness,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
decidable__or,
equal-wf-base,
int_subtype_base,
or_wf,
intformor_wf,
int_formula_prop_or_lemma,
length-remove-first,
itermSubtract_wf,
int_term_value_subtract_lemma
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
independent_functionElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
productElimination,
cumulativity,
because_Cache,
functionExtensionality,
applyEquality,
setEquality,
instantiate,
dependent_functionElimination,
natural_numberEquality,
addEquality,
sqequalRule,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
functionEquality,
universeEquality,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}].
no\_repeats(T;remove-first(P;L)) supposing no\_repeats(T;L)
Date html generated:
2017_04_17-AM-08_34_28
Last ObjectModification:
2017_02_27-PM-04_56_02
Theory : list_1
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