Nuprl Lemma : length-remove-first
∀[T:Type]
∀L:T List. ∀P:{x:T| (x ∈ L)} ⟶ 𝔹.
(((∀x∈L.¬↑(P x)) ∧ (remove-first(P;L) ~ L)) ∨ ((∃x∈L. ↑(P x)) ∧ (||remove-first(P;L)|| = (||L|| - 1) ∈ ℤ)))
Proof
Definitions occuring in Statement :
remove-first: remove-first(P;L),
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
length: ||as||,
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
and: P ∧ Q,
cand: A c∧ B,
remove-first: remove-first(P;L),
prop: ℙ,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
so_apply: x[s1;s2;s3],
top: Top,
subtract: n - m,
subtype_rel: A ⊆r B,
l_exists: (∃x∈L. P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
iff: P ⇐⇒ Q,
l_all: (∀x∈L.P[x]),
less_than: a < b,
squash: ↓T,
nat: ℕ,
ge: i ≥ j ,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
decidable__l_exists,
assert_wf,
decidable__assert,
list_wf,
bool_wf,
list_induction,
l_exists_wf,
l_member_wf,
equal_wf,
length_wf,
list_ind_wf,
nil_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
cons_wf,
subtract_wf,
list_ind_nil_lemma,
istype-void,
length_of_nil_lemma,
l_exists_wf_nil,
list_ind_cons_lemma,
length_of_cons_lemma,
istype-int,
int_subtype_base,
int_seg_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
l_exists_cons,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
subtract-is-int-iff,
false_wf,
l_all_wf,
not_wf,
select_wf,
decidable__le,
decidable__lt,
int_seg_wf,
nat_properties,
ge_wf,
less_than_wf,
list-cases,
l_all_wf_nil,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
le_wf,
subtract-1-ge-0,
nat_wf,
set_subtype_base,
spread_cons_lemma,
l_all_cons,
list-subtype
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
functionExtensionality,
cumulativity,
hypothesis,
Error :universeIsType,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
Error :functionIsType,
universeEquality,
Error :inrFormation_alt,
independent_pairFormation,
functionEquality,
setElimination,
rename,
because_Cache,
Error :setIsType,
intEquality,
Error :inhabitedIsType,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
Error :dependent_pairFormation_alt,
Error :equalityIsType1,
promote_hyp,
instantiate,
voidElimination,
natural_numberEquality,
Error :isect_memberEquality_alt,
Error :equalityIsType3,
minusEquality,
approximateComputation,
int_eqEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
Error :productIsType,
Error :inlFormation_alt,
imageElimination,
intWeakElimination,
axiomSqEquality,
Error :functionIsTypeImplies,
hypothesis_subsumption,
Error :dependent_set_memberEquality_alt,
applyLambdaEquality,
Error :equalityIsType4,
addEquality,
setEquality,
lambdaFormation,
isect_memberFormation
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}.
(((\mforall{}x\mmember{}L.\mneg{}\muparrow{}(P x)) \mwedge{} (remove-first(P;L) \msim{} L))
\mvee{} ((\mexists{}x\mmember{}L. \muparrow{}(P x)) \mwedge{} (||remove-first(P;L)|| = (||L|| - 1))))
Date html generated:
2019_06_20-PM-01_42_36
Last ObjectModification:
2018_10_03-PM-11_00_45
Theory : list_1
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