Nuprl Lemma : select-remove-first
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ 𝔹]. ∀[i:ℕ||remove-first(P;L)||].
(remove-first(P;L)[i] ~ L[i] supposing ∀j:ℕi + 1. (¬↑(P L[j]))
∧ remove-first(P;L)[i] ~ L[i + 1] supposing ∃j:ℕi + 1. (↑(P L[j])))
Proof
Definitions occuring in Statement :
remove-first: remove-first(P;L),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
remove-first: remove-first(P;L),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
l_all: (∀x∈L.P[x])
Lemmas referenced :
length-remove-first-le,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
list-cases,
list_ind_nil_lemma,
length_of_nil_lemma,
stuck-spread,
istype-base,
int_seg_properties,
int_seg_wf,
le_wf,
l_member_wf,
nil_wf,
bool_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
list_ind_cons_lemma,
length_of_cons_lemma,
cons_member,
cons_wf,
subtype_rel_sets,
eqtt_to_assert,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
assert-bnot,
nat_wf,
length_wf,
remove-first_wf,
list_wf,
list-subtype,
select_wf,
assert_wf,
not_wf,
lelt_wf,
decidable__lt,
false_wf,
select_cons_tl_sq,
int_seg_cases,
int_seg_subtype,
add-is-int-iff,
length_wf_nat,
subtract-add-cancel,
select_member,
add-member-int_seg2,
add-subtract-cancel,
select-cons-tl,
all_wf,
length-remove-first,
subtract-is-int-iff,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
Error :lambdaFormation_alt,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
productElimination,
independent_pairEquality,
axiomSqEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
unionElimination,
baseClosed,
Error :functionIsType,
Error :setIsType,
promote_hyp,
hypothesis_subsumption,
Error :equalityIsType1,
because_Cache,
Error :dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
Error :equalityIsType4,
baseApply,
closedConclusion,
applyEquality,
intEquality,
Error :inlFormation_alt,
Error :functionExtensionality_alt,
Error :inrFormation_alt,
equalityElimination,
Error :equalityIsType3,
cumulativity,
universeEquality,
Error :productIsType,
setEquality,
functionExtensionality,
addEquality,
lambdaFormation,
voidEquality,
isect_memberEquality,
lambdaEquality,
dependent_pairFormation,
dependent_set_memberEquality,
inlFormation,
pointwiseFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}]. \mforall{}[i:\mBbbN{}||remove-first(P;L)||].
(remove-first(P;L)[i] \msim{} L[i] supposing \mforall{}j:\mBbbN{}i + 1. (\mneg{}\muparrow{}(P L[j]))
\mwedge{} remove-first(P;L)[i] \msim{} L[i + 1] supposing \mexists{}j:\mBbbN{}i + 1. (\muparrow{}(P L[j])))
Date html generated:
2019_06_20-PM-01_42_54
Last ObjectModification:
2018_10_15-PM-05_48_05
Theory : list_1
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