Nuprl Lemma : select-firstn
∀[L:Top List]. ∀[n:ℕ]. ∀[x:ℕn]. (firstn(n;L)[x] ~ L[x])
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
select: L[n],
list: T List,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
le: A ≤ B,
prop: ℙ,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
unit: Unit,
bool: 𝔹,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
colength: colength(L),
cons: [a / b],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
select: L[n],
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
firstn: firstn(n;as),
guard: {T},
subtype_rel: A ⊆r B,
ge: i ≥ j ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
cand: A c∧ B,
int_iseg: {i...j}
Lemmas referenced :
list_wf,
nat_wf,
int_seg_wf,
top_wf,
length_wf,
decidable__le,
equal_wf,
lelt_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
int_seg_properties,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
list_ind_cons_lemma,
length_of_cons_lemma,
decidable__equal_int,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
le_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
spread_cons_lemma,
product_subtype_list,
base_wf,
stuck-spread,
list_ind_nil_lemma,
length_of_nil_lemma,
list-cases,
less_than_irreflexivity,
less_than_transitivity1,
colength_wf_list,
equal-wf-T-base,
less_than_wf,
ge_wf,
int_term_value_constant_lemma,
itermConstant_wf,
nat_properties,
subtract-add-cancel,
iff_weakening_equal,
false_wf,
add-is-int-iff,
full-omega-unsat,
length_firstn_eq,
true_wf,
squash_wf,
firstn_wf,
select_cons_tl_sq
Rules used in proof :
isect_memberEquality,
sqequalRule,
natural_numberEquality,
sqequalAxiom,
unionElimination,
because_Cache,
rename,
setElimination,
hypothesisEquality,
hypothesis,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_isectElimination,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
lambdaFormation,
productElimination,
computeAll,
voidEquality,
voidElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_pairFormation,
dependent_set_memberEquality,
equalityElimination,
imageElimination,
cumulativity,
instantiate,
addEquality,
applyLambdaEquality,
hypothesis_subsumption,
promote_hyp,
baseClosed,
applyEquality,
intWeakElimination,
universeEquality,
imageMemberEquality,
productEquality,
closedConclusion,
baseApply,
pointwiseFunctionality,
approximateComputation
Latex:
\mforall{}[L:Top List]. \mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbN{}n]. (firstn(n;L)[x] \msim{} L[x])
Date html generated:
2018_05_21-PM-00_40_01
Last ObjectModification:
2018_05_18-PM-04_18_05
Theory : list_1
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