Nuprl Lemma : firstn_is_iseg
∀[T:Type]. ∀L1,L2:T List. (L1 ≤ L2 ⇐⇒ ∃n:ℕ||L2|| + 1. (L1 = firstn(n;L2) ∈ (T List)))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
firstn: firstn(n;as),
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
iseg: l1 ≤ l2,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
false: False,
le: A ≤ B,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
guard: {T},
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
uiff: uiff(P;Q),
int_iseg: {i...j},
cand: A c∧ B,
firstn: firstn(n;as),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
less_than: a < b,
squash: ↓T,
sq_type: SQType(T),
true: True
Lemmas referenced :
exists_wf,
list_wf,
equal_wf,
append_wf,
int_seg_wf,
length_wf,
firstn_wf,
non_neg_length,
decidable__le,
full-omega-unsat,
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,
lelt_wf,
int_seg_subtype,
false_wf,
length_append,
subtype_rel_list,
top_wf,
length-append,
int_seg_properties,
list_induction,
length_of_nil_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
list_ind_cons_lemma,
length_zero,
length_firstn_eq,
le_wf,
lt_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
less_than_wf,
le_int_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
add-is-int-iff,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
cons_wf,
squash_wf,
true_wf,
nth_tl_wf,
append_firstn_lastn,
subtype_rel_sets
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaEquality,
natural_numberEquality,
addEquality,
setElimination,
rename,
Error :universeIsType,
universeEquality,
dependent_pairFormation,
dependent_set_memberEquality,
dependent_functionElimination,
because_Cache,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
cumulativity,
hyp_replacement,
productEquality,
baseClosed,
equalityElimination,
instantiate,
pointwiseFunctionality,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
imageMemberEquality,
setEquality
Latex:
\mforall{}[T:Type]. \mforall{}L1,L2:T List. (L1 \mleq{} L2 \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}||L2|| + 1. (L1 = firstn(n;L2)))
Date html generated:
2019_06_20-PM-01_28_19
Last ObjectModification:
2018_09_26-PM-05_39_29
Theory : list_1
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