Nuprl Lemma : fseg_select
∀[T:Type]
∀l1,l2:T List.
(fseg(T;l1;l2) ⇐⇒ (||l1|| ≤ ||l2||) c∧ (∀i:ℕ. l1[i] = l2[(||l2|| - ||l1||) + i] ∈ T supposing i < ||l1||))
Proof
Definitions occuring in Statement :
fseg: fseg(T;L1;L2),
select: L[n],
length: ||as||,
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
subtract: n - m,
add: n + m,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
cand: A c∧ B,
member: t ∈ T,
uimplies: b supposing a,
le: A ≤ B,
nat: ℕ,
prop: ℙ,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
less_than: a < b,
squash: ↓T,
fseg: fseg(T;L1;L2),
subtype_rel: A ⊆r B,
true: True,
guard: {T},
uiff: uiff(P;Q),
int_seg: {i..j-},
lelt: i ≤ j < k,
int_iseg: {i...j}
Lemmas referenced :
fseg_length,
le_witness_for_triv,
istype-less_than,
length_wf,
istype-nat,
fseg_wf,
istype-le,
select_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
subtract_wf,
itermAdd_wf,
itermSubtract_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
equal_wf,
add_nat_wf,
subtract_nat_wf,
append_wf,
le_wf,
squash_wf,
true_wf,
length_append,
subtype_rel_list,
top_wf,
subtype_rel_self,
iff_weakening_equal,
non_neg_length,
length-append,
add-is-int-iff,
subtract-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf,
istype-universe,
select_append_back,
add_functionality_wrt_eq,
less_than_wf,
list_wf,
decidable__equal_int,
firstn_wf,
list_extensionality,
nth_tl_wf,
length_nth_tl,
select-nth_tl,
append_firstn_lastn
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
productElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
universeIsType,
sqequalRule,
productIsType,
functionIsType,
isectIsType,
because_Cache,
equalityIstype,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
voidElimination,
addEquality,
imageElimination,
dependent_set_memberEquality_alt,
inhabitedIsType,
hyp_replacement,
applyLambdaEquality,
applyEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
productEquality,
intEquality
Latex:
\mforall{}[T:Type]
\mforall{}l1,l2:T List.
(fseg(T;l1;l2)
\mLeftarrow{}{}\mRightarrow{} (||l1|| \mleq{} ||l2||) c\mwedge{} (\mforall{}i:\mBbbN{}. l1[i] = l2[(||l2|| - ||l1||) + i] supposing i < ||l1||))
Date html generated:
2020_05_20-AM-08_06_18
Last ObjectModification:
2019_12_31-PM-05_00_01
Theory : general
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