Nuprl Lemma : iseg_select
∀[T:Type]. ∀l1,l2:T List. (l1 ≤ l2 ⇐⇒ (||l1|| ≤ ||l2||) c∧ (∀i:ℕ. l1[i] = l2[i] ∈ T supposing i < ||l1||))
Proof
Definitions occuring in Statement :
iseg: 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,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
le: A ≤ B,
squash: ↓T,
less_than: a < b,
and: P ∧ Q,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
uimplies: b supposing a,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
select: L[n],
uiff: uiff(P;Q),
iseg: l1 ≤ l2,
guard: {T},
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
sq_type: SQType(T),
subtype_rel: A ⊆r B,
true: True,
less_than': less_than'(a;b),
nat_plus: ℕ+,
cons: [a / b],
subtract: n - m
Lemmas referenced :
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
select_wf,
equal_wf,
less_than_wf,
isect_wf,
nat_wf,
length_wf,
le_wf,
iseg_wf,
iff_wf,
list_wf,
all_wf,
list_induction,
equal-wf-base-T,
nil_iseg,
nil_wf,
less_than'_wf,
non_neg_length,
base_wf,
stuck-spread,
length_of_nil_lemma,
false_wf,
int_term_value_add_lemma,
itermAdd_wf,
add-is-int-iff,
length_of_cons_lemma,
cons_wf,
equal-wf-T-base,
iseg_weakening,
length-append,
btrue_neq_bfalse,
bfalse_wf,
null_cons_lemma,
null_wf,
and_wf,
append_is_nil,
btrue_wf,
null_nil_lemma,
list_ind_cons_lemma,
length_wf_nat,
cons_iseg,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
iff_weakening_equal,
select_cons_hd,
true_wf,
squash_wf,
select_cons_tl,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
full-omega-unsat,
istype-int,
istype-void,
istype-le,
add_nat_plus,
istype-less_than,
nat_plus_properties,
istype-nat,
add-associates,
add-swap,
add-commutes,
zero-add,
istype-universe,
subtype_rel_self
Rules used in proof :
universeEquality,
independent_functionElimination,
productElimination,
imageElimination,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
natural_numberEquality,
dependent_functionElimination,
independent_isectElimination,
rename,
setElimination,
productEquality,
because_Cache,
hypothesis,
cumulativity,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_pairEquality,
baseClosed,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
addEquality,
dependent_set_memberEquality,
applyLambdaEquality,
instantiate,
imageMemberEquality,
applyEquality,
Error :dependent_set_memberEquality_alt,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
Error :universeIsType,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
Error :equalityIstype,
Error :isect_memberFormation_alt
Latex:
\mforall{}[T:Type]
\mforall{}l1,l2:T List. (l1 \mleq{} l2 \mLeftarrow{}{}\mRightarrow{} (||l1|| \mleq{} ||l2||) c\mwedge{} (\mforall{}i:\mBbbN{}. l1[i] = l2[i] supposing i < ||l1||))
Date html generated:
2019_06_20-PM-01_28_45
Last ObjectModification:
2019_03_27-PM-01_24_20
Theory : list_1
Home
Index