Nuprl Lemma : segment_factor
∀T:Type. ∀as:T List. ∀i:{0...||as||}. ∀j:{i...||as||}. ((as[i..j-]) = (Π i ≤ k < j. [as[k]]) ∈ (T List))
Proof
Definitions occuring in Statement :
lapp_imon: <T List,@>,
segment: as[m..n-],
select: L[n],
length: ||as||,
cons: [a / b],
nil: [],
list: T List,
int_iseg: {i...j},
all: ∀x:A. B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
mon_itop: Π lb ≤ i < ub. E[i]
Definitions unfolded in proof :
all: ∀x:A. B[x],
segment: as[m..n-],
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_iseg: {i...j},
squash: ↓T,
prop: ℙ,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
so_apply: x[s],
imon: IMonoid,
list: T List,
grp_car: |g|,
pi1: fst(t),
lapp_imon: <T List,@>,
subtract: n - m
Lemmas referenced :
int_iseg_wf,
length_wf,
list_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
firstn_factor,
nth_tl_wf,
subtract_wf,
int_iseg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
length_nth_tl,
iff_weakening_equal,
mon_itop_wf,
lapp_imon_wf,
cons_wf,
select_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
nil_wf,
int_seg_wf,
subtype_rel_self,
less_than_wf,
grp_car_wf,
imon_wf,
select_nth_tl,
imon_subtype_grp_sig,
itermAdd_wf,
int_term_value_add_lemma,
add-associates,
minus-one-mul,
add-swap,
add-commutes,
add-mul-special,
zero-mul,
zero-add,
mon_itop_shift
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
universeEquality,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
dependent_functionElimination,
dependent_set_memberEquality_alt,
because_Cache,
productElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productIsType,
instantiate,
functionIsType,
addEquality,
multiplyEquality,
minusEquality
Latex:
\mforall{}T:Type. \mforall{}as:T List. \mforall{}i:\{0...||as||\}. \mforall{}j:\{i...||as||\}. ((as[i..j\msupminus{}]) = (\mPi{} i \mleq{} k < j. [as[k]]))
Date html generated:
2019_10_16-PM-01_05_36
Last ObjectModification:
2018_10_08-AM-10_53_43
Theory : list_2
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