Nuprl Lemma : NextNonZero_wf
∀[T:Type]. ∀[L:(T × ℤ) List].
(NextNonZero(L) ∈ {L':(T × ℤ) List|
∃Z:(T × {z:ℤ| z = 0 ∈ ℤ} ) List
((L = (Z @ L') ∈ ((T × ℤ) List)) ∧ (0 < ||L'|| ⇒ (¬((snd(hd(L'))) = 0 ∈ ℤ))))} )
Proof
Definitions occuring in Statement :
NextNonZero: NextNonZero(L),
length: ||as||,
append: as @ bs,
hd: hd(l),
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
pi2: snd(t),
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
NextNonZero: NextNonZero(L),
nil: [],
it: ⋅,
cons: [a / b],
decidable: Dec(P),
colength: colength(L),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
cand: A c∧ B,
append: as @ bs,
list_ind: list_ind,
length: ||as||,
pi2: snd(t),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
le_wf,
istype-nat,
list_wf,
istype-universe,
nil_wf,
equal-wf-base,
length_wf,
append_wf,
subtype_rel_list,
subtype_rel_product,
length_of_nil_lemma,
length-append,
cons_wf,
length_of_cons_lemma,
list_ind_cons_lemma,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
reduce_hd_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
productEquality,
intEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
instantiate,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
sqequalBase,
isectIsTypeImplies,
universeEquality,
setEquality,
voidEquality,
productIsType,
setIsType,
functionIsType,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
imageMemberEquality,
independent_pairEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:(T \mtimes{} \mBbbZ{}) List].
(NextNonZero(L) \mmember{} \{L':(T \mtimes{} \mBbbZ{}) List|
\mexists{}Z:(T \mtimes{} \{z:\mBbbZ{}| z = 0\} ) List
((L = (Z @ L')) \mwedge{} (0 < ||L'|| {}\mRightarrow{} (\mneg{}((snd(hd(L'))) = 0))))\} )
Date html generated:
2019_10_31-AM-06_22_09
Last ObjectModification:
2019_02_19-PM-00_28_21
Theory : reals_2
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