Nuprl Lemma : wellfounded-llex
∀[A:Type]. ∀[<:A ⟶ A ⟶ ℙ].
((∀a,b:A. SqStable(<[a;b]))
⇒ WellFnd{i}(A;a,b.<[a;b])
⇒ WellFnd{i}(Des(A;a,b.<[a;b]);L1,L2.L1 llex(A;a,b.<[a;b]) L2))
Proof
Definitions occuring in Statement :
llex: llex(A;a,b.<[a; b]),
Des: Des(A;a,b.<[a; b]),
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
Des: Des(A;a,b.<[a; b]),
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
guard: {T},
so_lambda: λ2x.t[x],
uimplies: b supposing a,
infix_ap: x f y,
or: P ∨ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
ge: i ≥ j ,
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
cand: A c∧ B,
iff: P ⇐⇒ Q,
cons: [a / b],
less_than': less_than'(a;b),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
nat_plus: ℕ+,
true: True,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
descending: descending(a,b.<[a; b];L),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
select: L[n],
subtract: n - m,
llex: llex(A;a,b.<[a; b]),
nil: [],
it: ⋅,
istype: istype(T),
nat: ℕ
Lemmas referenced :
descending_wf,
append_wf,
cons_wf,
istype-universe,
Des_wf,
infix_ap_wf,
llex_wf,
wellfounded_wf,
sq_stable_wf,
all_wf,
subtype_rel_dep_function,
list_wf,
subtype_rel_universe1,
subtype_rel_self,
nil_wf,
llex-append1,
less_than_wf,
length_wf,
hd_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_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_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
descending-append,
decidable__lt,
list-cases,
length_of_nil_lemma,
product_subtype_list,
reduce_hd_cons_lemma,
length_of_cons_lemma,
list_ind_cons_lemma,
list_ind_nil_lemma,
append-nil,
subtype_rel_list,
top_wf,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
append_assoc_sq,
istype-false,
non_neg_length,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_wf,
squash_wf,
true_wf,
stuck-spread,
istype-base,
subtype_rel-equal,
nat_properties,
int_seg_properties,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
because_Cache,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
universeIsType,
setEquality,
functionIsType,
instantiate,
cumulativity,
universeEquality,
independent_functionElimination,
functionEquality,
closedConclusion,
independent_isectElimination,
dependent_set_memberEquality_alt,
setIsType,
dependent_functionElimination,
unionElimination,
inlFormation_alt,
productIsType,
equalityIsType1,
natural_numberEquality,
imageElimination,
productElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
inrFormation_alt,
hyp_replacement,
applyLambdaEquality,
promote_hyp,
hypothesis_subsumption,
imageMemberEquality,
baseClosed,
pointwiseFunctionality,
baseApply,
addEquality,
minusEquality,
voidEquality
Latex:
\mforall{}[A:Type]. \mforall{}[<:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
((\mforall{}a,b:A. SqStable(<[a;b]))
{}\mRightarrow{} WellFnd\{i\}(A;a,b.<[a;b])
{}\mRightarrow{} WellFnd\{i\}(Des(A;a,b.<[a;b]);L1,L2.L1 llex(A;a,b.<[a;b]) L2))
Date html generated:
2019_10_15-AM-11_12_05
Last ObjectModification:
2018_10_11-PM-11_08_12
Theory : general
Home
Index