Nuprl Lemma : closure-well-founded-total
A well-founded, one-one, decidable relation
which is "retracable" (in that everything
has only finitely many predecessors)
with at most one minimal element has the
property that its transitive closure totally
orders its field.⋅
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(WellFnd{i}(T;x,y.R x y)
⇒ (∀x,y:T. Dec(R x y))
⇒ (∀y:T. ∃L:T List. ∀x:T. ((R x y) ⇒ (x ∈ L)))
⇒ (∀a,b:T. (((R^*) a b) ∨ ((R^*) b a))) supposing
((∀y,z:T. ((∀x:T. ((¬(R x y)) ∧ (¬(R x z)))) ⇒ (y = z ∈ T))) and
one-one(T;T;R)))
Proof
Definitions occuring in Statement :
one-one: one-one(A;B;R),
l_member: (x ∈ l),
list: T List,
rel_star: R^*,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
not: ¬A,
cand: A c∧ B,
exists: ∃x:A. B[x],
so_apply: x[s],
and: P ∧ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
one-one: one-one(A;B;R),
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
rel_star: R^*,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
infix_ap: x f y,
iff: P ⇐⇒ Q,
subtract: n - m
Lemmas referenced :
wellfounded_wf,
decidable_wf,
l_member_wf,
list_wf,
exists_wf,
one-one_wf,
equal_wf,
rel_star_wf,
wellfounded-minimal-field,
not_wf,
all_wf,
decidable__le,
subtract_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_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_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
rel_exp_wf,
subtype_rel_self,
rel-exp-add-iff,
minus-one-mul,
add-swap,
add-mul-special,
zero-mul,
add-zero,
rel_exp-one-one,
add-commutes,
add-associates,
zero-add
Rules used in proof :
functionEquality,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
independent_pairFormation,
productElimination,
independent_functionElimination,
productEquality,
isectElimination,
extract_by_obid,
rename,
because_Cache,
universeEquality,
cumulativity,
functionExtensionality,
applyEquality,
hypothesis,
axiomEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
setElimination,
unionElimination,
inlFormation_alt,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
universeIsType,
voidElimination,
productIsType,
inhabitedIsType,
instantiate,
inrFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(WellFnd\{i\}(T;x,y.R x y)
{}\mRightarrow{} (\mforall{}x,y:T. Dec(R x y))
{}\mRightarrow{} (\mforall{}y:T. \mexists{}L:T List. \mforall{}x:T. ((R x y) {}\mRightarrow{} (x \mmember{} L)))
{}\mRightarrow{} (\mforall{}a,b:T. ((rel\_star(T; R) a b) \mvee{} (rel\_star(T; R) b a))) supposing
((\mforall{}y,z:T. ((\mforall{}x:T. ((\mneg{}(R x y)) \mwedge{} (\mneg{}(R x z)))) {}\mRightarrow{} (y = z))) and
one-one(T;T;R)))
Date html generated:
2020_05_20-AM-08_10_00
Last ObjectModification:
2020_01_28-PM-00_09_22
Theory : general
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