Nuprl Lemma : non-forking-wellfounded-linorder
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
  (decidable-non-minimal(T;x,y.R[x;y])
  
⇒ WellFnd{i}(T;x,y.R[x;y])
  
⇒ (∀m:T. (unique-minimal(T;x,y.R[x;y];m) 
⇒ non-forking(T;x,y.R[x;y]) 
⇒ WeakLinorder(T;x,y.x (R^*) y))))
Proof
Definitions occuring in Statement : 
non-forking: non-forking(T;x,y.R[x; y])
, 
decidable-non-minimal: decidable-non-minimal(T;x,y.R[x; y])
, 
unique-minimal: unique-minimal(T;x,y.R[x; y];m)
, 
rel_star: R^*
, 
weak-linorder: WeakLinorder(T;x,y.R[x; y])
, 
wellfounded: WellFnd{i}(A;x,y.R[x; y])
, 
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
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
weak-linorder: WeakLinorder(T;x,y.R[x; y])
, 
and: P ∧ Q
, 
infix_ap: x f y
, 
so_apply: x[s1;s2]
, 
prop: ℙ
, 
so_lambda: λ2x y.t[x; y]
, 
weak-connex: weak-connex(T; x,y.R[x; y])
, 
squash: ↓T
, 
true: True
, 
subtype_rel: A ⊆r B
, 
uimplies: b supposing a
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
rel_star: R^*
, 
exists: ∃x:A. B[x]
, 
nat: ℕ
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
ge: i ≥ j 
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
false: False
, 
not: ¬A
, 
top: Top
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
subtract: n - m
, 
non-forking: non-forking(T;x,y.R[x; y])
Lemmas referenced : 
unique-minimal-wellfounded-implies, 
rel_star_order, 
non-forking_wf, 
unique-minimal_wf, 
wellfounded_wf, 
decidable-non-minimal_wf, 
rel_star_wf, 
equal_wf, 
squash_wf, 
true_wf, 
eta_conv, 
iff_weakening_equal, 
decidable__lt, 
subtract_wf, 
nat_properties, 
decidable__le, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermSubtract_wf, 
itermVar_wf, 
intformless_wf, 
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_less_lemma, 
int_formula_prop_wf, 
le_wf, 
rel_exp_add_iff, 
infix_ap_wf, 
rel_exp_wf, 
exists_wf, 
nat_wf, 
minus-one-mul, 
add-swap, 
add-mul-special, 
zero-mul, 
add-zero, 
non-forking-rel_exp, 
add-commutes, 
add-associates, 
zero-add
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
because_Cache, 
hypothesisEquality, 
independent_functionElimination, 
hypothesis, 
dependent_functionElimination, 
independent_pairFormation, 
sqequalRule, 
cumulativity, 
lambdaEquality, 
applyEquality, 
functionExtensionality, 
functionEquality, 
universeEquality, 
imageElimination, 
imageMemberEquality, 
baseClosed, 
isect_memberEquality, 
hyp_replacement, 
equalitySymmetry, 
instantiate, 
equalityTransitivity, 
natural_numberEquality, 
independent_isectElimination, 
productElimination, 
setElimination, 
rename, 
unionElimination, 
inlFormation, 
dependent_pairFormation, 
dependent_set_memberEquality, 
int_eqEquality, 
intEquality, 
voidElimination, 
voidEquality, 
computeAll, 
applyLambdaEquality, 
inrFormation
Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    (decidable-non-minimal(T;x,y.R[x;y])
    {}\mRightarrow{}  WellFnd\{i\}(T;x,y.R[x;y])
    {}\mRightarrow{}  (\mforall{}m:T
                (unique-minimal(T;x,y.R[x;y];m)
                {}\mRightarrow{}  non-forking(T;x,y.R[x;y])
                {}\mRightarrow{}  WeakLinorder(T;x,y.x  rel\_star(T;  R)  y))))
Date html generated:
2018_05_21-PM-09_05_34
Last ObjectModification:
2017_07_26-PM-06_28_24
Theory : general
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