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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