Nuprl Lemma : decidable__rel_plus
∀[T:Type]
((∀x,y:T. Dec(x = y ∈ T))
⇒ (∀[R:T ⟶ T ⟶ ℙ]. (SWellFounded(x R y) ⇒ rel_finite(T;R) ⇒ (∀x,y:T. Dec(x R y)) ⇒ (∀x,y:T. Dec(x R+ y)))))
Proof
Definitions occuring in Statement :
strongwellfounded: SWellFounded(R[x; y]),
rel_finite: rel_finite(T;R),
rel_plus: R+,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
pi1: fst(t),
prop: ℙ,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
uimplies: b supposing a,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
rel_plus: R+,
iff: P ⇐⇒ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
rev_implies: P ⇐ Q
Lemmas referenced :
less_than_wf,
rel_finite_wf,
decidable__rel_exp_finite,
decidable__exists_int_seg,
rel_plus_wf,
decidable_functionality,
infix_ap_wf,
int_seg_wf,
int_seg_subtype_nat_plus,
exists_wf,
false_wf,
int_seg_subtype_nat,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
nat_plus_wf,
nat_plus_subtype_nat,
rel_exp_wf,
int_formula_prop_wf,
int_term_value_add_lemma,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
itermAdd_wf,
intformnot_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
le_wf,
nat_properties,
decidable__le,
nat_plus_properties,
equal_wf,
decidable_wf,
all_wf,
strongwellfounded_wf,
strongwellfounded_rel_exp
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
promote_hyp,
rename,
productElimination,
sqequalRule,
lambdaEquality,
applyEquality,
universeEquality,
functionEquality,
cumulativity,
independent_isectElimination,
setElimination,
dependent_functionElimination,
because_Cache,
unionElimination,
equalityTransitivity,
equalitySymmetry,
setEquality,
intEquality,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
introduction,
addEquality,
instantiate,
independent_functionElimination
Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]
(SWellFounded(x R y)
{}\mRightarrow{} rel\_finite(T;R)
{}\mRightarrow{} (\mforall{}x,y:T. Dec(x R y))
{}\mRightarrow{} (\mforall{}x,y:T. Dec(x R\msupplus{} y)))))
Date html generated:
2016_05_14-PM-03_52_29
Last ObjectModification:
2016_01_14-PM-11_11_07
Theory : relations2
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