Nuprl Lemma : diamond-implies-TC-confluent
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(rel-diamond-property(T;x,y.R[x;y])
⇒ (∃m:T ⟶ ℕ. ∀x,y:T. (R[x;y] ⇒ m y < m x))
⇒ rel-confluent(T;x,y.λx,y. R[x;y]^* x y))
Proof
Definitions occuring in Statement :
rel-confluent: rel-confluent(T;x,y.R[x; y]),
rel-diamond-property: rel-diamond-property(T;x,y.R[x; y]),
transitive-reflexive-closure: R^*,
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
rel-confluent: rel-confluent(T;x,y.R[x; y]),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x y.t[x; y],
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
infix_ap: x f y,
cand: A c∧ B,
transitive-reflexive-closure: R^*,
rel-diamond-property: rel-diamond-property(T;x,y.R[x; y]),
less_than: a < b,
squash: ↓T
Lemmas referenced :
istype-nat,
subtype_rel_self,
istype-less_than,
rel-diamond-property_wf,
istype-universe,
transitive-reflexive-closure_wf,
subtract_wf,
istype-int,
primrec-wf2,
less_than_wf,
add_nat_wf,
istype-void,
istype-le,
nat_properties,
decidable__le,
add-is-int-iff,
set_subtype_base,
le_wf,
int_subtype_base,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
transitive-reflexive-closure-cases,
transitive-closure_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
transitive-reflexive-closure-base-case,
transitive-reflexive-closure_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
because_Cache,
applyEquality,
thin,
instantiate,
sqequalHypSubstitution,
isectElimination,
universeEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
productElimination,
natural_numberEquality,
setIsType,
functionEquality,
productEquality,
dependent_functionElimination,
dependent_set_memberEquality_alt,
addEquality,
independent_pairFormation,
voidElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
pointwiseFunctionality,
promote_hyp,
intEquality,
independent_isectElimination,
baseClosed,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
equalityIstype,
hyp_replacement,
inlFormation_alt,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(rel-diamond-property(T;x,y.R[x;y])
{}\mRightarrow{} (\mexists{}m:T {}\mrightarrow{} \mBbbN{}. \mforall{}x,y:T. (R[x;y] {}\mRightarrow{} m y < m x))
{}\mRightarrow{} rel-confluent(T;x,y.\mlambda{}x,y. R[x;y]\^{}* x y))
Date html generated:
2019_10_15-AM-10_24_43
Last ObjectModification:
2019_08_16-PM-03_15_24
Theory : relations2
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