Nuprl Lemma : rel_plus_transitivity
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[x,y,z:T]. ((R+ x y) ⇒ (R+ y z) ⇒ (R+ x z))
Proof
Definitions occuring in Statement :
rel_plus: R+,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
rel_plus: R+,
exists: ∃x:A. B[x],
member: t ∈ T,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
infix_ap: x f y,
nat: ℕ,
subtype_rel: A ⊆r B
Lemmas referenced :
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
rel_exp_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
subtype_rel_self,
rel_plus_wf,
istype-universe,
rel_exp_add,
nat_plus_subtype_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
Error :dependent_pairFormation_alt,
Error :dependent_set_memberEquality_alt,
addEquality,
setElimination,
rename,
cut,
hypothesisEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
applyEquality,
instantiate,
because_Cache,
functionExtensionality,
functionEquality,
cumulativity,
universeEquality,
Error :inhabitedIsType,
Error :functionIsType
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[x,y,z:T]. ((R\msupplus{} x y) {}\mRightarrow{} (R\msupplus{} y z) {}\mRightarrow{} (R\msupplus{} x z))
Date html generated:
2019_06_20-PM-02_01_46
Last ObjectModification:
2019_02_26-AM-11_34_24
Theory : relations2
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