Nuprl Lemma : rel-star-rel-plus
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀x,y,z:T. ((x (R^*) y) ⇒ (y R z) ⇒ (x R+ z))
Proof
Definitions occuring in Statement :
rel_plus: R+,
rel_star: R^*,
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
Definitions unfolded in proof :
rel_plus: R+,
rel_star: R^*,
infix_ap: x f y,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
cand: A c∧ B
Lemmas referenced :
equal_wf,
and_wf,
int_formula_prop_less_lemma,
intformless_wf,
subtract_wf,
infix_ap_wf,
add-subtract-cancel,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
rel_exp_iff,
nat_plus_subtype_nat,
less_than_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
false_wf,
decidable__lt,
rel_exp_wf,
nat_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
applyEquality,
hypothesisEquality,
cut,
lemma_by_obid,
isectElimination,
hypothesis,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
int_eqEquality,
computeAll,
inlFormation,
productEquality,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}x,y,z:T. ((x rel\_star(T; R) y) {}\mRightarrow{} (y R z) {}\mRightarrow{} (x R\msupplus{} z))
Date html generated:
2016_05_14-PM-03_53_35
Last ObjectModification:
2016_01_14-PM-11_10_45
Theory : relations2
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