Nuprl Lemma : transitive-closure-cases
∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀x,y:A. ((x TC(R) y) ⇒ ((x R y) ∨ (∃z:A. ((x R z) ∧ (z TC(R) y)))))
Proof
Definitions occuring in Statement :
transitive-closure: TC(R),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
transitive-closure: TC(R),
infix_ap: x f y,
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
cons: [a / b],
false: False,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
length: ||as||,
list_ind: list_ind,
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
rel_path: rel_path(A;L;x;y),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
pi1: fst(t),
pi2: snd(t),
guard: {T},
exists: ∃x:A. B[x],
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
le: A ≤ B,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A
Lemmas referenced :
list-cases,
product_subtype_list,
transitive-closure_wf,
exists_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
equal_wf,
cons_wf,
non_neg_length,
decidable__lt,
length_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
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_le_lemma,
int_formula_prop_wf,
rel_path_wf,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
rename,
setElimination,
thin,
cut,
productEquality,
hypothesisEquality,
applyEquality,
hypothesis,
lambdaEquality,
universeEquality,
introduction,
extract_by_obid,
isectElimination,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
cumulativity,
functionExtensionality,
functionEquality,
imageElimination,
voidElimination,
inlFormation,
because_Cache,
isect_memberEquality,
voidEquality,
hyp_replacement,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
instantiate,
inrFormation,
dependent_pairFormation,
independent_pairFormation,
dependent_set_memberEquality,
dependent_pairEquality,
natural_numberEquality,
addEquality,
independent_isectElimination,
int_eqEquality,
intEquality,
computeAll
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. \mforall{}x,y:A. ((x TC(R) y) {}\mRightarrow{} ((x R y) \mvee{} (\mexists{}z:A. ((x R z) \mwedge{} (z TC(R) y)))))
Date html generated:
2017_04_17-AM-09_25_49
Last ObjectModification:
2017_02_27-PM-05_26_27
Theory : relations2
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