Nuprl Lemma : transitive-closure-transitive
∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ]. UniformlyTrans(A;x,y.x TC(R) y)
Proof
Definitions occuring in Statement :
transitive-closure: TC(R),
utrans: UniformlyTrans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
utrans: UniformlyTrans(T;x,y.E[x; y]),
implies: P ⇒ Q,
transitive-closure: TC(R),
infix_ap: x f y,
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
and: P ∧ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
rel_path: rel_path(A;L;x;y),
pi1: fst(t),
pi2: snd(t),
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
guard: {T}
Lemmas referenced :
transitive-closure_wf,
append_wf,
list_induction,
all_wf,
rel_path_wf,
list_wf,
list_ind_nil_lemma,
and_wf,
equal_wf,
list_ind_cons_lemma,
length-append,
decidable__lt,
length_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_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_wf,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
sqequalHypSubstitution,
sqequalRule,
setElimination,
thin,
applyEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
functionExtensionality,
hypothesis,
functionEquality,
universeEquality,
dependent_set_memberEquality,
productEquality,
lambdaEquality,
productElimination,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
addLevel,
hyp_replacement,
equalitySymmetry,
independent_pairFormation,
applyLambdaEquality,
levelHypothesis,
natural_numberEquality,
addEquality,
unionElimination,
imageElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. UniformlyTrans(A;x,y.x TC(R) y)
Date html generated:
2017_04_17-AM-09_25_45
Last ObjectModification:
2017_02_27-PM-05_26_20
Theory : relations2
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