Nuprl Lemma : transitive-closure-minimal
∀[A:Type]. ∀[R,Q:A ⟶ A ⟶ ℙ]. (R => Q ⇒ Trans(A;x,y.x Q y) ⇒ TC(R) => Q)
Proof
Definitions occuring in Statement :
transitive-closure: TC(R),
rel_implies: R1 => R2,
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
rel_implies: R1 => R2,
all: ∀x:A. B[x],
trans: Trans(T;x,y.E[x; y]),
transitive-closure: TC(R),
infix_ap: x f y,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
and: P ∧ Q,
subtype_rel: A ⊆r B,
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
cons: [a / b],
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),
hd: hd(l),
spreadn: spread3,
sq_exists: ∃x:{A| B[x]},
nat: ℕ,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
guard: {T},
colength: colength(L),
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
cand: A c∧ B
Lemmas referenced :
transitive-closure_wf,
trans_wf,
rel_implies_wf,
list-cases,
length_of_nil_lemma,
product_subtype_list,
list_ind_cons_lemma,
reduce_tl_cons_lemma,
equal_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
rel_path_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list_wf,
list_ind_nil_lemma,
list_accum_nil_lemma,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
list_accum_cons_lemma,
subtype_rel_self,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
rename,
applyEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
functionExtensionality,
hypothesis,
lambdaEquality,
functionEquality,
universeEquality,
setElimination,
productElimination,
productEquality,
dependent_functionElimination,
unionElimination,
imageElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
hyp_replacement,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
instantiate,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
axiomEquality,
because_Cache,
dependent_set_memberEquality,
addEquality,
baseClosed,
dependent_pairEquality
Latex:
\mforall{}[A:Type]. \mforall{}[R,Q:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. (R => Q {}\mRightarrow{} Trans(A;x,y.x Q y) {}\mRightarrow{} TC(R) => Q)
Date html generated:
2017_04_17-AM-09_25_53
Last ObjectModification:
2017_02_27-PM-05_26_53
Theory : relations2
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