Nuprl Lemma : prec-sub_wf
∀[P:Type]. ∀[a:Atom ⟶ P ⟶ ((P + P + Type) List)]. ∀[j:P]. ∀[x:prec(lbl,p.a[lbl;p];j)]. ∀[i:P].
∀[y:prec(lbl,p.a[lbl;p];i)].
(prec-sub(P;lbl,p.a[lbl;p];j;x;i;y) ∈ ℙ)
Proof
Definitions occuring in Statement :
prec-sub: prec-sub(P;lbl,p.a[lbl; p];j;x;i;y),
prec: prec(lbl,p.a[lbl; p];i),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
member: t ∈ T,
function: x:A ⟶ B[x],
union: left + right,
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prec-sub: prec-sub(P;lbl,p.a[lbl; p];j;x;i;y),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
let: let,
prec-arg-types: prec-arg-types(lbl,p.a[lbl; p];i;lbl),
prop: ℙ,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff
Lemmas referenced :
dest-prec_wf,
istype-atom,
int_seg_wf,
length_wf,
select-tuple_wf,
map_wf,
prec_wf,
list_wf,
int_seg_subtype_nat,
istype-false,
map-length,
istype-void,
int_seg_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
select-map,
subtype_rel_list,
top_wf,
assert_wf,
or_wf,
equal_wf,
select_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
true_wf,
bfalse_wf,
btrue_wf,
btrue_neq_bfalse,
l_member_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
Error :inhabitedIsType,
hypothesis,
Error :lambdaFormation_alt,
productElimination,
setElimination,
rename,
productEquality,
natural_numberEquality,
instantiate,
unionEquality,
cumulativity,
universeEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
Error :equalityIstype,
dependent_functionElimination,
independent_functionElimination,
Error :unionIsType,
independent_isectElimination,
independent_pairFormation,
imageElimination,
Error :isect_memberEquality_alt,
voidElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :universeIsType,
because_Cache,
applyLambdaEquality,
Error :inlEquality_alt,
hyp_replacement,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
Error :inrEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
Error :functionIsType
Latex:
\mforall{}[P:Type]. \mforall{}[a:Atom {}\mrightarrow{} P {}\mrightarrow{} ((P + P + Type) List)]. \mforall{}[j:P]. \mforall{}[x:prec(lbl,p.a[lbl;p];j)]. \mforall{}[i:P].
\mforall{}[y:prec(lbl,p.a[lbl;p];i)].
(prec-sub(P;lbl,p.a[lbl;p];j;x;i;y) \mmember{} \mBbbP{})
Date html generated:
2019_06_20-PM-02_05_38
Last ObjectModification:
2019_02_22-PM-07_07_59
Theory : tuples
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