Nuprl Lemma : prec-sub+-size
∀[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)].
||j;x|| < ||i;y|| supposing prec_sub+(P;lbl,p.a[lbl;p]) <j, x> <i, y>
Proof
Definitions occuring in Statement :
prec_sub+: prec_sub+(P;lbl,p.a[lbl; p]),
prec-size: ||i;x||,
prec: prec(lbl,p.a[lbl; p];i),
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
union: left + right,
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
infix_ap: x f y,
rel_plus: R+,
exists: ∃x:A. B[x],
prec_sub+: prec_sub+(P;lbl,p.a[lbl; p]),
less_than: a < b,
squash: ↓T,
prop: ℙ,
nat: ℕ,
trans: Trans(T;x,y.E[x; y]),
decidable: Dec(P),
or: P ∨ Q,
and: P ∧ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prec_sub: prec_sub(P;lbl,p.a[lbl; p])
Lemmas referenced :
rel_plus_closure,
prec_wf,
istype-atom,
prec_sub_wf,
less_than_wf,
prec-size_wf,
prec_sub+_wf,
subtype_rel_self,
member-less_than,
list_wf,
istype-universe,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
prec-sub-size
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
Error :inhabitedIsType,
hypothesis,
spreadEquality,
because_Cache,
Error :productIsType,
Error :universeIsType,
independent_functionElimination,
Error :lambdaFormation_alt,
dependent_functionElimination,
Error :dependent_pairEquality_alt,
instantiate,
universeEquality,
Error :isect_memberEquality_alt,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
Error :isectIsTypeImplies,
Error :functionIsType,
unionEquality,
cumulativity,
productElimination,
unionElimination,
imageElimination,
natural_numberEquality,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation
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)].
||j;x|| < ||i;y|| supposing prec\_sub+(P;lbl,p.a[lbl;p]) <j, x> <i, y>
Date html generated:
2019_06_20-PM-02_14_25
Last ObjectModification:
2019_02_23-PM-05_06_12
Theory : tuples
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