Nuprl Lemma : llex_transitivity
∀[A:Type]. ∀[<:A ⟶ A ⟶ ℙ].
(Trans(A;a,b.<[a;b])
⇒ (∀as,bs,cs:A List. ((as llex(A;a,b.<[a;b]) bs) ⇒ (bs llex(A;a,b.<[a;b]) cs) ⇒ (as llex(A;a,b.<[a;b]) cs))))
Proof
Definitions occuring in Statement :
llex: llex(A;a,b.<[a; b]),
list: T List,
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
llex: llex(A;a,b.<[a; b]),
infix_ap: x f y,
or: P ∨ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
cand: A c∧ B,
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
ge: i ≥ j ,
label: ...$L... t,
trans: Trans(T;x,y.E[x; y])
Lemmas referenced :
llex_wf,
subtype_rel_self,
list_wf,
trans_wf,
istype-universe,
decidable__lt,
length_wf,
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,
equal_wf,
squash_wf,
true_wf,
select_wf,
int_seg_properties,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
iff_weakening_equal,
istype-le,
istype-less_than,
int_seg_wf,
istype-nat,
nat_properties,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
sqequalRule,
unionElimination,
thin,
universeIsType,
cut,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
lambdaEquality_alt,
inhabitedIsType,
hypothesis,
instantiate,
universeEquality,
because_Cache,
functionIsType,
inlFormation_alt,
productElimination,
dependent_functionElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
productIsType,
equalityIstype,
inrFormation_alt,
functionEquality,
cumulativity,
hyp_replacement,
productEquality
Latex:
\mforall{}[A:Type]. \mforall{}[<:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
(Trans(A;a,b.<[a;b])
{}\mRightarrow{} (\mforall{}as,bs,cs:A List.
((as llex(A;a,b.<[a;b]) bs) {}\mRightarrow{} (bs llex(A;a,b.<[a;b]) cs) {}\mRightarrow{} (as llex(A;a,b.<[a;b]) cs))))
Date html generated:
2020_05_20-AM-08_07_38
Last ObjectModification:
2019_12_31-PM-06_54_48
Theory : general
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