Nuprl Lemma : l_before_l_index_le
∀[T:Type]
∀dT:EqDecider(T). ∀L:T List. ∀x,y:T.
((x ∈ L) ⇒ (y ∈ L) ⇒ x before y ∈ L ∨ (x = y ∈ T) supposing index(L;x) ≤ index(L;y))
Proof
Definitions occuring in Statement :
l_index: index(L;x),
l_before: x before y ∈ l,
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
prop: ℙ,
squash: ↓T,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
cand: A c∧ B,
ge: i ≥ j ,
guard: {T},
nat: ℕ,
lelt: i ≤ j < k,
true: True,
iff: P ⇐⇒ Q
Lemmas referenced :
le_witness_for_triv,
decidable__lt,
istype-le,
l_index_wf,
l_member_wf,
list_wf,
deq_wf,
istype-universe,
l_before_l_index,
l_before_wf,
select_wf,
squash_wf,
le_wf,
less_than_wf,
length_wf,
istype-int,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
intformless_wf,
intformle_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
non_neg_length,
decidable__le,
length_wf_nat,
int_seg_properties,
nat_properties,
itermConstant_wf,
int_term_value_constant_lemma,
equal_wf,
true_wf,
select_l_index,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_isectElimination,
rename,
dependent_functionElimination,
because_Cache,
unionElimination,
hypothesisEquality,
applyEquality,
Error :lambdaEquality_alt,
setElimination,
Error :inhabitedIsType,
sqequalRule,
Error :universeIsType,
instantiate,
universeEquality,
Error :inlFormation_alt,
independent_functionElimination,
Error :equalityIstype,
Error :inrFormation_alt,
imageElimination,
productEquality,
natural_numberEquality,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
applyLambdaEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]
\mforall{}dT:EqDecider(T). \mforall{}L:T List. \mforall{}x,y:T.
((x \mmember{} L) {}\mRightarrow{} (y \mmember{} L) {}\mRightarrow{} x before y \mmember{} L \mvee{} (x = y) supposing index(L;x) \mleq{} index(L;y))
Date html generated:
2019_06_20-PM-01_56_56
Last ObjectModification:
2019_01_13-PM-02_23_43
Theory : decidable!equality
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