Nuprl Lemma : l-ordered-remove-repeats-fun
∀[A,B:Type].
∀R:A ⟶ A ⟶ ℙ. ∀eq:EqDecider(B). ∀f:A ⟶ B. ∀L:A List.
(l-ordered(A;x,y.R[x;y];L) ⇒ l-ordered(A;x,y.R[x;y];remove-repeats-fun(eq;f;L)))
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
remove-repeats-fun: remove-repeats-fun(eq;f;L),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
prop: ℙ,
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],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
remove-repeats-fun: remove-repeats-fun(eq;f;L),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
and: P ∧ Q,
cand: A c∧ B,
deq: EqDecider(T),
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
nat: ℕ,
ge: i ≥ j ,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
l-ordered_wf,
remove-repeats-fun_wf,
list_wf,
list_ind_nil_lemma,
nil_wf,
list_ind_cons_lemma,
l-ordered-filter,
bnot_wf,
member_filter_2,
l_member_wf,
remove-repeats-fun-member,
int_seg_subtype_nat,
length_wf,
false_wf,
int_seg_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
less_than_wf,
equal_wf,
select_wf,
nat_properties,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
filter_wf5,
all_wf,
l-ordered-cons,
cons_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
applyEquality,
functionExtensionality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
productElimination,
because_Cache,
setElimination,
independent_pairFormation,
setEquality,
dependent_pairFormation,
natural_numberEquality,
independent_isectElimination,
unionElimination,
imageElimination,
int_eqEquality,
intEquality,
computeAll,
equalitySymmetry,
productEquality,
addLevel,
impliesFunctionality,
universeEquality
Latex:
\mforall{}[A,B:Type].
\mforall{}R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}. \mforall{}eq:EqDecider(B). \mforall{}f:A {}\mrightarrow{} B. \mforall{}L:A List.
(l-ordered(A;x,y.R[x;y];L) {}\mRightarrow{} l-ordered(A;x,y.R[x;y];remove-repeats-fun(eq;f;L)))
Date html generated:
2018_05_21-PM-07_38_54
Last ObjectModification:
2017_07_26-PM-05_13_07
Theory : general
Home
Index