Nuprl Lemma : pairwise-mapl-no-repeats
∀[T,T':Type].
∀L:T List. ∀f:{x:T| (x ∈ L)} ⟶ T'.
∀[P:T' ⟶ T' ⟶ ℙ']
(∀x,y:T. ((x ∈ L) ⇒ (y ∈ L) ⇒ P[f x;f y] supposing ¬(x = y ∈ T))) ⇒ (∀x,y∈mapl(f;L). P[x;y])
supposing no_repeats(T;L)
Proof
Definitions occuring in Statement :
mapl: mapl(f;l),
pairwise: (∀x,y∈L. P[x; y]),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
implies: P ⇒ Q,
so_apply: x[s1;s2],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
mapl: mapl(f;l),
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
true: True,
or: P ∨ Q,
guard: {T},
cand: A c∧ B,
uiff: uiff(P;Q),
not: ¬A,
false: False,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
less_than: a < b,
squash: ↓T
Lemmas referenced :
list_induction,
all_wf,
l_member_wf,
uall_wf,
isect_wf,
no_repeats_wf,
not_wf,
equal_wf,
pairwise_wf2,
mapl_wf,
list_wf,
no_repeats_witness,
nil_wf,
map_nil_lemma,
pairwise-nil,
cons_wf,
map_cons_lemma,
pairwise-cons,
cons_member,
subtype_rel_dep_function,
subtype_rel_sets,
set_wf,
no_repeats_cons,
member-mapl,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
select_member,
int_seg_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
setEquality,
because_Cache,
hypothesis,
applyEquality,
universeEquality,
setElimination,
rename,
isectEquality,
functionExtensionality,
dependent_set_memberEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
natural_numberEquality,
inlFormation,
independent_isectElimination,
inrFormation,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
imageElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T,T':Type].
\mforall{}L:T List. \mforall{}f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} T'.
\mforall{}[P:T' {}\mrightarrow{} T' {}\mrightarrow{} \mBbbP{}']
(\mforall{}x,y:T. ((x \mmember{} L) {}\mRightarrow{} (y \mmember{} L) {}\mRightarrow{} P[f x;f y] supposing \mneg{}(x = y))) {}\mRightarrow{} (\mforall{}x,y\mmember{}mapl(f;L). P[x;y])
supposing no\_repeats(T;L)
Date html generated:
2017_04_17-AM-08_41_39
Last ObjectModification:
2017_02_27-PM-05_00_20
Theory : list_1
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