Nuprl Lemma : no-repeats-pairwise
∀[T:Type]
∀L:T List
∀[P:{x:T| (x ∈ L)} ⟶ {x:T| (x ∈ L)} ⟶ ℙ']
(∀x,y:{x:T| (x ∈ L)} . P[x;y] supposing ¬(x = y ∈ T)) ⇒ (∀x,y∈L. P[x;y]) supposing no_repeats(T;L)
Proof
Definitions occuring in Statement :
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]} ,
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],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
pairwise: (∀x,y∈L. P[x; y]),
no_repeats: no_repeats(T;l),
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j
Lemmas referenced :
int_seg_wf,
nat_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
le_wf,
nat_properties,
length_wf,
false_wf,
int_seg_subtype_nat,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_seg_properties,
list-subtype,
select_wf,
list_wf,
no_repeats_wf,
equal_wf,
not_wf,
isect_wf,
l_member_wf,
all_wf,
no_repeats_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
instantiate,
setEquality,
cumulativity,
applyEquality,
lambdaEquality,
universeEquality,
sqequalRule,
setElimination,
because_Cache,
dependent_set_memberEquality,
dependent_functionElimination,
functionEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}']
(\mforall{}x,y:\{x:T| (x \mmember{} L)\} . P[x;y] supposing \mneg{}(x = y)) {}\mRightarrow{} (\mforall{}x,y\mmember{}L. P[x;y])
supposing no\_repeats(T;L)
Date html generated:
2016_05_14-PM-03_07_47
Last ObjectModification:
2016_01_15-AM-07_18_33
Theory : list_1
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