Nuprl Lemma : pairwise-iff2
∀[T:Type]
∀L:T List
∀[P:T ⟶ T ⟶ ℙ']
((∀x,y:T. (P[x;y] ⇒ P[y;x]))
⇒ no_repeats(T;L)
⇒ ((∀x,y∈L. P[x;y]) ⇐⇒ ∀x,y:T. ((x ∈ L) ⇒ (y ∈ L) ⇒ (¬(x = y ∈ T)) ⇒ P[x;y])))
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,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
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],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
member: t ∈ T,
false: False,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
or: P ∨ Q,
guard: {T},
pairwise: (∀x,y∈L. P[x; y]),
int_seg: {i..j-},
uimplies: b supposing a,
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
no_repeats: no_repeats(T;l),
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
istype-void,
l_member_wf,
pairwise_wf2,
subtype_rel_self,
no_repeats_wf,
list_wf,
istype-universe,
pairwise-implies,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
length_wf,
intformless_wf,
int_formula_prop_less_lemma,
select_member,
istype-le,
istype-less_than,
int_seg_subtype_nat,
istype-false,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
istype-nat,
set_subtype_base,
lelt_wf,
int_subtype_base,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
sqequalRule,
functionIsType,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
hypothesis,
cut,
introduction,
extract_by_obid,
universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
cumulativity,
lambdaEquality_alt,
applyEquality,
because_Cache,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
voidElimination,
setElimination,
rename,
independent_isectElimination,
productElimination,
imageElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
dependent_set_memberEquality_alt,
productIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
intEquality,
sqequalBase
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[P:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}']
((\mforall{}x,y:T. (P[x;y] {}\mRightarrow{} P[y;x]))
{}\mRightarrow{} no\_repeats(T;L)
{}\mRightarrow{} ((\mforall{}x,y\mmember{}L. P[x;y]) \mLeftarrow{}{}\mRightarrow{} \mforall{}x,y:T. ((x \mmember{} L) {}\mRightarrow{} (y \mmember{} L) {}\mRightarrow{} (\mneg{}(x = y)) {}\mRightarrow{} P[x;y])))
Date html generated:
2020_05_19-PM-09_43_24
Last ObjectModification:
2019_10_21-PM-10_25_04
Theory : list_1
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