Nuprl Lemma : pairwise-implies
∀[T:Type]
∀L:T List
∀[P:T ⟶ T ⟶ ℙ']. ((∀x,y∈L. P[x;y]) ⇒ (∀x,y:T. ((x ∈ L) ⇒ (y ∈ L) ⇒ ((x = y ∈ T) ∨ P[x;y] ∨ P[y;x]))))
Proof
Definitions occuring in Statement :
pairwise: (∀x,y∈L. P[x; y]),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
l_member: (x ∈ l),
pairwise: (∀x,y∈L. P[x; y]),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
nat: ℕ,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
le: A ≤ B
Lemmas referenced :
or_wf,
equal_wf,
exists_wf,
nat_wf,
less_than_wf,
length_wf,
select_wf,
nat_properties,
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,
all_wf,
int_seg_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
list_wf,
lelt_wf,
squash_wf,
le_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
instantiate,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
equalityTransitivity,
applyEquality,
functionExtensionality,
because_Cache,
lambdaEquality,
productEquality,
setElimination,
rename,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
universeEquality,
imageElimination,
functionEquality,
inrFormation,
inlFormation,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[P:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}']
((\mforall{}x,y\mmember{}L. P[x;y]) {}\mRightarrow{} (\mforall{}x,y:T. ((x \mmember{} L) {}\mRightarrow{} (y \mmember{} L) {}\mRightarrow{} ((x = y) \mvee{} P[x;y] \mvee{} P[y;x]))))
Date html generated:
2017_04_17-AM-07_44_13
Last ObjectModification:
2017_02_27-PM-04_16_29
Theory : list_1
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