Nuprl Lemma : pairwise-cons
∀[T:Type]. ∀x:T. ∀L:T List. ∀[P:T ⟶ T ⟶ ℙ']. ((∀x,y∈[x / L]. P[x;y]) ⇐⇒ (∀x,y∈L. P[x;y]) ∧ (∀y∈L.P[x;y]))
Proof
Definitions occuring in Statement :
pairwise: (∀x,y∈L. P[x; y]),
l_all: (∀x∈L.P[x]),
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
pairwise: (∀x,y∈L. P[x; y]),
member: t ∈ T,
int_seg: {i..j-},
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
uimplies: b supposing a,
lelt: i ≤ j < k,
top: Top,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
subtract: n - m,
ge: i ≥ j ,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
l_all: (∀x∈L.P[x]),
less_than': less_than'(a;b),
select: L[n],
cons: [a / b],
sq_type: SQType(T)
Lemmas referenced :
int_formula_prop_eq_lemma,
intformeq_wf,
subtract-add-cancel,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
select-cons-tl,
false_wf,
select_cons_tl_sq,
add-subtract-cancel,
lelt_wf,
int_term_value_add_lemma,
itermAdd_wf,
decidable__lt,
non_neg_length,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
itermSubtract_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
subtract_wf,
decidable__le,
int_seg_properties,
length_of_cons_lemma,
add-member-int_seg2,
list_wf,
l_member_wf,
l_all_wf,
and_wf,
cons_wf,
pairwise_wf2,
length_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
cut,
lemma_by_obid,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
instantiate,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
productElimination,
setEquality,
functionEquality,
universeEquality,
dependent_functionElimination,
because_Cache,
independent_isectElimination,
dependent_set_memberEquality,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
addEquality,
imageElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]
\mforall{}x:T. \mforall{}L:T List.
\mforall{}[P:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}']. ((\mforall{}x,y\mmember{}[x / L]. P[x;y]) \mLeftarrow{}{}\mRightarrow{} (\mforall{}x,y\mmember{}L. P[x;y]) \mwedge{} (\mforall{}y\mmember{}L.P[x;y]))
Date html generated:
2016_05_14-PM-01_49_46
Last ObjectModification:
2016_01_15-AM-08_19_22
Theory : list_1
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