Nuprl Lemma : word-rel-diamond
∀[X:Type]
∀x,y,z:(X + X) List.
(word-rel(X;x;y)
⇒ word-rel(X;x;z)
⇒ ((y = z ∈ ((X + X) List)) ∨ (∃w:(X + X) List. (word-rel(X;y;w) ∧ word-rel(X;z;w)))))
Proof
Definitions occuring in Statement :
word-rel: word-rel(X;w1;w2),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
word-rel: word-rel(X;w1;w2),
so_lambda: λ2x.t[x],
so_apply: x[s],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
uiff: uiff(P;Q),
cons: [a / b],
true: True,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
int_seg_subtype,
false_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
length_wf,
non_neg_length,
nat_properties,
decidable__lt,
lelt_wf,
less_than_wf,
word-rel_wf,
all_wf,
list_wf,
or_wf,
equal_wf,
exists_wf,
set_wf,
primrec-wf2,
nat_wf,
itermAdd_wf,
int_term_value_add_lemma,
length_wf_nat,
append_wf,
list-cases,
list_ind_nil_lemma,
list_ind_cons_lemma,
cons_one_one,
cons_wf,
product_subtype_list,
reduce_tl_cons_lemma,
and_wf,
tl_wf,
squash_wf,
true_wf,
reduce_hd_cons_lemma,
hd_wf,
ge_wf,
length_cons_ge_one,
subtype_rel_list,
top_wf,
inverse-letters_wf,
inverse-inverse-letters,
nil_wf,
length-append,
nil-append,
length_of_cons_lemma,
length_cons,
length_append,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
because_Cache,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
unionElimination,
addLevel,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
unionEquality,
cumulativity,
imageElimination,
independent_functionElimination,
functionEquality,
productEquality,
addEquality,
universeEquality,
hyp_replacement,
promote_hyp,
inlFormation,
imageMemberEquality,
baseClosed,
inrFormation,
instantiate,
equalityUniverse
Latex:
\mforall{}[X:Type]
\mforall{}x,y,z:(X + X) List.
(word-rel(X;x;y)
{}\mRightarrow{} word-rel(X;x;z)
{}\mRightarrow{} ((y = z) \mvee{} (\mexists{}w:(X + X) List. (word-rel(X;y;w) \mwedge{} word-rel(X;z;w)))))
Date html generated:
2017_10_05-AM-00_44_36
Last ObjectModification:
2017_07_28-AM-09_18_36
Theory : free!groups
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