Nuprl Lemma : rel-path-between-append
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
∀L1,L2:T List. ∀x,y,z:T.
(rel-path-between(T;R;x;y;L1)
⇒ rel-path-between(T;R;y;z;L2)
⇒ Refl(T;v1,v2.R v1 v2)
⇒ rel-path-between(T;R;x;z;L1 @ L2))
Proof
Definitions occuring in Statement :
rel-path-between: rel-path-between(T;R;x;y;L),
append: as @ bs,
list: T List,
refl: Refl(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
rel-path-between: rel-path-between(T;R;x;y;L),
and: P ∧ Q,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
listp: A List+,
subtype_rel: A ⊆r B,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
less_than': less_than'(a;b),
cons: [a / b],
bfalse: ff,
rel-path: rel-path(R;L),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
guard: {T},
infix_ap: x f y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
ge: i ≥ j ,
sq_type: SQType(T),
bnot: ¬bb,
so_lambda: λ2x.t[x],
so_apply: x[s],
last: last(L),
subtract: n - m,
refl: Refl(T;x,y.E[x; y]),
gt: i > j
Lemmas referenced :
refl_wf,
rel-path-between_wf,
list_wf,
istype-universe,
length-append,
decidable__lt,
length_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
hd-append-sq,
subtype_rel_list,
top_wf,
istype-less_than,
last_append,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
istype-void,
int_trichot,
subtract_wf,
int_seg_wf,
append_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-le,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
select_append_front,
subtype_rel_self,
iff_weakening_equal,
intformeq_wf,
int_formula_prop_eq_lemma,
select_append,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
non_neg_length,
length_append,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
decidable__equal_int,
select-as-hd,
add-associates,
minus-one-mul,
add-swap,
add-mul-special,
zero-add,
zero-mul,
add-member-int_seg2,
add-commutes,
select_wf,
select_append_back,
squash_wf,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
hypothesis,
functionIsType,
because_Cache,
universeEquality,
instantiate,
productElimination,
independent_pairFormation,
Error :memTop,
dependent_functionElimination,
natural_numberEquality,
addEquality,
unionElimination,
imageElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
dependent_set_memberEquality_alt,
promote_hyp,
hypothesis_subsumption,
setElimination,
rename,
productIsType,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
equalityIstype,
cumulativity,
intEquality,
multiplyEquality,
hyp_replacement,
closedConclusion,
minusEquality,
productEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}L1,L2:T List. \mforall{}x,y,z:T.
(rel-path-between(T;R;x;y;L1)
{}\mRightarrow{} rel-path-between(T;R;y;z;L2)
{}\mRightarrow{} Refl(T;v1,v2.R v1 v2)
{}\mRightarrow{} rel-path-between(T;R;x;z;L1 @ L2))
Date html generated:
2020_05_19-PM-09_52_53
Last ObjectModification:
2020_02_06-PM-09_36_48
Theory : relations2
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