Nuprl Lemma : append-tuple-one-one
∀[L1,L2:Type List].
∀[x1,x2:tuple-type(L1)]. ∀[y1,y2:tuple-type(L2)].
{(x1 = x2 ∈ tuple-type(L1)) ∧ (y1 = y2 ∈ tuple-type(L2))}
supposing append-tuple(||L1||;||L2||;x1;y1) = append-tuple(||L1||;||L2||;x2;y2) ∈ tuple-type(L1 @ L2)
supposing 0 < ||L2||
Proof
Definitions occuring in Statement :
append-tuple: append-tuple(n;m;x;y),
tuple-type: tuple-type(L),
length: ||as||,
append: as @ bs,
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
and: P ∧ Q,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
false: False,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
subtype_rel: A ⊆r B,
int_iseg: {i...j},
cand: A c∧ B
Lemmas referenced :
split-tuple_wf,
append_wf,
non_neg_length,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
length-append,
decidable__lt,
length_wf,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
istype-le,
istype-less_than,
tuple-type_wf,
append-tuple_wf,
list_wf,
split-tuple-append-tuple,
pi1_wf,
firstn_wf,
nth_tl_wf,
pi2_wf,
subtype_rel_tuple-type,
nth_tl_append,
subtype_rel-equal,
select_wf,
int_seg_properties,
length_append,
subtype_rel_list,
top_wf,
istype-universe,
length_nth_tl,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_seg_wf,
equal_functionality_wrt_subtype_rel2,
firstn_append,
firstn_length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
applyLambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
closedConclusion,
universeEquality,
hypothesisEquality,
hypothesis,
Error :dependent_set_memberEquality_alt,
because_Cache,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
productElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
Error :universeIsType,
addEquality,
imageElimination,
Error :productIsType,
independent_pairEquality,
axiomEquality,
Error :equalityIstype,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
setElimination,
rename,
applyEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[L1,L2:Type List].
\mforall{}[x1,x2:tuple-type(L1)]. \mforall{}[y1,y2:tuple-type(L2)].
\{(x1 = x2) \mwedge{} (y1 = y2)\}
supposing append-tuple(||L1||;||L2||;x1;y1) = append-tuple(||L1||;||L2||;x2;y2)
supposing 0 < ||L2||
Date html generated:
2019_06_20-PM-02_03_42
Last ObjectModification:
2018_12_07-PM-06_36_19
Theory : tuples
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