Nuprl Lemma : proper-iseg-append
∀[T:Type]. ∀L1,L2,L3,L4:T List. L1 @ L3 < L2 @ L4 ⇐⇒ L3 < L4 ∧ (L1 = L2 ∈ (T List)) supposing ||L1|| = ||L2|| ∈ ℤ
Proof
Definitions occuring in Statement :
proper-iseg: L1 < L2,
length: ||as||,
append: as @ bs,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
implies: P ⇒ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
cand: A c∧ B
Lemmas referenced :
list_induction,
all_wf,
list_wf,
isect_wf,
equal_wf,
length_wf,
iff_wf,
proper-iseg_wf,
append_wf,
equal-wf-base-T,
nil_wf,
length_of_nil_lemma,
list_ind_nil_lemma,
equal-wf-base,
length_of_cons_lemma,
list_ind_cons_lemma,
length_of_null_list,
non_neg_length,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
iff_weakening_equal,
cons_wf,
null_nil_lemma,
btrue_wf,
and_wf,
null_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
equal-wf-T-base,
cons-proper-iseg,
decidable__equal_int,
add-is-int-iff,
intformnot_wf,
int_formula_prop_not_lemma,
false_wf,
squash_wf,
true_wf,
reduce_tl_cons_lemma,
tl_wf,
reduce_hd_cons_lemma,
hd_wf,
ge_wf,
length_cons_ge_one,
subtype_rel_list,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
because_Cache,
intEquality,
productEquality,
independent_functionElimination,
baseClosed,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
rename,
independent_pairFormation,
productElimination,
independent_isectElimination,
applyEquality,
imageElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
imageMemberEquality,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
addEquality,
universeEquality,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]. \mforall{}L1,L2,L3,L4:T List. L1 @ L3 < L2 @ L4 \mLeftarrow{}{}\mRightarrow{} L3 < L4 \mwedge{} (L1 = L2) supposing ||L1|| = ||L2||
Date html generated:
2017_04_17-AM-08_47_19
Last ObjectModification:
2017_02_27-PM-05_05_18
Theory : list_1
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