hcju/isabelleps huggingface dataset:
rPar2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and A\<^sub>Q :: "name list"   and \<Psi>\<^sub>Q :: "'b"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' Q') A\<^sub>Q \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* M"   and "A\<^sub>Q \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* M"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' (P \<parallel> Q')) (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
isgcd_def: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and a :: "'a"   and b :: "'a" shows "x gcdof\<^bsub>G\<^esub> a b = (x divides\<^bsub>G\<^esub> a \<and> x divides\<^bsub>G\<^esub> b \<and> (\<forall>y\<in>carrier G. y divides\<^bsub>G\<^esub> a \<and> y divides\<^bsub>G\<^esub> b \<longrightarrow> y divides\<^bsub>G\<^esub> x))"
model_C: shows "\<forall>n. \<not> falsifies\<^sub>c (f n) C"
errMOD_igConsIPresIGWlsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igConsIPresIGWls MOD"   and "igWlsAllDisj MOD"   and "igWlsAbsIsInBar MOD" shows "igConsIPresIGWlsSTR (errMOD MOD)"
cases: fixes a1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and a2 :: "'Var \<Rightarrow> 'Val"   and a3 :: "nat"   and a4 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and a5 :: "'Var \<Rightarrow> 'Val" assumes "mevalp a1 a2 a3 a4 a5" obtains cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list" and n :: "nat" and mem :: "'Var \<Rightarrow> 'Val" and cm' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)" and mem' :: "'Var \<Rightarrow> 'Val" where "a1 = cms" and "a2 = mem" and "a3 = n" and "a4 = cms[n := cm']" and "a5 = mem'" and "(cms ! n, mem) \<leadsto> (cm', mem')" and "n < length cms"
4: fixes l1 :: "'lbl"   and l2 :: "'lbl" assumes "l1 \<in> set (L A)"   and "l2 \<in> set (L A)"   and "l1 \<noteq> l2" shows "GSMP_disjoint (N2 (proj_unl l1 B)) (N2 (proj_unl l2 B)) (f (set C) - {m. {} \<turnstile>\<^sub>c m})"
unfold1_dtor: shows "unfold1 dtor1 dtor2 = id"
magQ_qconv: fixes c :: "'f \<Rightarrow>\<^sub>U 'e"   and q :: "'c['d, 'f]" shows "\<lbrakk>qconv c q\<rbrakk>\<^sub>Q = of_rat (dconvfactor c (dimQ q)) \<cdot> \<lbrakk>q\<rbrakk>\<^sub>Q"
sets_eq_imp_space_eq: fixes M :: "'a measure"   and M' :: "'a measure" assumes "sets M = sets M'" shows "space M = space M'"
weak_ranking_index_def: fixes x :: "'a" shows "weak_ranking_index x = find_index ((\<in>) x) (weak_ranking le)"
toStates_def: fixes steps :: "step list" shows "toStates steps = (let (states, pos, lbls) = step_fold stepToState steps (lm.empty ()) 0 1 (Index 0) None False; pos = case pos of Index x \<Rightarrow> pos | LabelJump l x \<Rightarrow> Index (resolveLabel l lbls); states = endState # states; states = map (resolveLabels states lbls) states; states = calculatePrios states in case pos of Index s \<Rightarrow> if s < length states then (IArray states, pos, lbls) else abort STR ''Start index out of bounds'' (\<lambda>_. (IArray states, Index 0, lbls)))"
Lim_measure_decseq: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range A \<subseteq> sets M"   and "decseq A"   and "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "(\<lambda>n. Sigma_Algebra.measure M (A n)) \<longlonglongrightarrow> Sigma_Algebra.measure M (\<Inter> (range A))"
first_pd_no_cd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat" assumes "is_path \<pi>"   and "\<pi> n pd\<rightarrow> \<pi> 0"   and "\<forall>l<n. \<pi> l \<noteq> \<pi> n" shows "\<forall>l. \<not> n cd\<^bsup>\<pi>\<^esup>\<rightarrow> l"
polynomial_function_def: fixes p :: "'c \<Rightarrow> 'd" shows "polynomial_function p \<equiv> \<forall>f. bounded_linear f \<longrightarrow> real_polynomial_function (f \<circ> p)"
i_drop_i_take_1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "f \<Up> n \<Down> Suc 0 = [f n]"
healthy_bounded_byD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and b :: "real"   and P :: "'b \<Rightarrow> real" assumes "healthy t"   and "bounded_by b P"   and "nneg P" shows "bounded_by b (t P)"
eq_refl: fixes x :: "'c word"   and y :: "'c word" assumes "x = y" shows "x \<le>s y"
ag_pOp_commute: fixes A :: "('c, 'd) aGroup_scheme"   and x :: "'c"   and y :: "'c" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> y = y \<plusminus>\<^bsub>A\<^esub> x"
proper_designI: assumes "des.\<b> \<noteq> 0" shows "proper_design V B"
intros_26: fixes env :: "v sem_env"   and s1 :: "'a state"   and e :: "exp0"   and s2 :: "'a state"   and v1 :: "v"   and pes :: "(pat \<times> exp0) list"   and e' :: "exp0"   and env' :: "(char list \<times> v) list"   and bv :: "'a state \<times> (v, v) result" assumes "Big_Step_Unclocked_Single.evaluate env s1 e (s2, Rval v1)"   and "match_result env s2 v1 pes Bindv = Rval (e', env')"   and "Big_Step_Unclocked_Single.evaluate (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) s2 e' bv" shows "Big_Step_Unclocked_Single.evaluate env s1 (Mat e pes) bv"
prefixes_eq_snoc: fixes ys :: "'b list"   and xs :: "'b list list"   and x :: "'b list" shows "(prefixes ys = xs @ [x]) = ((ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = zs @ [z] \<and> xs = prefixes zs)) \<and> x = ys)"
le_add1: fixes n :: "nat"   and m :: "nat" shows "n \<le> n + m"
aHom_compos_assoc: fixes K :: "('c, 'd) aGroup_scheme"   and L :: "('e, 'f) aGroup_scheme"   and M :: "('g, 'h) aGroup_scheme"   and N :: "('i, 'j) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'g"   and h :: "'g \<Rightarrow> 'i" assumes "aGroup K"   and "aGroup L"   and "aGroup M"   and "aGroup N"   and "f \<in> aHom K L"   and "g \<in> aHom L M"   and "h \<in> aHom M N" shows "compos K h (compos K g f) = compos K (compos L h g) f"
list_choice: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list" shows "list_all (\<lambda>x. \<exists>y. P x y) xs = (\<exists>ys. list_all2 P xs ys)"
cinner_sum_right: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and x :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x (comm_monoid_add.sum plus zero f A) = (\<Sum>y\<in>A. cinner x (f y))"
simps_1: fixes d :: "'dyn" shows "norm_instr (Ubx.instr.IPush d) = Inca.instr.IPush d"
ivl_disj_un_one_5: fixes l :: "'a"   and u :: "'a" assumes "l \<le> u" shows "{l<..u} \<union> {u<..} = {l<..}"
case_eq_if: fixes f :: "'d match_expr \<Rightarrow> action \<Rightarrow> 'c"   and rule :: "'d rule" shows "(case rule of Rule x xa \<Rightarrow> f x xa) = f (get_match rule) (get_action rule)"
ChiRel_HAStates2: fixes a :: "'d"   and b :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "(a, b) \<in> ChiRel A" shows "b \<in> HAStates A"
simps_2: fixes k :: "nat"   and l :: "'b btree"   and a :: "'b"   and r :: "'b btree" shows "root_order_up\<^sub>i k (Up\<^sub>i l a r) = (order k l \<and> order k r)"
classesDefined: fixes C :: "qtname"   and c :: "class" assumes "class tprg C = Some c"   and "C \<noteq> Object" shows "\<exists>sc. class tprg (super c) = Some sc"
lhopital_left_at_top_at_top: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and g :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real" assumes "filterlim f at_top (at_left a)"   and "filterlim g at_top (at_left a)"   and "\<forall>\<^sub>F x in at_left a. (f has_real_derivative f' x) (at x)"   and "\<forall>\<^sub>F x in at_left a. (g has_real_derivative g' x) (at x)"   and "LIM x at_left a. f' x / g' x :> at_top" shows "LIM x at_left a. f x / g x :> at_top"
seqr_or_distr: fixes P :: "('b, 'd) urel"   and Q :: "('d, 'c) urel"   and R :: "('d, 'c) urel" shows "P ;; (Q \<or> R) = (P ;; Q \<or> P ;; R)"
\<chi>\<sigma>\<tau>: shows "instInpP \<chi> 0 \<sigma>\<tau> = exi z' (cnj (subst \<sigma>\<tau> (Var z') out) (subst \<chi> (Var z') inp))"
hb_irrefl: fixes m1 :: "'msgid \<times> 'op"   and m2 :: "'msgid \<times> 'op" assumes "hb m1 m2" shows "m1 \<noteq> m2"
OclIsAbsent_def: fixes X :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "X.oclIsAbsent() \<equiv> \<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> then \<lfloor>\<lfloor>oid_of (X \<tau>) \<notin> dom (heap (fst \<tau>)) \<and> oid_of (X \<tau>) \<notin> dom (heap (snd \<tau>))\<rfloor>\<rfloor> else invalid \<tau>"
mult_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a ; b ; c = a ; (b ; c)"
UNION_in_sets: fixes A :: "nat \<Rightarrow> 'a set"   and n :: "nat" assumes "range A \<subseteq> lambda_system \<Omega> M f" shows "\<Union> (A ` {0..<n}) \<in> lambda_system \<Omega> M f"
comp_null_1: fixes f :: "'a \<times> 'a" shows "CC.comp CC.null f = CC.null"
nat_mult_le_cancel_disj: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "(k * m \<le> k * n) = (0 < k \<longrightarrow> m \<le> n)"
abs_eq: fixes x :: "\<nu> option" shows "Semantics.d\<^sub>\<kappa> (make\<kappa> x) = x"
inf_Sup: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and a :: "'c"   and B :: "'c set" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "inf a (Sup B) = Sup (inf a ` B)"
exhaust: fixes y :: "'b err" obtains "y = Err"   | x2 :: "'b" where "y = OK x2"
HPhase1or2ReadElse_HInv1: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2ReadElse s s' p d q"   and "HInv1 s" shows "HInv1 s'"
nipd': shows "\<forall>j\<in>{k'..l'}. \<pi>' j \<noteq> ipd (\<pi>' k')"
cod_is_obj: fixes x :: "bool" assumes "x \<in> {False, True}" shows "True \<in> {False, True}"
countable_UN: fixes A :: "'e \<Rightarrow> 'd set"   and X :: "'e set"   and M :: "'d measure" assumes "A ` X \<subseteq> sets M" shows "\<Union> (A ` X) \<in> sets M"
ide_char: fixes a :: "'a \<times> 'a" shows "S.VV.ide a = (S.VV.arr a \<and> S.VxV.ide a)"
tl_append2: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<noteq> []" shows "tl (xs @ ys) = tl xs @ ys"
less_imp_of_nat_less: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "m < n" shows "less (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
converse_unfold: fixes r :: "('e \<times> 'd) set" shows "r\<inverse> = {(y, x). (x, y) \<in> r}"
primerec_terminate: fixes f :: "recf"   and x :: "nat"   and xs :: "nat list" assumes "primerec f x"   and "length xs = x" shows "terminate f xs"
Throw: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and n :: "nat"   and a' :: "val"   and s1 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<midarrow>n\<rightarrow> s1" shows "G\<turnstile>Norm s0 \<midarrow>Throw e\<midarrow>n\<rightarrow> abupd (throw a') s1"
intro: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e" assumes "monad return bind"   and "monad_duplicate_axioms bind" shows "monad_duplicate return bind"
IH: fixes ucl :: "('U \<times> 'C) list"   and ss' :: "('St, 'U, 'C) state" assumes "list_all isState sl"   and "map getGMUserCom (tl sl) = ucl" shows "\<exists>sl'. hd sl' = ss' \<and> wffp sl' \<and> list_all2 f (tl sl) (tl sl') \<and> map getGMUserCom (purgeIdle (tl sl')) = purge GH ucl"
multiset_inter_def: fixes A :: "'a multiset"   and B :: "'a multiset" shows "A \<inter># B = A - (A - B)"
simps_3: fixes f :: "'d \<Rightarrow> 'd option"   and v :: "nat" shows "option_comp_pow f (Suc (Suc v)) = (option_comp_pow f (Suc v) \<Lleftarrow> f)"
target: fixes T1 :: "'a"   and T2 :: "'a"   and TRel :: "('a \<times> 'a) set"   and Enc :: "'b \<Rightarrow> 'a" assumes "(T1, T2) \<in> TRel" shows "(TargetTerm T1, TargetTerm T2) \<in> encoding.indRelLTPO Enc TRel"
id_map_two: fixes list :: "('a \<times> 'b) list" shows "map (\<lambda>(x, y). (x, y)) list = list"
sums_add: fixes f :: "nat \<Rightarrow> 'b"   and a :: "'b"   and g :: "nat \<Rightarrow> 'b"   and b :: "'b" assumes "f sums a"   and "g sums b" shows "(\<lambda>n. f n + g n) sums (a + b)"
finite_Collect_conjI: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "finite {x. P x} \<or> finite {x. Q x}" shows "finite {x. P x \<and> Q x}"
bres_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<rightarrow> x \<le> z \<rightarrow> y"
preserves_comp_1: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and f' :: "'d"   and f :: "'d" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A (A f' f)" shows "\<tau> (A f' f) = B (G f') (\<tau> f)"
lsorted_lhdD: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c llist" assumes "ord.lsorted less_eq xs"   and "\<not> lnull xs"   and "\<not> lnull (ltl xs)" shows "less_eq (lhd xs) (lhd (ltl xs))"
qSwapAll_preserves_qFreshAll: fixes ys :: "'d"   and zs :: "'d"   and z1 :: "'e"   and z2 :: "'e"   and y :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and A :: "('f, 'g, 'd, 'e, 'h) qAbs" shows "qFresh ys (y @ys[z1 \<and> z2]_zs) (X #[[z1 \<and> z2]]_zs) = qFresh ys y X \<and> qFreshAbs ys (y @ys[z1 \<and> z2]_zs) (A $[[z1 \<and> z2]]_zs) = qFreshAbs ys y A"
IsoCompInv: fixes f :: "'b" assumes "ciso f" shows "f \<approx>> Cinv f"
llast_LCons: fixes r :: "'b llist"   and A :: "'b set"   and a :: "'b" assumes "r \<in> A\<^sup>\<star>" shows "LList2.llast (a ## r) = (if r = LNil then a else LList2.llast r)"
monotone_const: fixes ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'a" shows "monotone ord (\<le>) (\<lambda>_. c)"
rel_inject_2: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x21 :: "'f"   and x22 :: "'h"   and y21 :: "'g"   and y22 :: "'i" shows "rel_generat R1 R2 R3 (IO x21 x22) (IO y21 y22) = (R2 x21 y21 \<and> R3 x22 y22)"
mat_to_cols_list_to_mat: fixes A :: "complex Matrix.mat" shows "Tensor.mat_of_cols_list (dim_row A) (mat_to_cols_list A) = A"
fac_dvd: fixes q :: "nat"   and n :: "nat" assumes "0 < q"   and "q \<le> n" shows "q dvd n!"
test_axiom_def: shows "test_axiom \<equiv> [[? ($\<phi> vid2 local.empty)]]$\<phi> vid1 local.empty \<leftrightarrow> ($\<phi> vid2 local.empty \<rightarrow> $\<phi> vid1 local.empty)"
UN_absorb: fixes k :: "'b"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "k \<in> I" shows "A k \<union> \<Union> (A ` I) = \<Union> (A ` I)"
error_free_sxalloc: fixes G :: "prog"   and s0 :: "state"   and s1 :: "state" assumes "G\<turnstile>s0 \<midarrow>sxalloc\<rightarrow> s1"   and "error_free s0" shows "error_free s1"
that: shows "c \<subseteq>\<^sub>g Guard f g c'"
assms_5: shows "B = to_int_poly_i ff_ops b"
dp_def: shows "dp = last (prio_emb_pos prio (prio_emb_step prio t s) s)"
LstSeqP_SUCC: fixes H :: "fm set"   and r :: "tm"   and d :: "tm"   and y :: "tm"   and z :: "tm" assumes "H \<turnstile> LstSeqP r (SUCC d) y"   and "H \<turnstile> HPair d z IN r" shows "H \<turnstile> LstSeqP r d z"
norm_uminus_minus: fixes x :: "'a"   and y :: "'a" shows "norm (- x - y) = norm (x + y)"
elims_3: fixes x :: "'b list"   and xa :: "'b list"   and xb :: "nat" assumes "\<not> sublist_at x xa xb" obtains x :: "'b" and xs :: "'b list" and y :: "'b" and ys :: "'b list" where "xa__ = x # xsa__" and "xa = y__ # ysa__" and "xb = 0" and "\<not> (x = y__ \<and> sublist_at xsa__ ysa__ 0)"   | xs :: "'b list" and y :: "'b" and ys :: "'b list" and i :: "nat" where "xa__ = xs" and "xa = y # ys" and "xb = Suc i" and "\<not> sublist_at xs ys i"   | v :: "'b" and va :: "'b list" and uv_ :: "nat" where "xa__ = v__ # va" and "xa = []" and "xb = uv_"   | uu_ :: "'b list" and v :: "nat" where "xa__ = uu_" and "xa = []" and "xb = Suc v"
inf_hidden: fixes A :: "'b set"   and s :: "'b event list"   and P :: "'b process" assumes "\<forall>t. trace_hide t (ev ` A) = trace_hide s (ev ` A) \<longrightarrow> (t, ev ` A) \<notin> F P"   and "s \<in> T P" shows "\<exists>f. isInfHiddenRun f P A \<and> s \<in> range f"
gbm_succ_list_it_impl: shows "graph_succ_it gbm_\<alpha> gbm_invar gbm_succ_list_it"
map_ide: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "Hom.map (b, a) = S.mkIde (Hom.set (b, a))"
fls_X_inv_times_conv_shift_2: fixes f :: "'a fls" shows "f * fls_X_inv = fls_shift 1 f"
limit_iff_frequent: fixes a :: "'b"   and x :: "nat \<Rightarrow> 'b" shows "(a \<in> limit x) = (\<exists>\<^sub>\<infinity>n. x n = a)"
vec_nth_eucl_of_list_eq2: fixes M :: "real list"   and i :: "'b"   and j :: "'a" assumes "length M = CARD('a) * CARD('b)" shows "eucl_of_list M $ i $ j = M ! index Basis_list (axis i (axis j 1))"
B2_2: fixes p :: "int"   and F :: "int poly"   and l :: "nat" assumes "prime p"   and "0 < degree F"   and "2 ^ (degree F)\<^sup>2 * B2_LLL F ^ (2 * degree F) \<le> (p ^ l)\<^sup>2" shows "2 \<le> B2_LLL F"
intro: fixes a :: "real"   and v :: "real" assumes "a < 0"   and "0 \<le> v" shows "braking_movement a v"
inducts: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'c list"   and x2 :: "'d list"   and P :: "'c list \<Rightarrow> 'd list \<Rightarrow> bool" assumes "for_all2 r x1 x2"   and "P [] []"   and "\<And>x y xs ys. \<lbrakk>r x y; for_all2 r xs ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P x1 x2"
preserves_cod_def: fixes G :: "('o, 'a, 'a set, 'a set_arrow) functor" shows "preserves_cod G \<equiv> \<forall>f\<in>Ar. G \<^bsub>\<o>\<^esub> (Cod f) = Cod\<^bsub>BB\<^esub> G \<^bsub>\<a>\<^esub> f"
obj_trg: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "B.obj (trg\<^sub>B \<mu>)"
exec_instr_xcpt_h: fixes ins :: "bytecode"   and pc :: "nat"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and vars :: "val list"   and C :: "char list"   and M :: "char list"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and xcp :: "nat"   and T :: "ty"   and mxs :: "nat"   and xt :: "ex_table"   and \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and loc :: "val list" assumes "fst (exec_instr (ins ! pc) P h stk vars C M pc ics frs sh) = \<lfloor>xcp\<rfloor>"   and "P,T,mxs,length ins,xt \<turnstile> ins ! pc,pc :: \<Phi> C M"   and "P,\<Phi> |- (None, h, (stk, loc, C, M, pc, ics) # frs, sh) [ok]" shows "\<exists>obj. h xcp = \<lfloor>obj\<rfloor>"
f6: fixes A :: "'a set"   and Aa :: "'a set" shows "A - (- A - Aa) = A"
properties_distinct_contrapos: fixes i :: "nat"   and j :: "nat" assumes "seq (i + j) = seq i" shows "j \<notin> {0<..<lambda}"
cINF_lower: fixes f :: "'c \<Rightarrow> 'b multiset"   and A :: "'c set"   and x :: "'c" assumes "subset_mset.bdd_below (f ` A)"   and "x \<in> A" shows "Inf (f ` A) \<subseteq># f x"
coboundedI: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "a \<in> A" shows "a \<le> Max A"
clkp_set_def: fixes A :: "('d \<times> ('a, 'b) cconstraint \<times> ('a set \<times> 'd) pmf) set \<times> ('d \<Rightarrow> ('a, 'b) cconstraint)" shows "PTA.clkp_set A = PTA.collect_clki (PTA.inv_of A) \<union> PTA.collect_clkt (PTA.trans_of A)"
f_2: shows "AE x in M. 0 \<le> f x"
kruskal_checked_def: fixes L :: "(nat \<times> int \<times> nat) list" shows "kruskal_checked L = (if uGraph_from_list_invar L then kruskal' L \<bind> (\<lambda>F. return (Some F)) else return None)"
simps_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b tree"   and m :: "'b"   and r :: "'b tree" assumes "class.linorder less_eq less" shows "linorder.heap less_eq \<langle>l, m, r\<rangle> = (Ball (set_tree l \<union> set_tree r) (less_eq m) \<and> linorder.heap less_eq l \<and> linorder.heap less_eq r)"
split_paired_all: fixes P :: "'d \<times> 'e \<Rightarrow> prop" shows "(\<And>x. PROP P x) \<equiv> (\<And>a b. PROP P (a, b))"
f_join_append: fixes xs :: "'b list"   and ys :: "'b list"   and I :: "nat set" shows "(xs @ ys) \<Join>\<^sub> I = xs \<Join>\<^sub> I @ ys \<Join>\<^sub> (I \<oplus>- length xs)"
minus: fixes x :: "'a" assumes "vectorspace E"   and "x \<in> E" shows "p (- x) = p x"
cfcomp1: fixes f :: "'d \<rightarrow> 'c"   and g :: "'b \<rightarrow> 'd" shows "f oo g = (\<Lambda> x. f\<cdot>(g\<cdot>x))"
inject_1: fixes x2 :: "expr"   and y2 :: "expr" shows "(Expr x2 = Expr y2) = (x2 = y2)"
d2: shows "\<forall>p\<in>set Ps'. \<exists>p'\<in>set ps. p = extend ( \<Phi>1 + \<Gamma>' \<Rightarrow>* \<Psi> + \<Delta>') p'"
matrix_to_iarray_fst_Gauss_Jordan_upt_k_PA: fixes k :: "nat"   and A :: "(('b, 'c) vec, 'd) vec" assumes "k < ncols A" shows "matrix_to_iarray (fst (Gauss_Jordan_upt_k_PA A k)) = fst (Gauss_Jordan_upt_k_iarrays_PA (matrix_to_iarray A) k)"
simps: fixes domain :: "'l list"   and codom :: "'l"   and \<Sigma> :: "'l list" shows "Subx.sp_gen_pop_push (domain, codom) \<Sigma> = (let ar = length domain in if ar \<le> length \<Sigma> \<and> take ar \<Sigma> = domain then Ok (codom # drop ar \<Sigma>) else Error ())"
list_all2_rev: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'h list" shows "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
segment_as_ball_2: fixes a :: "'b"   and b :: "'b" shows "open_segment a b = affine hull {a, b} \<inter> ball ((a + b) /\<^sub>R 2) (norm (b - a) / 2)"
terminalNFormCases: fixes f :: "'a \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and n :: "'a" shows "terminal subs (f n) \<or> (\<exists>i A nAs. nforms (f n) = (i, A) # nAs)"