hcju/isabelleps huggingface dataset:
step: fixes X :: "'d set"   and v :: "'d \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'d"   and k :: "'d \<Rightarrow> nat"   and A :: "('e \<times> ('d, real) cconstraint \<times> 'f \<times> 'd list \<times> 'e) set \<times> ('e \<Rightarrow> ('d, real) cconstraint)"   and l :: "'e"   and Z :: "('d \<Rightarrow> real) set"   and l' :: "'e"   and Z' :: "('d \<Rightarrow> real) set"   and l'' :: "'e"   and Z'' :: "('d \<Rightarrow> real) set" assumes "Regions X v n not_in_X"   and "Regions.steps_z_beta X k v n not_in_X A l Z l' Z'"   and "Regions.step_z_beta X k v n not_in_X A l' Z' l'' Z''" shows "Regions.steps_z_beta X k v n not_in_X A l Z l'' Z''"
elims_2: fixes x :: "lock_action list" assumes "must_acquire_lock x" obtains las :: "lock_action list" where "x = Lock # lasa__"   | las :: "lock_action list" where "x = UnlockFail # lasb__" and "must_acquire_lock lasb__"   | las :: "lock_action list" where "x = ReleaseAcquire # las" and "must_acquire_lock las"
transitive_asymmetric_irreflexive: fixes x :: "'a" assumes "transitive x" shows "asymmetric x = irreflexive x"
r_def: shows "r \<equiv> Cn 2 r_nth [Cn 2 r_prefixes [Cn 2 S [recf.Id 2 1], recf.Id 2 0], recf.Id 2 1]"
GB_implies_unique_nf_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "G \<subseteq> dgrad_p_set d m"   and "is_Groebner_basis G" shows "\<exists>!h. (red G)\<^sup>*\<^sup>* f h \<and> \<not> is_red G h"
psubstInp_preserves_good: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option"   and inp :: "'e \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option" assumes "goodEnv rho"   and "goodInp inp" shows "goodInp inp %[rho]"
antipar_1: shows "trg r\<^sub>0\<^sup>* = src r\<^sub>0"
square_free_of_int_poly: fixes p :: "int poly" assumes "square_free p" shows "square_free (of_int_poly p)"
f9: shows "snd x \<otimes> r \<in> carrier R"
mk_next_pow_simp_2: fixes x :: "'a ltln" shows "X\<^sub>n' 1 x = mk_next x"
nonzerohyp: shows "Const (real (i + 1)) * Var x ^ i \<noteq> 0"
ALG'_def: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and qs :: "'c list"   and init :: "'c list"   and i :: "nat"   and x :: "'c" shows "ALG' A qs init i x = E (map_pmf (\<lambda>x. real (ALG x qs i x)) (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) (take i qs)))"
code_dec_morph: fixes \<C> :: "'b list set"   and x :: "'b list"   and y :: "'b list" assumes "code \<C>"   and "x \<in> \<langle>\<C>\<rangle>"   and "y \<in> \<langle>\<C>\<rangle>" shows "(Dec \<C> x) \<cdot> (Dec \<C> y) = Dec \<C> x \<cdot> y"
nth_scast: fixes x :: "'b word"   and n :: "nat" shows "scast x !! n = (n < LENGTH('a) \<and> (if n < LENGTH('b) - 1 then x !! n else x !! (LENGTH('b) - 1)))"
case_3: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case Protected of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f3"
SymSystem_comp: fixes u' :: "nat \<Rightarrow> 'a"   and init' :: "(nat \<Rightarrow> 'a) set"   and init :: "(nat \<Rightarrow> 'e) set"   and p :: "'e \<Rightarrow> 'e \<Rightarrow> 'c \<Rightarrow> bool"   and r :: "'e \<Rightarrow> 'e \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and p' :: "'a \<Rightarrow> 'a \<Rightarrow> 'd \<Rightarrow> bool"   and r' :: "'a \<Rightarrow> 'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool" assumes "u' \<in> init'" shows "SymSystem init p r \<circ> SymSystem init' p' r' = [: z \<leadsto> (u, x) . fst \<circ> u \<in> init \<and> snd \<circ> u \<in> init' \<and> z = x :] \<circ> {.u, x. prec_st p r (fst \<circ> u) x \<and> (\<forall>y. (\<box> lft_rel_st r) (fst \<circ> u) x y \<longrightarrow> prec_st p' r' (snd \<circ> u) y).} \<circ> [: id (\<lambda>(u, x). id ((\<box> lft_rel_st (\<lambda>(u, v) (u', v'). r u u' OO r' v v')) u x)) :]"
resT_induct: fixes P :: "'e\<cdot>'f resT \<Rightarrow> bool"   and r :: "'e\<cdot>'f resT" assumes "adm P"   and "P \<bottom>"   and "\<And>x. P (Done\<cdot>x)"   and "\<And>m f. (\<And>r. P (f\<cdot>r)) \<Longrightarrow> P (More\<cdot>(fmap\<cdot>f\<cdot>m))" shows "P r"
Labeling_axioms: shows "Labeling c s t (push_effect f e) l"
S4safe: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S4 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> (\<forall>l. $MemInv mm l) \<longrightarrow> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)"
div_plus_div_distrib_dvd_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "dvd.dvd times c a" shows "divide (plus a b) c = plus (divide a c) (divide b c)"
bin_nth_numeral_simps: fixes w :: "num"   and n :: "num" shows "bit (- numeral (num.Bit1 w)) (numeral n) = bit (- numeral (w + num.One)) (pred_numeral n)"
eff: fixes i :: "instr"   and P :: "'a prog"   and pc :: "nat"   and et :: "ex_table"   and t :: "(ty list \<times> ty err list) option" shows "eff i P pc et t = (case t of None \<Rightarrow> [] | \<lfloor>\<tau>\<rfloor> \<Rightarrow> norm_eff i P pc \<tau> @ xcpt_eff i P pc \<tau> et)"
natural_isomorphism_axioms: shows "natural_isomorphism hom.comp hom.comp hom.map GF.map \<phi>"
geodesic_segment_param_4: fixes G :: "'b set"   and x :: "'b"   and y :: "'b" assumes "geodesic_segment_between G x y" shows "isometry_on {0..dist x y} (geodesic_segment_param G x)"
norm_times_const: fixes a :: "complex"   and lst :: "complex list" shows "(\<Sum>y\<leftarrow>lst. (cmod (a * y))\<^sup>2) = (cmod a)\<^sup>2 * (\<Sum>y\<leftarrow>lst. (cmod y)\<^sup>2)"
learn_bc_wrt_goedel: fixes \<chi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "goedel_numbering \<chi>"   and "learn_bc \<psi> U s" shows "\<exists>t. learn_bc \<chi> U t"
assms_2: shows "B \<prec>\<^sub>\<circ> A"
span_add1: fixes U :: "'a vec set"   and v :: "'a vec"   and w :: "'a vec" assumes "U \<subseteq> carrier_vec n"   and "v \<in> local.span U"   and "w \<in> local.span U" shows "v + w \<in> local.span U"
traces_alt: fixes A :: "('e, 'd) ioa" shows "local.traces A = {tr. \<exists>e. is_exec_of A e \<and> tr = trace (ioa.asig A) e}"
z1: shows "cmod (winding_number ((linepath (a + complex_of_real kde) (a + complex_of_real e) +++ p +++ linepath (a - complex_of_real d) (a - complex_of_real kde)) +++ linepath (a - complex_of_real kde) (a + complex_of_real kde)) z) = 1"
pdigraphD_2: fixes \<alpha> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and J :: "V" assumes "pdigraph \<alpha> I \<AA>"   and "J \<subseteq>\<^sub>\<circ> I" shows "dg_prod J \<AA>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
bijective_reverse: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "bijective p"   and "bijective q" shows "(p \<le> r * q) = (q \<le> r\<^sup>T * p)"
i'_top: shows "i' < i * 2 ^ (l' - l)"
2: assumes "b = c"   and "a \<noteq> c" shows "\<exists>f. f a * a + f c * c = x * a + y * c + z * c"
72090: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<squnion> (x \<sqinter> y \<squnion> z)) = - (x \<squnion> z)"
b_pr0: shows "\<guillemotleft>b_pr0 : local.prod a b \<rightarrow> b\<guillemotright>"
hg: fixes x :: "(real, 'm) vec" assumes "x \<in> S" shows "continuous_on (g ` S) h \<and> h (g x) = x"
prems_2: shows "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"
jordan_matrix_char_poly: fixes n_as :: "(nat \<times> 'b) list" shows "char_poly (jordan_matrix n_as) = (\<Prod>(n, a)\<leftarrow>n_as. [:- a, 1::'b:] ^ n)"
assms: shows "mod_type_class.to_nat x + 1 < CARD('a)"
assms_2: shows "\<forall>y\<in>s. norm (f y) \<le> b"
IH: assumes "t \<in> pair ` setops\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)" shows "\<exists>s \<delta>. s \<in> pair ` setops\<^sub>s\<^sub>s\<^sub>t S \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t = s \<cdot> \<delta>"
inject_4: fixes x6 :: "'b rexp"   and y6 :: "'b rexp" shows "(Star x6 = Star y6) = (x6 = y6)"
right_minus_eq: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "(minus a b = zero) = (a = b)"
wf_not_dmd_box_decrease: fixes r :: "('a \<times> 'a) set"   and v :: "state \<Rightarrow> 'a" assumes "wf r" shows "\<turnstile> \<diamond>\<box>[(v$, $v) \<in> #r]_v \<longrightarrow> \<diamond>\<box>[#False]_v"
ord_pres_def: fixes f :: "'b \<Rightarrow> 'c" shows "ord_pres f = (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
adopt_node_impl: shows "adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = CD.a_adopt_node"
reduce_pres_denot: fixes e :: "exp"   and e' :: "exp" assumes "e \<longrightarrow> e'" shows "E e = E e'"
split_sel: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> 'c option \<Rightarrow> 'c tree \<Rightarrow> 'c tree \<Rightarrow> 'b"   and tree :: "'c tree" shows "P (case tree of Leaf \<Rightarrow> f1 | Node x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd) = ((tree = Leaf \<longrightarrow> P f1) \<and> (tree = Node (l tree) (r tree) (val tree) (lchild tree) (rchild tree) \<longrightarrow> P (f2 (l tree) (r tree) (val tree) (lchild tree) (rchild tree))))"
sum_two_elements': fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and d :: "'a" assumes "(\<Sum>i\<in>{a, b}. f i * i) = d" obtains p :: "'a" and q :: "'a" where "d = p * a + q * b"
P'_prop: shows "P' = (\<lambda>X. X \<union> {x \<in> B - B'. f x = X}) ` P"
components_are_equivalences: fixes a :: "'d" assumes "D.obj a" shows "D.equivalence_map (\<eta>\<^sub>0 a)"
Nonce_analz_image_Key_lemma: fixes P :: "bool"   and N :: "nat"   and KK :: "nat set"   and H :: "msg set" assumes "P \<longrightarrow> Nonce N \<in> analz (Key ` KK \<union> H) \<longrightarrow> Nonce N \<in> analz H" shows "P \<longrightarrow> (Nonce N \<in> analz (Key ` KK \<union> H)) = (Nonce N \<in> analz H)"
image_iff: fixes z :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(z \<in> f ` A) = (\<exists>x\<in>A. z = f x)"
unique_calc_x_axis_intersection: fixes H :: "circline"   and x :: "complex_homo" assumes "is_poincare_line H"   and "H \<noteq> x_axis"   and "x \<in> unit_disc"   and "x \<in> circline_set H \<inter> circline_set x_axis" shows "x = calc_x_axis_intersection H"
SUP_cong: fixes A :: "'k set"   and B :: "'k set"   and C :: "'k \<Rightarrow> 'l"   and D :: "'k \<Rightarrow> 'l"   and Sup :: "'l set \<Rightarrow> 'l" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
sendmsg_def: fixes P :: "'e \<Rightarrow> bool"   and a :: "'e seq_action" shows "sendmsg P a \<equiv> case a of send m \<Rightarrow> P m | _ \<Rightarrow> True"
fdomainE: fixes a :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> r" obtains b :: "V" where "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
gfp_exp_bound: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P. unitary P \<Longrightarrow> unitary (t P)" shows "bounded_by 1 (gfp_exp t)"
1: shows "\<guillemotleft>cmp\<^sub>D\<^sub>N \<mu>\<nu> : DN (src (snd \<mu>\<nu>)) \<rightarrow>\<^sub>B DN (trg (fst \<mu>\<nu>))\<guillemotright>"
cr_af_def: shows "cr_af \<equiv> \<lambda>x y. x = ??.Abstract_AF.af.Rep_af y"
communicatingD1: fixes C :: "'a set"   and a :: "'a"   and b :: "'a" assumes "C \<in> UNIV // K1.communicating"   and "(a, b) \<in> K1.communicating"   and "a \<in> C" shows "b \<in> C"
G_eq_suminf: fixes x :: "'s"   and y :: "'s" shows "G x y = (\<Sum>i. ennreal (p x y i))"
compact_AR: fixes S :: "'a set" shows "(compact S \<and> AR S) = (compact S \<and> S retract_of UNIV)"
valid_path_aux_Append: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and cs :: "'a list"   and as :: "'a list"   and as' :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.valid_path_aux kind get_return_edges cs as"   and "CFG.valid_path_aux kind get_return_edges (CFG.upd_cs kind cs as) as'" shows "CFG.valid_path_aux kind get_return_edges cs (as @ as')"
resl_galois: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "(x \<le> z \<leftarrow> y) = (x \<cdot> y \<le> z)"
inj_dual: fixes dual :: "'b \<Rightarrow> 'b" assumes "class.dual dual" shows "inj dual"
list_set_autoref_empty: fixes R :: "('a \<times> 'b) set" shows "([], {}) \<in> \<langle>R\<rangle>list_set_rel"
minus_mult_minus: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) (uminus b) = times a b"
subst_clsscomp_subst: fixes CC :: "'a literal multiset set"   and \<tau> :: "'s"   and \<sigma> :: "'s" shows "CC \<cdot>cs (\<tau> \<odot> \<sigma>) = CC \<cdot>cs \<tau> \<cdot>cs \<sigma>"
"3_5": fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y" shows "rel x y \<or> rel y x"
inj_on_map_lists: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "inj_on (map f) (lists A)"
ta_tb_wf_3: shows "fv ta \<inter> fv tb = {}"
surj_iff: fixes f :: "'b \<Rightarrow> 'a" shows "surj f = (f \<circ> inv' f = id)"
set_less_aux_Compl_iff: fixes ao :: "'a option"   and xs :: "'a list"   and ys :: "'a list" shows "set_less_aux_Compl ao xs ys = (set_less_eq_aux_Compl ao xs ys \<and> \<not> Compl_set_less_eq_aux ao ys xs)"
oalist_eq_def: fixes xs :: "'x"   and ys :: "'x" shows "oalist_eq xs ys = oalist_eq_raw (list_of_oalist xs) (list_of_oalist ys)"
product_rule: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier (UP R)"   and "g \<in> carrier (UP R)" shows "pderiv (f \<otimes>\<^bsub>UP R\<^esub> g) = pderiv f \<otimes>\<^bsub>UP R\<^esub> g \<oplus>\<^bsub>UP R\<^esub> f \<otimes>\<^bsub>UP R\<^esub> pderiv g"
tendsto_discrete: fixes "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and y :: "'a"   and F :: "'b filter" assumes "class.discrete_topology open" shows "filterlim f (topological_space.nhds open y) F = (\<forall>\<^sub>F x in F. f x = y)"
Rep_bit0_def: fixes x :: "'a bit0" shows "Rep x = Rep_bit0 x"
sup1E: fixes A :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "sup A B x" obtains "A x"   | "B x"
not_le: fixes y :: "'d"   and x :: "'d" shows "(\<not> y \<le> x) = (x < y)"
distinct_5: fixes x2 :: "char list"   and x3 :: "bool" shows "B x2 \<noteq> C x3"
cartesian_product_infiniteL_imp_infinite: fixes A :: "'a set"   and B :: "'b set" assumes "infinite A"   and "B \<noteq> {}" shows "infinite (A \<times> B)"
evalFAll: fixes M :: "model"   and phi :: "vbl \<Rightarrow> object"   and z :: "signs"   and body :: "formula" shows "evalF M phi (FAll z body) = sign z (\<forall>x\<in>objects M. sign z (evalF M (\<lambda>v. case v of zeroX \<Rightarrow> x | nextX v \<Rightarrow> phi v) body))"
VA: shows "Vagree \<nu> \<omega> (\<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGF (InContext x1 x2))))"
R: shows "P \<Longrightarrow>\<^sub>A R"
lookup_proj_poly: fixes k :: "unit"   and p :: "'b \<Rightarrow>\<^sub>0 'a"   and t :: "'b" shows "lookup (punit.proj_poly k p) t = lookup p (fst (t, k))"
simps_1: fixes f :: "'b \<rightarrow> 'b \<rightarrow> 'b" shows "foldl1\<cdot>f\<cdot>[] = \<bottom>"
power_mod: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "(a mod b) ^ n mod b = a ^ n mod b"
simps_2: fixes h :: "'c \<Rightarrow> 'd list \<Rightarrow> bool"   and xs :: "'d list" shows "interpret h FalseF xs = False"
py: shows "(x, y) \<in> r\<^sup>* \<and> y \<in> NF r"
simps: fixes a :: "'e"   and r :: "('e \<times> 'f) set" shows "(a \<in> Domain r) = (\<exists>aa b. a = aa \<and> (aa, b) \<in> r)"
extending_constrains: fixes F :: "'a program"   and A :: "'a set"   and B :: "'a set" shows "extending (\<lambda>G. UNIV) h F (extend_set h A co extend_set h B) (A co B)"
Inf_enat_def: fixes A :: "enat set" shows "Inf A = (if A = {} then \<infinity> else LEAST x. x \<in> A)"
assms: shows "k \<in>\<^sub>\<circ> vcard xs"
proj_mcont: fixes c :: "'id" shows "mcont lSup lprefix lSup lprefix (proj c)"
index_mat_multcol_2: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and a :: "'a" assumes "i < dim_row A"   and "j < dim_col A" shows "multcol j a A $$ (i, j) = a * A $$ (i, j)"
inversions_imp_less_1: fixes z :: "nat \<times> nat"   and xs :: "'a list" assumes "z \<in> inversions xs" shows "fst z < snd z"
am4: fixes x :: "'a"   and y :: "'a" shows "ad (x \<cdot> y) \<cdot> ad (x \<cdot> ad y) = ad x"
lex_pm_zero_min: fixes s :: "'c \<Rightarrow>\<^sub>0 'd" shows "lex_pm 0 s"
hyps_1: shows "P \<turnstile> y \<prec>\<^sup>1 z"
is_extensional: fixes f :: "(bool, unit) J.arr" assumes "\<not> J.arr f" shows "\<chi> f = C.null"
fv_r\<^sub>e\<^sub>q_def: fixes ac :: "poscheckvariant"   and x :: "('d, 'c) strand_step" shows "fv_r\<^sub>e\<^sub>q ac x \<equiv> case x of \<langle>ac': s \<doteq> t\<rangle>\<^sub>s\<^sub>t \<Rightarrow> if ac = ac' then fv t else {} | _ \<Rightarrow> {}"
assms_1: shows "\<Gamma> \<turnstile>\<^sub>c e1 : dist_param_type dst"
intro: fixes get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" assumes "\<And>ptr h h'. reads (get_child_nodes_locs ptr) (get_child_nodes ptr) h h'"   and "\<And>h ptr. \<lbrakk>type_wf h; known_ptr ptr; ptr |\<in>| object_ptr_kinds h\<rbrakk> \<Longrightarrow> h \<turnstile> ok get_child_nodes ptr"   and "\<And>h ptr. h \<turnstile> ok get_child_nodes ptr \<Longrightarrow> ptr |\<in>| object_ptr_kinds h"   and "\<And>ptr h. pure (get_child_nodes ptr) h" shows "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"
r1: shows "r1 = 0 \<or> degree r1 < degree g"
dvd_div_iff_mult: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c \<noteq> (0::'a)"   and "c dvd b" shows "(a dvd b div c) = (a * c dvd b)"
lin_indpt_list_nonzero: fixes n :: "nat"   and G :: "'a vec list" assumes "cof_vec_space.lin_indpt_list n G" shows "0\<^sub>v n \<notin> set G"
mapping_of_pmfI: fixes m :: "('b, real) mapping"   and p :: "'b pmf" assumes "\<And>x. x \<in> Mapping.keys m \<Longrightarrow> Mapping.lookup m x = Some (pmf p x)"   and "Mapping.keys m = set_pmf p" shows "mapping_of_pmf p = m"
itrm_brackets_all: fixes extra_rule :: "nat itrm \<Rightarrow> nat itrm \<Rightarrow> bool"   and itrm_bracket :: "nat \<Rightarrow> nat itrm \<Rightarrow> nat itrm option"   and x :: "nat itrm"   and vs :: "nat list"   and x' :: "nat itrm" assumes "itrm_abstraction extra_rule itrm_bracket"   and "set (opaque x) \<subseteq> set vs"   and "itrm_abstraction.itrm_brackets itrm_bracket vs x = Some x'" shows "opaque x' = []"
inf_sup_aci_8: fixes 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 x :: "'c"   and y :: "'c" assumes "class.lattice inf less_eq less sup" shows "sup x (sup x y) = sup x y"
Q''TransQ''': shows "Q'' \<longmapsto> \<tau> \<prec> Q'''"
residuated_conjr2: fixes y :: "'a" shows "residuated (\<lambda>x. x \<rhd> y)"
ds_ord_eq: fixes x :: "'a" shows "(x \<sqsubseteq> d x) = (x = d x)"
conforms_RetD: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lname \<Rightarrow> ty option" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "x = Some (Jump Ret)" shows "locals s Result \<noteq> None"
se_star_mono_for_sub: fixes c1 :: "('d, 'e) conf"   and ls :: "('d, 'e) label list"   and c1' :: "('d, 'e) conf"   and c2 :: "('d, 'e) conf"   and c2' :: "('d, 'e) conf" assumes "se_star c1 ls c1'"   and "se_star c2 ls c2'"   and "c2 \<sqsubseteq> c1" shows "c2' \<sqsubseteq> c1'"
unity: fixes G1 :: "'b monoid"   and G2 :: "'b monoid"   and f :: "'b \<Rightarrow> 'b" assumes "group_hom G1 G2 f" shows "exact_seq ([G2, G1], [f])"
IH: assumes "i < length ps - 1"   and "poly (ps ! i) x = 0"   and "poly (ps ! (i + 1)) x = 0"   and "\<And>i x. \<lbrakk>i < length ps - 2; poly (ps ! (i + 1)) x = 0\<rbrakk> \<Longrightarrow> sgn (poly (ps ! (i + 2)) x) = - sgn (poly (ps ! i) x)" shows "\<forall>j\<le>i + 1. poly (ps ! j) x = 0"
add_diff_inverse: fixes a :: "'c"   and b :: "'c" assumes "\<not> a < b" shows "b + (a - b) = a"
kpower_inductr_var: fixes h :: "'c \<Rightarrow> 'd set"   and g :: "'c \<Rightarrow> 'd set"   and f :: "'d \<Rightarrow> 'd set"   and i :: "nat" assumes "h \<squnion> g \<circ>\<^sub>K f \<le> g" shows "h \<circ>\<^sub>K kpower f i \<le> g"
assms: shows "fmdom' (snd a) \<inter> vs = {}"
arr_assoc': fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "D.ide a"   and "D.ide b"   and "D.ide c" shows "D.arr (D.assoc' a b c)"
Lxy_in: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> S" shows "Lxy [x] S = [x]"
simps_2: fixes a :: "'b"   and b :: "'b trm"   and al :: "('b \<times> 'b trm) list"   and bl :: "('b \<times> 'b trm) list" shows "(a, b) # al \<lozenge> bl = (a, b \<lhd> bl) # (al \<lozenge> bl)"
eq_transitive_ax_def: shows "eq_transitive_ax \<equiv> mk_eq Core.x Core.y \<longmapsto> mk_eq Core.y Core.z \<longmapsto> mk_eq Core.x Core.z"
pred_def: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b measure" shows "Measurable.pred M P = ({x \<in> space M. P x} \<in> sets M)"
simps_7: fixes bs :: "real list"   and c :: "int"   and a :: "Ferrack.num" shows "Inum bs (Mul c a) = real_of_int c * Inum bs a"
get_shadow_root_impl: shows "get_shadow_root = get_shadow_root"
bal_list_def: fixes n :: "nat"   and xs :: "'b list" shows "Root_Balanced_Tree.bal_list n xs = Time_Monad.val (bal_list_tm n xs)"
nondecF: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "F x \<le> F y"
vsvI: fixes r :: "V" assumes "vbrelation r"   and "\<And>a b c. \<lbrakk>\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r; \<langle>a, c\<rangle> \<in>\<^sub>\<circ> r\<rbrakk> \<Longrightarrow> b = c" shows "vsv r"
cUn_iff: fixes c :: "'f"   and A :: "'f cset"   and B :: "'f cset" shows "cin c (cUn A B) = (cin c A \<or> cin c B)"
ia: shows "ia < IArray.length (matrix_to_iarray (Gram_Schmidt_column_k A k) !! i)"
restrictI: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'c \<Rightarrow> 'd set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x" shows "restrict f A \<in> Pi A B"
Compl_insert: fixes x :: "'c"   and A :: "'c set" shows "- insert x A = - A - {x}"
assms_1: shows "(\<lambda>s. P s \<and> T s) \<le> |X] Q"
lunit_commutes_with_L: fixes a :: "'a" assumes "ide a" shows "\<l>[\<I> \<otimes> a] = \<I> \<otimes> \<l>[a]"