hcju/isabelleps huggingface dataset:
z_min: fixes y :: "('t \<Rightarrow>\<^sub>0 'b) list \<times> 't list \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" assumes "length (snd (snd y)) < length (snd (snd z))" shows "y \<notin> Q0"
not_lex: fixes x :: "real \<times> real"   and y :: "real \<times> real" shows "(\<not> Counterclockwise_2D_Arbitrary.lex x y) = lexs y x"
equation_minus_iff: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "(a = uminus b) = (b = uminus a)"
case_prodI2: fixes p :: "'c \<times> 'd"   and c :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> c a b" shows "case p of (a, b) \<Rightarrow> c a b"
g_def: shows "g = (\<lambda>n. \<Sum>d | d dvd n. moebius_mu d * f d)"
smatch_smatch'_eq: fixes pat :: "Term_Class.term"   and t :: "sterm" assumes "linear pat"   and "is_value t" shows "match pat t = smatch' (mk_pat pat) t"
listrel_iff_nth: fixes xs :: "'b list"   and ys :: "'c list"   and r :: "('b \<times> 'c) set" shows "((xs, ys) \<in> listrel r) = (length xs = length ys \<and> (\<forall>n<length xs. (xs ! n, ys ! n) \<in> r))"
semiring_mat: fixes n :: "nat"   and b :: "'b" shows "semiring (ring_mat TYPE('a) n b)"
assms_2: shows "f = monom P (f (deg R f)) (deg R f)"
mkbot_def: fixes p :: "alt \<Rightarrow> real"   and b :: "alt" shows "mkbot p b \<equiv> p(b := Min (range p) - 1)"
pickBinp_alphaBinp_iff_equal: fixes binp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs set option"   and binp' :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs set option" assumes "goodBinp binp"   and "goodBinp binp'" shows "((pickBinp binp, pickBinp binp') \<in> alphaBinp) = (binp = binp')"
rerename_subst_noop: fixes "from" :: "nat"   and V :: "'var set"   and s :: "'subst"   and to :: "nat"   and f :: "'var \<Rightarrow> 'var" assumes "freshenLC from ` V \<inter> subst_lconsts s = {}" shows "subst_renameLCs (rerename V from to f) s = subst_renameLCs f s"
strip_while_eq_Cons_rec: fixes P :: "'f \<Rightarrow> bool"   and x :: "'f"   and xs :: "'f list" shows "(strip_while P (x # xs) = x # strip_while P xs) = (\<not> (P x \<and> (\<forall>x\<in>set xs. P x)))"
map_poly_pCons: fixes f :: "'d \<Rightarrow> 'c"   and c :: "'d"   and p :: "'d poly" assumes "f (0::'d) = (0::'c)" shows "map_poly f (pCons c p) = pCons (f c) (map_poly f p)"
rel_spmf_bindI': fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c spmf"   and g :: "'b \<Rightarrow> 'd spmf" assumes "rel_spmf A p q"   and "\<And>x y. \<lbrakk>A x y; x \<in> set_spmf p; y \<in> set_spmf q\<rbrakk> \<Longrightarrow> rel_spmf B (f x) (g y)" shows "rel_spmf B (p \<bind> f) (q \<bind> g)"
bval_eq_if_eq_on_vars: fixes b :: "bexp"   and s\<^sub>1 :: "char list \<Rightarrow> int"   and s\<^sub>2 :: "char list \<Rightarrow> int" assumes "s\<^sub>1 = s\<^sub>2 on vars b" shows "bval b s\<^sub>1 = bval b s\<^sub>2"
get_dom_component_parent_inside: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and node_ptr :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr \<in> set c"   and "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r Some parent" shows "parent \<in> set c"
csq_1: shows "Maps.commutative_square SPN_fgh.\<nu>.leg0 SPN_fgh.\<pi>.leg1 SPN_fgh.Prj\<^sub>0\<^sub>1 SPN_fgh.Prj\<^sub>0"
f_last_message_hold_append_mod: fixes xs :: "'b message_af list"   and k :: "nat"   and ys :: "'b message_af list" assumes "length xs mod k = 0" shows "(xs @ ys) \<longmapsto> k = xs \<longmapsto> k @ ys \<longmapsto> k"
assms: shows "(i, j) \<in> (b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>) \<inter> (e \<union> b\<inverse> \<union> m\<inverse> \<union> ov\<inverse> \<union> ov \<union> s\<inverse> \<union> s \<union> f\<inverse> \<union> f \<union> d\<inverse> \<union> d) \<inter> (b \<union> m \<union> ov \<union> s \<union> d)"
size_normalise_decomp: fixes a :: "'b"   and x :: "'b"   and y :: "'b"   and n :: "nat" assumes "a \<in> set (snd (normalise_decomp x y n))"   and "y \<noteq> (0::'b)" shows "euclidean_size a < euclidean_size y"
arctan_lower_11_def: shows "arctan_lower_11 \<equiv> \<lambda>x. - (pi / 2) - 3 * x / (1 + 3 * x\<^sup>2)"
trace_eq_simcl_bind2_scale: fixes q :: "'a spmf"   and S :: "'b spmf \<Rightarrow> 'c spmf \<Rightarrow> bool"   and p :: "'b spmf"   and f :: "'a \<Rightarrow> 'c spmf" assumes "\<forall>x\<in>set_spmf q. trace_eq_simcl S p (f x)" shows "trace_eq_simcl S (scale_spmf (weight_spmf q) p) (q \<bind> f)"
pos': shows "1 < bs ! i * x - hb * x / ln x powr (1 + e)"
FreeUltrafilterNat_HInfinite: fixes X :: "nat \<Rightarrow> 'a" assumes "\<And>u. \<forall>\<^sub>F n in \<U>. u < norm (X n)" shows "star_n X \<in> HInfinite"
distinct_21: fixes x31 :: "('a, 'b) MWLfCom"   and x32 :: "('a, 'b) MWLfCom"   and x51 :: "'a"   and x52 :: "('a, 'b) MWLfCom" shows "x31;x32 \<noteq> while x51 do x52 od"
xy_def: shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& x \<^bold>\<noteq> y \<^bold>& (\<^bold>\<forall>F. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>) in v]"
cong2_conga_cong: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A B C CongA A' B' C'"   and "Cong A B A' B'"   and "Cong B C B' C'" shows "Cong A C A' C'"
assms: shows "is_partition (us @ xs @ ys @ zs @ vs)"
lt_p'': shows "punit.lt (rep_list p') = punit.lt (rep_list p)"
semantics_tm_id: fixes t :: "'a tm" shows "\<lparr>\<^bold>#, \<^bold>\<dagger>\<rparr> t = t"
self_gen: fixes G :: "'b list set" shows "\<langle>\<langle>G\<rangle>\<rangle> = \<langle>G\<rangle>"
asymp_equiv_add_rightI: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd" assumes "f \<sim>[F] g"   and "h \<in> o[F](g)" shows "(\<lambda>x. f x + h x) \<sim>[F] g"
sym_iff: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "symmetric A equiv"   and "x \<in> A"   and "y \<in> A" shows "equiv x y = equiv y x"
next_performopD_7: fixes c0 :: "('b, 'loc \<times> 't) Exchange.configuration"   and c1 :: "('b, 'loc \<times> 't) Exchange.configuration"   and p :: "'b"   and \<Delta>neg :: "('loc \<times> 't) multiset"   and \<Delta>mint_msg :: "('b \<times> 'loc \<times> 't) multiset"   and \<Delta>mint_self :: "('loc \<times> 't) multiset" assumes "cri.next_performop' c0 c1 p \<Delta>neg \<Delta>mint_msg \<Delta>mint_self" shows "c_glob c1 = c_glob c0"
float_val_gt_threshold: fixes a :: "('a, 'b) IEEE.float" assumes "1 < LENGTH('a)"   and "is_finite a" shows "- threshold TYPE(('a, 'b) IEEE.float) < valof a"
update_convs_1: fixes I' :: "nat \<Rightarrow> nat"   and I :: "nat"   and M :: "nat"   and N :: "nat"   and S :: "nat"   and more :: "'b" shows "vars.I_update I' \<lparr>vars.I = I, M = M, N = N, S = S, \<dots> = more\<rparr> = \<lparr>vars.I = I' I, M = M, N = N, S = S, \<dots> = more\<rparr>"
TESL_sem_decreases_head: fixes \<gamma> :: "'a constr"   and \<Gamma> :: "'a constr list" shows "\<lbrakk>\<lbrakk> \<gamma> # \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<subseteq> \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
is_natural_transformation: shows "natural_transformation A B F H \<sigma>\<tau>.map"
finsum_cong: fixes A :: "'g set"   and B :: "'g set"   and f :: "'g \<Rightarrow> 'c"   and g :: "'g \<Rightarrow> 'c" assumes "A = B"   and "(f \<in> B \<rightarrow> carrier V) = True"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "finsum V f A = finsum V g B"
strict_implies_not_eq: fixes b :: "'d multiset"   and a :: "'d multiset" assumes "b \<subset># a" shows "a \<noteq> b"
mHom_lin: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and m :: "'e"   and f :: "'e \<Rightarrow> 'i"   and a :: "'f" assumes "R module M"   and "R module N"   and "m \<in> carrier M"   and "f \<in> mHom R M N"   and "a \<in> carrier R" shows "f (a \<cdot>\<^sub>s\<^bsub>M\<^esub> m) = a \<cdot>\<^sub>s\<^bsub>N\<^esub> f m"
Int_UNIV_right: fixes A :: "'d set" shows "A \<inter> UNIV = A"
m_div_def: fixes G :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" shows "x \<oslash>\<^bsub>G\<^esub> y = (if y = \<zero>\<^bsub>G\<^esub> then \<zero>\<^bsub>G\<^esub> else x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> y)"
totalize_def: fixes F :: "'c program" shows "totalize F \<equiv> mk_program (Init F, totalize_act ` Acts F, AllowedActs F)"
ebmovovissiffiddibmovsd: shows "(e \<union> b \<union> m \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse> \<union> d \<union> d\<inverse>) O (b \<union> m \<union> ov \<union> s \<union> d) \<subseteq> b \<union> m \<union> ov \<union> s \<union> d \<union> f\<inverse> \<union> d\<inverse> \<union> ov\<inverse> \<union> s\<inverse> \<union> f \<union> e"
unique_factorization_m_def: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "p.unique_factorization_m f cfs = (p.Mf ` Collect (p.factorization_m f) = {p.Mf cfs})"
pdevs_val_degree_cong: fixes b :: "'a pdevs"   and d :: "'a pdevs"   and a :: "nat \<Rightarrow> real"   and c :: "nat \<Rightarrow> real" assumes "b = d"   and "\<And>i. i < degree b \<Longrightarrow> a i = c i" shows "pdevs_val a b = pdevs_val c d"
SN_rel_on_Id: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set"   and T :: "'a set" shows "SN_rel_on R (S\<^sup>=) T = SN_rel_on R S T"
outstanding_volatile_writes_unowned_by_others_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "outstanding_volatile_writes_unowned_by_others ts \<equiv> \<forall>i p\<^sub>i is\<^sub>i \<O>\<^sub>i \<R>\<^sub>i \<D>\<^sub>i \<theta>\<^sub>i sb\<^sub>i j p\<^sub>j is\<^sub>j \<O>\<^sub>j \<R>\<^sub>j \<D>\<^sub>j \<theta>\<^sub>j sb\<^sub>j. i < length ts \<longrightarrow> j < length ts \<longrightarrow> i \<noteq> j \<longrightarrow> ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i) \<longrightarrow> ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j) \<longrightarrow> (\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) \<inter> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb\<^sub>i = {}"
invertible_right_inverse2: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c"   and v :: "'c" assumes "monoid M composition unit"   and "monoid.invertible M composition unit u"   and "u \<in> M"   and "v \<in> M" shows "composition u (composition (monoid.inverse M composition unit u) v) = v"
square_part_dvd': fixes n :: "nat" shows "square_part n dvd n"
is_projection_on_zero: shows "is_projection_on (\<lambda>_. 0::'b) {0::'b}"
map_disj_com: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" shows "h\<^sub>0 \<bottom> h\<^sub>1 = h\<^sub>1 \<bottom> h\<^sub>0"
WtNFT_def: fixes cl :: "('a, 'b, 'c) cmd list" shows "WtNFT cl \<equiv> \<Sum>n\<in>theNFT cl. 1 / real (length cl)"
1: fixes t :: "real" assumes "t * 2 \<le> 1" shows "u + t * (v * 2) \<le> v + t * (u * 2)"
pdivmod_eq_pdivmod_monic: fixes g :: "'a poly"   and f :: "'a poly" assumes "monic g" shows "pdivmod f g = pdivmod_monic f g"
in_multiset_of_conv_nth: fixes x :: "'b"   and l :: "'b list" shows "(x \<in># mset l) = (\<exists>i<length l. x = l ! i)"
algebraic_csqrt: fixes x :: "complex" assumes "algebraic x" shows "algebraic (csqrt x)"
K5_msg_in_parts_spies: fixes Tgs' :: "agent"   and A :: "agent"   and authK :: "nat"   and servK :: "msg"   and B :: "agent"   and Ts :: "msg"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs' A (Crypt authK \<lbrace>servK, Agent B, Ts, servTicket\<rbrace>) \<in> set evs" shows "servTicket \<in> parts (knows Spy evs)"
rel_fun_def_butlast: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and T :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'g \<Rightarrow> 'h"   and g :: "'f \<Rightarrow> 'i \<Rightarrow> 'j" shows "(R ===> S ===> T) f g = (\<forall>x y. R x y \<longrightarrow> (S ===> T) (f x) (g y))"
obs_neq_some_contradicts': fixes \<sigma> :: "'var \<Rightarrow> 'val"   and i :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and i' :: "nat"   and k :: "nat" assumes "obsp \<sigma> i \<noteq> obsp \<sigma>' i' \<or> cs\<^bsup>path \<sigma>\<^esup> i \<noteq> cs\<^bsup>path \<sigma>'\<^esup> i'"   and "is_kth_obs (path \<sigma>) k i"   and "is_kth_obs (path \<sigma>') k i'" shows "\<exists>k k'. (\<sigma>', k') \<cc> (\<sigma>, k) \<and> path \<sigma> k \<in> dom local.att \<or> (\<sigma>, k) \<cc> (\<sigma>', k') \<and> path \<sigma>' k' \<in> dom local.att"
i_nr: shows "\<forall>i<length (pivot_positions A). i < dim_row A"
vimageI: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and b :: "'b"   and B :: "'b set" assumes "f a = b"   and "b \<in> B" shows "a \<in> f -` B"
disc_15: fixes x3 :: "'b" shows "is_Atom (Atom x3)"
limited_family_union: fixes B :: "'a set set"   and k :: "nat" assumes "finite B"   and "\<forall>P\<in>B. card P \<le> k" shows "card (\<Union> B) \<le> card B * k"
assms_1: shows "A \<in> carrier_mat nr1 nc1"
Morera_local_triangle_ball: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex" assumes "\<And>z. z \<in> S \<Longrightarrow> \<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and> (\<forall>b c. closed_segment b c \<subseteq> ball a e \<longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0)" shows "f analytic_on S"
Janiszewski: fixes S :: "complex set"   and T :: "complex set"   and a :: "complex"   and b :: "complex" assumes "compact S"   and "closed T"   and "connected (S \<inter> T)"   and "connected_component (- S) a b"   and "connected_component (- T) a b" shows "connected_component (- (S \<union> T)) a b"
get_exp_fulfills: fixes a :: "'c" assumes "a \<in> carrier H" shows "a = h [^]\<^bsub>H\<^esub> H.get_exp h a"
concat_chop: fixes n :: "nat"   and xs :: "'a list" assumes "0 < n" shows "concat (chop n xs) = xs"
swapInp_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and inp :: "'b \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm set option" shows "inp %[x \<and> y]_xs \<equiv> lift (swap xs x y) inp"
borel_eq_box: shows "borel = sigma UNIV (range (\<lambda>(a, b). box a b))"
prj_simps_5: shows "C.dom \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0\<^sub>1 = HH\<mu>\<nu>H\<pi>\<rho>.chine"
rel_gpv''D: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and gpv :: "('d, 'f, 'h) gpv"   and gpv' :: "('e, 'g, 'i) gpv" assumes "rel_gpv'' A C R gpv gpv'" shows "rel_spmf (rel_generat A C (rel_fun R (rel_gpv'' A C R))) (the_gpv gpv) (the_gpv gpv')"
sqn_nsqn: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat" shows "sqn rt dip - 1 \<le> nsqn rt dip"
le_infE: fixes x :: "'b \<times> 'c"   and a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" assumes "less_eq_prod x (inf_prod a b)" obtains "less_eq_prod x a" and "less_eq_prod x b"
Zleast_mL_I: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat"   and x :: "'a" assumes "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "j \<le> n"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "x \<in> I" shows "ant (mL K P I j) \<le> (\<nu>\<^bsub>K P j\<^esub>) x"
subnets_host_attributes_def: shows "subnets_host_attributes \<equiv> [''v11'' \<mapsto> Subnet 1, ''v12'' \<mapsto> Subnet 1, ''v13'' \<mapsto> Subnet 1, ''v1b'' \<mapsto> BorderRouter 1, ''v21'' \<mapsto> Subnet 2, ''v22'' \<mapsto> Subnet 2, ''v23'' \<mapsto> Subnet 2, ''v2b'' \<mapsto> BorderRouter 2, ''v3b'' \<mapsto> BorderRouter 3]"
suff_def: fixes A :: "'b set"   and s :: "'b llist" shows "suff A s \<equiv> {r \<in> A\<^sup>\<infinity>. s \<le> r}"
evals\<^sub>1_preserves_len: fixes P :: "nat exp prog"   and es\<^sub>0 :: "nat exp list"   and h\<^sub>0 :: "heap"   and ls\<^sub>0 :: "val list"   and sh\<^sub>0 :: "sheap"   and es\<^sub>1 :: "nat exp list"   and h\<^sub>1 :: "heap"   and ls\<^sub>1 :: "val list"   and sh\<^sub>1 :: "sheap" assumes "P \<turnstile>\<^sub>1 \<langle>es\<^sub>0,(h\<^sub>0, ls\<^sub>0, sh\<^sub>0)\<rangle> [\<Rightarrow>] \<langle>es\<^sub>1,(h\<^sub>1, ls\<^sub>1, sh\<^sub>1)\<rangle>" shows "length ls\<^sub>0 = length ls\<^sub>1"
insert_commute: fixes x :: "'b"   and y :: "'b"   and A :: "'b set" shows "x \<triangleright> y \<triangleright> A = y \<triangleright> x \<triangleright> A"
joinable_connected_component_eq: fixes T :: "'c set"   and S :: "'c set"   and x :: "'c"   and y :: "'c" assumes "connected T"   and "T \<subseteq> S"   and "connected_component_set S x \<inter> T \<noteq> {}"   and "connected_component_set S y \<inter> T \<noteq> {}" shows "connected_component_set S x = connected_component_set S y"
op_nres_CHECK_bnd_def: fixes f :: "unit \<Rightarrow> 'd"   and \<Phi> :: "bool"   and m :: "'c nres" shows "op_nres_CHECK_bnd f \<Phi> m \<equiv> CHECK f \<Phi> \<bind> (\<lambda>_. m)"
def_hashmap_size_literal_def: fixes uu :: "String.literal itself" shows "def_hashmap_size uu = 10"
scaleC_left: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and r :: "complex"   and a :: "'d"   and b :: "'e" assumes "bounded_sesquilinear prod" shows "prod (r *\<^sub>C a) b = cnj r *\<^sub>C prod a b"
ifp_compatible_with_ipc: shows "\<forall>a b c x. Policy.sp_spec_subj_subj a b \<and> Policy.sp_spec_subj_obj b (PAGE x) WRITE \<and> Policy.sp_spec_subj_obj c (PAGE x) READ \<longrightarrow> Policy.ifp a c"
case_prod_mono: fixes p :: "'f \<times> 'g"   and f :: "'f \<Rightarrow> 'g \<Rightarrow> 'h"   and f' :: "'f \<Rightarrow> 'g \<Rightarrow> 'h" assumes "\<And>a b. p = (a, b) \<Longrightarrow> f a b \<le> f' a b" shows "(case p of (x, xa) \<Rightarrow> f x xa) \<le> (case p of (x, xa) \<Rightarrow> f' x xa)"
mat_eye_l: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "eye *\<^sub>m\<^sub>m A = A"
lens_compat_def: fixes X :: "'b \<Longrightarrow> 'c"   and Y :: "'d \<Longrightarrow> 'c" shows "(X ##\<^sub>L Y) = (\<forall>s\<^sub>1 s\<^sub>2. s\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>2 = s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2)"
brnL_Int: fixes n1 :: "nat"   and n2 :: "nat"   and cl :: "('test, 'atom, 'choice) cmd list" assumes "n1 \<noteq> n2"   and "n1 < length cl"   and "n2 < length cl" shows "{brnL cl n1..<brnL cl n1 + brn (cl ! n1)} \<inter> {brnL cl n2..<brnL cl n2 + brn (cl ! n2)} = {}"
eq: shows "[0..<length (x # xs)] = 0 # [Suc 0..<Suc (length xs)]"
WhileTrue: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list" assumes "s \<in> b" shows "\<Gamma>\<turnstile> (While b c # cs, css, Normal s) \<rightarrow> (c # While b c # cs, css, Normal s)"
keys_def: shows "Mapping.keys \<equiv> map_fun mapping.rep id dom"
monom_mult_eq_zero_iff: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and c :: "'f"   and t :: "'d"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "term_powerprod pair_of_term term_of_pair" shows "(term_powerprod.monom_mult pair_of_term term_of_pair c t p = 0) = (c = (0::'f) \<or> p = 0)"
meet_glb: fixes z :: "('a \<times> 'a set) set"   and x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set" assumes "z \<subseteq> x \<and> z \<subseteq> y" shows "z \<subseteq> x \<inter> y"
range_sbintrunc: fixes n :: "nat" shows "range (signed_take_bit n) = {i. - (2 ^ n) \<le> i \<and> i < 2 ^ n}"
step_Normal_elim_cases_3: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'e \<Rightarrow> 'e"   and s :: "'e"   and u :: "('e, 'd, 'f) com \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Basic f, Normal s) \<rightarrow> u" obtains "u = (Skip, Normal (f s))"
scaleRat_linear_poly_def: shows "(*R) \<equiv> map_fun id (map_fun Rep_linear_poly Abs_linear_poly) fun_scale"
\<D>_4: fixes K :: "'n set" assumes "K \<in> \<D>" shows "\<exists>a b. K = cbox a b"
simps_1: fixes C_info :: "(name, dt_def) fmap"   and name :: "name"   and vs :: "value list" shows "special_constants.vwelldefined' C_info (Vconstr name vs) = list_all (special_constants.vwelldefined' C_info) vs"
2: fixes x :: "complex" assumes "x \<in> {w. - pi < Im w \<and> Im w < pi}" shows "(exp has_derivative blinfun_apply (Blinfun ((*) (exp x)))) (at x)"
icdi_path_swap: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi>' :: "nat \<Rightarrow> 'a"   and j :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>'"   and "IFC_def.is_icdi prob j \<pi> k"   and "IFC_def.eq_up_to \<pi> j \<pi>'" shows "IFC_def.is_icdi prob j \<pi>' k"
perm_restrict_def: fixes f :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and x :: "'a" shows "perm_restrict f S x \<equiv> if x \<in> S then f x else x"
map_of_subst: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and y :: "Vars.var"   and k :: "Vars.var" shows "map_of \<Gamma>[x::h=y] k = map_option (\<lambda>e. e[x::=y]) (map_of \<Gamma> k)"
eq_newpkt_def: fixes m :: "msg" shows "eq_newpkt m \<equiv> case m of Pkt nat1 nat2 \<Rightarrow> False | Newpkt d did \<Rightarrow> True"
init_s: fixes t :: "(nat \<times> linear_poly) list" assumes "\<triangle> t" shows "checked_s t {} (init_s t)"
k0_def: shows "k0 = local.Min ((fst ` set xs \<union> fst ` set ys) \<inter> {k. f k (lookup_pair xs k) (lookup_pair ys k) \<noteq> Some Eq})"
from_nat_inj: fixes x :: "nat"   and xa :: "nat" assumes "mod_type_class.from_nat x = mod_type_class.from_nat xa"   and "x < CARD('d)"   and "xa < CARD('d)" shows "x = xa"
cA: shows "char_poly A = (\<Prod>a\<leftarrow>as. [:- a, 1:])"
literalNotInEarlierLevelsThanItsLevel: fixes level :: "nat"   and e :: "'a"   and M :: "('a \<times> bool) list" assumes "level < elementLevel e M" shows "e \<notin> set (elements (prefixToLevel level M))"
dfa_is_node_def: fixes A :: "nat bdd list \<times> bool list" shows "dfa_is_node A = (\<lambda>q. q < length (fst A))"
fls_divide_X_inv_power': fixes f :: "'a fls"   and n :: "nat" assumes "inverse (1::'a) = (1::'a)" shows "f / fls_X_inv ^ n = fls_shift (- int n) f"
0: shows "ProblemIkTpartG_GEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl eintT eintF eintP"
is_ft_dghm_def: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB> \<equiv> \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB> \<and> is_ft_dghm_axioms \<AA> \<FF>"
elims: fixes x :: "nat"   and xa :: "(nat \<times> nat) list"   and y :: "nat list" assumes "pathVertices x xa = y" obtains u :: "nat" where "x = u__" and "xa = []" and "y = [u__]"   | u :: "nat" and e :: "nat \<times> nat" and es :: "(nat \<times> nat) list" where "x = u" and "xa = e # es" and "y = fst e # pathVertices (snd e) es"