hcju/isabelleps huggingface dataset:
diff_cancel: fixes A :: "'a multiset" shows "A - A = {#}"
1: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option" shows "uninst.redundant (uses(g := u g)) (phis(g := p g)) g = uninst.redundant u p g"
adjacent_nodes_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat" shows "Graph.adjacent_nodes c u \<equiv> Graph.E c `` {u} \<union> (Graph.E c)\<inverse> `` {u}"
empt_invariant: shows "prod_invariant A B (prod_empt A B)"
pendingReqs_def: fixes s :: "('a, 'b, 'c) SLin_state" shows "pendingReqs s \<equiv> {pending s p |p. status s p \<in> {Pending, Aborted}}"
prems_2: shows "CT \<turnstile> D \<not><: Da"
distinct_17: fixes x31 :: "com"   and x32 :: "com"   and x51 :: "expr"   and x52 :: "com" shows "x31;; x32 \<noteq> while (x51) x52"
q_in_bs: shows "q_in_bs \<longrightarrow> q \<in> set gs \<union> set bs"
least_fixpoint_char: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "is_least_fixpoint f x = (has_least_fixpoint f \<and> x = \<mu> f)"
set_borel_integral_eq_integral_2: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "set_integrable lborel S f" shows "set_lebesgue_integral lborel S f = integral S f"
semiring_norm_166: fixes v :: "num"   and w :: "num"   and y :: "'a" shows "numeral v + (- numeral w + y) = neg_numeral_class.sub v w + y"
MakeNTApp: fixes X :: "'d"   and f :: "('d, 'g, 'e, 'h, 'f, 'i) NatTrans" assumes "X \<in> obj\<^bsub>NTCatDom (MakeNT f)\<^esub>" shows "f $$ X = MakeNT f $$ X"
get_parent_parent_in_heap: fixes h :: "(_) heap"   and child_node :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "h \<turnstile> Shadow_DOM.get_parent child_node \<rightarrow>\<^sub>r Some parent" shows "parent |\<in>| object_ptr_kinds h"
simps_2: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and U :: "'c set"   and x :: "'d"   and xs :: "'d list" shows "sources_aux I D U (x # xs) = (if \<exists>v\<in>sources_aux I D U xs. (D x, v) \<in> I then insert (D x) (sources_aux I D U xs) else sources_aux I D U xs)"
mat_kernel_def: fixes A :: "'a mat" shows "mat_kernel A = {v \<in> carrier_vec (dim_col A). A *\<^sub>v v = 0\<^sub>v (dim_row A)}"
assms: fixes u :: "'a" assumes "u \<in> keys_pp s \<union> keys_pp t" shows "lookup_pp s u = lookup_pp t u"
same_cs_if_not_recv: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and cid :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "\<not> isRecv ev" shows "fst (cs c cid) = fst (cs c' cid)"
prefix_butlastD: fixes xs :: "'a list"   and ys :: "'a list" assumes "prefix xs (butlast ys)" shows "prefix xs ys"
sel_2: fixes r :: "real"   and x :: "complex" shows "Im (r *\<^sub>R x) = r * Im x"
rep_eq: fixes x :: "'a fset" shows "fBall x = Ball (fset x)"
proj_subst: fixes l :: "'a"   and A :: "('a strand_label \<times> ('b, 'c) stateful_strand_step) list"   and \<delta> :: "'c \<Rightarrow> ('b, 'c) Term.term" shows "proj l (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) = proj l A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>"
inpt_lemma: fixes xs :: "nat list"   and vl :: "nat"   and m :: "nat" assumes "Suc (length xs) = vl" shows "rec_exec (rec_inpt vl) (m # xs) = inpt m xs"
ev_signal_weakly_step_consistent: fixes u :: "partition_id_t"   and s1 :: "state_t"   and s2 :: "state_t"   and tid :: "thread_id_t"   and ipt :: "int_point_t"   and partner :: "thread_id_t" assumes "vpeq u s1 s2"   and "vpeq (Step.partition tid) s1 s2"   and "atomic_step_invariant s1"   and "atomic_step_invariant s2"   and "atomic_step_precondition s1 (current s1) ipt"   and "atomic_step_precondition s1 (current s1) ipt" shows "vpeq u (atomic_step_ev_signal tid partner s1) (atomic_step_ev_signal tid partner s2)"
p_prod_distl: fixes X :: "'a list set"   and Y :: "'a list set"   and Z :: "'a list set" shows "p_prod X (Y \<union> Z) = p_prod X Y \<union> p_prod X Z"
prim_invar2_def: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option" shows "Prim2.prim_invar2 w g r Q \<pi> \<equiv> Prim2.prim_invar2_init r Q \<pi> \<or> Prim2.prim_invar2_ctd w g r Q \<pi>"
unit_id: fixes f :: "unit \<Rightarrow> unit" shows "f = id"
lin_indpt_cone_def: shows "lin_indpt_cone \<equiv> \<Union> {finite_cone B |B. B \<subseteq> A \<and> card B = n \<and> lin_indpt B}"
p_def: fixes B' :: "'b set" shows "p B' \<equiv> B \<subseteq> B' \<and> independent B' \<and> span B' = UNIV"
isLim_supr: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r"   and "i \<in> Field r"   and "isLim r i" shows "i = supr r (underS r i)"
rbt_delete_is_rbt: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt"   and k :: "'c" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.is_rbt less (ord.rbt_delete less k t)"
elims_3: fixes x :: "'c \<Rightarrow> bool"   and xa :: "'c stream" assumes "\<not> holds x xa" obtains P :: "'c \<Rightarrow> bool" and xs :: "'c stream" where "x = P" and "xa = xs" and "\<not> P (shd xs)"
assms_2: shows "\<Psi> \<turnstile> \<phi>"
existence_ivl_subset: fixes t0 :: "real"   and x0 :: "'a" shows "existence_ivl t0 x0 \<subseteq> T"
cil_length: fixes w :: "'d list"   and wa :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and wb :: "'d list" assumes "w \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> wb" shows "length w = length wa + length wb"
atomic_step_ev_wait_one_def: fixes tid :: "thread_id_t"   and s :: "state_t" shows "atomic_step_ev_wait_one tid s = s\<lparr>thread := (thread s)(tid := thread s tid\<lparr>ev_counter := ev_counter (thread s tid) - 1\<rparr>)\<rparr>"
dot_cross_self_3: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "x \<times> y \<bullet> y = 0"
assms_3: fixes x :: "real"   and y :: "real" assumes "x \<in> T"   and "y \<in> T"   and "x < y" shows "x + d \<le> y"
good_imp_qGood_pick: fixes X :: "('e, 'f, 'varSort, 'var, 'g) qTerm set" assumes "good X" shows "qGood (pick X)"
nproj1: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "norm (proj x) = 1"
induct: fixes P :: "('b \<times> nat) list \<Rightarrow> bool"   and a0 :: "('b \<times> nat) list" assumes "\<And>x p. P [(x, p)]"   and "\<And>x p v va. P (v # va) \<Longrightarrow> P ((x, p) # v # va)"   and "P []" shows "P a0"
holomorphic_on_paste_across_line: fixes S :: "complex set"   and d :: "complex"   and f :: "complex \<Rightarrow> complex"   and k :: "real" assumes "open S"   and "d \<noteq> 0"   and "f holomorphic_on S \<inter> {z. d \<bullet> z < k}"   and "f holomorphic_on S \<inter> {z. k < d \<bullet> z}"   and "continuous_on S f" shows "f holomorphic_on S"
y: shows "y = ys1 @ y1 # y2 # ys2"
leg1_simps_5: shows "local.dom p\<^sub>1 = p\<^sub>1"
W2: shows "\<forall>(\<phi>, \<delta>)\<in>Is. \<exists>(\<phi>', \<delta>')\<in>insert (Q, \<delta>0) Is. \<forall>t. (\<box>(\<lambda>s. \<phi> s \<and> \<lceil>\<lambda>s. \<delta> s = t\<rceil> s \<longrightarrow> (\<diamond>(\<lambda>s. \<phi>' s \<and> \<lceil>\<lambda>s. (\<delta>' s, t) \<in> R\<rceil> s)) s)) \<sigma>"
sum_moebius_mu_divisors': fixes n :: "nat" shows "(\<Sum>d | d dvd n. moebius_mu d) = (if n = 1 then 1::'a else (0::'a))"
seqI: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g1 :: "'a"   and f1 :: "'a"   and g2 :: "'b"   and f2 :: "'b" assumes "product_category C1 C2"   and "partial_magma.arr C1 (C1 g1 f1) \<and> partial_magma.arr C2 (C2 g2 f2)" shows "partial_magma.arr (product_category.comp C1 C2) (product_category.comp C1 C2 (g1, g2) (f1, f2))"
round_count_inv: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and q :: "nat"   and t :: "nat \<times> nat list \<times> 'c list"   and n :: "nat"   and f :: "'c \<Rightarrow> nat"   and r :: "nat" assumes "index_less index key"   and "bn_inv p q t"   and "add_inv n t"   and "count_inv f t" shows "count_inv f (round index key p q r t)"
Exe2E: fixes v :: "i"   and F :: "\<Pi>\<^sub>2"   and x :: "\<kappa>"   and y :: "\<kappa>" assumes "[\<lparr>F,x,y\<rparr> in v]" shows "\<exists>r o\<^sub>1 o\<^sub>2. Some r = d\<^sub>2 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = d\<^sub>\<kappa> y \<and> (o\<^sub>1, o\<^sub>2) \<in> ex2 r v"
map_def: fixes f :: "'c" shows "Ko\<tau>'_Ko\<tau>.map f = (if \<tau>.A.arr f then C ((K \<circ> \<tau>') (\<tau>.A.cod f)) ((K \<circ> \<tau>) f) else K.B.null)"
Order: fixes D :: "('c, 'd) Order_scheme" assumes "Torder D" shows "Order D"
map_sel_1: fixes a :: "'c llist"   and f :: "'c \<Rightarrow> 'd" assumes "\<not> lnull a" shows "lhd (lmap f a) = f (lhd a)"
resource_of_oracle_def: fixes "oracle" :: "'g \<Rightarrow> 'h \<Rightarrow> ('i \<times> 'g) spmf" shows "resource_of_oracle oracle \<equiv> corec_resource (\<lambda>s a. map_spmf (map_prod id Inr) (oracle s a))"
gen_dfs_post_def: fixes E :: "('a \<times> 'a) set"   and U :: "'b"   and S :: "'a set"   and V0 :: "'a set"   and u0 :: "'a"   and V :: "'a set"   and brk :: "bool" shows "gen_dfs_post E U S V0 u0 V brk \<equiv> (\<not> brk \<longrightarrow> E `` (V - S) \<subseteq> V) \<and> u0 \<in> V \<and> V0 \<subseteq> V \<and> V \<subseteq> V0 \<union> (E - UNIV \<times> S)\<^sup>* `` {u0}"
C: shows "(Owner n, \<lbrace>Num 1, PubKey A\<rbrace>) \<in> s \<longrightarrow> PriKey A \<notin> spied s"
remdups_id_iff_distinct: fixes xs :: "'b list" shows "(remdups xs = xs) = distinct xs"
R1_R2_commute: shows "R1 \<circ> R2 = R2 \<circ> R1"
r_neg1: fixes x :: "'a vec"   and y :: "'a vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n" shows "\<ominus>\<^bsub>V\<^esub> x + (x + y) = y"
of_nat_power: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "nat"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (m ^ n) = power.power one times (semiring_1.of_nat one plus zero m) n"
x3: shows "\<forall>e\<in>edges G - F. P e"
isometry_inverse_1: fixes f :: "'b \<Rightarrow> 'c" assumes "isometry f" shows "isometry (inv f)"
Body: fixes K :: "key"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> parts H" shows "X \<in> parts H"
cUnCI: fixes c :: "'f"   and B :: "'f cset"   and A :: "'f cset" assumes "\<not> cin c B \<Longrightarrow> cin c A" shows "cin c (cUn A B)"
fpxs_supp_def: shows "fpxs_supp \<equiv> map_fun fpxs_nth id Formal_Puiseux_Series.supp"
qFreshInp_def: fixes xs :: "'varSort"   and x :: "'var"   and qinp :: "'c \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm option" shows "qFreshInp xs x qinp \<equiv> liftAll (qFresh xs x) qinp"
lr_fps_denominator_def: fixes cs :: "'a list" shows "lr_fps_denominator cs = Poly (rev cs)"
one_app_congR: fixes t1 :: "lam"   and t2 :: "lam"   and s :: "lam" assumes " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2" shows " App s t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* App s t2"
VcompNml_HcompNml: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term"   and w :: "'c local.term" assumes "Nml t"   and "Nml u"   and "Nml v"   and "Nml w"   and "VSeq t v"   and "VSeq u w"   and "Src v = Trg w" shows "(t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> (v \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> w) = t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> v \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> w"
remdups_rev_fst: fixes r :: "'b"   and rs :: "'b list" shows "remdups_rev (r # rs) = (if r \<in> set rs then r # remdups_rev (removeAll r rs) else r # remdups_rev rs)"
FlexRayArch_def: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list" shows "FlexRayArch n nReturn nC nStore nGet \<equiv> CorrectSheaf n \<and> ((\<forall>i<n. msg 1 (nReturn i)) \<and> DisjointSchedules n nC \<and> IdenticCycleLength n nC \<longrightarrow> FlexRayArchitecture n nReturn nC nStore nGet)"
distinct_2: fixes x2 :: "name" shows "Var x2 \<noteq> zer"
interpretation_substI: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "\<And>v. fv (\<theta> v) = {}" shows "interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
bij_betw_compose_left_perm_PiE: fixes \<pi> :: "'a \<Rightarrow> 'a"   and B :: "'a set"   and A :: "'b set" assumes "\<pi> permutes B" shows "bij_betw (\<lambda>f. restrict (\<pi> \<circ> f) A) (A \<rightarrow>\<^sub>E B) (A \<rightarrow>\<^sub>E B)"
preorder_on_def: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "preorder_on A r \<equiv> refl_on A r \<and> trans r"
minus_exp_eq_not_mask: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "uminus (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = not (mask n)"
pivot_funD_2: fixes A :: "'c mat"   and nr :: "nat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat" assumes "dim_row A = nr"   and "pivot_fun A f nc"   and "i < nr"   and "j < f i" shows "A $$ (i, j) = (0::'c)"
translation_apply: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c"   and b :: "'c" assumes "left_translations_of_monoid M composition unit"   and "a \<in> M"   and "b \<in> M" shows "left_translations_of_monoid.translation M composition a b = composition a b"
label_incr_start_Node_smaller: fixes l :: "nat"   and n :: "w_node"   and i :: "nat" assumes "(_ l _) = n \<oplus> i" shows "n = (_ l - i _)"
dg_prod_ObjD_1: fixes a :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Obj\<rparr>" shows "vsv a"
F1: fixes i :: "real"   and j :: "real"   and \<pp> :: "'a set" assumes "i \<in> I"   and "j \<in> I"   and "\<pp> \<in> V i"   and "\<pp> \<in> V j" shows "s i \<pp> = s j \<pp>"
flatten1_im: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite A"   and "finite B" shows "\<Sum> ((\<lambda>x. \<Sum> (f x ` B)) ` A) = \<Sum> ((\<lambda>p. f (fst p) (snd p)) ` (A \<times> B))"
word_minus_one_le_leq: fixes x :: "'b word"   and y :: "'b word" assumes "x - 1 < y" shows "x \<le> y"
normNum_def: shows "normNum = (\<lambda>(a, b). if a = 0 \<or> b = 0 then 0\<^sub>N else let g = gcd a b in if 0 < b then (a div g, b div g) else (- (a div g), - (b div g)))"
vector_sadd_rdistrib: fixes a :: "'a"   and b :: "'a"   and x :: "('a, 'b) Finite_Cartesian_Product.vec" shows "(a + b) *s x = a *s x + b *s x"
fin_B: shows "finite {l. length l = m \<and> sum_list l + 1 = N}"
difimov: shows "(d\<inverse> \<union> f\<inverse> \<union> ov \<union> e \<union> f \<union> m \<union> b \<union> s\<inverse> \<union> s) O (m \<union> ov \<union> s \<union> d \<union> b \<union> f\<inverse> \<union> f \<union> e) \<subseteq> 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>"
vec_nth: fixes x :: "('c, 'b) vec" shows "($) x \<in> UNIV"
tendsto_complex_iff: fixes f :: "'a \<Rightarrow> complex"   and x :: "complex"   and F :: "'a filter" shows "(f \<longlongrightarrow> x) F = (((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F \<and> ((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F)"
digraph_if_digraph: fixes \<beta> :: "V"   and \<CC> :: "V"   and \<alpha> :: "V" assumes "digraph \<beta> \<CC>"   and "\<Z> \<alpha>"   and "\<CC>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "\<And>A B. \<lbrakk>A \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>; B \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>; A \<in>\<^sub>\<circ> Vset \<alpha>; B \<in>\<^sub>\<circ> Vset \<alpha>\<rbrakk> \<Longrightarrow> \<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>A. \<Union>\<^sub>\<circ> (VLambda B (Hom \<CC> a) `\<^sub>\<circ> B)) `\<^sub>\<circ> A) \<in>\<^sub>\<circ> Vset \<alpha>" shows "digraph \<alpha> \<CC>"
assms: shows "Im x < 0 \<or> Im x = 0 \<and> 0 \<le> Re x"
less_def: fixes x :: "'a"   and y :: "'a" shows "(x \<sqsubset> y) = (x \<sqsubseteq> y \<and> \<not> y \<sqsubseteq> x)"
ThrowRedsThrow: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and r :: "nat \<times> char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>" shows "P,E \<turnstile> \<langle>throw e,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>"
horizontal_chop_empty_right: shows "\<forall>v. \<exists>u. v=v\<parallel>u"
lexordp_conv_lexord: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "class.linorder less_eq less" shows "ord.lexordp less xs ys = ((xs, ys) \<in> lexord {(x, y). less x y})"
fct_gen_subalgebra_sets_mem: fixes B :: "'c set"   and N :: "'c measure"   and X :: "'d \<Rightarrow> 'c"   and M :: "'d measure" assumes "B \<in> sets N" shows "X -` B \<inter> space M \<in> sets (fct_gen_subalgebra M N X)"
eq_implies: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "reflexive A less_eq"   and "x = y"   and "x \<in> A" shows "less_eq x y"
assms_3: shows "qs \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x))])"
is_methdD: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig" assumes "is_methd G C sig" shows "is_class G C \<and> methd G C sig \<noteq> None"
r_cons_else_def: shows "r_cons_else \<equiv> Cn 2 S [Cn 2 r_prod_encode [Cn 2 r_length [recf.Id 2 1], Cn 2 r_prod_encode [recf.Id 2 0, Cn 2 r_pdec2 [Cn 2 r_dec [recf.Id 2 1]]]]]"
simps: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and a :: "'a stream" shows "ev \<phi> a = ((\<exists>xs. a = xs \<and> \<phi> xs) \<or> (\<exists>xs. a = xs \<and> ev \<phi> (stl xs)))"
f2: shows "cpu_reg_val PC s1 = cpu_reg_val PC s2"
sqrt_unique: fixes m :: "nat"   and n :: "nat" assumes "m\<^sup>2 \<le> n"   and "n < (Suc m)\<^sup>2" shows "local.sqrt n = m"
sub_conjD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "sub_conj t"   and "sound P"   and "sound Q" shows "t P && t Q \<tturnstile> t (P && Q)"
il_commutative: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
arr_cod: fixes f :: "'a" assumes "Left_a.arr f" shows "Left_a.arr (Left_a.cod f)"
extends_subst_cong_lit: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a) Term.term option"   and L :: "('b, 'a) Term.term literal" assumes "extends_subst \<sigma> \<tau>"   and "vars_lit L \<subseteq> dom \<sigma>" shows "L \<cdot>lit subst_of_map Var \<sigma> = L \<cdot>lit subst_of_map Var \<tau>"
M3: fixes s :: "'a"   and t :: "'a" assumes "s \<in> M"   and "t \<in> S \<union> N"   and "t \<notin> M" shows "\<Delta> t < \<Delta> s"
append_one_prefix: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le>p ys"   and "\<^bold>|xs\<^bold>| < \<^bold>|ys\<^bold>|" shows "xs \<cdot> [ys ! \<^bold>|xs\<^bold>|] \<le>p ys"
string_to_nat'_def: fixes s :: "char list" shows "string_to_nat' s \<equiv> the (string_to_nat s)"
change_eval_lt: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "real"   and y :: "real" assumes "aEvalUni (LessUni (a, b, c)) x \<noteq> aEvalUni (LessUni (a, b, c)) y \<and> x < y" shows "\<exists>w\<ge>x. w \<le> y \<and> a * w\<^sup>2 + b * w + c = 0"
UP_mapsto: shows "UP \<in> UNIV \<rightarrow> Univ"
globally_consistent_var_def: fixes A :: "'Var \<Rightarrow> ('Val \<times> 'Val) option"   and mds :: "Mode \<Rightarrow> 'Var set"   and x :: "'Var" shows "globally_consistent_var A mds x \<equiv> case A x of None \<Rightarrow> True | Some (v, v') \<Rightarrow> x \<notin> mds AsmNoWrite \<and> (dma x = Low \<longrightarrow> v = v')"
Equiv_Exec_equiv_statesI': fixes equiv_states :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and localState1 :: "'f \<Rightarrow> 'd"   and c1 :: "'f"   and localState2 :: "'g \<Rightarrow> 'e"   and c2 :: "'g"   and m :: "'h"   and input_fun1 :: "'h \<Rightarrow> 'i message_af"   and input_fun2 :: "'h \<Rightarrow> 'j message_af"   and output_fun1 :: "'f \<Rightarrow> 'k message_af"   and output_fun2 :: "'g \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and trans_fun2 :: "'j message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and k1 :: "nat"   and k2 :: "nat" assumes "equiv_states (localState1 c1) (localState2 c2)"   and "Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2" shows "equiv_states (localState1 (f_Exec_Comp trans_fun1 NoMsg\<^bsup>k1 - Suc 0\<^esup> (trans_fun1 (input_fun1 m) c1))) (localState2 (f_Exec_Comp trans_fun2 NoMsg\<^bsup>k2 - Suc 0\<^esup> (trans_fun2 (input_fun2 m) c2)))"
not_moebius_conjugate: shows "\<not> is_moebius conjugate"
ring_simprules_19: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = \<ominus>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
eq_\<rho>: fixes U :: "'a set set"   and V :: "'a set set"   and W :: "'a set set"   and s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and t :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and W' :: "'a set set" assumes "is_zariski_open U"   and "is_zariski_open V"   and "is_zariski_open W"   and "W \<subseteq> U \<inter> V"   and "s \<in> \<O> U"   and "t \<in> \<O> V"   and "sheaf_spec_morphisms U W s = sheaf_spec_morphisms V W t"   and "is_zariski_open W'"   and "W' \<subseteq> W" shows "sheaf_spec_morphisms U W' s = sheaf_spec_morphisms V W' t"
Re_Im_le_cmod: fixes w :: "complex"   and \<phi> :: "real" shows "Im w * sin \<phi> + Re w * cos \<phi> \<le> cmod w"
simps_3: fixes x2 :: "'f" shows "\<lfloor>x2\<rfloor> \<noteq> None"
assms_2: shows "callee_invariant_on right Ir \<I>r"
wtPB_def: fixes \<Phi> :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list set" shows "GE.wtPB \<Phi> \<equiv> Ball \<Phi> GE.wtC"
peek_and_ign_res: fixes P :: "'b assn"   and p :: "state \<Rightarrow> bool" shows "(P \<and>. p)\<down> = (P\<down> \<and>. p)"
f'_def: shows "f' = (\<lambda>x. s * x powr (s - 1))"