hcju/isabelleps huggingface dataset:
rupture_surj_hom_2: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "ring_hom_ring (K [X]\<^bsub>K [X]\<^esub>) (Rupt\<^bsub>K [X]\<^esub> K p) (UP.rupture_surj K p)"
find_map_filter_Some: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'a \<Rightarrow> bool"   and as :: "'b list"   and b :: "'a" assumes "find_map_filter f p as = Some b" shows "p b \<and> b \<in> f ` set as"
scc_disc: fixes y :: "'v" assumes "y \<in> scc"   and "y \<in> dom (discovered s)"   and "y \<noteq> v" shows "\<delta> s v < \<delta> s y"
degree_mult_eq: fixes p :: "'a Polynomial.poly"   and q :: "'a Polynomial.poly" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "Polynomial.degree (p * q) = Polynomial.degree p + Polynomial.degree q"
fresh_subst_E2: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y"   and "fresh zs z (X #[Y / y]_ys)" shows "fresh ys y X \<or> fresh zs z Y"
gen_llength_code_1: fixes n :: "nat" shows "gen_llength n LNil = enat n"
coeff_inverse: fixes x :: "'a poly" shows "Abs_poly (poly.coeff x) = x"
simps: fixes g :: "'g"   and v :: "'var"   and n :: "'node"   and phis :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping" shows "readVariableRecursive g v n phis = (if v \<in> defs g n then ((v, n, SimpleDef), phis) else case predecessors g n of [] \<Rightarrow> ((v, n, PhiDef), Mapping.update (n, v) [] phis) | [m] \<Rightarrow> readVariableRecursive g v m phis | m # ab # lista \<Rightarrow> case Mapping.lookup phis (n, v) of None \<Rightarrow> let phis = Mapping.update (n, v) [] phis; (phis, args) = readArgs g v n phis (m # ab # lista) in ((v, n, PhiDef), Mapping.update (n, v) args phis) | Some x \<Rightarrow> ((v, n, PhiDef), phis))"
field_le_mult_one_interval: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "\<And>z. \<lbrakk>less zero z; less z one\<rbrakk> \<Longrightarrow> less_eq (times z x) y" shows "less_eq x y"
measure_Union_le: fixes F :: "'a set set"   and M :: "'a measure" assumes "finite F"   and "\<And>S. S \<in> F \<Longrightarrow> S \<in> sets M" shows "Sigma_Algebra.measure M (\<Union> F) \<le> sum (Sigma_Algebra.measure M) F"
prems_1: shows "pred_stream (\<lambda>s. \<phi> (reps (abss s)) \<longrightarrow> \<phi> s) x"
cases: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'a list"   and a3 :: "'a" assumes "rtrancl_path r a1 a2 a3" obtains x :: "'a" where "a1 = x__" and "a2 = []" and "a3 = x__"   | x :: "'a" and y :: "'a" and ys :: "'a list" and z :: "'a" where "a1 = x" and "a2 = y # ys" and "a3 = z" and "r x y" and "rtrancl_path r y ys z"
m_closed: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> carrier (R Quot I)"   and "y \<in> carrier (R Quot I)" shows "x \<oplus>\<^bsub>R Quot I\<^esub> y \<in> carrier (R Quot I)"
oPrec_oSuc: fixes x :: "ordinal" shows "oPrec (oSuc x) = x"
f_simps_1: fixes x :: "'a"   and i :: "nat" assumes "x \<in> W i" shows "f i x = H (apply_chart (\<psi> i) x)"
Rep: fixes x :: "'a" shows "Rep x \<in> {a. sort_of a \<in> range sort_fun}"
inj': fixes x :: "'a star"   and y :: "'b star"   and z :: "'a star"   and w :: "'b star" assumes "(*f2* f) x y = (*f2* f) z w" shows "x = z \<and> y = w"
sba2_associative: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = x \<squnion> y \<squnion> z"
addc_density_measurable: fixes f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> 'a"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and t :: "pdf_type"   and e' :: "expr" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (state_measure V' \<Gamma> \<Otimes>\<^sub>M stock_measure t)"   and "t = REAL \<or> t = INTEG"   and "\<Gamma> \<turnstile> e' : t"   and "randomfree e'"   and "free_vars e' \<subseteq> V'" shows "(\<lambda>(\<rho>, x). f \<rho> (op_sem Add <|x, expr_sem_rf \<rho> (Minus $$ e')|>)) \<in> borel_measurable (state_measure V' \<Gamma> \<Otimes>\<^sub>M stock_measure t)"
a: shows "a \<otimes> h [^] e = \<^bold>g [^] z"
Top_Left_Zero: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel"   and utp_unit :: "'a hrel"   and P :: "'a hrel" assumes "utp_theory_unital_zerol hcond utp_unit"   and "P is hcond" shows "\<top>\<^bsub>utp_order hcond\<^esub> ;; P = \<top>\<^bsub>utp_order hcond\<^esub>"
case_cong: fixes "term" :: "('b, 'c) Term.term"   and term' :: "('b, 'c) Term.term"   and f1 :: "'c \<Rightarrow> 'd"   and g1 :: "'c \<Rightarrow> 'd"   and f2 :: "'b \<Rightarrow> ('b, 'c) Term.term list \<Rightarrow> 'd"   and g2 :: "'b \<Rightarrow> ('b, 'c) Term.term list \<Rightarrow> 'd" assumes "term = term'"   and "\<And>x1. term' = Var x1 \<Longrightarrow> f1 x1 = g1 x1"   and "\<And>x21 x22. term' = Fun x21 x22 \<Longrightarrow> f2 x21 x22 = g2 x21 x22" shows "(case term of Var x1 \<Rightarrow> f1 x1 | Fun x21 x22 \<Rightarrow> f2 x21 x22) = (case term' of Var x1 \<Rightarrow> g1 x1 | Fun x21 x22 \<Rightarrow> g2 x21 x22)"
cfg_wgreibach_axioms: shows "cfg_wgreibach \<lparr>init = (), prod = \<lambda>_. {|[], [\<aa>, S, \<bb>, S]|}\<rparr>"
of_int_ceiling_cancel: fixes x :: "'a" shows "(of_int \<lceil>x\<rceil> = x) = (\<exists>n. x = of_int n)"
ev\<^sub>B_eq': fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and P :: "'BExp set"   and e :: "'BExp"   and t :: "'BExp set" assumes "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2"   and "pred P mem\<^sub>1"   and "\<Gamma> \<turnstile>\<^sub>b e \<in> t "   and "P \<turnstile> t" shows "ev\<^sub>B mem\<^sub>1 e = ev\<^sub>B mem\<^sub>2 e"
Collect_enabled_S: fixes K :: "'c \<Rightarrow> 'c pmf"   and x :: "'c" shows "Collect (MC_syntax.enabled K x) \<in> sets (stream_space (count_space UNIV))"
substClosed_def: fixes Rel :: "(pi \<times> pi) set" shows "substClosed Rel \<equiv> {(P, Q) |P Q. \<forall>\<sigma>. (P[<\<sigma>>], Q[<\<sigma>>]) \<in> Rel}"
Diff_disjoint: fixes A :: "'d set"   and B :: "'d set" shows "A \<inter> (B - A) = {}"
abs_ereal_pos: fixes x :: "ereal" shows "0 \<le> \<bar>x\<bar>"
lex_pm_strict_alt: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "lex_pm_strict s t = (\<not> lex_pm t s)"
subst_not_in_dom: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and x :: "'d"   and \<sigma> :: "'d \<Rightarrow> 'e option"   and mem :: "'d \<Rightarrow> 'e" assumes "sifum_security stop eval some_val some_val'"   and "x \<notin> dom \<sigma>" shows "subst \<sigma> mem x = mem x"
inv_closed: fixes x :: "'c" assumes "x \<in> carrier S" shows "\<ominus>\<^bsub>S\<^esub> x \<in> carrier S"
order: fixes p :: "'b poly"   and a :: "'b" assumes "p \<noteq> 0" shows "[:- a, 1::'b:] ^ order a p dvd p \<and> \<not> [:- a, 1::'b:] ^ Suc (order a p) dvd p"
convex_hull_eq_real_cbox: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "convex hull {x, y} = cbox x y"
less_asym: fixes x :: "'a \<times> ('a, 'b) operation"   and y :: "'a \<times> ('a, 'b) operation"   and P :: "bool" assumes "hb x y"   and "\<not> P \<Longrightarrow> hb y x" shows "P"
\<ddagger>: shows "inv_into H (ordermap H VWF) ` elts \<beta> \<subseteq> H"
\<b>_def: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and i :: "nat" shows "\<b> ps i = (LEAST d. \<a> ps \<le> d \<and> (\<forall>(h, U)\<in>set ps. i \<le> card U \<longrightarrow> poly_deg h < d))"
word_clr_le: fixes w :: "'a word"   and n :: "nat" shows "set_bit_class.set_bit w n False \<le> w"
number_partition_of_2: fixes A :: "'c set"   and B :: "'d set"   and F :: "('c \<Rightarrow> 'd) set" assumes "finite A"   and "finite B"   and "F \<in> (A \<rightarrow>\<^sub>E B) // domain_and_range_permutation A B" shows "size (number_partition_of A B F) \<le> card B"
MGT_def: fixes c :: "com" shows "{=}.c.{->} = {(=)}. c .{evalc c}"
to_Arity_cases: fixes x :: "Arity" obtains y :: "nat" where "x = to_Arity y" and "y \<in> UNIV"
hext_new: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a :: "nat"   and x :: "char list \<times> (char list \<times> char list \<Rightarrow> val option)" assumes "h a = None" shows "h \<unlhd> h(a \<mapsto> x)"
A_mc_k: shows "A $ mc $ mod_type_class.from_nat k \<noteq> (0::'a)"
subclsR_subcls1: fixes C :: "char list"   and D :: "char list"   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" assumes "P \<turnstile> C \<prec>\<^sub>R D" shows "P \<turnstile> C \<prec>\<^sup>1 D"
assms_1: shows "nrelated.P_env \<Gamma> env nenv"
nop_onto_vdomain: shows "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"
\<psi>\<^sub>2_to_\<psi>\<^sub>3: shows "(H \<Otimes> Quantum.Id 1) * Tensor.mat_of_cols_list 4 (map (map complex_of_real) [[real (1 - f 0) / 2 - real (f 0) / 2, real (f 0) / 2 - real (1 - f 0) / 2, real (1 - f 1) / 2 - real (f 1) / 2, real (f 1) / 2 - real (1 - f 1) / 2]]) = Tensor.mat_of_cols_list 4 (map (map complex_of_real) [[real (1 - f 0) / (2 * sqrt 2) - real (f 0) / (2 * sqrt 2) + real (1 - f 1) / (2 * sqrt 2) - real (f 1) / (2 * sqrt 2), real (f 0) / (2 * sqrt 2) - real (1 - f 0) / (2 * sqrt 2) + (real (f 1) / (2 * sqrt 2) - real (1 - f 1) / (2 * sqrt 2)), real (1 - f 0) / (2 * sqrt 2) - real (f 0) / (2 * sqrt 2) - real (1 - f 1) / (2 * sqrt 2) + real (f 1) / (2 * sqrt 2), real (f 0) / (2 * sqrt 2) - real (1 - f 0) / (2 * sqrt 2) - real (f 1) / (2 * sqrt 2) + real (1 - f 1) / (2 * sqrt 2)]])"
induced_function_birkhoff_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real"   and n :: "nat"   and x :: "'a" assumes "A \<in> sets MA" shows "A.birkhoff_sum f (qmpt.birkhoff_sum (A.induced_map A) (A.return_time_function A) n x) x = qmpt.birkhoff_sum (A.induced_map A) (A.induced_function A f) n x"
final_simulation1: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and final1 :: "'g \<Rightarrow> bool"   and final2 :: "'i \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'i"   and tls1 :: "'h list"   and s1' :: "'g" assumes "delay_bisimulation_diverge_final trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2 final1 final2"   and "bisim s1 s2"   and "\<tau>trsys.\<tau>rtrancl3p trsys1 \<tau>move1 s1 tls1 s1'"   and "final1 s1'" shows "\<exists>s2' tls2. \<tau>trsys.\<tau>rtrancl3p trsys2 \<tau>move2 s2 tls2 s2' \<and> bisim s1' s2' \<and> final2 s2' \<and> list_all2 tlsim tls1 tls2"
24647: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - y \<squnion> - (- x \<sqinter> - y) = top"
col_rw: shows "cols (mat_of_cols n (cols A)) = cols A"
disjoint_eq_subset_Compl: fixes A :: "'b set"   and B :: "'b set" shows "(A \<inter> B = {}) = order.greater_eq (- B) A"
flow_of_cf_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'a"   and e :: "nat \<times> nat" shows "flow_of_cf cf e \<equiv> if e \<in> E then c e - cf e else (0::'a)"
Rep_quantale_homset_cases: fixes y :: "'c \<Rightarrow> 'd" assumes "y \<in> quantale_homset" obtains x :: "('c, 'd) quantale_homset" where "y = Rep_quantale_homset x"
is_volatile_Write\<^sub>s\<^sub>b_simps_3: fixes p :: "'b"   and p' :: "'b"   and "is" :: "instr list" shows "is_volatile_Write\<^sub>s\<^sub>b (Prog\<^sub>s\<^sub>b p p' is) = False"
I2a: fixes s :: "state"   and s' :: "state" assumes "HNext s s'"   and "HInv1 s" shows "HInv1 s'"
switch_vchop2: fixes v' :: "view"   and v1' :: "view"   and v2' :: "view"   and v :: "view"   and c :: "cars" shows "v'=v1'--v2' \<and> v = c > v' \<longrightarrow> (\<exists>v1 v2. v1 = c > v1' \<and> v2 = c > v2' \<and> v=v1--v2)"
norm_dist_dot: fixes x :: "'b"   and y :: "'b" shows "(norm_dist x y)\<^sup>2 = (x - y) \<bullet> (x - y)"
cobounded1: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" shows "less_eq_prod (inf_prod a b) a"
cong_pow_1_nat: fixes a :: "nat"   and b :: "nat"   and x :: "nat" assumes "[a = 1] (mod b)" shows "[a ^ x = 1] (mod b)"
simps_3: fixes x :: "'c"   and y :: "'c" shows "Num_arc_delta x * Num_arc_delta y = Num_arc_delta (x + y)"
hcomplex_i_mult_left: fixes z :: "complex star" shows "iii * (iii * z) = - z"
tab_simps_4: shows "local.dom \<rho>\<sigma>.tab = r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1"
cases: fixes x :: "'b list" obtains "x = []"   | xa :: "'b" where "x = [xa__]"   | xa :: "'b" and y :: "'b" and xs :: "'b list" where "x = xa # y # xs"
uminus_Inf: fixes A :: "'b set" shows "- Inf A = Sup (uminus ` A)"
Inf_trans_def: fixes S :: "(('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real) set" shows "Inf_trans S = (\<lambda>P. Inf_exp {t P |t. t \<in> S})"
tensor_codomain_wall_additivity: fixes xs :: "wall"   and ys :: "wall" shows "codomain_wall (xs \<otimes> ys) = codomain_wall xs + codomain_wall ys"
order_refl: fixes x :: "'c \<times> 'd" shows "less_eq_prod x x"
eval_simps_1: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide b"   and "ide c" shows "arr (eval b c)"
intro: fixes bisim :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and trsys1 :: "'a \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> bool"   and \<tau>move1 :: "'a \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> bool"   and trsys2 :: "'b \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and \<tau>move2 :: "'b \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>s1 s2 s1'. \<lbrakk>bisim s1 s2; \<tau>trsys.silent_move trsys1 \<tau>move1 s1 s1'\<rbrakk> \<Longrightarrow> \<exists>s2'. (\<tau>trsys.silent_move trsys2 \<tau>move2)\<^sup>*\<^sup>* s2 s2' \<and> bisim s1' s2'"   and "\<And>s1 s2 s2'. \<lbrakk>bisim s1 s2; \<tau>trsys.silent_move trsys2 \<tau>move2 s2 s2'\<rbrakk> \<Longrightarrow> \<exists>s1'. (\<tau>trsys.silent_move trsys1 \<tau>move1)\<^sup>*\<^sup>* s1 s1' \<and> bisim s1' s2'"   and "\<And>s1 s2. bisim s1 s2 \<Longrightarrow> \<tau>trsys.\<tau>diverge trsys1 \<tau>move1 s1 = \<tau>trsys.\<tau>diverge trsys2 \<tau>move2 s2" shows "delay_bisimulation_diverge_axioms trsys1 trsys2 bisim \<tau>move1 \<tau>move2"
P_of_state_Liminf: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" shows "P_of_state (Liminf_state Sts) = Liminf_llist (lmap P_of_state Sts)"
pdevs_of_ivl_def: fixes l :: "'a"   and u :: "'a" shows "pdevs_of_ivl l u = scaleR_pdevs (1 / 2) (inner_scaleR_pdevs (u - l) One_pdevs)"
encode_operator_precondition_def: fixes \<Pi> :: "'a strips_problem"   and t :: "nat"   and op :: "'a strips_operator" shows "encode_operator_precondition \<Pi> t op \<equiv> let vs = \<Pi>\<^sub>\<V>; ops = \<Pi>\<^sub>\<O> in \<^bold>\<And>map (\<lambda>v. \<^bold>\<not> (Atom (Operator t (index ops op))) \<^bold>\<or> Atom (State t (index vs v))) (strips_operator.precondition_of op)"
n_k_in_set: fixes k :: "nat"   and n :: "nat" assumes "0 < k"   and "0 < n"   and "fact k = (\<Prod>i<n. 2 ^ n - 2 ^ i)" shows "(n, k) \<in> {(1, 1), (2, 3)}"
greaterThanAtMost_eq_atLeastAtMost_diff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.order less_eq less" shows "ord.greaterThanAtMost less_eq less a b = ord.atLeastAtMost less_eq a b - {a}"
elims_2: fixes x :: "'fsym trm" assumes "Ik.wt x" obtains xa :: "Preliminaries.var" where "x = Var xa"   | f :: "'fsym" and Tl :: "'fsym trm list" where "x = Fn f Tl" and "wtFsym f \<and> list_all Ik.wt Tl \<and> arOf f = map Ik.tpOf Tl"
induct: fixes P :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree \<Rightarrow> 'c \<times> 'd \<Rightarrow> (('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree \<Rightarrow> bool"   and a0 :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a1 :: "'c \<times> 'd"   and a2 :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "\<And>t1 x uu_ t2 y t3. P t1 x (R uu_ t2 y t3)"   and "\<And>t1 x uv_ t2 y. P (B uv_ t1 x t2) y \<langle>\<rangle>"   and "\<And>t1 x uv_ t2 y v vc vf vb. P (B uv_ t1 x t2) y (B vf v vc vb)"   and "\<And>t1 x uw_ t2 y ux_ t3 z. P (R uw_ t1 x (B ux_ t2 y t3)) z \<langle>\<rangle>"   and "\<And>t1 x uw_ t2 y ux_ t3 z v vc vf vb. P (R uw_ t1 x (B ux_ t2 y t3)) z (B vf v vc vb)"   and "\<And>x. P \<langle>\<rangle> x \<langle>\<rangle>"   and "\<And>v vc vf x. P (R vf v vc \<langle>\<rangle>) x \<langle>\<rangle>"   and "\<And>v vc vf va vg vj ve x. P (R vf v vc (R vj va vg ve)) x \<langle>\<rangle>"   and "\<And>x v vc vf vb. P \<langle>\<rangle> x (B vf v vc vb)"   and "\<And>va vg vj x v vc vf vb. P (R vj va vg \<langle>\<rangle>) x (B vf v vc vb)"   and "\<And>va vg vj vd vk vn vi x v vc vf vb. P (R vj va vg (R vn vd vk vi)) x (B vf v vc vb)" shows "P a0 a1 a2"
assms_2: shows "f \<in> S \<rightarrow>\<^sub>E T - {a}"
conseq: fixes \<Gamma> :: "'f \<Rightarrow> ('g, 'f, 'h) com option"   and \<Theta> :: "('g set \<times> 'f \<times> 'g set \<times> 'g set) set"   and F :: "'h set"   and P' :: "'e \<Rightarrow> 'g set"   and c :: "('g, 'f, 'h) com"   and Q' :: "'e \<Rightarrow> 'g set"   and A' :: "'e \<Rightarrow> 'g set"   and P :: "'g set"   and Q :: "'g set"   and A :: "'g set" assumes "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' Z) c (Q' Z),(A' Z)"   and "\<forall>s. s \<in> P \<longrightarrow> (\<exists>Z. s \<in> P' Z \<and> Q' Z \<subseteq> Q \<and> A' Z \<subseteq> A)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
subring_def: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme" shows "subring H R \<equiv> subgroup H (add_monoid R) \<and> submonoid H R"
extremum_uniqueI: fixes a :: "'c" assumes "a \<le> (0::'c)" shows "a = (0::'c)"
nonzero_octo_of_real_divide: fixes y :: "real"   and x :: "real" assumes "y \<noteq> 0" shows "octo_of_real (x / y) = octo_of_real x / octo_of_real y"
comp_fun_commute_on_axioms: shows "comp_fun_commute_on f A"
induct: fixes P :: "typ \<Rightarrow> typ \<Rightarrow> (variable \<times> String.literal set \<Rightarrow> typ option) \<Rightarrow> bool"   and a0 :: "typ"   and a1 :: "typ"   and a2 :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "\<And>v S T subs. P (Tv v S) T subs"   and "\<And>a Ts b Us subs. (\<And>x xa y xb aa. \<lbrakk>a = b \<and> length Ts = length Us; x \<in> set (zip Ts Us); (xa, y) = x; xb = Some aa\<rbrakk> \<Longrightarrow> P xa y aa) \<Longrightarrow> P (Ty a Ts) (Ty b Us) subs"   and "\<And>v va vb vc subs. P (Ty v va) (Tv vb vc) subs" shows "P a0 a1 a2"
intro: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and top :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>x. inf x (uminus x) = bot"   and "\<And>x. sup x (uminus x) = top"   and "\<And>x y. minus x y = inf x (uminus y)" shows "class.boolean_algebra_axioms minus uminus inf sup bot top"
mem_typings: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and es :: "exp list"   and Cs :: "nat list"   and ei :: "exp" assumes "CT;\<Gamma> \<turnstile>+ es : Cs"   and "ei \<in> set es" shows "\<exists>Ci. CT;\<Gamma> \<turnstile> ei : Ci"
f1: shows "low_equal s1 s2 \<and> t1 = snd (fst (add_instr_sub1 (fst instr) (if fst instr = arith_type ADD \<or> fst instr = arith_type ADDcc then user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1))) + get_operand2 (snd instr) s1 else user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1))) + get_operand2 (snd instr) s1 + ucast (get_icc_C (cpu_reg_val PSR (snd (fst (get_curr_win () s1)))))) (user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1)))) (get_operand2 (snd instr) s1) (snd (fst (write_reg (if get_operand_w5 (snd instr ! 3) \<noteq> 0 then if fst instr = arith_type ADD \<or> fst instr = arith_type ADDcc then user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1))) + get_operand2 (snd instr) s1 else user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1))) + get_operand2 (snd instr) s1 + ucast (get_icc_C (cpu_reg_val PSR (snd (fst (get_curr_win () s1))))) else user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s1)))) (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s1)))))))) \<and> t2 = snd (fst (add_instr_sub1 (fst instr) (if fst instr = arith_type ADD \<or> fst instr = arith_type ADDcc then user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2))) + get_operand2 (snd instr) s2 else user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2))) + get_operand2 (snd instr) s2 + ucast (get_icc_C (cpu_reg_val PSR (snd (fst (get_curr_win () s2)))))) (user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2)))) (get_operand2 (snd instr) s2) (snd (fst (write_reg (if get_operand_w5 (snd instr ! 3) \<noteq> 0 then if fst instr = arith_type ADD \<or> fst instr = arith_type ADDcc then user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2))) + get_operand2 (snd instr) s2 else user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2))) + get_operand2 (snd instr) s2 + ucast (get_icc_C (cpu_reg_val PSR (snd (fst (get_curr_win () s2))))) else user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s2)))) (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s2))))))))"
simps_8: fixes v :: "'d" shows "generalize_instr (ILoad v) = ILoad v"
axioms_2: fixes G :: "('c, 'd) graph" assumes "tree G" shows "connected_graph G"
tr: shows "set tr \<subseteq> E <+> IA \<times> UNIV <+> IU \<times> UNIV"
Lim_transform_eventually: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'd" assumes "(f \<longlongrightarrow> l) F"   and "\<forall>\<^sub>F x in F. f x = g x" shows "(g \<longlongrightarrow> l) F"
to_fun_def: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int" shows "(\<bullet>) f \<equiv> to_function Zp f"
ChiRel_def: fixes A :: "('d, 'e, 'f) hierauto" shows "ChiRel A \<equiv> {(S, S'). S \<in> HAStates A \<and> S' \<in> HAStates A \<and> S' \<in> Chi A S}"