hcju/isabelleps huggingface dataset:
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F' \<Phi>\<^sub>F'"
exists_inj_ge_index: fixes S :: "nat set"   and n :: "nat"   and k :: "nat" assumes "S \<subseteq> {0..<n}"   and "card S = k" shows "\<exists>f. inj_on f {0..<k} \<and> f ` {0..<k} = S \<and> (\<forall>i\<in>{0..<k}. i \<le> f i)"
le_less_trans: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less y z" shows "less x z"
costBarrierDecreases: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and iA :: "nat \<times> formula"   and n :: "nat" assumes "branch subs gamma f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "contains f iA n"   and "\<not> considers f iA n" shows "costBarrier iA (f (Suc n)) < costBarrier iA (f n)"
subst_cls_add_mset: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and L :: "'b literal"   and C :: "'b literal multiset"   and \<sigma> :: "'c" shows "substitution_ops.subst_cls subst_atm (add_mset L C) \<sigma> = add_mset (substitution_ops.subst_lit subst_atm L \<sigma>) (substitution_ops.subst_cls subst_atm C \<sigma>)"
st_vec_transition_vec_of_st_mat: fixes A :: "'a st_mat"   and a :: "'a"   and i :: "'a" shows "st_vec (transition_vec_of_st_mat A a) $h i = st_mat A $h i $h a"
simps: fixes \<epsilon> :: "'a"   and n :: "'a"   and x :: "'a" assumes "sqrt_approximation \<epsilon> n" shows "sqrt_approximation.sqrt_approx_main \<epsilon> n x = (if (0::'a) < x then if x * x - n < \<epsilon> then x else sqrt_approximation.sqrt_approx_main \<epsilon> n ((n / x + x) / (2::'a)) else (0::'a))"
conjugate_dist_add: fixes conjugate :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "conjugate (plus a b) = plus (conjugate a) (conjugate b)"
eclose_hinsert: fixes a :: "hf"   and b :: "hf" shows "eclose (b \<triangleleft> a) = eclose b \<squnion> (eclose a \<triangleleft> a)"
vimage_diff: fixes fa :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set"   and f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and C :: "'e set" assumes "func fa S T"   and "f \<in> A \<rightarrow> B" shows "A - f -` (B - C) = A \<inter> f -` C"
swap_msgs_Send_Recv: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and d :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event"   and e :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration"   and e' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSend ev"   and "isRecv ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
mkCone_def: fixes p0 :: "'c"   and p1 :: "'c" shows "mkCone p0 p1 \<equiv> \<lambda>j. if j = J.MkIde False then p0 else if j = J.MkIde True then p1 else C.null"
openI: fixes m :: "'a set" assumes "m \<in> T" shows "m open"
higher_differentiable_on_real_Suc: fixes S :: "real set"   and f :: "real \<Rightarrow> 'b"   and n :: "nat" assumes "open S" shows "higher_differentiable_on S f (Suc n) = ((\<forall>x\<in>S. f differentiable at x) \<and> higher_differentiable_on S (\<lambda>x. frechet_derivative f (at x) 1) n)"
set_subset_Univ: fixes b :: "'d"   and a :: "'d" assumes "D.ide b"   and "D.ide a" shows "HomD.set (b, a) \<subseteq> S.Univ"
zero_spaceN_iff': fixes x :: "'b"   and N :: "'b quasinorm" shows "(x \<in> zero_space\<^sub>N N) = (x \<in> space\<^sub>N N \<and> Norm N x = 0)"
sum_list_nonneg_eq_0_iff: fixes xs :: "'a list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> (0::'a) \<le> x" shows "(sum_list xs = (0::'a)) = (\<forall>x\<in>set xs. x = (0::'a))"
open_subopen: fixes S :: "'a set" shows "open S = (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
disjoint: shows "{p. p partitions Suc m \<and> sum p {..Suc m} = Suc k \<and> p 1 \<noteq> 0} \<inter> {p. p partitions Suc m \<and> sum p {..Suc m} = Suc k \<and> p 1 = 0} = {}"
map_cong: fixes x :: "'a pvar"   and ya :: "'a pvar"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x = ya"   and "\<And>z. z \<in> set_pvar ya \<Longrightarrow> f z = g z" shows "map_pvar f x = map_pvar g ya"
cancel_suffix: fixes a :: "'a list"   and c :: "'a list"   and b :: "'a list"   and d :: "'a list" assumes "a @ c = b @ d"   and "length c \<le> length d" shows "a = b @ take (length d - length c) d"
infinite: fixes A :: "'e set" assumes "infinite A" shows "card A = 0"
elims_3: fixes x :: "nat \<times> nat list"   and xa :: "nat \<times> cell list \<times> cell list"   and xb :: "cell list" assumes "\<not> inv_check_left_moving x xa xb" obtains as :: "nat" and lm :: "nat list" and s :: "nat" and l :: "cell list" and r :: "cell list" and ires :: "cell list" where "x = (as, lm)" and "xa = (s, l, r)" and "xb = ires" and "\<not> (inv_check_left_moving_on_leftmost (as, lm) (s, l, r) ires \<or> inv_check_left_moving_in_middle (as, lm) (s, l, r) ires)"
monoseq_def: fixes X :: "nat \<Rightarrow> 'a" shows "monoseq X = ((\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n) \<or> (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m))"
xor_inverse: fixes x :: "'e" assumes "x \<in> carrier L" shows "x \<oplus> x = \<bottom>"
snd_divmod_nat: fixes m :: "nat"   and n :: "nat" shows "snd (divmod_nat m n) = m mod n"
carrier_polynomial: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list" assumes "ring R"   and "subring K R"   and "polynomial\<^bsub>R\<^esub> K p" shows "polynomial\<^bsub>R\<^esub> (carrier R) p"
end_smult_distrib_left: fixes S :: "'v \<Rightarrow> 'v"   and T :: "'v \<Rightarrow> 'v"   and a :: "'f" assumes "range S \<subseteq> V"   and "range T \<subseteq> V" shows "a \<cdot>\<cdot> (S + T) = a \<cdot>\<cdot> S + a \<cdot>\<cdot> T"
walk_2: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Graph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w" shows "Graph.walk G [v, w]"
cases: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and a :: "'d stream" assumes "alw \<phi> a" obtains xs :: "'d stream" where "a = xs" and "\<phi> xs" and "alw \<phi> (stl xs)"
prems_2: shows "list.set \<beta>s \<subseteq> ON"
mm_equiv_elim: fixes c\<^sub>1 :: "'Com"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "'Com"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" obtains \<R> :: "((('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" where "strong_low_bisim_mm \<R>" and "(\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>) \<in> \<R>"
RPred1_subseteq_R1: shows "\<R>\<^sub>0\<^sub>1 \<subseteq> \<R>"
RBTi1_axioms: shows "RBTi1 TYPE('b) (\<lambda>x xa. int xa \<le> \<lceil>c * log 2 (real x + 1)\<rceil>)"
infsetsum'_converges_Diff: fixes f :: "'c \<Rightarrow> 'd"   and B :: "'c set"   and A :: "'c set" assumes "infsetsum'_converges f B"   and "infsetsum'_converges f A"   and "A \<subseteq> B" shows "infsetsum'_converges f (B - A)"
fin: fixes j :: "nat" assumes "i \<le> j" shows "emeasure M (A j) < \<infinity>"
length_ci_dec: fixes ly :: "nat list"   and ss :: "nat"   and n :: "nat"   and e :: "nat" shows "length (ci ly ss (Dec n e)) = 4 * n + 32"
comp_suffix: fixes f :: "'c \<Rightarrow> 'b"   and \<sigma> :: "nat \<Rightarrow> 'c"   and i :: "nat" shows "f \<circ> \<sigma> |\<^sub>s i = (f \<circ> \<sigma>) |\<^sub>s i"
fwi_middle_id: fixes j' :: "nat"   and j :: "nat"   and i' :: "nat"   and i :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat" assumes "j' < j"   and "i' \<le> i" shows "fwi m n k i' j' i j = m i j"
phidefNodes_def: fixes g :: "'g" shows "phidefNodes g = fold (\<lambda>(n, v). Mapping.update v n) (sorted_list_of_set (dom (phis g))) Mapping.empty"
list_order_trans: fixes op_list :: "('c \<times> 'c option) list"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "insert_ops op_list"   and "Insert_Spec.list_order op_list x y"   and "Insert_Spec.list_order op_list y z" shows "Insert_Spec.list_order op_list x z"
intro: fixes M :: "'b measure"   and F :: "nat \<Rightarrow> 'b measure" assumes "prob_space M"   and "disc_filtr_prob_space_axioms M F" shows "disc_filtr_prob_space M F"
tllist_ord_partial_function_definitions: shows "partial_function_definitions local.tllist_ord local.tSup"
flow_initial_time: fixes t0 :: "real"   and x0 :: "'a" assumes "t0 \<in> UNIV"   and "x0 \<in> X" shows "flow0 x0 (t0 - t0) = x0"
oseq_deliverTE: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>d\<^sub>a\<^sub>t\<^sub>a :: "'e \<Rightarrow> nat"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, {l}deliver(s\<^sub>d\<^sub>a\<^sub>t\<^sub>a) . p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i" obtains "a = deliver (s\<^sub>d\<^sub>a\<^sub>t\<^sub>a (\<sigma> i))" and "\<sigma>' i = \<sigma> i" and "q = p"
nth_least_lift_2: fixes n :: "nat"   and A :: "nat set" assumes "enat n < esize A" shows "nth_least (lift A) (Suc n) = Suc (nth_least A n)"
assms_1: shows "boa \<le> vimage2p rha rha (=)"
linear_primitive_irreducible: fixes f :: "'a poly" assumes "degree f = 1"   and "primitive f" shows "irreducible f"
obs_consistent_def: fixes R :: "('b \<times> 'c) set"   and pi :: "'d \<Rightarrow> 'e"   and Sa :: "('b, 'e) spec"   and Sc :: "('c, 'd) spec" shows "obs_consistent R pi Sa Sc \<equiv> \<forall>s t. (s, t) \<in> R \<longrightarrow> pi (obs Sc t) = obs Sa s"
rep_eq: fixes x :: "float"   and xa :: "float" shows "(x < xa) = (real_of_float x < real_of_float xa)"
simps_11: fixes i :: "int"   and a :: "Cooper.num" shows "zlfm (Dvd i a) = (if i = 0 then zlfm (Eq a) else let (c, r) = zsplit0 a in if c = 0 then Dvd \<bar>i\<bar> r else if 0 < c then Dvd \<bar>i\<bar> (CN 0 c r) else Dvd \<bar>i\<bar> (CN 0 (- c) (Neg r)))"
snd_not_one_fst: fixes a :: "('d, 'c) OrdSum" assumes "snd (Rep_OrdSum a) \<noteq> (1::'c)" shows "fst (Rep_OrdSum a) = (1::'d)"
lowner_le_minus: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat"   and D :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "D \<in> carrier_mat n n"   and "A \<le>\<^sub>L B"   and "C \<le>\<^sub>L D" shows "A - D \<le>\<^sub>L B - C"
flowD_outside: fixes g :: "'v option \<times> 'v option \<Rightarrow> ennreal"   and e :: "'v option \<times> 'v option" assumes "flow \<Psi> g"   and "e \<notin> \<^bold>E\<^bsub>\<Psi>\<^esub>" shows "g e = 0"
C_disj_lt: fixes i :: "nat"   and j :: "nat"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat" assumes "i < j" shows "C M2 M1 \<Omega> V m i \<inter> C M2 M1 \<Omega> V m j = {}"
comm_ring_axioms: shows "comm_ring {0} ring0.subtraction ring0.subtraction 0 0"
sel: fixes x :: "'c \<Rightarrow> 'd \<times> 'c" shows "run_state (State x) = x"
eNorm_def: fixes N :: "'c quasinorm"   and x :: "'c" shows "eNorm N x = snd (Rep_quasinorm N) x"
hyps: fixes B :: "'a multiset" assumes "A \<subset># B" shows "size A < size B"
Quotient_alt_def2: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and T :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> (\<forall>x y. R x y = (T x (Abs y) \<and> T y (Abs x))))"
meet_related2: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "dlat le"   and "le y x" shows "dlat.meet le x y = y"
DmdOr: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" shows "\<turnstile> (\<diamond>(F \<or> G)) = (\<diamond>F \<or> \<diamond>G)"
simps_3: fixes lx :: "'b tree"   and uu :: "'b"   and uv :: "'b tree" shows "cost Del_min [\<langle>lx, uu, uv\<rangle>] = T\<^sub>p\<^sub>a\<^sub>s\<^sub>s\<^sub>2 (pass\<^sub>1 lx) + T\<^sub>p\<^sub>a\<^sub>s\<^sub>s\<^sub>1 lx"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) ((\<otimes>\<^sub>D) \<I>\<^sub>D) D.map D.\<ll>"
qe_cooper\<^sub>1_def: fixes \<phi> :: "atom fm" shows "qe_cooper\<^sub>1 \<phi> = (let as = atoms\<^sub>0 \<phi>; d = zlcms (map divisor as); ls = lbounds as in or (Disj [0..d - 1] (\<lambda>n. subst n [] (inf\<^sub>- \<phi>))) (Disj ls (\<lambda>(i, ks). Disj [0..d - 1] (\<lambda>n. subst (i + n) (- ks) \<phi>))))"
ee_factorsD: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list"   and bs :: "'c list"   and a :: "'c"   and b :: "'c" assumes "comm_monoid_cancel G"   and "essentially_equal G as bs"   and "factors G as a"   and "factors G bs b"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "a \<sim>\<^bsub>G\<^esub> b"
that_3: fixes x :: "'a"   and y :: "'a"   and ys :: "'a list" assumes "xs = x # y # ys" shows "thesis"
FactGroup_iso: assumes "f ` carrier G = carrier H" shows "G Mod kernel G H f \<cong> H"
has_integral_bound: fixes B :: "real"   and f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and a :: "'a"   and b :: "'a" assumes "0 \<le> B"   and "(f has_integral i) (cbox a b)"   and "\<And>x. x \<in> cbox a b \<Longrightarrow> norm (f x) \<le> B" shows "norm i \<le> B * content (cbox a b)"
in_sub_in1: shows "inside (path_image c1 \<union> path_image c) \<subseteq> inside (path_image c1 \<union> path_image c2)"
diff: fixes x :: "'a"   and y :: "'a" shows "h (x - y) = h x - h y"
simps_17: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and I :: "\<I>"   and r :: "formula Regex.regex" shows "sat \<sigma> V v i (MatchF I r) = (\<exists>j\<ge>i. mem (\<tau> \<sigma> j - \<tau> \<sigma> i) I \<and> Regex.match (sat \<sigma> V v) r i j)"
P_def: shows "P = {q. prime q \<and> q - 1 dvd 2 * p}"
rtranclp_into_tranclp2: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "r a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
Pareto_strict_iff: fixes x :: "'b"   and y :: "'b" shows "x \<prec>[Pareto R] y = ((\<forall>i\<in>dom. R i x y) \<and> (\<exists>i\<in>dom. x \<prec>[R i] y))"
K_def: fixes x :: "'a"   and uu :: "'b" shows "K x uu = x"
positive_def: fixes A :: "complex mat" shows "positive A = (A \<in> carrier_mat (dim_col A) (dim_col A) \<and> (\<forall>v. dim_vec v = dim_col A \<longrightarrow> 0 \<le> inner_prod v (A *\<^sub>v v)))"
assms_1: shows "fBall M (\<lambda>x. x = y)"
ordLess_Well_order_simp: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "r <o r'" shows "Well_order r \<and> Well_order r'"
conj_mono: fixes c\<^sub>0 :: "'b::refinement_lattice"   and d\<^sub>0 :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d\<^sub>1 :: "'b::refinement_lattice" assumes "(c\<^sub>0::'b::refinement_lattice) \<sqsubseteq> (d\<^sub>0::'b::refinement_lattice)"   and "(c\<^sub>1::'b::refinement_lattice) \<sqsubseteq> (d\<^sub>1::'b::refinement_lattice)" shows "(c\<^sub>0::'b::refinement_lattice) \<iinter> (c\<^sub>1::'b::refinement_lattice) \<sqsubseteq> (d\<^sub>0::'b::refinement_lattice) \<iinter> (d\<^sub>1::'b::refinement_lattice)"
1: shows "\<forall>f. null \<star> f = null \<and> f \<star> null = null"
mtf2_effect: fixes q :: "'a"   and xs :: "'a list"   and i :: "nat"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs" shows "(index xs q < i \<and> i < length xs \<longrightarrow> index (mtf2 n q xs) (xs ! i) = index xs (xs ! i) \<and> index xs q < index (mtf2 n q xs) (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) < length xs) \<and> (index xs q = i \<longrightarrow> index (mtf2 n q xs) (xs ! i) = index xs q - n \<and> index (mtf2 n q xs) (xs ! i) = index xs q - n) \<and> (index xs q - n \<le> i \<and> i < index xs q \<longrightarrow> index (mtf2 n q xs) (xs ! i) = Suc (index xs (xs ! i)) \<and> index xs q - n < index (mtf2 n q xs) (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) \<le> index xs q) \<and> (i < index xs q - n \<longrightarrow> index (mtf2 n q xs) (xs ! i) = index xs (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) < index xs q - n)"
isomorphicE: fixes a :: "'a"   and a' :: "'a" assumes "B.isomorphic a a'" obtains f :: "'a" where "\<guillemotleft>f : a \<Rightarrow>\<^sub>B a'\<guillemotright> \<and> B.iso f"
ith_mem: fixes i :: "nat"   and es :: "'a list" assumes "i < length es" shows "es ! i \<in> set es"
aform_err_trunc_bound_eucl: fixes y :: "real"   and e :: "nat \<Rightarrow> real"   and X :: "real"   and xs :: "real pdevs"   and d :: "real"   and p :: "nat" assumes "y \<in> aform_err e ((X, xs), d)" shows "y \<in> aform_err e ((fst (trunc_bound_eucl p X), xs), snd (trunc_bound_eucl p X) + d)"
bij_2: shows "bij_betw f\<^sub>2 (uverts G\<^sub>2) (uverts G\<^sub>3)"
lappend_snocL1_conv_LCons2: fixes xs :: "'a llist"   and y :: "'a"   and ys :: "'a llist" shows "xs @\<^sub>l (y #\<^sub>l []\<^sub>l) @\<^sub>l ys = xs @\<^sub>l (y #\<^sub>l ys)"
rep_eq: fixes x :: "'a"   and xa :: "'a" shows "(\<langle>$\<rangle>) \<langle>x \<leftrightarrow> xa\<rangle> = Fun.swap x xa id"
mimic_single_step: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>1' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>1 \<approx> s\<^sub>1'"   and "s\<^sub>1 \<leadsto> s\<^sub>2" shows "\<exists>s\<^sub>2''. s\<^sub>2 \<approx> s\<^sub>2'' \<and> s\<^sub>1' \<leadsto> s\<^sub>2''"
1: shows "\<guillemotleft>local.inv \<a>[f, g, h] : f \<star> g \<star> h \<Rightarrow> (f \<star> g) \<star> h\<guillemotright>"
is_process4: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" assumes "is_process P" shows "\<forall>s X Y. (s, Y) \<notin> FAILURES P \<or> \<not> X \<subseteq> Y \<or> (s, X) \<in> FAILURES P"
a_inv_closed: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "\<ominus>\<^bsub>P\<^esub> x \<in> carrier P"
lit_atoms_cases: fixes xa :: "'a literal" shows "atoms_of_lit xa = (case xa of xda\<^sup>+ \<Rightarrow> xda | xea\<inverse> \<Rightarrow> xea)"
lset_ldrop_subset: fixes n :: "enat"   and xs :: "'c llist" shows "lset (ldrop n xs) \<subseteq> lset xs"
sorted_remove1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and a :: "'a" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sorted less_eq (remove1 a xs)"
q_gt_0: fixes q :: "nat" assumes "rivest q" shows "0 < q"
lhd_LCons: fixes x21 :: "'a"   and x22 :: "'a llist" shows "lhd (x21 #\<^sub>l x22) = x21"
A_def: shows "A = {q \<in> set (local.shift qs). m < card (snd q)}"
prime_int_not_dvd: fixes p :: "int"   and n :: "int" assumes "prime p"   and "n < p"   and "1 < n" shows "\<not> n dvd p"
s_def: shows "s = enn2real (integral\<^sup>N (count_space B) g)"
tendsto_mult: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and b :: "'b" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x * g x) \<longlongrightarrow> a * b) F"
leq_zero_imp: fixes r :: "('c \<times> 'c) set"   and x :: "'c" assumes "wo_rel r"   and "(x, zero r) \<in> r" shows "x = zero r"
distr: fixes a :: "real" shows "emeasure M {x \<in> space M. X x \<le> a} = emeasure lborel (A \<inter> {..a}) / ennreal r"
simps_3: fixes a :: "atom" shows "atoms\<^sub>0 (Atom a) = (if hd_coeff a \<noteq> 0 then [a] else [])"
bisim1AAssFail: fixes P :: "'a prog"   and h :: "'heap"   and a :: "'addr expr1"   and i :: "'addr expr1"   and e :: "'addr expr1"   and ad :: "'addr"   and xs :: "'addr val list"   and v' :: "'addr val"   and v :: "'addr val"   and v'' :: "'addr val" shows "P,a\<lfloor>i\<rceil> := e,h \<turnstile> (Throw ad, xs) \<leftrightarrow> ([v', v, v''], xs, length (compE2 a) + length (compE2 i) + length (compE2 e), \<lfloor>ad\<rfloor>)"
Collect_mono: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "Collect P \<subseteq> Collect Q"
simps_1: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "upd_locks l t [] = l"
pmult_Cons: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and h :: "'b"   and t :: "'b list"   and l2 :: "'b list" assumes "class.semiring_0 plus zero times" shows "semiring_0.pmult plus zero times (h # t) l2 = (if t = [] then semiring_0.cmult times h l2 else semiring_0.padd plus (semiring_0.cmult times h l2) (zero # semiring_0.pmult plus zero times t l2))"
domintros_5: fixes e :: "exp0"   and uz :: "t" assumes "is_value_dom e" shows "is_value_dom (Tannot e uz)"
dual\<^sub>s\<^sub>t_self_inverse: fixes S :: "('a, 'b) strand_step list" shows "dual\<^sub>s\<^sub>t (dual\<^sub>s\<^sub>t S) = S"
mat_tensorlist_mult_def: fixes A :: "real mat"   and Ts :: "real tensor vec"   and ds :: "nat list" shows "mat_tensorlist_mult A Ts ds = Matrix.vec (dim_row A) (\<lambda>j. tensor_from_lookup ds (\<lambda>is. (A *\<^sub>v map_vec (\<lambda>T. lookup T is) Ts) $ j))"
epred_llength: fixes xs :: "'b llist" shows "epred (llength xs) = llength (ltl xs)"
blocks_2: fixes be :: "nat" assumes "be \<notin> snd ` set ((i_begin, i_end) # bs)"   and "be \<notin> {l, k}"   and "be < n" shows "A $$ (be, k) = (0::'a)"
distinct_3: fixes x1 :: "('c, 'd) Term.term"   and x31 :: "poscheckvariant"   and x32 :: "('c, 'd) Term.term"   and x33 :: "('c, 'd) Term.term" shows "send\<langle>x1\<rangle>\<^sub>s\<^sub>t \<noteq> \<langle>x31: x32 \<doteq> x33\<rangle>\<^sub>s\<^sub>t"
fmlookup_drop_set: fixes A :: "'b set"   and m :: "('b, 'a) fmap"   and x :: "'b" shows "fmlookup (fmdrop_set A m) x = (if x \<notin> A then fmlookup m x else None)"
aborting_error_update: shows "\<forall>s d a' a. current s \<noteq> d \<and> aborting s d a \<longrightarrow> aborting (set_error_code s a') d a"
Nlt0_def: shows "0>\<^sub>N = (\<lambda>(a, b). a < 0)"
poly2_monom: fixes a :: "'b poly"   and n :: "nat"   and x :: "'b"   and y :: "'b" shows "poly2 (monom a n) x y = poly a x * y ^ n"
bij_from_nat: shows "bij_betw mod_type_class.from_nat {0..<CARD('c)} UNIV"
liftAll2_def: fixes phi :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and inp1 :: "'c \<Rightarrow> 'a option"   and inp2 :: "'c \<Rightarrow> 'b option" shows "liftAll2 phi inp1 inp2 \<equiv> \<forall>i v1 v2. inp1 i = Some v1 \<and> inp2 i = Some v2 \<longrightarrow> phi v1 v2"
l3_secrecy_def: shows "l3_secrecy \<equiv> l1_secrecy"