hcju/isabelleps huggingface dataset:
poly_map_poly_of_real_cnj: fixes p :: "real poly"   and z :: "complex" shows "poly (map_poly complex_of_real p) (cnj z) = cnj (poly (map_poly complex_of_real p) z)"
sij_def: fixes a :: "nat list"   and b :: "nat list"   and i :: "nat"   and j :: "nat" shows "Linear_Diophantine_Equations.sij a b i j = ((zeroes (length a))[i := Linear_Diophantine_Equations.dij a b i j], (zeroes (length b))[j := Linear_Diophantine_Equations.eij a b i j])"
blinding_of_T: fixes rh :: "'e \<Rightarrow> 'f"   and bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "blinding_of_on UNIV rh bo" shows "blinding_of_on UNIV (root_hash_T rh) (blinding_of_T rh bo)"
mset_subset_eq_add_mset_cancel: fixes a :: "'k"   and A :: "'k multiset"   and B :: "'k multiset" shows "(add_mset a A \<subseteq># add_mset a B) = (A \<subseteq># B)"
nres_order_simps_20: fixes x :: "'a"   and y :: "'a" shows "(RETURN x \<le> RETURN y) = (x = y)"
sup_commute: fixes sup :: "'ap \<Rightarrow> 'ap \<Rightarrow> 'ap"   and less_eq :: "'ap \<Rightarrow> 'ap \<Rightarrow> bool"   and less :: "'ap \<Rightarrow> 'ap \<Rightarrow> bool"   and x :: "'ap"   and y :: "'ap" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
integrable_fst': fixes f :: "'b \<times> 'a \<Rightarrow> 'c" assumes "integrable (M2 \<Otimes>\<^sub>M M1) f" shows "integrable M2 (\<lambda>x. LINT y|M1. f (x, y))"
ring_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<cdot>\<^sub>r (y \<plusminus> z) = x \<cdot>\<^sub>r y \<plusminus> x \<cdot>\<^sub>r z"
simps_6: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "'a list" shows "set1_ta_rule x1 \<rightarrow> x2 x3 = insert x1 (set x3)"
assms_6: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M" shows "interpret_slp slp xs ! slp_index_lookup slp M f = interpret_floatarith f xs"
nonneg: shows "0 \<le> s * (s - a) * (s - b) * (s - c)"
Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e_inverse: fixes y :: "\<langle>\<langle>'d set\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "y \<in> {X. X = \<bottom> \<or> X = null \<or> (\<forall>x\<in>\<lceil>\<lceil>X\<rceil>\<rceil>. x \<noteq> \<bottom>)}" shows "Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e y) = y"
Finite_imp_cardinal_cons: fixes A :: "V set"   and a :: "V" assumes "finite A"   and "a \<notin> A" shows "vcard (set (insert a A)) = csucc (vcard (set A))"
wt_interpretation_exists: obtains \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" where "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)" and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>" and "subst_range \<I> \<subseteq> public_ground_wf_terms"
DISJNT2: shows "P\<^sub>2 \<inter> wrap B\<^sub>2 = {}"
le_supE: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd"   and x :: "'c \<times> 'c \<Rightarrow> 'd" assumes "a \<oplus> b \<preceq> x" obtains "a \<preceq> x" and "b \<preceq> x"
csk: shows "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'"
run_owner_ii_unused: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "\<exists>B. PriKey B \<notin> used (fst (run_owner_ii n s))"
VA: shows "Vagree \<nu> \<omega> (\<Union>y\<in>{y. Inl (Inr y) \<in> SIGF (InContext x1 x2)}. FVT (\<sigma> y))"
diff_monom: fixes a :: "'b"   and n :: "nat"   and b :: "'b" shows "monom a n - monom b n = monom (a - b) n"
shuffles_commutes: fixes xs :: "'a list"   and ys :: "'a list" shows "shuffles xs ys = shuffles ys xs"
cast_simps_4: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isOp s X" shows "wlsInp (fst3 (castOp s X)) (snd3 (castOp s X))"
fac_work_final_body3_eq: shows "fac_work_final = wrapA\<cdot>(fix\<cdot>fac_acc_body3)"
cis_pm_pi_1: fixes x :: "real" shows "cis (x - pi) = - cis x"
induct: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x1 :: "'c"   and x2 :: "'c"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* x1 x2"   and "\<And>a. P a a"   and "\<And>a b c. \<lbrakk>r\<^sup>*\<^sup>* a b; P a b; r b c\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
sup_right_dist_inf: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "- y \<sqinter> - z \<squnion> - x = (- y \<squnion> - x) \<sqinter> (- z \<squnion> - x)"
B: shows "Esigma (a * r * c) = a * u * q + a * r * c"
default_prefix_def: shows "default_prefix = PrefixMatch 0 0"
left_uvar_indep_right_uvar: fixes x :: "'d \<Longrightarrow> 'f"   and y :: "'h \<Longrightarrow> 'g" shows "0-x \<bowtie> 1-y"
simps: fixes a :: "msg"   and H :: "msg set" shows "(a \<in> parts H) = ((\<exists>X. a = X \<and> X \<in> H) \<or> (\<exists>K X. a = X \<and> Crypt K X \<in> parts H) \<or> (\<exists>X Y. a = X \<and> \<lbrace>X, Y\<rbrace> \<in> parts H) \<or> (\<exists>X Y. a = Y \<and> \<lbrace>X, Y\<rbrace> \<in> parts H))"
simps: fixes b :: "bool"   and v :: "nat" shows "negate (Lit b v) = Lit (\<not> b) v"
final_deterministic_RP_step: fixes St :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat" assumes "is_final_dstate St" shows "deterministic_RP_step St = St"
image_iff: fixes z :: "'k"   and f :: "'l \<Rightarrow> 'k"   and A :: "'l set" shows "(z \<in> f ` A) = (\<exists>x\<in>A. z = f x)"
ifs: shows "\<not> p - Suc 0 \<le> lo"
brnL_0: fixes cl :: "('test, 'atom, 'choice) cmd list" shows "brnL cl 0 = 0"
meet_comm: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y = y \<sqinter> x"
collapse_2: fixes state :: "('a, 'b, 'c) state" assumes "isState state" shows "State (state.getGMState state) (getGMUser state) (getGMCom state) = state"
not_eq_order_implies_strict: fixes a :: "'a Word.word"   and b :: "'a Word.word" assumes "a \<noteq> b"   and "a \<le>s b" shows "a <s b"
fun_rel_def: fixes A :: "('a \<times> 'c) set"   and B :: "('b \<times> 'f) set" shows "A \<rightarrow> B \<equiv> {(f, f'). \<forall>(a, a')\<in>A. (f a, f' a') \<in> B}"
simps_11: fixes c\<^sub>1 :: "('a, 'b, 'c) com"   and c\<^sub>2 :: "('a, 'b, 'c) com" shows "redex (Catch c\<^sub>1 c\<^sub>2) = redex c\<^sub>1"
open_path_connected_component: fixes S :: "'a set"   and x :: "'a" assumes "open S" shows "path_component S x = connected_component S x"
irreducible\<^sub>d_factor: fixes p :: "'a poly" assumes "0 < degree p" shows "\<exists>q r. irreducible\<^sub>d q \<and> p = q * r \<and> degree r < degree p"
Pd_shuffle_to_top_Field: fixes Y :: "'a set"   and ds' :: "'b set"   and d :: "'b" assumes "inj_on Xd Y" shows "Field (TruePref.Pd_shuffle_to_top ds' Y d) = Field (Pd d) \<union> (if d \<in> ds' then ManiPref'.dX Y d else {})"
nb\<^sub>2: shows "distinct (map (\<lambda>op. Operator k (index (\<Pi>\<^sub>\<O>) op)) (remdups (\<Pi>\<^sub>\<O>)))"
bisim1CondElse: fixes P :: "'a prog"   and h :: "'heap"   and e2 :: "'addr expr1"   and exs :: "'addr expr1 \<times> 'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e :: "'addr expr1"   and e1 :: "'addr expr1" assumes "P,e2,h \<turnstile> exs \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,if (e) e1 else e2,h \<turnstile> exs \<leftrightarrow> (stk, loc, Suc (Suc (length (compE2 e) + length (compE2 e1) + pc)), xcp)"
sel: fixes x :: "'a \<Rightarrow> 'b" shows "run_env (EnvT x) = x"
s0g_secrecyI: fixes x :: "'a s0g_state" assumes "kn x \<subseteq> az x \<union> lkr x" shows "x \<in> s0g_secrecy"
simplex_likeD_finite: fixes P :: "'a set" assumes "domain.simplex_like P" shows "finite P"
start_nth_lt0_all: shows "\<forall>p\<in>{1..<length start}. start ! p = rev [0..<n]"
\<tau>inf_step_table_into_\<tau>inf_step: fixes s :: "'s2"   and sstls :: "('s2 \<times> 's2 \<times> 'tl2 \<times> 's2) llist" assumes "s -\<tau>2-sstls\<rightarrow>*t \<infinity>" shows "s -\<tau>2-lmap (fst \<circ> snd \<circ> snd) sstls\<rightarrow>* \<infinity>"
unitary_gen_real: fixes M :: "complex \<times> complex \<times> complex \<times> complex" assumes "unitary_gen M" shows "\<exists>k>0. mat_adj M *\<^sub>m\<^sub>m M = cor k *\<^sub>s\<^sub>m eye"
strategyproofness_1: shows "(pmf (sds R27) a < pmf (sds R13) a \<or> pmf (sds R27) a + pmf (sds R27) c < pmf (sds R13) a + pmf (sds R13) b \<or> pmf (sds R27) a + (pmf (sds R27) c + pmf (sds R27) d) < pmf (sds R13) a + (pmf (sds R13) b + pmf (sds R13) d)) \<or> pmf (sds R27) a = pmf (sds R13) a \<and> pmf (sds R27) a + pmf (sds R27) c = pmf (sds R13) a + pmf (sds R13) b \<and> pmf (sds R27) a + (pmf (sds R27) c + pmf (sds R27) d) = pmf (sds R13) a + (pmf (sds R13) b + pmf (sds R13) d)"
simps_2: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'a list"   and x :: "'a" shows "ipurge_tr_aux I D U (xs @ [x]) = (if \<exists>v\<in>sinks_aux I D U xs. (v, D x) \<in> I then ipurge_tr_aux I D U xs else ipurge_tr_aux I D U xs @ [x])"
oalist_inv_rawD2: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "sorted_wrt (lt_of_comp comp) (map fst xs)"
minus_leq: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat" shows "less_eq a b = less_eq (minus a b) zero"
lift_append: fixes f :: "'d \<Rightarrow> 'a"   and gens :: "'d set"   and x :: "(bool \<times> 'd) list"   and y :: "(bool \<times> 'd) list" assumes "f \<in> gens \<rightarrow> carrier G"   and "x \<in> lists (UNIV \<times> gens)"   and "y \<in> lists (UNIV \<times> gens)" shows "lift f (x @ y) = lift f x \<otimes> lift f y"
UP\<^sub>0_simps_1: fixes a :: "'a" assumes "B.obj a" shows "ide (UP\<^sub>0 a)"
res: shows "(if mo = 0 then remove_prime_factor i n' (i # ps) else (n, ps)) = (m, qs)"
gbinomial_rec: fixes a :: "'a"   and k :: "nat" shows "a + (1::'a) gchoose Suc k = (a gchoose k) * ((a + (1::'a)) / of_nat (Suc k))"
power_series_conv_imp_absconv_weak: fixes a :: "nat \<Rightarrow> 'a"   and z :: "'a"   and w :: "'a" assumes "summable (\<lambda>n. a n * z ^ n)"   and "norm w < norm z" shows "summable (\<lambda>n. of_real (norm (a n)) * w ^ n)"
smap_sconst: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "smap f (sconst x) = sconst (f x)"
steps_empty: fixes f :: "'a \<Rightarrow> nat"   and l :: "nat"   and up :: "nat"   and left :: "nat" shows "steps {} f l up left = 0"
mult_ac_2: fixes a :: "'g"   and b :: "'g" shows "a * b = b * a"
assms: shows "sp_instr F x \<Sigma> = Ok \<Sigma>'"
set_sort_oalist: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list" assumes "comparator comp"   and "distinct (map fst xs)" shows "set (comparator.sort_oalist comp xs) = {kv \<in> set xs. snd kv \<noteq> (0::'b)}"
space_lim_stream: fixes x :: "'a" shows "space (lim_stream x) = streams (space M)"
cblinfun_compose_scaleC_left: fixes a :: "complex"   and A :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "a *\<^sub>C A o\<^sub>C\<^sub>L B = a *\<^sub>C (A o\<^sub>C\<^sub>L B)"
aligned_shiftr_mask_shiftl: fixes x :: "'a word"   and n :: "nat"   and v :: "nat" assumes "is_aligned x n" shows "(x >> n) && mask v << n = x && mask (v + n)"
List_explode_no_empty: fixes xss :: "'b list list" assumes "[] \<notin> set xss" shows "List_explode xss = map Some (concat xss)"
color_paint_Black: fixes t :: "('c \<times> color) tree" shows "color (paint Black t) = Black"
simps: fixes \<Sigma> :: "'b set set"   and \<phi> :: "'b ltl"   and \<G> :: "'b ltl set" shows "ltl_to_rabin \<Sigma> \<phi> \<G> = (semi_mojmir_def.step \<Sigma> \<up>af\<^sub>G (Abs \<phi>), semi_mojmir_def.initial (Abs \<phi>), {(mojmir_to_rabin_def.fail\<^sub>R \<Sigma> \<up>af\<^sub>G (Abs \<phi>) {q. \<G> \<Turnstile>\<^sub>P Rep q} \<union> mojmir_to_rabin_def.merge\<^sub>R \<up>af\<^sub>G (Abs \<phi>) {q. \<G> \<Turnstile>\<^sub>P Rep q} i, mojmir_to_rabin_def.succeed\<^sub>R \<up>af\<^sub>G (Abs \<phi>) {q. \<G> \<Turnstile>\<^sub>P Rep q} i) |i. i < semi_mojmir_def.max_rank \<Sigma> \<up>af\<^sub>G (Abs \<phi>)})"
deflation: fixes y :: "'b" assumes "y \<in> carrier \<Y>" shows "\<pi>\<^sup>* (\<pi>\<^sub>* y) \<sqsubseteq>\<^bsub>\<Y>\<^esub> y"
satPB_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi> :: "('fsym, 'psym) lit list set" shows "Ik.satPB \<xi> \<Phi> \<equiv> \<forall>c\<in>\<Phi>. Ik.satC \<xi> c"
nb\<^sub>2: fixes v :: "'a" assumes "v \<in> set (\<Pi>\<^sub>\<V>)" shows "cnf (encode_state_variable 0 (index (\<Pi>\<^sub>\<V>) v) ((\<Pi>\<^sub>I) v)) = {{literal_formula_to_literal (encode_state_variable 0 (index (\<Pi>\<^sub>\<V>) v) ((\<Pi>\<^sub>I) v))}}"
HStartBallot_HInv4a2: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HStartBallot s s' p"   and "HInv4a2 s p" shows "HInv4a2 s' p"
diff_dvd_imp_mod_eq: fixes m :: "nat"   and b :: "nat"   and a :: "nat" assumes "m dvd b - a"   and "a \<le> b" shows "a mod m = b mod m"
ldropn_eq_LConsD: fixes n :: "nat"   and xs :: "'g llist"   and y :: "'g"   and ys :: "'g llist" assumes "ldropn n xs = LCons y ys" shows "enat n < llength xs"
hb_consistent_remove1: fixes xs :: "'a list"   and x :: "'a" assumes "hb_consistent xs" shows "hb_consistent (remove1 x xs)"
Cauchy_integral_formula_global: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and z :: "complex"   and \<gamma> :: "real \<Rightarrow> complex" assumes "open S"   and "f holomorphic_on S"   and "z \<in> S"   and "valid_path \<gamma>"   and "path_image \<gamma> \<subseteq> S - {z}"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0" shows "((\<lambda>w. f w / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
access_matrix_nodes_defined: fixes V :: "('a word \<times> 'a wordinterval) list"   and E :: "('a word \<times> 'a word) list"   and c :: "parts_connection"   and rs :: "'a simple_rule list"   and s :: "'a word"   and d :: "'a word" assumes "(V, E) = access_matrix c rs"   and "(s, d) \<in> set E" shows "s \<in> dom (map_of V)"
mergeable_1: shows "\<forall>v1\<in>S. \<forall>v2\<in>S. \<forall>A B f. (\<forall>x\<in>A. x < n \<and> v1 ! x = f x) \<and> (\<forall>x\<in>B. x < n \<and> v2 ! x = f x) \<longrightarrow> (\<exists>v\<in>S. \<forall>x\<in>A \<union> B. v ! x = f x)"
linorder_neqE_linordered_idom: fixes x :: "'b"   and y :: "'b" assumes "x \<noteq> y" obtains "x < y"   | "y < x"
eval_semantics: fixes pi :: "'c \<Rightarrow> char list \<Rightarrow> bool"   and s :: "'c"   and r :: "'b \<Rightarrow> 'c \<Rightarrow> 'c set"   and p :: "'b fm" shows "eval (pi s) (semantics (Kripke pi r) s) p = (Kripke pi r, s \<Turnstile> p)"
intro: assumes "\<And>x. P x x"   and "\<And>x y z. \<lbrakk>P x y; P y z\<rbrakk> \<Longrightarrow> P x z"   and "\<And>x y. O x y = (\<exists>z. P z x \<and> P z y)" shows "PM"
disjoint_family_on_def: fixes A :: "'b \<Rightarrow> 'c set"   and S :: "'b set" shows "disjoint_family_on A S = (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
weakly_equivalent_iff_not_distinguished: fixes P :: "'state"   and Q :: "'state" shows "(P \<equiv>\<cdot> Q) = (\<nexists>x. weak_formula x \<and> x distinguishes P from Q)"
elims: fixes x :: "'e \<Rightarrow> 'e"   and xa :: "'d \<Rightarrow> 'd"   and xb :: "('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr"   and y :: "('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr" assumes "\<R>\<^sub>L x xa xb = y" obtains \<alpha> :: "'e \<Rightarrow> 'e" and \<beta> :: "'d \<Rightarrow> 'd" and \<sigma> :: "'d \<Rightarrow> ('e, 'd, 'f) val option" and \<tau> :: "'d \<Rightarrow> ('e, 'd, 'f) val option" and e :: "('e, 'd, 'f) expr" where "x = \<alpha>" and "xa = \<beta>" and "xb = (\<sigma>, \<tau>, e)" and "y = (\<R>\<^sub>S \<alpha> \<beta> \<sigma>, \<R>\<^sub>S \<alpha> \<beta> \<tau>, \<R>\<^sub>E \<alpha> \<beta> e)"
homologous_rel_trans: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "homologous_rel p X S b c"   and "homologous_rel p X S a b" shows "homologous_rel p X S a c"
assms: shows "\<phi> \<sim>\<^sub>Q \<psi>"
has_white_path_to_refl: fixes x :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state" shows "(x has_white_path_to x) s"
le_imp_less_or_eq: fixes D :: "('d, 'e) Order_scheme"   and a :: "'d"   and b :: "'d" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq>\<^bsub>D\<^esub> b = (a \<prec>\<^bsub>D\<^esub> b \<or> a = b)"
fmap_update_eqvt: fixes p :: "perm"   and a :: "'a"   and b :: "'b"   and f :: "('a, 'b) fmap" shows "p \<bullet> f(a $$:= b) = (p \<bullet> f)(p \<bullet> a $$:= p \<bullet> b)"
inject_3: fixes x31 :: "expr"   and x32 :: "bop"   and x33 :: "expr"   and y31 :: "expr"   and y32 :: "bop"   and y33 :: "expr" shows "(x31 \<guillemotleft>x32\<guillemotright> x33 = y31 \<guillemotleft>y32\<guillemotright> y33) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
poly_funE: fixes f :: "int \<Rightarrow> int" assumes "poly_fun f" obtains p :: "rat poly" where "\<And>a. rat_of_int (f a) = poly p (rat_of_int a)"
taylor_deg_1_eval: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and fa :: "'a"   and f'a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c = to_fun (shift 2 (T\<^bsub>a\<^esub> f)) b"   and "fa = to_fun f a"   and "f'a = deriv f a" shows "to_fun f (b \<oplus> a) = fa \<oplus> f'a \<otimes> b \<oplus> c \<otimes> b [^] 2"
all_channels_done_in_final_state: fixes t :: "('a, 'b, 'c) event list"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "channel cid = Some (p, q)" shows "snd (cs final cid) = Done"
induct: fixes x1 :: "char list \<Rightarrow> com option"   and x2 :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and x3 :: "(com \<times> (char list \<Rightarrow> int \<Rightarrow> int)) option"   and P :: "(char list \<Rightarrow> com option) \<Rightarrow> com \<times> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (com \<times> (char list \<Rightarrow> int \<Rightarrow> int)) option \<Rightarrow> bool" assumes "small_steps x1 x2 x3"   and "\<And>\<pi> cs. P \<pi> cs (Some cs)"   and "\<And>\<pi> cs. small_step \<pi> cs = None \<Longrightarrow> P \<pi> cs None"   and "\<And>\<pi> cs cs1 cs2. \<lbrakk>small_step \<pi> cs = Some cs1; small_steps \<pi> cs1 cs2; P \<pi> cs1 cs2\<rbrakk> \<Longrightarrow> P \<pi> cs cs2" shows "P x1 x2 x3"
lincomb_del2: fixes S :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and v :: "'c" assumes "finite S"   and "S \<subseteq> carrier V"   and "a \<in> S \<rightarrow> carrier K"   and "v \<in> S" shows "lincomb a S = a v \<odot>\<^bsub>V\<^esub> v \<oplus>\<^bsub>V\<^esub> lincomb a (S - {v})"
system_if_mat_kernel: fixes Q :: "'c mat"   and v :: "'c vec" assumes "dim_row Q = dim_col Q"   and "dim_row Q = dim_vec v" shows "(Q\<^sup>T *\<^sub>v v = v) = (v \<in> mat_kernel (Q - 1\<^sub>m (dim_row Q))\<^sup>T)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> bool \<Rightarrow> 'a"   and new_thread_action :: "('b, 'c, 'd) new_thread_action" shows "P (case new_thread_action of NewThread x xa xb \<Rightarrow> f1 x xa xb | ThreadExists x xa \<Rightarrow> f2 x xa) = ((\<forall>x11 x12 x13. new_thread_action = NewThread x11 x12 x13 \<longrightarrow> P (f1 x11 x12 x13)) \<and> (\<forall>x21 x22. new_thread_action = ThreadExists x21 x22 \<longrightarrow> P (f2 x21 x22)))"
vec_vec_Tensor_right_id: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and v :: "'b list" assumes "Matrix_Tensor.mult id f" shows "mult.vec_vec_Tensor f v [id] = v"
rotate_is_Nil_conv: fixes n :: "nat"   and xs :: "'b list" shows "(rotate n xs = []) = (xs = [])"
fds_exp_minus: fixes f :: "'a fds" shows "fds_exp (- f) = inverse (fds_exp f)"
ipath_mkTree_ev: fixes s :: "'state"   and rs :: "'rule stream"   and steps :: "('state \<times> 'rule) stream"   and r :: "'rule" assumes "s \<in> S"   and "fair rs"   and "ipath (mkTree rs s) steps"   and "r \<in> R"   and "alw (holds (enabledAtStep r)) steps" shows "ev (holds (\<lambda>step. snd step = r)) steps"
ordIso_symmetric: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "r =o r'" shows "r' =o r"
factors_mult_single: fixes a :: "'a"   and fb :: "'a list"   and b :: "'a" assumes "irreducible G a"   and "factors G fb b"   and "a \<in> carrier G" shows "factors G (a # fb) (a \<otimes> b)"
sqrt: shows "ma_coeff sqrt = 0 \<and> ma_base sqrt = 0 \<or> ma_coeff sqrt \<noteq> 0 \<and> prime_product (ma_base sqrt)"
cases: fixes a1 :: "'a list \<times> (nat \<Rightarrow> 'a option) \<times> 'b"   and a2 :: "'a list \<times> (nat \<Rightarrow> 'a option) \<times> 'b" assumes "step a1 a2" obtains ws :: "nat \<Rightarrow> 'a option" and i :: "nat" and t :: "'a" and ts :: "'a list" and s :: "'b" where "a1 = (t # ts__, ws__, s__)" and "a2 = (ts__, ws__(i__ \<mapsto> t), s__)" and "ws__ i__ = None" and "i__ < n"   | ws :: "nat \<Rightarrow> 'a option" and i :: "nat" and a :: "'a" and ts :: "'a list" and s :: "'b" where "a1 = (ts, ws, s)" and "a2 = (ts, ws(i := None), f a s)" and "ws i = Some a" and "i < n"
exec_all_finalD: fixes P :: "jvm_prog"   and x :: "nat option"   and h :: "heap"   and sh :: "sheap"   and \<sigma> :: "jvm_state" assumes "P \<turnstile> (x, h, [], sh) -jvm\<rightarrow> \<sigma>" shows "\<sigma> = (x, h, [], sh)"
entail_all_bot: fixes N :: "'a set" assumes "\<exists>B\<in>Bot. N \<Turnstile> {B}" shows "\<forall>B'\<in>Bot. N \<Turnstile> {B'}"
assms_4: fixes i :: "'a" assumes "i \<in> Basis" shows "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i \<or> b \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> a \<bullet> i"
simps_3: fixes v :: "'b list"   and \<phi> :: "'b formula"   and e :: "char list \<times> 'b list" shows "matches v (Neg \<phi>) e = matches v \<phi> e"
T_on_rand'_as_sum: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and t :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> nat"   and A :: "('a \<Rightarrow> 'd pmf) \<times> ('a \<times> 'd \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) pmf)"   and s0 :: "('a \<times> 'd) pmf"   and rs :: "'b list" shows "On_Off.T_on_rand' step t A s0 rs = (\<Sum>n<length rs. On_Off.T_on_rand' step t A (On_Off.config'_rand step A s0 (take n rs)) [rs ! n])"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "'a" assumes "ide a"   and "ide a'"   and "\<guillemotleft>g : a' \<otimes> \<I> \<rightarrow> a \<otimes> \<I>\<guillemotright>" shows "\<exists>f. \<guillemotleft>f : a' \<rightarrow> a\<guillemotright> \<and> f \<otimes> \<I> = g"
borel_measurable_LIMSEQ_metric: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>i. f i \<in> borel_measurable M"   and "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) \<longlonglongrightarrow> g x" shows "g \<in> borel_measurable M"
assms: shows "r\<inverse>\<^sub>\<bullet> \<circ>\<^sub>\<bullet> r = r"
lindep_same: fixes S :: "'a vec set" assumes "S \<subseteq> mat_kernel B" shows "KB.lin_dep S = KB.NC.lin_dep S"
Some_Sup: fixes A :: "'b set" assumes "A \<noteq> {}" shows "Some (\<Squnion> A) = \<Squnion> (Some ` A)"