hcju/isabelleps huggingface dataset:
power_mult: fixes a :: "'a"   and m :: "nat"   and n :: "nat" shows "power.power 1' (;) a (m * n) = power.power 1' (;) (power.power 1' (;) a m) n"
commute: fixes a :: "'a DBMEntry"   and b :: "'a DBMEntry" shows "dbm_add a b = dbm_add b a"
simps: fixes x :: "'a extT"   and y :: "'a extT" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
E_def: shows "E = some_third_chamber C D (C \<inter> D)"
zero_cap_simp: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<notin> E" shows "c (u, v) = (0::'a)"
pk: fixes s :: "'a"   and t :: "'a"   and a :: "'b"   and u :: "'a"   and b :: "'b" assumes "(s, t) \<in> L a"   and "(s, u) \<in> R b" shows "(t, u) \<in> local.conversion' (under r a) O (R b)\<^sup>= O local.conversion' (under r a \<union> under r b) O ((L a)\<inverse>)\<^sup>= O local.conversion' (under r b)"
Tree_wf\<^sub>pE: fixes a\<^sub>p :: "('a, 'b, 'c, 'd) Tree\<^sub>p"   and b\<^sub>p :: "('a, 'b, 'c, 'd) Tree\<^sub>p" assumes "(a\<^sub>p, b\<^sub>p) \<in> Tree_wf\<^sub>p" obtains a :: "('a, 'b, 'c, 'd) Tree" and b :: "('a, 'b, 'c, 'd) Tree" where "a\<^sub>p = abs_Tree\<^sub>p a" and "b\<^sub>p = abs_Tree\<^sub>p b" and "(a, b) \<in> Tree_wf"
vrrestriction_vcomp: fixes r :: "V"   and s :: "V"   and A :: "V" shows "(r \<circ>\<^sub>\<circ> s) \<restriction>\<^sup>r\<^sub>\<circ> A = r \<restriction>\<^sup>r\<^sub>\<circ> A \<circ>\<^sub>\<circ> s"
mat_delete_dim_2: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" shows "dim_col (mat_delete A i j) = dim_col A - 1"
inv_def: shows "inv = (\<lambda>(res, V, ws). case res of None \<Rightarrow> dom ws \<subseteq> V \<and> finite (dom ws) \<and> V \<subseteq> E\<^sup>* `` U0 \<and> E `` (V - dom ws) \<subseteq> V \<and> (\<forall>v\<in>V - dom ws. \<not> P v) \<and> U0 \<subseteq> V \<and> (\<forall>v p. ws v = Some p \<longrightarrow> (\<forall>v\<in>set p. \<not> P v) \<and> (\<exists>u0\<in>U0. path E u0 (rev p) v)) | Some (p, v) \<Rightarrow> \<exists>u0\<in>U0. path E u0 p v \<and> P v \<and> (\<forall>v\<in>set p. \<not> P v))"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(Inl x1 = Inl y1) = (x1 = y1)"
inverse_map_def: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and T :: "'a set" shows "inverse_map f S T \<equiv> restrict (inv_into S f) T"
frechet_derivative_inverse: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "f differentiable at x"   and "f x \<noteq> (0::'d)" shows "frechet_derivative (\<lambda>x. inverse (f x)) (at x) = (\<lambda>h. - (1::'d) / (f x)\<^sup>2 * frechet_derivative f (at x) h)"
addN'_def: fixes g :: "'g"   and n :: "'node" shows "addN' g n = fold (\<lambda>v m. m(v \<mapsto> case m v of None \<Rightarrow> {n} | Some x \<Rightarrow> insert n x)) (sorted_list_of_set (uses g n))"
mult_left_dist_sup: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.idempotent_left_zero_semiring one times sup less_eq less bot" shows "times x (sup y z) = sup (times x y) (times x z)"
4: shows "list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p (unlabel \<A>)"
dvd_div_eq_0_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 b :: "'b"   and a :: "'b" assumes "class.semidom_divide divide plus minus zero times one"   and "dvd.dvd times b a" shows "(divide a b = zero) = (a = zero)"
devExt_nopt_leq: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a"   and n' :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "\<not> Option.is_none (Blockchain.devBC active cmp bc mining t n nid n')" shows "n' \<le> n"
and_eq_0_is_nth: fixes y :: "'b word"   and n :: "nat"   and x :: "'b word" assumes "y = 1 << n" shows "(x AND y = 0) = (\<not> x !! n)"
Bet_cases: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C \<or> Bet C B A" shows "Bet A B C"
swing_neg_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "swing_neg w1 w2" shows "kauff_mat w1 = kauff_mat w2"
differentiable_at_imp_differentiable_on: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> s \<Longrightarrow> f differentiable at x" shows "f differentiable_on s"
order_refl: fixes x :: "'e word" shows "x \<le>s x"
pmdl_closed_trd: fixes p :: "'t \<Rightarrow>\<^sub>0 'd"   and B :: "('t \<Rightarrow>\<^sub>0 'd) set"   and fs :: "('t \<Rightarrow>\<^sub>0 'd) list" assumes "p \<in> pmdl B"   and "set fs \<subseteq> pmdl B" shows "trd fs p \<in> pmdl B"
fps_of_poly_add: fixes p :: "'b poly"   and q :: "'b poly" shows "fps_of_poly (p + q) = fps_of_poly p + fps_of_poly q"
sets_measure_spmf: fixes p :: "'b spmf" shows "sets (measure_spmf p) = sets (count_space UNIV)"
gbm_to_list_impl: shows "graph_to_list gbm_\<alpha> gbm_invar gga_to_list"
Some_INF: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "Some (\<Sqinter> (f ` A)) = (\<Sqinter>x\<in>A. Some (f x))"
assms_4: fixes ys :: "bool list" assumes "xs = False # ys @ [True]" shows "P"
assms_1: shows "x \<in> carrier (DirProds G I)"
exposed_face_of_polyhedron: fixes S :: "'b set"   and F :: "'b set" assumes "polyhedron S" shows "(F exposed_face_of S) = (F face_of S)"
prod_emb_empty: fixes M :: "'d set"   and L :: "'d \<Rightarrow> 'e measure"   and K :: "'d set" shows "prod_emb M L K {} = {}"
codomains_def: fixes f :: "'a" shows "C.codomains f \<equiv> {b. C.ide b \<and> b \<cdot>\<^sub>C f \<noteq> C.null}"
separable_def: fixes k :: "nat" shows "principal.separable k = (principal.gauss_sum k = cnj (principal_dchar n k) * principal.gauss_sum 1)"
cor_def: fixes T :: "'b set set"   and s :: "'b set" shows "core\<^bsub>T\<^esub> s = \<Union> {m. m open\<^bsub>T\<^esub> \<and> m \<subseteq> s}"
mod_diff_right_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus a (modulo b c)) c = modulo (minus a b) c"
power_inductr: fixes z :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and n :: "nat" assumes "z \<union> y O x \<subseteq> y" shows "z O rel_dioid.power x n \<subseteq> y"
tranclp_rtranclp_absorb: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "R\<^sup>+\<^sup>+\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
NewArray_\<tau>red1r_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and T :: "ty" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (newA T\<lfloor>e\<rceil>, xs) (newA T\<lfloor>e'\<rceil>, xs')"
collision_within_less: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o"   and "s\<^sub>o \<le> movement.s a\<^sub>e v\<^sub>e s\<^sub>e (movement.t_stop a\<^sub>e v\<^sub>e)"   and "movement.s a\<^sub>e v\<^sub>e s\<^sub>e (movement.t_stop a\<^sub>e v\<^sub>e) < movement.s a\<^sub>o v\<^sub>o s\<^sub>o (movement.t_stop a\<^sub>o v\<^sub>o)" shows "safe_distance.collision a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o {0..} = safe_distance.collision a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o {0<..<min (movement.t_stop a\<^sub>e v\<^sub>e) (movement.t_stop a\<^sub>o v\<^sub>o)}"
cod_src: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "cod (src \<mu>) = src \<mu>"
strict_trans: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "a \<prec> b"   and "b \<prec> c" shows "a \<prec> c"
M_rel_i_def: fixes f :: "'i"   and F :: "int" shows "M_rel_i f F = (arith_ops_record.to_int ff_ops f = M F)"
basis_exists: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "vector_space scale" obtains B :: "'c set" where "B \<subseteq> V" and "\<not> module.dependent scale B" and "V \<subseteq> module.span scale B" and "card B = vector_space.dim scale V"
ci_def: shows "ci \<equiv> \<lambda>p. contour_integral (circlepath p (get_e p)) ff"
pos_literal_in_imp_true_cls: fixes A :: "'a"   and C :: "'a literal multiset"   and I :: "'a set" assumes "Pos A \<in># C"   and "A \<in> I" shows "I \<Turnstile> C"
arrE: fixes f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr \<times> 'a2" assumes "A2_BxA2.arr f" obtains "A2_B.arr (fst f) \<and> A2.arr (snd f)"
ring_mat_simps_5: fixes ty :: "'c itself"   and n :: "nat"   and b :: "'d" shows "carrier (ring_mat ty n b) = carrier_mat n n"
emptyrem2taut: fixes A :: "'a set"   and \<phi> :: "'a" assumes "A .\<bottom>. \<phi> = {}" shows "\<tturnstile> \<phi>"
interior_of_subset: fixes X :: "'d topology"   and S :: "'d set" shows "X interior_of S \<subseteq> S"
rtranclp_mono': fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and s :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* a b"   and "r \<le> s" shows "s\<^sup>*\<^sup>* a b"
3: assumes "\<lbrakk>M; D; A\<rbrakk>\<^sub>s \<I>"   and "\<lbrakk>M \<union> (ik\<^sub>s\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>); dbupd\<^sub>s\<^sub>s\<^sub>t A \<I> D; B\<rbrakk>\<^sub>s \<I>" shows "\<lbrakk>M; D; A @ B\<rbrakk>\<^sub>s \<I>"
sel_3: fixes x21 :: "'b tree"   and x22 :: "'b"   and x23 :: "'b tree" shows "un_Node3 \<langle>x21, x22, x23\<rangle> = x23"
matrix_to_iarray_ncols: fixes A :: "(('c, 'd) vec, 'e) vec" shows "ncols A = ncols_iarray (matrix_to_iarray A)"
abs_model_check_def: fixes uu :: "'b itself"   and uv :: "'c itself"   and uw :: "'d itself"   and ux :: "'e itself"   and sys :: "('a, 'f set, 'g) sa_rec_scheme"   and \<phi> :: "'f ltlc" shows "abs_model_check uu uv uw ux sys \<phi> \<equiv> ltl_to_gba_spec (not\<^sub>c \<phi>) \<bind> (\<lambda>gba. ASSERT (igba gba) \<bind> (\<lambda>_. ASSERT (sa sys) \<bind> (\<lambda>_. inter_spec sys gba \<bind> (\<lambda>(Gprod, map_state). ASSERT (igb_graph Gprod) \<bind> (\<lambda>_. find_ce_spec Gprod \<bind> (\<lambda>ce. case ce of None \<Rightarrow> RETURN None | Some None \<Rightarrow> RETURN (Some None) | Some (Some r) \<Rightarrow> RETURN (Some (Some (map_state \<circ> r)))))))))"
convex_hull_caratheodory: fixes p :: "'a set" shows "convex hull p = {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y}"
ss: shows "{b. \<exists>a\<in>elts A. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r} \<subseteq> (\<lambda>x. x\<lparr>1\<^sub>\<nat>\<rparr>) ` elts r"
cases: fixes x :: "'a list cstate \<times> auth_query" obtains m :: "'a list" where "x = (Store m__, Forward)"   | m :: "'a list" where "x = (Store ma__, Drop)"   | m :: "'a list" where "x = (Store m, Look)"   | uu_ :: "auth_query" where "x = (Void, uu___)"   | uu_ :: "auth_query" where "x = (cstate.Fail, uua___)"   | v :: "'a list" and uu_ :: "auth_query" where "x = (cstate.Collect v__, uu_)"   | "x = (Void, Look)"   | "x = (cstate.Fail, Look)"   | v :: "'a list" where "x = (cstate.Collect va__, Look)"   | "x = (Void, Drop)"   | "x = (cstate.Fail, Drop)"   | v :: "'a list" where "x = (cstate.Collect v, Drop)"
ex_x_axis_poincare_distance_negative': fixes d :: "real" assumes "0 \<le> d" shows "let x = (1 - exp d) / (1 + exp d) in is_real (cor x) \<and> Re (cor x) \<le> 0 \<and> - 1 < Re (cor x) \<and> of_complex (cor x) \<in> unit_disc \<and> of_complex (cor x) \<in> circline_set x_axis \<and> poincare_distance 0\<^sub>h (of_complex (cor x)) = d"
blocking_on_def: fixes ds :: "D2 set"   and X :: "X5 set"   and h :: "H2"   and X' :: "X5 set" shows "BSI.blocking_on ds X h X' = (X' \<noteq> CX5h h X \<and> X' = CX5h h (X \<union> X') \<and> X' \<subseteq> BSI.CD_on ds (X \<union> X'))"
a1: shows "low_equal s1 s2 \<and> t1 = dcache_mod c v s1 \<and> t2 = dcache_mod c v s2"
dom_MkArr: fixes A :: "'j \<Rightarrow> 'c"   and B :: "'j \<Rightarrow> 'c"   and F :: "'j \<Rightarrow> 'c" assumes "J_C.arr (J_C.MkArr A B F)" shows "J_C.dom (J_C.MkArr A B F) = J_C.MkIde A"
enc: shows "fin_cut_same (z n) (map \<pi> x) @ replicate m (z n) \<in> enc (w, I)"
SUCC_Subset_I: fixes H :: "fm set"   and x :: "tm"   and z :: "tm" assumes "H \<turnstile> x SUBS z"   and "H \<turnstile> x IN z" shows "H \<turnstile> SUCC x SUBS z"
eqspanS: shows "affine hull (+) (- a) ` S = span ((+) (- a) ` S)"
QCongAnFlat_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.QCongAnFlat Bet Cong a \<equiv> Tarski_neutral_dimensionless.QCongA Bet Cong a \<and> (\<forall>A B C. a A B C \<longrightarrow> \<not> Bet A B C)"
skl1_inv2_trans: shows "{skl1_inv2} TS.trans skl1 {> skl1_inv2}"
l8_15_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and X :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B X"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C X" shows "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C X"
map_cong: fixes x :: "'a option"   and ya :: "'a option"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x = ya"   and "\<And>z. z \<in> set_option ya \<Longrightarrow> f z = g z" shows "map_option f x = map_option g ya"
sams_lea_lta123_suma2__lta456: fixes A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p"   and G' :: "'p"   and H' :: "'p"   and I' :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and D' :: "'p"   and E' :: "'p"   and F' :: "'p" assumes "A' B' C' LtA A B C"   and "G H I LeA G' H' I'"   and "SAMS A B C D E F"   and "A B C D E F SumA G H I"   and "A' B' C' D' E' F' SumA G' H' I'" shows "D E F LtA D' E' F'"
translation_invertible: fixes a :: "'a" assumes "a \<in> G" shows "transformation.sub.invertible (a)\<^sub>R"
ultraD: fixes A :: "'a set" assumes "A \<subseteq> carrier"   and "A \<notin> F" shows "carrier - A \<in> F"
Compl_Int: fixes A :: "'b set"   and B :: "'b set" shows "- (A \<inter> B) = - A \<union> - B"
mC_ZOC_part_def: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'a"   and t :: "'b"   and I0 :: "nat set"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_ZOC_part c d s t I0 P F \<equiv> {} \<notin> P - {I0} \<and> {} \<notin> F ` (P - {I0}) \<and> I0 \<in> P \<and> part {..<brn c} P \<and> part {..<brn d} (F ` P)"
c_1: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" assumes "stateful_strand_step.is_Send stateful_strand_step" shows "send\<langle>the_msg stateful_strand_step\<rangle> = stateful_strand_step"
push_in_let_conv_2: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "Let x (RETURN \<circ> f) = RETURN (Let x f)"
shift_map_keys_punit_def: shows "shift_map_keys_punit = punit.shift_map_keys"
indets_monomial_single: fixes c :: "'b"   and k :: "nat"   and v :: "'c" assumes "c \<noteq> (0::'b)"   and "k \<noteq> 0" shows "indets (monomial c (monomial k v)) = {v}"
sba1_idempotent: fixes x :: "'a" shows "x \<squnion> x = x"
subalgebra_sigma_finite: fixes N :: "'a measure" assumes "subalgebra M N" shows "sigma_finite_subalgebra M N"
foundedMono: fixes subs :: "'a \<Rightarrow> 'a set"   and P :: "'a \<Rightarrow> bool"   and A :: "(nat \<times> 'a) set"   and Q :: "'a \<Rightarrow> bool" assumes "founded subs P A"   and "\<forall>x. P x \<longrightarrow> Q x" shows "founded subs Q A"
octo_inverse_mult: fixes x :: "octo"   and y :: "octo" shows "inverse (x * y) = inverse y * inverse x"
in_code_status_status_rel_iff_1: fixes b :: "status" shows "((CFOUND, b) \<in> code_status_status_rel) = (b = FOUND)"
insert1_def: fixes x :: "'b" shows "insert1 x \<equiv> \<lambda>(l, r, buf). (if l = r then grow1 (length buf + 1) (l, r, buf) else RETURN (l, r, buf)) \<bind> (\<lambda>(l, r, buf). ASSERT (l < length buf) \<bind> (\<lambda>_. RETURN (l + 1, r, buf[l := x])))"
rel_set_image_eq: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'a set"   and B :: "'b set"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "rel_set P A B"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B; P a b\<rbrakk> \<Longrightarrow> f a = g b" shows "f ` A = g ` B"
oseq_sendTE: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   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}send(s\<^sub>m\<^sub>s\<^sub>g) . p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i" obtains "a = send (s\<^sub>m\<^sub>s\<^sub>g (\<sigma> i))" and "\<sigma>' i = \<sigma> i" and "q = p"
subdag_parentdag_low: fixes not :: "dag"   and lt :: "dag"   and p :: "ref"   and rt :: "dag" assumes "not \<le> lt" shows "not \<le> Node lt p rt"
cobounded1: fixes a :: "'b"   and b :: "'b" shows "a \<le> a \<sqinter> b"
cases: fixes a :: "'b stream" assumes "sdistinct a" obtains x :: "'b" and xs :: "'b stream" where "a = x ## xs" and "x \<notin> sset xs" and "sdistinct xs"
that: assumes "arr_Rel \<alpha> T"   and "vsv (T\<lparr>ArrVal\<rparr>)" shows "thesis"
Diff_subset: fixes A :: "'q set"   and B :: "'q set" shows "A - B \<subseteq> A"
norm_scaleC: fixes scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and scaleC :: "complex \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and a :: "complex"   and x :: "'d" assumes "class.complex_normed_vector scaleR scaleC plus zero minus uminus dist norm sgn uniformity open" shows "norm (scaleC a x) = cmod a * norm x"
assms_1: shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
index_not_zero_impl_diff_group: fixes ps :: "'a set"   and G :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2"   and "int (\<B> index ps) = 1"   and "G \<in> \<G>" shows "\<not> ps \<subseteq> G"
compose_eq': fixes A :: "'b set"   and G :: "'d \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'd" shows "compose A G F = restrict (G \<circ> F) A"
abs_eq: fixes x :: "'b" shows "prod_left x = Blinfun (\<lambda>a. a ** x)"
VUnion_vsingleton: fixes a :: "V" shows "\<Union>\<^sub>\<circ> (set {a}) = a"
l1_iagreement_Resp_def: shows "l2_iagreement_Resp \<equiv> {s. \<forall>A B N. signalsResp s (Commit A B N) \<le> signalsResp s (Running A B N)}"
LAssReds: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and V :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>V:=e,s,b\<rangle> \<rightarrow>* \<langle>V:=e',s',b'\<rangle>"
Cons_2: fixes j :: "nat" assumes "j + 1 < length (x # xs)"   and "remdups_adj (x # xs) ! i = (x # xs) ! j"   and "remdups_adj (x # xs) ! (i + 1) = (x # xs) ! (j + 1)" shows "thesis"
Compl_Un: fixes A :: "'a set"   and B :: "'a set" shows "- (A \<union> B) = - A \<inter> - B"
simplex_sat0: fixes cs :: "('a \<times> constraint) list"   and v :: "(nat, rat) mapping" assumes "solve_exec_code cs = Inr v" shows "\<langle>v\<rangle> \<Turnstile>\<^sub>c\<^sub>s (snd ` set cs)"
awalkE': fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "awalk u p v" obtains "set (pawalk_verts u p) \<subseteq> pverts G" and "set p \<subseteq> parcs G" and "pcas u p v" and "hd (pawalk_verts u p) = u" and "last (pawalk_verts u p) = v" and "u \<in> pverts G" and "v \<in> pverts G"
unsynlr_leq: fixes x :: "('a, 'b) synlr"   and y :: "('a, 'b) synlr" shows "(unsynlr x \<subseteq> unsynlr y) = (x \<le> y)"
flip_at_base_simps_4: fixes a :: "'a"   and b :: "'a"   and x :: "'b" assumes "sort_of (atom a) \<noteq> sort_of (atom b)" shows "(a \<leftrightarrow> b) \<bullet> x = x"
vsv_vcard_vdomain: shows "vcard (\<AA>\<lparr>Obj\<rparr>) = vcard (\<FF>\<lparr>ObjMap\<rparr>)"
I: shows "Max {i. i < length ns \<and> offs ns 0 ! i < k} < length ns"
exec_appendL: fixes P :: "instr list"   and i :: "int"   and s :: "char list \<Rightarrow> int"   and stk :: "int list"   and i' :: "int"   and s' :: "char list \<Rightarrow> int"   and stk' :: "int list"   and P' :: "instr list" assumes "P \<turnstile> (i, s, stk) \<rightarrow>* (i', s', stk')" shows "P' @ P \<turnstile> (size P' + i, s, stk) \<rightarrow>* (size P' + i', s', stk')"
wf_acyclic: fixes r :: "('b \<times> 'b) set" assumes "wf r" shows "acyclic r"
closure_of_eq: fixes A :: "'a set"   and x :: "'a"   and x' :: "'a" assumes "A \<subseteq> carrier L"   and "x \<in> closure_of A"   and "x' \<in> carrier L"   and "x .= x'" shows "x' \<in> closure_of A"
Ord_succ_iff: fixes k :: "hf" shows "Ord (succ k) = Ord k"
HEndPhase1_HInv4d: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv4d s q"   and "Inv2b s"   and "HInv4c s p" shows "HInv4d s' q"
lcp_pref: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<and>\<^sub>p ys \<le>p xs"
member_fs_set_permute_iff: fixes p :: "perm"   and x :: "'a"   and X :: "'a fs_set" shows "(p \<bullet> x \<in>\<^sub>f\<^sub>s p \<bullet> X) = (x \<in>\<^sub>f\<^sub>s X)"
spec_ops_id_inc: fixes xs :: "('a \<times> 'b) list"   and oid :: "'a"   and oper :: "'b"   and deps :: "'b \<Rightarrow> 'a set"   and x :: "'a" assumes "spec_ops (xs @ [(oid, oper)]) deps"   and "x \<in> set (map fst xs)" shows "x < oid"
DDH_bound1: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "\<bar>spmf (DDH0 \<A>) True - spmf (DDH1 \<A>) True\<bar> \<le> \<bar>spmf (ddh_0 \<A>) True - spmf (ddh_1 \<A>) True\<bar> + \<bar>spmf (ddh_1 \<A>) True - spmf (DDH1 \<A>) True\<bar>"
wfTerminationLess: fixes decisionVars :: "nat set"   and F0 :: "Literal list list" assumes "finite decisionVars" shows "wf (terminationLess F0 decisionVars)"
imp_compat_eqvR: fixes \<phi> :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla"   and "prv (eqv \<chi>1 \<chi>2)" shows "prv (eqv (imp \<phi> \<chi>1) (imp \<phi> \<chi>2))"