hcju/isabelleps huggingface dataset:
finite_imp_ex_se_star_succ: fixes c :: "('d, 'e) conf"   and ls :: "('d, 'e) label list" assumes "finite (pred c)"   and "\<forall>e\<in>pred c. finite (Bexp.vars e)"   and "finite_labels ls" shows "\<exists>c'. se_star c ls c'"
ipresIGSubstAll_def: fixes h :: "'h \<Rightarrow> 'i"   and hA :: "'j \<Rightarrow> 'k"   and MOD :: "('l, 'm, 'varSort, 'sort, 'n, 'o, 'h, 'j, 'p) model_scheme"   and MOD' :: "('q, 'r, 'varSort, 's, 't, 'o, 'i, 'k, 'u) model_scheme" shows "ipresIGSubstAll h hA MOD MOD' \<equiv> ipresIGSubst h MOD MOD' \<and> ipresIGSubstAbs h hA MOD MOD'"
disc_rfr_proc_borel_adapted: fixes M :: "'b measure"   and F :: "nat \<Rightarrow> 'b measure"   and r :: "real" assumes "disc_filtr_prob_space M F" shows "borel_adapt_stoch_proc F (disc_rfr_proc r)"
le_supI2: fixes x :: "'b multiset"   and b :: "'b multiset"   and a :: "'b multiset" assumes "x \<subseteq># b" shows "x \<subseteq># a \<union># b"
assms: shows "x \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>"
enumdom_nonempty_domain_2: fixes i :: "nat"   and x\<^sub>0 :: "nat"   and x :: "nat" assumes "\<phi> i x\<^sub>0 \<down>" shows "(\<phi> i x \<down>) = (\<exists>y. eval r_enumdom [i, y] \<down>= x)"
rcosets_part_G: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.group G"   and "subgroup H G" shows "\<Union> (rcosets\<^bsub>G\<^esub> H) = carrier G"
hta_has_idx_s_def: fixes H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_has_idx_s H \<equiv> hta_idx_s H \<noteq> None"
less_name_code_3: fixes x :: "char"   and xs :: "char list"   and y :: "char"   and ys :: "char list" shows "(Name (x # xs) < Name (y # ys)) = (of_char x < of_char y \<or> x = y \<and> Name xs < Name ys)"
eval_body'_eval_body_eq: shows "eval_body' = unwrapC oo eval_body oo wrapC"
mult_sv_mv: fixes k :: "complex"   and A :: "complex \<times> complex \<times> complex \<times> complex"   and v :: "complex \<times> complex" shows "k *\<^sub>s\<^sub>v (A *\<^sub>m\<^sub>v v) = A *\<^sub>m\<^sub>v (k *\<^sub>s\<^sub>v v)"
Atm_transT_invert: fixes atm :: "'atom"   and s :: "'state"   and s' :: "'state" assumes "(Atm atm, s) \<rightarrow>t s'" obtains "s' = aval atm s"
exp_one_2pi_iff: fixes x :: "real" shows "(exp (2 * complex_of_real pi * \<i> * complex_of_real x) = 1) = (x \<in> \<int>)"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G G' \<Phi>\<^sub>G' \<tau>\<^sub>0 \<tau>\<^sub>1"
EQ_2: shows "C\<^sub>1 = c\<^sub>1"
3: shows "\<forall>u v a w'. w \<noteq> u @- a ## v @- a ## w'"
r_prefixes_Suc: fixes s :: "nat \<Rightarrow> nat option"   and n :: "nat"   and j :: "nat" assumes "r1_bc s" shows "eval (r1_bc.r_prefixes s) [Suc n, j] \<down>= (let b = the (eval (r1_bc.r_prefixes s) [n, j]) in e_snoc (e_append_zeros b (the (eval (r1_bc.r_find_k s) [b]))) 1)"
simps: fixes x :: "'b"   and xs :: "'b list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
assms_1: shows "\<exists>n'\<ge>n. \<parallel>tid\<parallel>\<^bsub>t n'\<^esub>"
pmf_of_alist_def: fixes xs :: "('b \<times> real) list" shows "pmf_of_alist xs = embed_pmf (\<lambda>x. case map_of xs x of None \<Rightarrow> 0 | Some p \<Rightarrow> p)"
transpE: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "transp r"   and "r x y"   and "r y z" obtains "r x z"
parents_finite: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list" shows "finite (parents d b p)"
pref_profile_unique_favorites_def: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "pref_profile_unique_favorites agents alts R \<equiv> pref_profile_wf agents alts R \<and> pref_profile_unique_favorites_axioms R"
convergent_add_const_iff: fixes c :: "'b"   and f :: "nat \<Rightarrow> 'b" shows "convergent (\<lambda>n. c + f n) = convergent f"
simps_10: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "rewrite_X_enat (\<phi> M\<^sub>n \<psi>) = combine mk_strong_release (rewrite_X_enat \<phi>) (rewrite_X_enat \<psi>)"
p_decomp: shows "p = q @ [(v, l, u)] @ r"
args_FnCall: fixes F :: "nat"   and Xs :: "exp list" shows "args (FnCall F Xs) = Xs"
disconnected_nodes_eq_h3: fixes doc_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" shows "h3 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes = h' \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes"
ss_lt_ssa_q: shows "\<forall>(q, \<tau>)\<in>set (step (hd w) (ss ! hd w)). ss ! q \<sqsubseteq>\<^bsub>r\<^esub> ssa ! q"
not_before_Cons: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "\<not> x < y in y # xs"
A_def: shows "A = cbox (- 1 - complex_of_real pi * \<i>) (1 + complex_of_real pi * \<i>) - ball 0 r"
bottomI: fixes z :: "'a"   and P :: "'a set" assumes "z \<in> P"   and "\<And>y. y \<in> P \<Longrightarrow> y \<succeq> z" shows "z = domain.bottom P"
uniformly_continuous_onE: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and e :: "real" assumes "uniformly_continuous_on s f"   and "0 < e" obtains d :: "real" where "0 < d" and "\<And>x x'. \<lbrakk>x \<in> s; x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
deformation_retract_imp_homotopy_eqv: fixes S :: "'b set"   and r :: "'b \<Rightarrow> 'b"   and T :: "'b set" assumes "homotopic_with_canon (\<lambda>x. True) S S id r"   and "retraction S T r" shows "S homotopy_eqv T"
inverse_arrows_runit: fixes a :: "'a" assumes "ide a" shows "inverse_arrows (runit a) (runit' a)"
suffix_remdups_adj: fixes xs :: "'c list"   and ys :: "'c list" assumes "suffix xs ys" shows "suffix (remdups_adj xs) (remdups_adj ys)"
any_has_not_element_type': fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "element_type \<tau> \<sigma>"   and "OclVoid[?] \<le> \<tau>" shows "False"
token_run_P: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and P :: "'d \<Rightarrow> bool"   and x :: "nat"   and n :: "nat"   and m :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "\<not> P (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n)"   and "P (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x (Suc (n + m)))" shows "\<exists>m'\<le>m. \<not> P (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x (n + m')) \<and> P (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x (Suc (n + m')))"
uminus_dvd_conv_2: fixes d :: "int"   and t :: "int" shows "d dvd t \<equiv> d dvd - t"
prems_2: fixes x :: "'a" assumes "x \<in> S" shows "f x = g x"
DERIV_pos_imp_increasing_open: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "a < b"   and "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> \<exists>y. (f has_real_derivative y) (at x) \<and> 0 < y"   and "continuous_on {a..b} f" shows "f a < f b"
defined_offending': fixes G :: "'v graph" assumes "wf_graph G"   and "\<not> c_sinvar m G" shows "c_offending_flows m G \<noteq> {}"
prems: shows "higher_differentiable_on (S \<times> cbox a b) (\<lambda>a. case a of (a, b) \<Rightarrow> f a b) (Suc k)"
osum_congL: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set"   and t :: "('f \<times> 'f) set" assumes "r =o s"   and "Well_order t" shows "r +o t =o s +o t"
subset_setmonotone: fixes f :: "'b set \<Rightarrow> 'b set"   and X :: "'b set" assumes "setmonotone f" shows "X \<subseteq> f X"
semigroup_add_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "semigroup_add_hom hom \<equiv> plus_hom hom"
leD: fixes y :: "'a"   and x :: "'a" assumes "y \<preceq> x" shows "\<not> x \<prec> y"
echelon_form_Hermite_of_upt_row_i: fixes A :: "(('a, 'b) vec, 'c) vec"   and ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "nat" assumes "echelon_form A"   and "ass_function ass"   and "res_function res" shows "echelon_form (Hermite_of_upt_row_i A k ass res)"
Qeq\<Psi>: shows "insertAssertion (extractFrame Q) \<Psi> \<hookrightarrow>\<^sub>F \<langle>\<epsilon>, \<Psi> \<otimes> \<one>\<rangle>"
IH: assumes "(t, s) \<in> set (db'\<^sub>s\<^sub>s\<^sub>t A I D)" shows "\<exists>B C t' s'. A = B @ insert\<langle>t',s'\<rangle> # C \<and> t = t' \<cdot> I \<and> s = s' \<cdot> I \<and> (\<forall>t'' s''. delete\<langle>t'',s''\<rangle> \<in> set C \<longrightarrow> t \<noteq> t'' \<cdot> I \<or> s \<noteq> s'' \<cdot> I)"
vconn_implies_hpar_3: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright>" shows "src\<^sub>C g = src\<^sub>C f"
matrix_add_assoc: fixes A :: "('c, 'd, 'e) matrix"   and B :: "('c, 'd, 'e) matrix"   and C :: "('c, 'd, 'e) matrix" shows "A + B + C = A + (B + C)"
assms_3: shows "num_params (tm_poly t) \<le> length I"
order_strictI: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. less_eq a b = (less a b \<or> a = b)"   and "\<And>a b. less a b \<Longrightarrow> \<not> less b a"   and "\<And>a. \<not> less a a"   and "\<And>a b c. \<lbrakk>less a b; less b c\<rbrakk> \<Longrightarrow> less a c" shows "class.order less_eq less"
restrict_total_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "restrict_total f A = to_partial f |` A"
simps: fixes a :: "'a LDCFG_node"   and S :: "'a LDCFG_node set" shows "(a \<in> CFGExit_wf.wod_backward_slice S) = ((\<exists>n. a = n \<and> CFGExit_wf.valid_node n \<and> n \<in> S) \<or> (\<exists>n' n\<^sub>1 n\<^sub>2. a = n' \<and> CFGExit_wf.weak_order_dependence n' n\<^sub>1 n\<^sub>2 \<and> n\<^sub>1 \<in> CFGExit_wf.wod_backward_slice S \<and> n\<^sub>2 \<in> CFGExit_wf.wod_backward_slice S) \<or> (\<exists>n' V n''. a = n' \<and> CFGExit_wf.data_dependence n' V n'' \<and> n'' \<in> CFGExit_wf.wod_backward_slice S))"
simps: fixes A :: "'a mat" shows "Smith_mxn A = (if dim_row A = 0 \<or> dim_col A = 0 then (1\<^sub>m (dim_row A), A, 1\<^sub>m (dim_col A)) else if dim_row A = 1 then (1\<^sub>m 1, Smith_1xn A) else if dim_row A = 2 then Smith_2xn A else if dim_col A = 1 then let (P, S) = Smith_nx1 A in (P, S, 1\<^sub>m 1) else if dim_col A = 2 then Smith_nx2 A else let A1 = mat_of_row (Matrix.row A 0); A2 = mat_of_rows (dim_col A) (map (Matrix.row A) [1..<dim_row A]); (P1, D1, Q1) = Smith_mxn A2; C = A1 * Q1 @\<^sub>r P1 * A2 * Q1; D = mat_of_rows (dim_col A) [Matrix.row C 0, Matrix.row C 1]; E = mat_of_rows (dim_col A) (map (Matrix.row C) [2..<dim_row A]); (P2, F, Q2) = Smith_2xn D; H = P2 * D * Q2 @\<^sub>r E * Q2; (P_H2, H2) = reduce_column div_op H; (H2_UL, H2_UR, H2_DL, H2_DR) = split_block H2 1 1; (P3, S', Q3) = Smith_mxn H2_DR; S = four_block_mat (Matrix.mat 1 1 (\<lambda>(a, b). H $$ (0, 0))) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_row A - 1) 1) S'; P1' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_row A - 1)) (0\<^sub>m (dim_row A - 1) 1) P1; P2' = four_block_mat P2 (0\<^sub>m 2 (dim_row A - 2)) (0\<^sub>m (dim_row A - 2) 2) (1\<^sub>m (dim_row A - 2)); P3' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_row A - 1)) (0\<^sub>m (dim_row A - 1) 1) P3; Q3' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_col A - 1) 1) Q3 in (P3' * P_H2 * P2' * P1', S, Q1 * Q2 * Q3'))"
type_low_vars_low: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma> :: "'b \<Rightarrow> Sec option"   and e :: "'a"   and mds :: "Mode \<Rightarrow> 'b set"   and x :: "'b" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "sifum_types.type_aexpr aexp_vars dma \<Gamma> e Low"   and "sifum_types.mds_consistent dma mds \<Gamma>"   and "x \<in> aexp_vars e" shows "sifum_types.to_total dma \<Gamma> x = Low"
quasi_isometry_on_bounded: fixes lambda :: "real"   and C :: "real"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "lambda C -quasi_isometry_on X f"   and "bounded X" shows "bounded (f ` X)"
Gauss_Jordan: fixes A :: "(('a, 'c) vec, 'b) vec" shows "\<exists>P. invertible P \<and> Gauss_Jordan A = P ** A \<and> reduced_row_echelon_form (Gauss_Jordan A)"
isDiagonal_upt_k_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" shows "isDiagonal_upt_k A k = (\<forall>a b. to_nat a \<noteq> to_nat b \<and> (to_nat a < k \<or> to_nat b < k) \<longrightarrow> A $ a $ b = (0::'d))"
ln_exp_cf4_lower_bound_pos: fixes x :: "real" assumes "0 \<le> x" shows "ln (exp_cf4 x) \<le> x"
ERETURN_def: fixes x :: "'d" shows "ERETURN x \<equiv> RETURN (Inr x)"
pos: fixes x :: "real" assumes "1 \<le> x" shows "0 < x + sqrt (x\<^sup>2 - 1)"
standardize_apart: fixes C :: "fterm literal set"   and Cs :: "fterm literal set set"   and C' :: "fterm literal set" assumes "C \<in> Cs"   and "var_renaming_of C C'" shows "mresolution_step Cs (Cs \<union> {C'})"
is_Done_bind_resumption: fixes x :: "('i, 'g, 'h) Resumption.resumption"   and f :: "'i \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "Resumption.resumption.is_Done (x \<bind> f) = (Resumption.resumption.is_Done x \<and> (Resumption.resumption.result x \<noteq> None \<longrightarrow> Resumption.resumption.is_Done (f (the (Resumption.resumption.result x)))))"
fast_ppoint_mult_closed: fixes a :: "int"   and b :: "int"   and q :: "int \<times> int \<times> int"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curvep a b q" shows "on_curvep a b (fast_ppoint_mult (int p) a n q)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set" assumes "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"   and "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_disconnected_nodes heap_is_wellformed parent_child_rel" shows "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel"
add_eq_0_iff_both_eq_0: fixes x :: "'k multiset"   and y :: "'k multiset" shows "(x + y = {#}) = (x = {#} \<and> y = {#})"
CFG_axioms: shows "CFG src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_get_proc get_proc Main) (lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind) procs Main"
r_prenum_prim: shows "prim_recfn 2 r_prenum"
induced_arrowI_2: fixes \<chi>' :: "J.Arr \<Rightarrow> 'c"   and a' :: "'c" assumes "\<chi>' \<in> D.cones a'" shows "D.cones_map (induced_arrow a' \<chi>') (D.mkCone p0 p1) = \<chi>'"
finite_enumerate_regs: fixes a :: "vname aexp" shows "finite (AExp.enumerate_regs a)"
assms: fixes s :: "'a stream" shows "f (stl s) = stl (f s)"
poly_mult_var'_1: fixes p :: "'a list" assumes "set p \<subseteq> carrier R" shows "poly_mult X p = local.normalize (p @ [\<zero>])"
rdv': fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(z \<sqinter> x \<cdot> \<top>) \<cdot> y = z \<cdot> y \<sqinter> x \<cdot> \<top>"
restrict1_the_inv_into_G_binop_G: fixes g :: "'a" assumes "g \<in> G" shows "restrict1 (the_inv_into G (binop g)) G = G_perm (\<ii> g)"
sets_T': fixes I :: "'s pmf" shows "sets (K.T' I) = sets (stream_space (count_space UNIV))"
disj_2: shows "h\<^sub>0' \<bottom> h\<^sub>1'"
cosh_dist_formula: fixes a :: "(real, 3) vec"   and b :: "(real, 3) vec" assumes "a \<noteq> 0"   and "b \<noteq> 0"   and "proj2_abs a \<in> hyp2"   and "proj2_abs b \<in> hyp2" shows "cosh_dist (proj2_abs a) (proj2_abs b) = \<bar>a \<bullet> (M *v b)\<bar> / sqrt (a \<bullet> (M *v a) * (b \<bullet> (M *v b)))"
case_prodD: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f" assumes "case (a, b) of (c, d) \<Rightarrow> f c d" shows "f a b"
step_Normal_elim_cases_8: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and p :: "'a"   and s :: "'b"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Call p, Normal s) \<rightarrow> u" obtains bdy :: "('b, 'a, 'c) com" where "u = (bdy, Normal s)" and "\<Gamma> p = Some bdy"   | "u = (Skip, Stuck)" and "\<Gamma> p = None"
Unf_lang_equiv: fixes \<phi> :: "'b ltln" shows "\<phi> \<sim>\<^sub>L Unf \<phi>"
freeultrafilter_def: fixes F :: "'b filter" shows "freeultrafilter F \<equiv> ultrafilter F \<and> freeultrafilter_axioms F"
mult_matrix_zero_right: fixes fadd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and fmul :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and A :: "'c matrix" assumes "fadd (0::'b) (0::'b) = (0::'b)"   and "\<forall>a. fmul a (0::'d) = (0::'b)" shows "mult_matrix fmul fadd A 0 = 0"
trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)"   and "g \<in> \<Theta>[F](h)" shows "f \<in> \<Theta>[F](h)"
ring_record_simps_12: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and zero :: "'a"   and add :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and more :: "'b" shows "(\<oplus>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub>) = add"
diameter_ge_0: fixes S :: "'b set" assumes "bounded S" shows "0 \<le> diameter S"
smc_1_is_arr_iff: fixes \<aa> :: "V"   and \<ff> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" shows "f : a \<mapsto>\<^bsub>smc_1 \<aa> \<ff>\<^esub> b = (a = \<aa> \<and> b = \<aa> \<and> f = \<ff>)"
snth_szip: fixes s1 :: "'b stream"   and s2 :: "'c stream"   and n :: "nat" shows "szip s1 s2 !! n = (s1 !! n, s2 !! n)"
Gr_insert: fixes x :: "'b"   and F :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "BNF_Def.Gr (insert x F) f = insert (x, f x) (BNF_Def.Gr F f)"
zero_adds: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c" assumes "class.comm_monoid_add plus zero" shows "plus.adds plus zero a"
exist_matrix'_implies_similar: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec"   and X :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'b) vec" assumes "linear (*s) (*s) f"   and "matrix' X X f = A"   and "matrix' Y Y f = B"   and "is_basis (set_of_vector X)"   and "is_basis (set_of_vector Y)" shows "similar_matrices A B"
smult_dvd_iff: fixes a :: "'c"   and p :: "'c poly"   and q :: "'c poly" shows "(smult a p dvd q) = (if a = (0::'c) then q = 0 else p dvd q)"
unset_break_idem: fixes M :: "'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option" shows "unset_break_status ;- unset_break_status ;- M = unset_break_status ;- M"
simps_9: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and p :: "('a, 'b) form" shows "subc c s (Forall p) = Forall (subc c (liftt s) p)"
field_frac1: fixes x :: "'a" assumes "1\<^sub>r \<noteq> \<zero>"   and "x \<in> carrier K" shows "x = x \<cdot>\<^sub>r 1\<^sub>r\<^bsup>\<hyphen> K\<^esup>"
list2policyR_End: fixes p :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators"   and list :: "((int \<times> int) set set, int) Combinators list" assumes "p \<notin> dom (C a)" shows "C (list2policyR (a # list)) p = (C a \<Oplus> list2policy (map C list)) p"
LIMSEQ_D: fixes X :: "nat \<Rightarrow> 'b"   and L :: "'b"   and r :: "real" assumes "X \<longlonglongrightarrow> L"   and "0 < r" shows "\<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
subset_ins_def: fixes qs :: "bool list" shows "subset_ins qs = (\<lambda>(bdd, qss). (bddinsert bdd qs (Some (length qss)), qss @ [qs]))"
column_def: fixes j :: "'e"   and A :: "(('c, 'e) vec, 'd) vec" shows "column j A = (\<chi>i. A $ i $ j)"
edr_characterization: shows "All admits_diagonal_reduction = (All admits_triangular_reduction \<and> (\<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'b} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'b})))"
boundOutputRemoveTau: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "\<tau>.P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P'"   and "x \<sharp> P" shows "P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P'"
fps_conv_radius_binomial: fixes c :: "'a" shows "fps_conv_radius (fps_binomial c) = (if c \<in> \<nat> then \<infinity> else 1)"
RBT_Impl_fold1_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and rbt :: "('b, unit) rbt" shows "RBT_Impl_fold1 f rbt = List.fold f (tl (RBT_Impl.keys rbt)) (hd (RBT_Impl.keys rbt))"
pre_def: shows "pre = (\<lambda>S (V, u0). gen_dfs_pre E U S V u0 \<and> E `` V \<inter> onstack = {})"
indets_monomial_single_subset: fixes v :: "'b"   and k :: "nat"   and c :: "'c" shows "indets (monomial c (monomial k v)) \<subseteq> {v}"
exhaust: fixes y :: "PreSimplyTyped.type" obtains "y = TUnit"   | x2 :: "nat" where "y = TVar x2"   | x31 :: "PreSimplyTyped.type" and x32 :: "PreSimplyTyped.type" where "y = TArr x31 x32"   | x41 :: "PreSimplyTyped.type" and x42 :: "PreSimplyTyped.type" where "y = TPair x41 x42"
simps: fixes xs :: "'c ltl_prop_equiv_quotient list" shows "\<up>And xs = foldl and_abs (Abs true) xs"
assms_1: shows "(f, g) \<in> [P]\<^sub>f R \<rightarrow> \<langle>S\<rangle>nres_rel"
list_all2_appendI: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d list"   and b :: "'e list"   and c :: "'d list"   and d :: "'e list" assumes "list_all2 P a b"   and "list_all2 P c d" shows "list_all2 P (a @ c) (b @ d)"
wf_same_fst: fixes P :: "'b \<Rightarrow> bool"   and R :: "'b \<Rightarrow> ('c \<times> 'c) set" assumes "\<And>x. P x \<Longrightarrow> wf (R x)" shows "wf (same_fst P R)"
less_not_sym: fixes x :: "'b DBMEntry"   and y :: "'b DBMEntry" assumes "x \<prec> y" shows "\<not> y \<prec> x"
element_ptr_kinds_eq_h2: shows "element_ptr_kinds h2 = element_ptr_kinds h'"
unfoldI: fixes P :: "pi"   and Q :: "pi" assumes "P \<leadsto>\<^sup>^<weakBisim> Q"   and "Q \<approx> P" shows "P \<approx> Q"
nu_def: fixes x :: "'a" shows "\<nu> x = x \<sqinter> nc"
zip_eq_conv: fixes xs :: "'b list"   and ys :: "'c list"   and zs :: "('b \<times> 'c) list" assumes "length xs = length ys" shows "(zip xs ys = zs) = (map fst zs = xs \<and> map snd zs = ys)"
finite_imageI: fixes F :: "'k set"   and h :: "'k \<Rightarrow> 'l" assumes "finite F" shows "finite (h ` F)"
conjugate_vec_0: fixes f :: "nat \<Rightarrow> 'a" shows "conjugate (vec 0 f) = vec 0 f"
l2_inv6_init: shows "init l2 \<subseteq> l2_inv6"
reducible\<^sub>dE: fixes p :: "'a poly" assumes "\<not> irreducible\<^sub>d p" obtains "degree p = 0"   | q :: "'a poly" and r :: "'a poly" where "0 < degree q" and "degree q < degree p" and "0 < degree r" and "degree r < degree p" and "p = q * r"
eventually_within_Un: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and s :: "'b set"   and t :: "'b set" shows "eventually P (at x within s \<union> t) = (eventually P (at x within s) \<and> eventually P (at x within t))"
fconst_onI: fixes a :: "V"   and A :: "V"   and c :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "[a, c]\<^sub>\<circ> \<in>\<^sub>\<circ> fconst_on A c"