hcju/isabelleps huggingface dataset:
I_minWeightBasisI: fixes T :: "'a set"   and E :: "'a set" assumes "indep T"   and "\<And>e. e \<in> carrier - E - T \<Longrightarrow> \<not> indep (insert e T)"   and "E \<subseteq> carrier"   and "\<And>x y. \<lbrakk>x \<in> T; y \<in> E\<rbrakk> \<Longrightarrow> weight x \<le> weight y"   and "T \<subseteq> carrier"   and "no_smallest_element_skipped E T" shows "I_minWeightBasis (T, E)"
cnj_zero: shows "Octonions.cnj 0 = 0"
square_free_factorizationD_2: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list"   and a :: "'a poly"   and i :: "nat" assumes "square_free_factorization p (c, bs)"   and "(a, i) \<in> set bs" shows "square_free a \<and> degree a \<noteq> 0"
simps: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" shows "simple_bochner_integrable M f = (simple_function M f \<and> emeasure M {y \<in> space M. f y \<noteq> (0::'d)} \<noteq> \<infinity>)"
skip_type: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" shows "\<turnstile> \<Gamma>,\<S>,P {Skip} \<Gamma>,\<S>,P"
less_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "lt x y"   and "lt y z" shows "lt x z"
transfer: shows "(pcr_residue_ring ===> pcr_residue_ring ===> pcr_residue_ring) (+) (+)"
primitive_reflect_poly: fixes f :: "'a poly" shows "primitive (reflect_poly f) = primitive f"
c_def: fixes i :: "nat" shows "local.c i = local.xij_f 0 i"
multcol_mat: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and k :: "nat"   and a :: "'a" assumes "A \<in> carrier_mat nr n" shows "multcol k a A = A * multrow_mat n k a"
of_nat_Suc: fixes n :: "nat" shows "\<guillemotleft>Suc n\<guillemotright>\<^sub>\<nat> = \<one> \<oplus> \<guillemotleft>n\<guillemotright>\<^sub>\<nat>"
card_length: fixes xs :: "'g list" shows "card (set xs) \<le> length xs"
New1ArrayThrow: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and T :: "ty"   and a :: "'addr"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>newA T\<lfloor>Throw a\<rceil>,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
Strong_Soundness: fixes c :: "'b ann_com"   and s :: "'b"   and co :: "'b ann_com option"   and t :: "'b"   and q :: "'b set" assumes "(Some c, s) -*\<rightarrow> (co, t)"   and "s \<in> pre c"   and "\<turnstile> c  q" shows "if co = None then t \<in> q else t \<in> pre (the co)"
ortho_lin_indpt: fixes W :: "'a vec set"   and X :: "'a vec set" assumes "W \<subseteq> carrier_vec n"   and "X \<subseteq> carrier_vec n"   and "\<And>w x. \<lbrakk>w \<in> W; x \<in> X\<rbrakk> \<Longrightarrow> x \<bullet> w = (0::'a)"   and "lin_indpt W"   and "lin_indpt X" shows "lin_indpt (W \<union> X)"
P_new_decomp: shows "P_new = new_pre @ x # new_post"
rconjby_order2_eq_lconjby: fixes s :: "'a" assumes "s + s = (0::'a)" shows "rconjby s = lconjby s"
R\<^sub>N_def: fixes n :: "nat"   and u :: "nat"   and l :: "nat" shows "R\<^sub>N n u l = R {..<n} u l"
add_strict_increasing2: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "{#} \<subseteq># a"   and "b \<subset># c" shows "b \<subset># a + c"
cycle_free_up_to_def: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat" shows "cycle_free_up_to m k n \<equiv> \<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..k} \<longrightarrow> (\<forall>j\<le>n. len m i j (rem_cycles i j xs) \<le> len m i j xs) \<and> (0::'a) \<le> len m i i xs"
nonneg_cexpr_map_vars: fixes f :: "nat \<Rightarrow> nat"   and V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" assumes "nonneg_cexpr (f -` V) (\<Gamma> \<circ> f) e" shows "nonneg_cexpr V \<Gamma> (map_vars f e)"
elims: fixes x :: "operation"   and xa :: "int"   and y :: "int option" assumes "counter_op x xa = y" obtains xb :: "int" where "x = Increment" and "xa = xb__" and "y = Some (xb__ + 1)"   | xb :: "int" where "x = Decrement" and "xa = xb" and "y = Some (xb - 1)"
wf_projectQuery: fixes I :: "nat set"   and n :: "nat"   and Q :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and QQ :: "(nat set \<times> 'a option list set) set" assumes "1 \<le> card I"   and "wf_query n I Q Qn"   and "non_empty_query Q"   and "covering I Q"   and "\<forall>X\<in>Q. 1 \<le> card (fst X \<inter> I)"   and "QQ = projectQuery I Q"   and "included I Qn"   and "non_empty_query Qn" shows "rwf_query n I QQ Qn"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "abel_semigroup f"   and "semilattice_axioms f" shows "semilattice f"
L_Cons: fixes \<omega> :: "('c \<times> 'd) stream"   and o :: "'d"   and os :: "'d list" shows "(\<exists>xs \<omega>'. \<omega> = xs @- \<omega>' \<and> map snd xs = o # os) = (snd (shd \<omega>) = o \<and> (\<exists>xs \<omega>'. stl \<omega> = xs @- \<omega>' \<and> map snd xs = os))"
map_fun_def: fixes f :: "'g \<Rightarrow> 'i"   and g :: "'j \<Rightarrow> 'h"   and h :: "'i \<Rightarrow> 'j" shows "(f ---> g) h = g \<circ> h \<circ> f"
zip_with_prev_altdef: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" shows "zip_with_prev f x xs = (if xs = [] then [] else f x (hd xs) # map (\<lambda>i. f (xs ! i) (xs ! (i + 1))) [0..<length xs - 1])"
simps_1: fixes D :: "ast_domain"   and ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and a :: "'a atom" shows "ast_domain.wf_fmla D ty_ent (Atom a) = ast_domain.wf_atom D ty_ent a"
imF_ne: shows "(\<lambda>h. cmod (deriv h 0)) ` F \<noteq> {}"
write_actionsp_write_actions_eq: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist" shows "write_actionsp E = (\<lambda>x. x \<in> write_actions E)"
sets_N: shows "sets N = sets (stream_space (count_space UNIV))"
ras_top_def: fixes s :: "'c array \<times> nat" shows "ras_top s \<equiv> ras_get s 0"
that: fixes g :: "real \<Rightarrow> real" assumes "g square_integrable S"   and "(\<lambda>n. l2norm S (\<lambda>x. f n x - g x)) \<longlonglongrightarrow> 0" shows "thesis"
in_language_to_language: fixes L :: "'b list set"   and w :: "'b list" shows "in_language (to_language L) w = (w \<in> L)"
h_minus_aux_le_h': fixes h' :: "'v \<times> 'v \<Rightarrow> ennreal"   and j :: "nat"   and e :: "'v \<times> 'v" assumes "\<And>e. h' e \<noteq> \<top>" shows "h_minus_aux h' j e \<le> h' e"
reciprocal_0_iff: fixes P :: "'b poly"   and p :: "nat" assumes "degree P \<le> p" shows "(reciprocal_poly p P = 0) = (P = 0)"
ContingentlyFalse_def: shows "ContingentlyFalse \<equiv> \<lambda>p. \<^bold>\<not>p \<^bold>& \<^bold>\<diamond>p"
that: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b"   and "f x = y"   and "\<And>x'. \<lbrakk>a \<le> x'; x' < x\<rbrakk> \<Longrightarrow> f x' < y" shows "thesis"
UN_simps_7: fixes A :: "'c set"   and B :: "'d \<Rightarrow> 'c set"   and C :: "'d set" shows "(\<Union>x\<in>C. A - B x) = A - \<Inter> (B ` C)"
RG_proj_mult_rightdelta: fixes g :: "'g"   and x :: "('a, 'g) aezfun"   and r :: "'a" assumes "g \<in> G" shows "RG_proj (x * (r \<delta>\<delta> g)) = RG_proj x * (r \<delta>\<delta> g)"
projs_4: fixes dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" shows "\<pi>\<^sub>5 (dsn, dsk, flag, hops, nhip) = hops"
split_paired_Ex: fixes P :: "'a \<times> 'b \<Rightarrow> bool" shows "(\<exists>x. P x) = (\<exists>a b. P (a, b))"
continuous_on_joinpaths_D2: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "continuous_on {0..1} (g1 +++ g2)"   and "pathfinish g1 = pathstart g2" shows "continuous_on {0..1} g2"
MM1_axioms_def: shows "MM1_axioms \<equiv> \<forall>y x. PP y x \<longrightarrow> (\<exists>z. PP z x \<and> \<not> O z y)"
take_bit_Suc_1: 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 division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "take_bit (Suc n) one = one"
sigma_sets_empty_eq: fixes A :: "'a set" shows "sigma_sets A {} = {{}, A}"
trivial_subgroup_is_normal: shows "{\<one>} \<lhd> G"
intro: fixes h :: "'c \<times> 'd \<Rightarrow> 'e" assumes "good_map h" shows "Extend h"
add_assoc_var: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "D y z \<and> D x (y \<oplus> z)" shows "x \<oplus> y \<oplus> z = x \<oplus> (y \<oplus> z)"
is_strongly_scdom_component_safe_def: shows "is_strongly_scdom_component_safe \<equiv> l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_scdom_component_safe get_owner_document get_disconnected_nodes to_tree_order"
word_arith_wis_6: fixes a :: "'b word" shows "word_pred a = word_of_int (uint a - 1)"
example_how_to_use_properties_in_proofs: shows "\<forall>p. Policy.ifp p p"
ipresIGSubst_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and MOD :: "('h, 'i, 'd, 'j, 'k, 'a, 'g, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.ipresIGSubst Delta h (FixSyn.termMOD Delta) MOD = FixSyn.ipresSubst Delta h MOD"
assms: shows "\<forall>x. P1 x = P2 x"
fst_gb_schema_dummy_components_init: fixes rc :: "nat"   and data :: "nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "struct_spec sel ap ab compl" shows "component_of_term ` Keys (fst ` set (fst (gb_schema_dummy (rc, data) D (ab gs [] bs data) (ap gs [] [] bs data)))) = component_of_term ` Keys (fst ` set (gs @ bs))"
assms_2: shows "trace_rest_eq rest1 rest2 (outs_\<I> \<I>RA) (outs_\<I> \<I>RU) (return_spmf s_rest) (return_spmf s_rest')"
field_axioms: shows "class.field (/) inverse (*) (1::'d) (+) (0::'d) (-) uminus"
assms_3: shows "\<forall>c. c \<in> Com X \<longrightarrow> vars (strip c) \<subseteq> X \<longrightarrow> f c \<in> Com X"
PDG_BS_def: fixes S :: "'node set" shows "PDG_BS S \<equiv> {n'. \<exists>n. PDG.PDG_path sourcenode targetnode valid_edge Def Use weak_control_dependence n' n \<and> n \<in> S \<and> valid_node n}"
image_Suc_fvi: fixes b :: "nat"   and \<phi> :: "'c MFOTL.formula" shows "Suc ` MFOTL.fvi (Suc b) \<phi> = MFOTL.fvi b \<phi> - {0}"
e_in_n_un_rf_n: shows "E \<in> N \<union> Rf N"
l_null: fixes x :: "'a list" assumes "x \<in> carrier (poly_ring R)" shows "\<zero>\<^bsub>poly_ring R\<^esub> \<otimes>\<^bsub>poly_ring R\<^esub> x = \<zero>\<^bsub>poly_ring R\<^esub>"
resumption_partial_function_definition: shows "partial_function_definitions resumption_ord resumption_lub"
homeomorphic_space_def: fixes X :: "'e topology"   and Y :: "'f topology" shows "X homeomorphic_space Y \<equiv> \<exists>f g. homeomorphic_maps X Y f g"
powser_def: fixes cs :: "real msllist"   and x :: "real" shows "powser cs x = (\<Sum>n. lcoeff cs n * x ^ n)"
cont2monofunE: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "cont f"   and "x \<sqsubseteq> y" shows "f x \<sqsubseteq> f y"
sup_fun_def: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" shows "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
i_Exec_Stream_nth_Suc: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d"   and n :: "nat" shows "i_Exec_Comp_Stream trans_fun input c (Suc n) = trans_fun (input (Suc n)) (i_Exec_Comp_Stream trans_fun input c n)"
fps_left_inverse: fixes x :: "'a"   and f :: "'a fps" assumes "x * f $ 0 = (1::'a)" shows "fps_left_inverse f x * f = 1"
simps_1: fixes i :: "int" shows "type (Iv i) = Ity"
p_max: fixes x :: "real" shows "ego.p x \<le> ego.p_max"
assms: shows "v \<equiv> Tensor.mat_of_cols_list 4 (map (map complex_of_real) [[1 / sqrt 2, 0, 1 / sqrt 2, 0], [0, 1 / sqrt 2, 0, 1 / sqrt 2], [1 / sqrt 2, 0, - 1 / sqrt 2, 0], [0, 1 / sqrt 2, 0, - 1 / sqrt 2]])"
agree_Cons: fixes x :: "'d"   and env :: "('d, bool) mapping"   and s :: "'d \<Rightarrow> bool"   and b :: "bool" assumes "x \<notin> Mapping.keys env" shows "agree s (Mapping.update x b env) = ((if b then s x else \<not> s x) \<and> agree s env)"
\<phi>_def: shows "\<phi> = (\<lambda>g. \<lambda>U\<in>rcosets P. U #> inv g)"
rfn_compseries_iMTr1: fixes r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and h :: "nat \<Rightarrow> 'a set" assumes "0 < r"   and "0 < s"   and "compseries G r f"   and "h \<in> wcsr_rfns G r f s" shows "f ` {i. i \<le> r} \<subseteq> h ` {i. i \<le> s * r}"
M_M: fixes m :: "int"   and x :: "int" shows "poly_mod.M m (poly_mod.M m x) = poly_mod.M m x"
limited_and_def: fixes x :: "'b word"   and y :: "'b word" shows "limited_and x y = (x AND y = x)"
weakSimulation_def: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs" shows "P \<leadsto>\<^sup>^<Rel> Q \<equiv> \<forall>a Q'. Q \<longmapsto>a \<prec> Q' \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sup>^a \<prec> P' \<and> (P', Q') \<in> Rel)"
drop_perp_def: fixes p :: "proj2"   and l :: "proj2_line" shows "drop_perp p l \<equiv> proj2_line_through p (pole l)"
gt_diff: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "vars_mset s \<subseteq># vars_mset t"   and "local.kbo_std.wt t = local.kbo_std.wt s"   and "head t >\<^sub>h\<^sub>d head s" shows "local.kbo_std.gt t s"
branchStops: fixes subs :: "'a \<Rightarrow> 'a set"   and Gamma :: "'a"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "branch subs Gamma f"   and "terminal subs (f n)" shows "f (Suc n) = f n"
dlex_pm_trans: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b"   and u :: "'a \<Rightarrow>\<^sub>0 'b" assumes "dlex_pm s t"   and "dlex_pm t u" shows "dlex_pm s u"
Phase1or2ReadElse_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" shows "Phase1or2ReadElse s s' p d q = (d \<in> disksWritten s p \<and> StartBallot s s' p)"
tracesp_traces_eq: fixes p :: "prog" shows "tracesp p = (\<lambda>x. x \<in> traces p)"
simps_3: shows "dist_param_type Gaussian = PRODUCT REAL REAL"
infsetsum_nonneg_is_SUPREMUM_ereal: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set" assumes "f abs_summable_on A"   and "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" shows "ereal (infsetsum f A) = (SUP F\<in>{F. finite F \<and> F \<subseteq> A}. ereal (sum f F))"
simps_1: fixes x1 :: "'c"   and x2 :: "'d"   and y1 :: "'c"   and y2 :: "'d" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
P1_ideal_ddh0_\<sigma>_true: fixes \<sigma> :: "bool"   and M :: "'a \<times> 'a"   and z :: "'f"   and \<A> :: "('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('g \<Rightarrow> 'e spmf)"   and D :: "'e \<times> 'a \<Rightarrow> bool spmf" assumes "\<sigma> = True" shows "P1_ideal_model M \<sigma> z \<A> \<bind> D = local.ddh.DDH0 (P1_DDH_mal_adv_\<sigma>_true M z \<A> D)"
prems_7: shows "Marker \<in> set (msgs (S t i) cid)"
pdivides_iff_shell: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subfield K (poly_ring R)"   and "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)" shows "p pdivides\<^bsub>poly_ring R\<^esub> q = p divides\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub> q"
less_multiset_def: fixes M' :: "'a multiset"   and M :: "'a multiset" shows "(M' < M) = ((M', M) \<in> mult {(x', x). x' < x})"
keys_default: fixes k :: "'a"   and v :: "'b"   and m :: "('a, 'b) mapping" shows "Mapping.keys (Mapping.default k v m) = insert k (Mapping.keys m)"
fv_subset_if_in_strand_ik: fixes t :: "('c, 'd) Term.term"   and S :: "('c, 'd) strand_step list" assumes "t \<in> ik\<^sub>s\<^sub>t S" shows "fv t \<subseteq> \<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v S))"
src_cod: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c" assumes "horizontal_homs V src trg"   and "partial_magma.arr V \<mu>" shows "src (partial_magma.cod V \<mu>) = src \<mu>"
similar_mat_witD_3: fixes n :: "nat"   and A :: "'c mat"   and B :: "'c mat"   and P :: "'c mat"   and Q :: "'c mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "A = P * B * Q"
A: fixes m :: "'s"   and P :: "'k \<times> 'v \<Rightarrow> bool" shows "g_sel m P = iterate_sel_no_map (iteratei m) P"
subtermI': fixes x :: "('a, 'b) Term.term"   and X :: "('a, 'b) Term.term list"   and f :: "'a" assumes "x \<in> set X" shows "x \<sqsubset> Fun f X"
calc_atm_2: fixes x :: "NominalSequents.var" shows "[] \<bullet> x = x"
gd_term_to_pair_unit: fixes cmp_term :: "'b nat_term_order" shows "gd_term to_pair_unit fst (\<lambda>s t. le_of_nat_term_order cmp_term (fst (s, the_min)) (fst (t, the_min))) (\<lambda>s t. lt_of_nat_term_order cmp_term (fst (s, the_min)) (fst (t, the_min))) (le_of_nat_term_order cmp_term) (lt_of_nat_term_order cmp_term)"
ilim: shows "(\<lambda>i. integral\<^sup>L M (f i)) \<longlonglongrightarrow> x"
x_in_dom_allow: shows "x \<in> dom (Cp (AllowPortFromTo c d po))"
canon_ang_minus_pi2: fixes \<alpha> :: "real" assumes "- 2 * pi < \<alpha>"   and "\<alpha> \<le> 0" shows "\<downharpoonright>\<alpha> - pi\<downharpoonleft> = \<alpha> + pi"
xs: fixes i :: "nat" assumes "i < n" shows "P (xs i) i"
Asm: fixes P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and p :: "pname"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool"   and C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set" assumes "(P, CALL p, Q) \<in> C" shows "C \<turnstile> {P} CALL p {Q}"
list_of_pdevs_eq: fixes X :: "'b pdevs" shows "rev (list_of_pdevs X) = filter ((\<noteq>) (0::'b) \<circ> snd) (map (\<lambda>i. (i, pdevs_apply X i)) [0..<degree X])"
simps_1: fixes t :: "'b btree" shows "inorder_up\<^sub>i (T\<^sub>i t) = inorder t"
IH: fixes m :: "nat" assumes "ev_at P m (stl \<omega>)" shows "n = m"
K0_def: shows "K0 = cone hull {1} \<times> C0"
ref_simps_2: fixes f :: "AttId" shows "ref (staticLoc f) = nullV"
minus_one_mult_self: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "times (power.power one times (uminus one) n) (power.power one times (uminus one) n) = one"
is_rrep_def: fixes \<xi> :: "'b state_scheme" shows "is_rrep \<xi> \<equiv> case msg \<xi> of Rrep hops' dip' dsn' oip' sip' \<Rightarrow> {\<xi>\<lparr>hops := hops', dip := dip', dsn := dsn', oip := oip', sip := sip'\<rparr>} | _ \<Rightarrow> {}"
simps_1: fixes uu :: "('b \<times> 'b) set"   and uv :: "'c \<Rightarrow> 'b"   and uw :: "'b set" shows "sinks_aux_less uu uv uw [] = {}"
transfer_operations_rl_2: shows "(product_appr ops, product_listset) \<in> appr_rell \<rightarrow> appr_rell \<rightarrow> appr_rell"
strict_coboundedI2: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "b < c" shows "inf a b < c"
bounded_iff: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word" shows "(a \<le>s signed.min b c) = (a \<le>s b \<and> a \<le>s c)"
gt_imp_wt: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "t >\<^sub>t s" shows "wt t \<ge>\<^sub>p wt s"
normalized_src_ips_def2: fixes ms :: "'b common_primitive match_expr" shows "normalized_src_ips ms = normalized_n_primitive (is_Src, src_sel) normalized_cidr_ip ms"
map_of_eq_None_iff: fixes xys :: "('c \<times> 'b) list"   and x :: "'c" shows "(map_of xys x = None) = (x \<notin> fst ` set xys)"
split_space_into_finite_sets_and_rest: fixes N :: "'a measure" assumes "absolutely_continuous M N"   and "sets N = sets M" shows "\<exists>B. disjoint_family B \<and> range B \<subseteq> sets M \<and> (\<forall>i. emeasure N (B i) \<noteq> \<infinity>) \<and> (\<forall>A\<in>sets M. A \<inter> \<Union> (range B) = {} \<longrightarrow> emeasure M A = 0 \<and> emeasure N A = 0 \<or> 0 < emeasure M A \<and> emeasure N A = \<infinity>)"
simps_4: fixes K :: "nat"   and X :: "freemsg" shows "freeright (DECRYPT K X) = freeright X"
Domain_iff: fixes a :: "'a"   and r :: "('a \<times> 'b) set" shows "(a \<in> Domain r) = (\<exists>y. (a, y) \<in> r)"
in_elts_\<omega>_sum: fixes \<delta> :: "V"   and \<alpha>s :: "V list" assumes "\<delta> \<in> elts (\<omega>_sum \<alpha>s)" shows "\<exists>k<length \<alpha>s. \<exists>\<gamma>\<in>elts (\<omega> \<up> \<alpha>s ! k). \<delta> = \<omega>_sum (take k \<alpha>s) + \<gamma>"
mkIde_set: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and a :: "'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.ide S a" shows "set_category_given_img.mkIde S img \<AA> (set_category_data.set S img a) = a"
vars_def: fixes p :: "'c mpoly" shows "vars p = \<Union> (keys ` keys (mapping_of p))"
N_prime_mono: fixes p :: "int"   and p' :: "int" assumes "prime p"   and "prime p'"   and "p \<le> p'" shows "N p \<le> N p'"
get_shadow_root_ptr_simp7: fixes document_ptr :: "(_) document_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and f :: "('m, 'n, 'o, 'w option) RShadowRoot_scheme"   and h :: "(_) heap" shows "get document_ptr (put shadow_root_ptr f h) = get document_ptr h"
ma_is_rat: fixes x :: "mini_alg" shows "is_rat (real_of x) = ma_is_rat x"
rotate_drop_take: fixes n :: "nat"   and xs :: "'d list" shows "List.rotate n xs = drop (n mod List.length xs) xs @ take (n mod List.length xs) xs"
that: fixes a :: "complex"   and c :: "complex" assumes "(\<lambda>z. if z = 1 then a else newman z + deriv zeta z + c * zeta z) analytic_on {s. 1 \<le> Re s}" shows "thesis"
simps_2: fixes f :: "'b"   and Tl :: "'b trm list" shows "nv2T (Fn f Tl) = {}"
chine_unit_in_hom: shows "\<guillemotleft>\<eta>.chine : f.dsrc \<rightarrow>\<^sub>C g.leg0 \<down>\<down> f.leg1\<guillemotright>"
subset_empty_fset: fixes xs :: "'b fset" shows "(xs |\<subseteq>| {||}) = (xs = {||})"
progress_set_def: fixes F :: "'b program"   and T :: "'b set"   and B :: "'b set" shows "progress_set F T B \<equiv> {L. lattice L \<and> B \<in> L \<and> T \<in> L \<and> closed F T B L}"
abs_inverse: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a set \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a set"   and c :: "'a set" assumes "quot_type R Abs Rep"   and "\<exists>x. R x x \<and> c = Collect (R x)" shows "Rep (Abs c) = c"
valid_strategy_in_V: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v :: "'a" assumes "G'.strategy p \<sigma>"   and "v \<in> G'.VV p"   and "\<not> G'.deadend v" shows "\<sigma> v \<in> V\<^bsub>G'\<^esub>"
star_assoc: fixes p :: "assertion"   and q :: "assertion"   and r :: "assertion" shows "(p \<star> q) \<star> r = p \<star> q \<star> r"
ex_Minf: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite S"   and "S \<noteq> {}" shows "\<exists>m\<in>S. \<forall>s\<in>S. f m \<le> f s"
surjective: fixes r :: "('c, 'd) ring_scheme" shows "r = \<lparr>carrier = carrier r, monoid.mult = (\<otimes>\<^bsub>r\<^esub>), one = \<one>\<^bsub>r\<^esub>, zero = \<zero>\<^bsub>r\<^esub>, add = (\<oplus>\<^bsub>r\<^esub>), \<dots> = ring.more r\<rparr>"