hcju/isabelleps huggingface dataset:
step: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and c' :: "'a"   and mds' :: "Mode \<Rightarrow> 'b set"   and mem' :: "'b \<Rightarrow> 'c"   and lc :: "('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)"   and c'' :: "'a"   and mds'' :: "Mode \<Rightarrow> 'b set"   and mem'' :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'"   and "((c', mds'), mem') \<in> sifum_security.loc_reach eval lc"   and "(((c', mds'), mem'), (c'', mds''), mem'') \<in> eval" shows "((c'', mds''), mem'') \<in> sifum_security.loc_reach eval lc"
satA_e: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and at :: "('fsym, 'psym) atm" shows "U.satA \<xi> at = M.satA \<xi> at"
sees_method_compPD: fixes f :: "String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'b \<Rightarrow> 'a"   and P :: "'b prog"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and fm :: "'a option"   and D :: "String.literal" assumes "compP f P \<turnstile> C sees M: Ts\<rightarrow>T = fm in D" shows "\<exists>m. P \<turnstile> C sees M: Ts\<rightarrow>T = m in D \<and> map_option (f D M Ts T) m = fm"
size_of_mat: fixes signs :: "rat list list"   and subsets :: "nat list list" shows "M_mat signs subsets \<in> carrier_mat (length subsets) (length signs)"
borel_measurable_ereal_iff_Iio: fixes f :: "'b \<Rightarrow> ereal"   and M :: "'b measure" shows "(f \<in> borel_measurable M) = (\<forall>a. f -` {..<a} \<inter> space M \<in> sets M)"
has_domain_iff_has_codomain: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "(domains f \<noteq> {}) = (codomains f \<noteq> {})"
into_from_WW_aux: fixes ns :: "nat list"   and k :: "nat" assumes "strict_sorted ns"   and "\<forall>n\<in>list.set ns. k \<le> n" shows "into_WW k (from_WW k ns) = ns"
sgn_neg_inf_def: fixes p :: "'a poly" shows "sgn_neg_inf p \<equiv> if even (degree p) then sgn (lead_coeff p) else - sgn (lead_coeff p)"
process_surj_pair: fixes P :: "('c event list \<times> 'c event set) set \<times> 'c event list set" shows "(FAILURES P, DIVERGENCES P) = P"
get_operand2_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and op_list :: "inst_operand list" assumes "low_equal s1 s2" shows "get_operand2 op_list s1 = get_operand2 op_list s2"
of_nat_dvd_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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "dvd.dvd times (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m dvd n)"
inject: fixes x1 :: "nat"   and y1 :: "nat" shows "((_ x1 _) = (_ y1 _)) = (x1 = y1)"
S3_def: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "S3 rmhist p = S rmhist True True False clkB rpcA histA histB p"
lim3: shows "\<forall>i<d. \<forall>j<d. (\<lambda>n. (complex_of_real c \<cdot>\<^sub>m f n) $$ (i, j)) \<longlonglongrightarrow> matrix_seq.lowner_lub (\<lambda>n. complex_of_real c \<cdot>\<^sub>m f n) $$ (i, j)"
r_into_trancl': fixes p :: "'d \<times> 'd"   and r :: "('d \<times> 'd) set" assumes "p \<in> r" shows "p \<in> r\<^sup>+"
c_failures_last: fixes xps :: "('a \<times> 'b) list"   and X :: "('a \<times> 'b) set"   and step :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and out :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and s\<^sub>0 :: "'c" assumes "(xps, X) \<in> c_failures step out s\<^sub>0"   and "xps \<noteq> []" shows "snd (last xps) = out (foldl step s\<^sub>0 (butlast (map fst xps))) (last (map fst xps))"
ap_spec_add_pairs_punit_canon: shows "punit.ap_spec add_pairs_punit_canon"
subset_step_R: fixes p :: "real poly"   and qs1 :: "real poly list"   and signs1 :: "rat list list"   and qs2 :: "real poly list"   and signs2 :: "rat list list" assumes "set (characterize_consistent_signs_at_roots p qs1) \<subseteq> set signs1"   and "set (characterize_consistent_signs_at_roots p qs2) \<subseteq> set signs2" shows "set (characterize_consistent_signs_at_roots p (qs1 @ qs2)) \<subseteq> set (signs_smash signs1 signs2)"
poly_cont: fixes e :: "real"   and z :: "'a"   and p :: "'a poly" assumes "0 < e" shows "\<exists>d>0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
meet_def: fixes x :: "int"   and y :: "int" shows "int.meet x y = (THE inf. int.is_inf x y inf)"
nucleus_def: fixes f :: "'b \<Rightarrow> 'b" shows "nucleus f = (clop f \<and> (\<forall>x y. f x \<cdot> f y \<le> f (x \<cdot> y)))"
assms_4: fixes u :: "'t" assumes "component_of_term u < j" shows "is_sig_GB_in d G u"
is_offending_flows_min_set_def: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and f :: "('b \<times> 'b) set"   and G :: "'b graph"   and nP :: "'b \<Rightarrow> 'c" shows "SecurityInvariant_withOffendingFlows.is_offending_flows_min_set sinvar f G nP \<equiv> SecurityInvariant_withOffendingFlows.is_offending_flows sinvar f G nP \<and> (\<forall>(e1, e2)\<in>f. \<not> sinvar (add_edge e1 e2 (delete_edges G f)) nP)"
set_card: fixes a :: "'s" assumes "ide a" shows "|local.set a| <o \<AA>"
equivP_def: fixes A :: "'b \<Rightarrow> bool"   and \<phi> :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "equivP A \<phi> \<equiv> reflP A \<phi> \<and> symP \<phi> \<and> transP \<phi>"
read_tmps_distinct_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "read_tmps_distinct ts \<equiv> \<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> distinct_read_tmps sb"
lemParallelTrans: fixes lineA :: "'a Line"   and lineB :: "'a Line"   and lineC :: "'a Line" assumes "lineA \<parallel>  lineB"   and "lineB \<parallel>  lineC"   and "direction lineB \<noteq> 0" shows "lineA \<parallel>  lineC"
mbufnt_take_add'_1: fixes f :: "event_data option list set list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and xss :: "event_data option list set list list"   and buf :: "event_data option list set list list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "event_data option list set list list"   and nts' :: "nat list"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and j' :: "nat"   and P' :: "String.literal \<Rightarrow> nat option"   and m :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set" assumes "mbufnt_take f z (mbufn_add xss buf) nts = (z', buf', nts')"   and "pred_mapping (\<lambda>x. x \<le> j) P"   and "pred_mapping (\<lambda>x. x \<le> j') P'"   and "rel_mapping (\<le>) P P'"   and "safe_regex m g r"   and "to_mregex r = (mr, \<phi>s)"   and "wf_mbufn' \<sigma> P V j n R r buf"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [progress_regex \<sigma> P r j..<j'] nts"   and "list_all3 list_all2 (map (\<lambda>\<phi> i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) \<phi>s) (map2 upt (map (\<lambda>\<phi>. Monitor.progress \<sigma> P \<phi> j) \<phi>s) (map (\<lambda>\<phi>. Monitor.progress \<sigma> P' \<phi> j') \<phi>s)) xss"   and "j \<le> j'" shows "wf_mbufn' \<sigma> P' V j' n R r buf'"
dom_eqI: fixes a :: "'a"   and f :: "'a" assumes "S.ide a"   and "S.seq f a" shows "S.dom f = a"
UP_cring_def: fixes R :: "('e, 'f) ring_scheme" shows "UP_cring R \<equiv> cring R"
mult_adjugate_det: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "A ** adjugate A = Finite_Cartesian_Product.mat (Determinants.det A)"
split_face_edges_f12_bet: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list"   and f12 :: "face"   and f21 :: "face" assumes "pre_split_face f ram1 ram2 vs"   and "(f12, f21) = split_face f ram1 ram2 vs"   and "vs \<noteq> []"   and "between (vertices f) ram1 ram2 = []" shows "\<E> f12 = {(hd vs, ram1), (ram1, ram2), (ram2, last vs)} \<union> Edges (rev vs)"
box_midpoint: fixes a :: "'b"   and b :: "'b" assumes "box a b \<noteq> {}" shows "(1 / 2) *\<^sub>R (a + b) \<in> box a b"
degree_m_eq_monic: fixes f :: "int poly" assumes "monic f"   and "1 < p ^ n" shows "q.degree_m f = degree f"
dirichlet_prod_code: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "dirichlet_prod f g n = (if n = 0 then 0::'b else fold_atLeastAtMost_nat (\<lambda>d acc. if d dvd n then f d * g (n div d) + acc else acc) 1 n (0::'b))"
wqo_on_trees: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "wqo_on P A" shows "wqo_on (tree_emb A P) (trees A)"
cross3_simps_14: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a + (b - c) = a + b - c"
rrts_def: shows "rrts = map (\<lambda>r. Complex r 0) rr"
prob_Max_IID_geometric_atMost: fixes n :: "nat"   and i :: "nat" assumes "p \<in> {0..1}" shows "measure_pmf.prob (H\<^sub>N n) {..i} = measure_pmf.prob (geometric_pmf p) {..i} ^ n"
mod_greater_zero_iff_not_dvd: fixes m :: "nat"   and n :: "nat" shows "(0 < m mod n) = (\<not> n dvd m)"
tag_name_eq2: fixes ptr' :: "(_) element_ptr" shows "|h \<turnstile> get_tag_name ptr'|\<^sub>r = |h' \<turnstile> get_tag_name ptr'|\<^sub>r"
index_mat_multcol_3: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and k :: "nat"   and a :: "'a" assumes "i < dim_row A"   and "j < dim_col A"   and "k \<noteq> j" shows "multcol k a A $$ (i, j) = A $$ (i, j)"
inj_on_mult': fixes x :: "nat"   and q :: "nat" assumes "coprime x q" shows "inj_on (\<lambda>b. x * b mod q) ({..<q} - {0})"
setops\<^sub>s\<^sub>s\<^sub>t_pair_image_cons_8: fixes X :: "'a list"   and F :: "(('fun, 'a) Term.term \<times> ('fun, 'a) Term.term) list"   and G :: "(('fun, 'a) Term.term \<times> ('fun, 'a) Term.term) list"   and S :: "('fun, 'a) stateful_strand_step list" shows "pair ` setops\<^sub>s\<^sub>s\<^sub>t (\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> # S) = pair ` set G \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t S"
continuous_map_coordinatewise_then_product: fixes I :: "'c set"   and T1 :: "'d topology"   and T :: "'c \<Rightarrow> 'e topology"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'e" assumes "\<And>i. i \<in> I \<Longrightarrow> continuous_map T1 (T i) (\<lambda>x. f x i)"   and "\<And>i x. \<lbrakk>i \<notin> I; x \<in> topspace T1\<rbrakk> \<Longrightarrow> f x i = undefined" shows "continuous_map T1 (product_topology T I) f"
prime_factorization_gcd: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)"   and "b \<noteq> (0::'a)" shows "prime_factorization (gcd a b) = prime_factorization a \<inter># prime_factorization b"
path_component_in_path_components_of: fixes X :: "'a topology"   and a :: "'a" shows "(path_component_of_set X a \<in> path_components_of X) = (a \<in> topspace X)"
set_mset_filter: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c multiset" shows "set_mset (filter_mset P M) = {a. a \<in># M \<and> P a}"
p_def: shows "p = take \<^bold>|s\<^bold>| w"
rep_eq: fixes x :: "'c prefix"   and xa :: "'c prefix" shows "(x \<le> xa) = (\<exists>r. Rep_prefix xa = Rep_prefix x @ r)"
Suc_6: fixes k :: "nat"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "i \<le> k"   and "k < j"   and "I k h"   and "effect (f k) h h' r" shows "I (Suc k) h'"
red_ord: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red F p q" shows "q \<prec>\<^sub>p p"
that: fixes t :: "real" assumes "0 \<le> t"   and "t < 2 * pi"   and "x = cos t"   and "y = sin t" shows "thesis"
preserves_hom: fixes f :: "'b \<times> 'b \<times> 'b"   and a :: "'b \<times> 'b \<times> 'b"   and b :: "'b \<times> 'b \<times> 'b" assumes "VVV.in_hom f a b" shows "\<guillemotleft>\<a>[fst f, fst (snd f), snd (snd f)] : HoHV a \<Rightarrow> HoVH b\<guillemotright>"
exec_serial_then_path_to': fixes s :: "nat \<Rightarrow> nat option"   and ops :: "(nat, nat) sas_plus_operator list" assumes "s \<in> valid_states"   and "\<forall>op\<in>set ops. \<exists>ast_op\<in>set ast\<delta>. op = abs_ast_operator ast_op"   and "sat_preconds_as s ops" shows "path_to s (map (fst \<circ> the \<circ> lookup_action) ops) (execute_serial_plan_sas_plus s ops)"
simps_4: fixes x2 :: "'b"   and x1 :: "'a" shows "Inr x2 \<noteq> Inl x1"
wlp_while_mono: fixes WS :: "complex mat \<Rightarrow> complex mat"   and M :: "nat \<Rightarrow> complex mat"   and P :: "complex mat"   and Q :: "complex mat" assumes "\<And>P. is_quantum_predicate P \<Longrightarrow> is_quantum_predicate (WS P)"   and "\<And>P Q. \<lbrakk>is_quantum_predicate P; is_quantum_predicate Q; P \<le>\<^sub>L Q\<rbrakk> \<Longrightarrow> WS P \<le>\<^sub>L WS Q"   and "measurement d 2 M"   and "is_quantum_predicate P"   and "is_quantum_predicate Q"   and "P \<le>\<^sub>L Q" shows "wlp_while (M 0) (M 1) WS P \<le>\<^sub>L wlp_while (M 0) (M 1) WS Q"
assms_2: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t"
class_methods_f_def: fixes cld :: "cld" shows "class_methods_f cld = (case cld of cld_def dcl cl fds mds \<Rightarrow> mds)"
poly_reflect_poly_nz: fixes x :: "'b"   and p :: "'b poly" assumes "x \<noteq> (0::'b)" shows "poly (reflect_poly p) x = x ^ degree p * poly p (inverse x)"
between_not_r1: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "distinct vs" shows "ram1 \<notin> set (between vs ram1 ram2)"
flip: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "bounded_bilinear prod" shows "bounded_bilinear (\<lambda>x y. prod y x)"
pair_is_static_simp: fixes x :: "'a"   and s :: "'b" shows "is_static (x, s) = is_static s"
nn_integral_eq_integral: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> real" assumes "integrable M f"   and "AE x in M. 0 \<le> f x" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal (integral\<^sup>L M f)"
conforms_xconf: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv"   and x' :: "abopt" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "\<forall>a. x' = Some (Xcpt (Loc a)) \<longrightarrow> G,s\<turnstile>Addr a\<Colon>\<preceq>Class (SXcpt Throwable)"   and "x' = Some (Jump Ret) \<longrightarrow> locals s Result \<noteq> None" shows "(x', s)\<Colon>\<preceq>(G, L)"
subset_self: fixes A :: "'c set" shows "A \<subseteq> A"
compare_self_neq_LT: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and compare :: "'b \<rightarrow> 'b \<rightarrow> Ordering"   and x :: "'b" assumes "class.Ord_linear eq compare" shows "compare\<cdot>x\<cdot>x \<noteq> LT"
distinct_9: fixes x2 :: "'b" shows "right x2 \<noteq> ack"
mono: fixes xs :: "'a llist" shows "mono_llist_lift (\<lambda>ldrop. F ldrop xs)"
remove_code_1: fixes x :: "'c"   and xs :: "'c list" shows "Set.remove x (set xs) = set (removeAll x xs)"
foldli_mono_dres_flat: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'g dres"   and f' :: "'f \<Rightarrow> 'g \<Rightarrow> 'g dres"   and l :: "'f list"   and c :: "'g \<Rightarrow> bool"   and \<sigma> :: "'g dres" assumes "\<And>a x. flat_ge (f a x) (f' a x)" shows "flat_ge (foldli l (case_dres False False c) (\<lambda>x s. s \<bind> f x) \<sigma>) (foldli l (case_dres False False c) (\<lambda>x s. s \<bind> f' x) \<sigma>)"
univ_poly_add: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set" shows "(\<oplus>\<^bsub>K [X]\<^bsub>R\<^esub>\<^esub>) = ring.poly_add R"
assms_2: shows "\<sigma>\<^bsup>k\<^esup> \<restriction> local.reads (path \<sigma> k) = \<sigma>'\<^bsup>k'\<^esup> \<restriction> local.reads (path \<sigma> k)"
elims: fixes x :: "'a fps"   and xa :: "'a"   and xb :: "nat"   and y :: "'a" assumes "fps_right_inverse_constructor x xa xb = y" obtains f :: "'a fps" and a :: "'a" where "x = f__" and "xa = a__" and "xb = 0" and "y = a__"   | f :: "'a fps" and a :: "'a" and v :: "nat" where "x = f" and "xa = a" and "xb = Suc v" and "y = - a * (\<Sum>i = 1..Suc v. fps_nth f i * fps_right_inverse_constructor f a (Suc v - i))"
step: fixes CC' :: "'b literal multiset multiset"   and D :: "'b literal multiset"   and E :: "'b literal multiset"   and \<Gamma> :: "'b inference set"   and CC :: "'b literal multiset set" assumes "Infer CC' D E \<in> \<Gamma>"   and "\<And>C'. C' \<in># CC' \<Longrightarrow> C' \<in> inference_system.saturate \<Gamma> CC"   and "D \<in> inference_system.saturate \<Gamma> CC" shows "E \<in> inference_system.saturate \<Gamma> CC"
k_def: shows "k \<equiv> card {p. p \<in># prime_factorization n \<and> odd p}"
Int_emptyI: fixes A :: "'c set"   and B :: "'c set" assumes "\<And>x. \<lbrakk>x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> False" shows "A \<inter> B = {}"
simps_10: fixes v :: "Term.var"   and e :: "expr" shows "constVal (v:=e) = None"
assms_2: fixes x :: "nat" shows "g (length vs + x) = f x"
assms_2: shows "(g, h) \<in> A\<^sub>2"
cup_dim: shows "rat_poly.row_length (brickmat cup) = 1"
vlrestriction_vunion: fixes r :: "V"   and s :: "V"   and A :: "V" shows "(r \<union>\<^sub>\<circ> s) \<restriction>\<^sup>l\<^sub>\<circ> A = r \<restriction>\<^sup>l\<^sub>\<circ> A \<union>\<^sub>\<circ> s \<restriction>\<^sup>l\<^sub>\<circ> A"
simps_1: fixes lc :: "'b"   and q :: "'b poly"   and r :: "'b poly"   and d :: "'b poly"   and dr :: "nat"   and n :: "nat" shows "pseudo_divmod_main lc q r d dr (Suc n) = (let rr = Polynomial.smult lc r; qq = coeff r dr; rrr = rr - monom qq n * d; qqq = Polynomial.smult lc q + monom qq n in pseudo_divmod_main lc qqq rrr d (dr - 1) n)"
cardSuc_ordLess_ordLeq: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Card_order r"   and "Card_order r'" shows "(r <o r') = (cardSuc r \<le>o r')"
that: shows "\<I>_full,\<I>_full \<turnstile>\<^sub>C conv1 \<sim> conv2"
ics_SDG_path_split: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n'' :: "'node SDG_node" assumes "n ics-ns\<rightarrow>\<^sub>d* n'"   and "n'' \<in> set ns" obtains ns' :: "'node SDG_node list" and ns'' :: "'node SDG_node list" where "ns = ns' @ ns''" and "n ics-ns'\<rightarrow>\<^sub>d* n''" and "n'' ics-ns''\<rightarrow>\<^sub>d* n'"
gen_def: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and cs :: "'c list"   and w :: "'d" shows "gen P cs w \<equiv> \<forall>c\<in>set cs. P c w"
"case": fixes f :: "String.literal \<Rightarrow> tm list \<Rightarrow> 'a"   and x1 :: "String.literal"   and x2 :: "tm list" shows "(case Rl x1 x2 of Rl x xa \<Rightarrow> f x xa) = f x1 x2"
rec_2: fixes f1 :: "brick list \<Rightarrow> 'a"   and f2 :: "brick list \<Rightarrow> wall \<Rightarrow> 'a \<Rightarrow> 'a"   and x21 :: "brick list"   and x22 :: "wall" shows "rec_wall f1 f2 (x21 * x22) = f2 x21 x22 (rec_wall f1 f2 x22)"
A_greater_zero: fixes p :: "nat"   and a :: "int"   and x :: "int" assumes "GAUSS p a"   and "x \<in> GAUSS.A p" shows "0 < x"
subgraph_comb_planar: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph" assumes "subgraph G H"   and "comb_planar H"   and "\<exists>rev. bidirected_digraph G rev" shows "comb_planar G"
ODE_sem_OProd: fixes I :: "('b, 'c, 'a) interp"   and ODE1 :: "('b, 'a) ODE"   and ODE2 :: "('b, 'a) ODE" shows "ODE_sem I (OProd ODE1 ODE2) = (\<lambda>\<nu>. ODE_sem I ODE1 \<nu> + ODE_sem I ODE2 \<nu>)"
normal_form_ideal_Polys: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "F \<subseteq> P[X]" shows "normal_form (ideal F \<inter> P[X]) = normal_form F"
NN_imp_xx': fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "\<chi> \<in> fmla" shows "bprv (imp (subst \<chi> \<langle>neg \<phi>\<rangle> xx') (all xx' (imp (NN \<langle>\<phi>\<rangle> (Var xx')) \<chi>)))"
divideC_field_simps_1: fixes c :: "complex"   and a :: "'d"   and b :: "'d" assumes "c \<noteq> 0" shows "(a = b /\<^sub>C c) = (c *\<^sub>C a = b)"
eq_sym_conv: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (b = a)"
filter_tame_succs: fixes P :: "graph \<Rightarrow> bool"   and succs :: "graph \<Rightarrow> graph list"   and ok :: "graph \<Rightarrow> bool"   and g :: "graph"   and g' :: "graph" assumes "invariant P succs"   and "\<And>g. final g \<Longrightarrow> succs g = []"   and "\<And>g. \<lbrakk>P g; \<not> ok g\<rbrakk> \<Longrightarrow> final g \<and> \<not> tame g"   and "g [succs]\<rightarrow>* g'"   and "P g"   and "final g'"   and "tame g'" shows "g [filter ok \<circ> succs]\<rightarrow>* g'"
MkIde_Cod': fixes a :: "(bool, 'arr list) concrete_category.arr" assumes "ide a" shows "MkIde (Cod a) = a"
star_mult_circ: fixes x :: "'a" shows "x\<^sup>\<star> * x\<^sup>\<circ> = x\<^sup>\<circ>"
minVertex_zero2: fixes z :: "nat" shows "minVertex (Face (rev [0..<Suc z]) Nonfinal) = 0"
subset_of_face_of_affine_hull: fixes T :: "'b set"   and S :: "'b set"   and U :: "'b set" assumes "T face_of S"   and "convex S"   and "U \<subseteq> S"   and "\<not> disjnt (affine hull T) (rel_interior U)" shows "U \<subseteq> T"