hcju/isabelleps huggingface dataset:
J: shows "f = d * (s' + a' * b') \<and> NonceS (S (Card n, n, run)) = Some s' \<and> IntMapK (S (Card n, n, run)) = Some b' \<and> ExtMapK (S (Card n, n, run)) = Some a' \<and> IntAgrK (S (Card n, n, run)) = Some d \<and> d \<noteq> 0 \<and> s' + a' * b' \<noteq> 0"
geodesic_segment_topology_7: fixes G :: "'b set" assumes "geodesic_segment G" shows "proper G"
resCap_gzero: fixes p :: "(nat \<times> nat) list" assumes "isAugmentingPath p" shows "(0::'capacity) < resCap p"
nonzero_mult_div_cancel_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.semidom_divide divide plus minus zero times one"   and "b \<noteq> zero" shows "divide (times a b) b = a"
nn_integral_Markov_inequality: fixes u :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and A :: "'a set"   and c :: "ennreal" assumes "u \<in> borel_measurable M"   and "A \<in> sets M" shows "emeasure M ({x \<in> space M. 1 \<le> c * u x} \<inter> A) \<le> c * set_nn_integral M A u"
wary_subst_ground_heads: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary_subst \<rho>" shows "local.ground_heads (head (subst \<rho> s)) \<subseteq> local.ground_heads (head s)"
degree_sum_unique: fixes I :: "'b set"   and i :: "'b"   and p :: "'b \<Rightarrow> 'c poly" assumes "finite I"   and "i \<in> I"   and "\<And>j. \<lbrakk>j \<in> I; j \<noteq> i\<rbrakk> \<Longrightarrow> degree (p j) < degree (p i)" shows "degree (sum p I) = degree (p i)"
Sigma_Int_Paircomp2: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and P :: "'b \<Rightarrow> bool" shows "Sigma A B \<inter> {(x, y). P y} = (SIGMA z:A. B z \<inter> {y. P y})"
sum_easier: fixes L :: "(nat \<times> int \<times> nat) list"   and E :: "(nat \<times> int \<times> nat) list" assumes "uGraph_from_list_invar L"   and "set E \<subseteq> set L" shows "sum (uGraph_from_list_\<alpha>_weight L) (uGraph_from_list_\<alpha>_edges E) = (\<Sum>(u, w, v)\<in>set E. w)"
set_intros_1: fixes x21 :: "'b"   and x22 :: "'b tree"   and x23 :: "'b tree"   and x24 :: "nat" shows "x21 \<in> set_of (MKT x21 x22 x23 x24)"
finite_ReZ_segments_def: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" shows "finite_ReZ_segments g z = finite_Psegments (\<lambda>t. Re (g t - z) = 0) 0 1"
segment_Expre: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Worder D"   and "a \<in> segment D b" shows "ExPre (Iod D (segment D b)) a = ExPre D a"
inj_def: fixes f :: "'h \<Rightarrow> 'i" shows "inj f = (\<forall>x y. f x = f y \<longrightarrow> x = y)"
butlast_append: fixes xs :: "'g list"   and ys :: "'g list" shows "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
NegI: fixes a :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "a # G \<turnstile> FF" shows "G \<turnstile> Neg a"
even_of_nat_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (semiring_1.of_nat one plus zero n) = even n"
acc_sim1: fixes G :: "('a, 'b) gb_graph_rec_scheme"   and f :: "'a \<Rightarrow> 'c"   and r :: "nat \<Rightarrow> 'a"   and ecnv :: "('a, 'b) gb_graph_rec_scheme \<Rightarrow> 'd" assumes "gb_rename_precond G f"   and "gb_graph.is_acc G r" shows "gb_graph.is_acc (gb_rename_ext ecnv f G) (f \<circ> r)"
sup_None_1: fixes y :: "'b option" shows "None \<squnion> y = y"
ternary_lift3: fixes tv :: "ternaryvalue" shows "(eval_ternary_Not tv = TernaryFalse) = (tv = TernaryTrue)"
isin_case: fixes b :: "bool"   and m :: "'b \<Rightarrow> 'b trie option"   and xs :: "'b list" shows "isin (Nd b m) xs = (case xs of [] \<Rightarrow> b | x # ys \<Rightarrow> case m x of None \<Rightarrow> False | Some t \<Rightarrow> isin t ys)"
cond_bind_pmf: fixes A :: "'h set"   and f :: "'i \<Rightarrow> 'h pmf"   and p :: "'i pmf" assumes "A \<inter> (\<Union>x\<in>set_pmf p. set_pmf (f x)) \<noteq> {}" shows "cond_pmf (p \<bind> f) A = cond_bind_pmf p f A \<bind> (\<lambda>x. cond_pmf (f x) A)"
scale: fixes r :: "'a"   and b :: "'a" shows "D (r * b) = r *s D b"
finite_UN: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and M :: "'b measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> fmeasurable M" shows "\<Union> (A ` I) \<in> fmeasurable M"
absorb1: fixes a :: "'a upred"   and b :: "'a upred" assumes "b \<sqsubseteq> a" shows "(a \<and> b) = a"
finsert_eq_iff: fixes a :: "'b"   and A :: "'b fset"   and b :: "'b"   and B :: "'b fset" assumes "a |\<notin>| A"   and "b |\<notin>| B" shows "(finsert a A = finsert b B) = (if a = b then A = B else \<exists>C. A = finsert b C \<and> b |\<notin>| C \<and> B = finsert a C \<and> a |\<notin>| C)"
Prod_any_power_lookup_eq: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b \<Rightarrow>\<^sub>0 nat" shows "(\<Prod>k. f k ^ lookup x k) = (\<Prod>k\<in>keys x. f k ^ lookup x k)"
base: fixes s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and e :: "((nat \<times> nat) \<times> transition) fset" shows "obtains s r e s r []"
inj_comm_ring_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_comm_ring_hom hom \<equiv> comm_ring_hom hom \<and> zero_hom_0 hom"
semiring_norm_3: fixes n :: "num" shows "num.One + num.Bit0 n = num.Bit1 n"
binomial: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "(x \<oplus> y) [^] n = (\<Oplus>i\<in>{..n}. [(n choose i)] \<cdot> (x [^] i \<otimes> y [^] (n - i)))"
Gromov_product_at_def: fixes e :: "'c"   and x :: "'c"   and y :: "'c" shows "Gromov_product_at e x y = (dist e x + dist e y - dist x y) / 2"
projection_insert: fixes S :: "'b set"   and a :: "'b"   and u :: "'b" assumes "\<And>s. s \<in> S \<Longrightarrow> is_orthogonal a s" shows "projection (closure (cspan (insert a S))) u = projection (cspan {a}) u + projection (closure (cspan S)) u"
horner_sum_eq_sum: fixes f :: "'c \<Rightarrow> 'a"   and a :: "'a"   and xs :: "'c list" shows "horner_sum f a xs = (\<Sum>n = 0..<length xs. f (xs ! n) * a ^ n)"
empty_path_nodes: fixes n :: "'b LDCFG_node"   and n' :: "'b LDCFG_node" assumes "CFGExit_wf.path n [] n'" shows "n = n'"
Cons_2: shows "\<forall>a\<in>set (a # as). 1 \<le> a"
Propagate_Black: shows "\<turnstile> Propagate_Black  \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> (\<acute>obc \<subset> Blacks \<acute>M \<or> \<acute>Safe)\<rbrace>"
assms_3: fixes b :: "'t \<Rightarrow>\<^sub>0 'b"   and c :: "'b"   and t :: "'a" assumes "b \<in> B" shows "monom_mult c t b \<in> B"
intros_19: fixes t1 :: "t"   and v :: "v"   and t2 :: "t"   and sx :: "sx option"   and v' :: "v" assumes "types_agree t1 v"   and "cvt t2 sx v = Some v'" shows "\<lparr>[$C v, $Cvtop t2 Convert t1 sx]\<rparr> \<leadsto> \<lparr>[$C v']\<rparr>"
co_path_nonempty: fixes e :: "'b \<times> 'b"   and w :: "'b"   and p :: "('b \<times> 'b) list" shows "(co_path e w p = []) = (p = [])"
Aizerman_onD: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set"   and C :: "'a set" assumes "Aizerman_on A f"   and "B \<subseteq> A"   and "C \<subseteq> B"   and "f B \<subseteq> C" shows "f C \<subseteq> f B"
fps_deriv_neg: fixes f :: "'a fps" shows "fps_deriv (- f) = - fps_deriv f"
Liminf_llist_def: fixes Xs :: "'a set llist" shows "Liminf_llist Xs = (\<Union>i\<in>{i. enat i < llength Xs}. \<Inter> (lnth Xs ` {j. i \<le> j \<and> enat j < llength Xs}))"
multiple_proper_design: fixes n :: "nat" assumes "0 < n" shows "proper_design \<V> (d.multiple_blocks n)"
add_diff_assoc2: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b" shows "b - a + c = b + c - a"
lmap_transfer: fixes A :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_fun (rel_fun A B) (rel_fun (llist_all2 A) (llist_all2 B)) lmap lmap"
bprules_3: shows "\<Union> (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<union> {{v} |v. v \<in> V}) = U"
exhaust: fixes y :: "cmd" obtains x1 :: "nat" and x2 :: "trm" where "y = <x1>x2"
ideD_1: fixes a :: "(bool, bool list) concrete_category.arr" assumes "J.ide a" shows "J.arr a"
notin_set_remove1: fixes x :: "'k"   and xs :: "'k list"   and y :: "'k" assumes "x \<notin> set xs" shows "x \<notin> set (remove1 y xs)"
select_convs_2: fixes mag :: "'b"   and dim :: "(int, 'a) dimvec"   and more :: "'c" shows "Quantity.dim \<lparr>mag = mag, dim = dim, \<dots> = more\<rparr> = dim"
pt_subseteq_eqvt: fixes pi :: "('b \<times> 'b) list"   and X :: "'a set"   and Y :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> (X \<subseteq> Y) = (pi \<bullet> X \<subseteq> pi \<bullet> Y)"
induct: fixes x1 :: "boolean_binop"   and x2 :: "'a OCL_Types.type"   and x3 :: "'a OCL_Types.type"   and x4 :: "'a OCL_Types.type"   and P :: "boolean_binop \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "boolean_binop_type x1 x2 x3 x4"   and "\<And>\<tau> \<sigma> \<rho>. \<lbrakk>\<tau> \<squnion> \<sigma> = \<rho>; \<rho> \<le> Boolean[?]\<rbrakk> \<Longrightarrow> P AndOp \<tau> \<sigma> \<rho>"   and "\<And>\<tau> \<sigma> \<rho>. \<lbrakk>\<tau> \<squnion> \<sigma> = \<rho>; \<rho> \<le> Boolean[?]\<rbrakk> \<Longrightarrow> P OrOp \<tau> \<sigma> \<rho>"   and "\<And>\<tau> \<sigma> \<rho>. \<lbrakk>\<tau> \<squnion> \<sigma> = \<rho>; \<rho> \<le> Boolean[?]\<rbrakk> \<Longrightarrow> P XorOp \<tau> \<sigma> \<rho>"   and "\<And>\<tau> \<sigma> \<rho>. \<lbrakk>\<tau> \<squnion> \<sigma> = \<rho>; \<rho> \<le> Boolean[?]\<rbrakk> \<Longrightarrow> P ImpliesOp \<tau> \<sigma> \<rho>" shows "P x1 x2 x3 x4"
no_updates_none_aux: fixes j :: "state stream"   and e :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat \<Rightarrow>f value option" assumes "\<exists>p i. j = make_full_observation e None r p i" shows "alw (\<lambda>x. datastate (shd x) = r) j"
msubstleneg: fixes vs :: "'b list"   and c :: "poly"   and a :: "poly"   and b :: "tm"   and x :: "'b"   and bs :: "'b list"   and t :: "tm" assumes "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> < (0::'b)"   and "islin (Le (CNP 0 a b))" shows "Ifm vs (x # bs) (msubstleneg c t a b) = Ifm vs (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> # bs) (Le (CNP 0 a b))"
linord_of_list_Linear_order: fixes xs :: "'a list"   and ys :: "'a set" assumes "distinct xs"   and "ys = set xs" shows "linear_order_on ys (linord_of_list xs)"
assms_2: shows "set (vwalk_arcs p) \<subseteq> arcs_ends G"
cons_to_append: fixes list :: "'a list"   and a :: "'a" shows "list \<noteq> [] \<longrightarrow> (\<exists>ls. a # list = ls @ [last list])"
that: fixes a :: "V"   and b :: "V" assumes "ab = [a, b]\<^sub>\<circ>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "thesis"
pred_set: fixes P1 :: "'d \<Rightarrow> bool"   and P2 :: "'e \<Rightarrow> bool" shows "pred_gpv P1 P2 = (\<lambda>x. Ball (results'_gpv x) P1 \<and> Ball (outs'_gpv x) P2)"
conjuncive_demonic: fixes r :: "'e \<Rightarrow> 'd" shows "conjunctive [: r :]"
HInit_HInv2: fixes s :: "state" assumes "HInit s" shows "HInv2 s"
simps_1: fixes i :: "nat"   and a :: "'a" shows "insertl i a [] = [a]"
OF_match_linear_append: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "('c, 'b) flow_entry_match list"   and b :: "('c, 'b) flow_entry_match list"   and p :: "'d" shows "OF_match_linear \<gamma> (a @ b) p = (case OF_match_linear \<gamma> a p of Action a \<Rightarrow> Action a | NoAction \<Rightarrow> OF_match_linear \<gamma> b p | Undefined \<Rightarrow> Undefined)"
continuous_bot: fixes f :: "'a \<Rightarrow> 'b" shows "continuous bot f"
leneq: shows "a * \<^bold>|x\<^bold>| + b * \<^bold>|y\<^bold>| = c * \<^bold>|z\<^bold>|"
assms_3: fixes x :: "'a" shows "0 \<le> f x"
AE_all_imp_countable: fixes Q :: "'a \<Rightarrow> bool"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'b measure" assumes "countable {x. Q x}" shows "(AE x in M. \<forall>y. Q y \<longrightarrow> P x y) = (\<forall>y. Q y \<longrightarrow> (AE x in M. P x y))"
x_eq: shows "x \<in> grid (start dm) ({0..<dm} - ds) \<and> p \<in> grid x ds"
token_run_step': fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "token_run x (Suc n) = \<delta> (token_run x n) (w n)"
assms_6: shows "fmul (0::'a) (0::'b) = (0::'c)"
Bseq_def: fixes X :: "nat \<Rightarrow> 'a" shows "Bseq X = (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
m_assoc: fixes x :: "rat"   and y :: "rat"   and z :: "rat" shows "x + y + z = x + (y + z)"
upd_row_transpose: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and p :: "('c, 'd) vec" shows "upd_row (Square_Matrix.transpose M) i p = Square_Matrix.transpose (upd_col M i p)"
bounded_gen_check_axioms: shows "bounded_gen_check C\<^sub>2 B\<^sub>2"
terminates_to_terminates_normalize: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>c \<down> s" shows "\<Gamma>\<turnstile>Language.normalize c \<down> s"
n_not_Suc_n: fixes n :: "nat" shows "n \<noteq> Suc n"
simps: fixes rs :: "(name \<times> pterm) fset"   and a1 :: "pterm"   and a2 :: "pterm" shows "(rs \<turnstile>\<^sub>p a1 \<longrightarrow> a2) = ((\<exists>name rhs. a1 = Pconst name \<and> a2 = rhs \<and> (name, rhs) |\<in>| rs) \<or> (\<exists>c cs t t'. a1 = Pabs cs $\<^sub>p t \<and> a2 = t' \<and> c |\<in>| cs \<and> c \<turnstile> t \<rightarrow> t') \<or> (\<exists>t t' u. a1 = t $\<^sub>p u \<and> a2 = t' $\<^sub>p u \<and> rs \<turnstile>\<^sub>p t \<longrightarrow> t') \<or> (\<exists>u u' t. a1 = t $\<^sub>p u \<and> a2 = t $\<^sub>p u' \<and> rs \<turnstile>\<^sub>p u \<longrightarrow> u'))"
funion_image_bind_eq: fixes f :: "'d \<Rightarrow> 'c fset"   and M :: "'d fset" shows "ffUnion (f |`| M) = fbind M f"
mfinal1_simulation: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and s1 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s2 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('g \<times> 'd \<Rightarrow>f nat) option) \<times> 'h) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list" assumes "FWdelay_bisimulation_final_base final1 r1 final2 r2 bisim \<tau>move1 \<tau>move2"   and "FWbisimulation_base.mbisim bisim bisim_wait s1 s2"   and "final_thread.mfinal final1 s1" shows "\<exists>s2'. (\<tau>trsys.silent_move (multithreaded_base.redT final2 r2 convert_RA) (\<tau>multithreaded.m\<tau>move \<tau>move2))\<^sup>*\<^sup>* s2 s2' \<and> FWbisimulation_base.mbisim bisim bisim_wait s1 s2' \<and> final_thread.mfinal final2 s2' \<and> shr s2' = shr s2"
lookup_update_by_fun: fixes k :: "'c"   and f :: "'b \<Rightarrow> 'b"   and xs :: "('c, 'b) oalist_ntm"   and k' :: "'c" shows "OAlist_lookup_ntm (OAlist_update_by_fun_ntm k f xs) k' = (if k = k' then f else id) (OAlist_lookup_ntm xs k')"
bidirected_digraph: shows "bidirected_digraph (with_proj G) (edge_rev M)"
f_nonneg: fixes x :: "real" assumes "a < ereal x"   and "ereal x < b" shows "0 \<le> f (g x)"
Decompose: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and T :: "('fun, 'var) Term.term list"   and t\<^sub>i :: "('fun, 'var) Term.term" assumes "M \<turnstile> t"   and "Ana t = (K, T)"   and "\<And>k. k \<in> set K \<Longrightarrow> M \<turnstile> k"   and "t\<^sub>i \<in> set T" shows "M \<turnstile> t\<^sub>i"
a_a_one: shows "d (1::'a) = (1::'a)"
Do: fixes f :: "state \<Rightarrow> state set"   and P :: "state \<Rightarrow> bool" shows "\<turnstile> {\<lambda>s. \<forall>t\<in>f s. P t} Do f {P}"
singleton_list_hd_last: fixes xs :: "'b list" assumes "length xs = 1" shows "hd xs = last xs"
add_strict_mono: fixes \<alpha> :: "V"   and \<beta> :: "V"   and \<gamma> :: "V"   and \<delta> :: "V" assumes "\<alpha> < \<beta>"   and "\<gamma> < \<delta>"   and "Ord \<alpha>"   and "Ord \<beta>"   and "Ord \<gamma>"   and "Ord \<delta>" shows "\<alpha> + \<gamma> < \<beta> + \<delta>"
strip_while_def: fixes P :: "'f \<Rightarrow> bool" shows "strip_while P = rev \<circ> dropWhile P \<circ> rev"
sum_mset_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.sum_mset plus zero \<equiv> comm_monoid_mset.F plus zero"
suc_def: fixes n :: "'n"   and \<sigma> :: "'var \<Rightarrow> 'val" shows "suc n \<sigma> = fst (local.step (n, \<sigma>))"
Lt_Le_dbm_lt_D: fixes a :: "'d"   and b :: "'d" assumes "Lt a \<prec> Le b" shows "a \<le> b"
sigma_prod: fixes A :: "'c set set"   and X :: "'c set"   and B :: "'d set set"   and Y :: "'d set" assumes "\<exists>E\<subseteq>A. countable E \<and> X = \<Union> E"   and "A \<subseteq> Pow X"   and "\<exists>E\<subseteq>B. countable E \<and> Y = \<Union> E"   and "B \<subseteq> Pow Y" shows "sigma X A \<Otimes>\<^sub>M sigma Y B = sigma (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
gbij_comp_bij: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and H :: "('i, 'j) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and g :: "'g \<Rightarrow> 'i" assumes "Group F"   and "Group G"   and "Group H"   and "gbij\<^bsub>F,G\<^esub> f"   and "gbij\<^bsub>G,H\<^esub> g" shows "gbij\<^bsub>F,H\<^esub> (g \<circ>\<^bsub>F\<^esub> f)"
closed_vimage_Int: fixes s :: "'c set"   and t :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "closed s"   and "continuous_on t f"   and "closed t" shows "closed (f -` s \<inter> t)"
sources_def: fixes \<mu> :: "'a" shows "sources \<mu> \<equiv> {a. weak_unit a \<and> \<mu> \<star> a \<noteq> null}"
norm_cauchy_schwarz_equal: fixes x :: "'b"   and y :: "'b" shows "(\<bar>x \<bullet> y\<bar> = norm x * norm y) = collinear {0::'b, x, y}"
rel_funI: fixes A :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and B :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'j"   and g :: "'i \<Rightarrow> 'k" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "rel_fun A B f g"
ordertype_eqpoll: fixes A :: "V set" assumes "small A" shows "elts (ordertype A VWF) \<approx> A"
proj_sum_list: fixes k :: "nat"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "proj_poly k (sum_list xs) = sum_list (map (proj_poly k) xs)"
drop_append_drop_minus_idem: fixes n :: "nat"   and XS :: "'a list" assumes "n < length XS" shows "map ((!) XS) [0..<n] @ drop n XS = XS"
af_subformulas\<^sub>\<mu>: fixes \<phi> :: "'b ltln"   and w :: "'b set list" shows "subformulas\<^sub>\<mu> (af \<phi> w) = subformulas\<^sub>\<mu> \<phi>"
assms: shows "consistent \<sigma> \<sigma>\<^sub>s\<^sub>y\<^sub>m s"
rank_iarray_def: fixes A :: "'a iarray iarray" shows "rank_iarray A = (let A' = Gauss_Jordan_iarrays A; nrows = IArray.length A' in card {i. i < nrows \<and> \<not> is_zero_iarray (A' !! i)})"
equivalence_in_bicategory_axioms: shows "internal_equivalence (trg \<mu>) (trg \<mu>) (local.inv \<i>[trg \<mu>]) \<i>[trg \<mu>]"
sinvar_valid_remove_minimalize_offending_overapprox: fixes m :: "'b SecurityInvariant_configured"   and nodesG :: "'b set"   and edgesG :: "('b \<times> 'b) set"   and Es :: "('b \<times> 'b) list"   and nP :: "'b \<Rightarrow> 'c" assumes "configured_SecurityInvariant m"   and "wf_graph \<lparr>nodes = nodesG, edges = edgesG\<rparr>"   and "\<not> c_sinvar m \<lparr>nodes = nodesG, edges = edgesG\<rparr>"   and "set Es = edgesG"   and "distinct Es" shows "c_sinvar m \<lparr>nodes = nodesG, edges = edgesG - set (SecurityInvariant_withOffendingFlows.minimalize_offending_overapprox (\<lambda>G nP. c_sinvar m G) Es [] \<lparr>nodes = nodesG, edges = edgesG\<rparr> nP)\<rparr>"
preserves_modes_mem_mds\<^sub>A_simp: fixes mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set" assumes "\<forall>m. Eg2_var\<^sub>C_of_Eg1 ` mds\<^sub>A m = range Eg2_var\<^sub>C_of_Eg1 \<inter> mds\<^sub>C m" shows "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"
diagonal_imp_triangular_1x2: fixes A :: "'b mat" assumes "A \<in> carrier_mat 1 2"   and "admits_diagonal_reduction A" shows "admits_triangular_reduction A"
assms_1: shows "h \<turnstile> assigned_nodes slot \<rightarrow>\<^sub>r nodes"
sup_bot_right: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup x bot = x"
irreducible\<^sub>d_m_def: fixes f :: "int poly" shows "irreducible\<^sub>d_m f \<equiv> 0 < degree_m f \<and> (\<forall>g h. degree_m g < degree_m f \<longrightarrow> degree_m h < degree_m f \<longrightarrow> Mp f \<noteq> Mp (g * h))"
filterlim_poly_at_infinity: fixes p :: "'a poly" assumes "0 < degree p" shows "filterlim (poly p) at_infinity at_infinity"
simps_19: fixes x31 :: "acom"   and x32 :: "acom"   and x41 :: "bexp"   and x42 :: "acom"   and x43 :: "acom" shows "x31;; x32 \<noteq> IF x41 THEN x42 ELSE x43"
vec_nth_inject: fixes x :: "('d, 'c) vec"   and y :: "('d, 'c) vec" shows "(($) x = ($) y) = (x = y)"
cval_add_def: fixes u :: "'e \<Rightarrow> 'f"   and d :: "'f" shows "u \<oplus> d = (\<lambda>x. u x + d)"
insert_Diff_single: fixes a :: "'b"   and A :: "'b set" shows "insert a (A\<setminus>{a}) = insert a A"
zimage_iff: fixes y :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b zet" shows "zin y (zimage f A) = (\<exists>x. zin x A \<and> y = f x)"
vars_uminus: fixes p :: "linear_poly" shows "vars (- p) = vars p"
apps_preserves_beta: fixes r :: "dB"   and s :: "dB"   and ss :: "dB list" assumes "r \<rightarrow>\<^sub>\<beta> s" shows "r \<degree>\<degree> ss \<rightarrow>\<^sub>\<beta> s \<degree>\<degree> ss"
compile_fd1: shows "compile_fundef \<A> (\<O> f) fd1 = Ok fd2"
elims_3: fixes x :: "'a ext"   and xa :: "'a ext" assumes "\<not> x < xa" obtains x :: "'a ext" and y :: "'a ext" where "xa__ = x" and "xa = y" and "\<not> (x \<le> y \<and> \<not> y \<le> x)"
cod_lunit: fixes a :: "'a" assumes "ide a" shows "cod \<l>[a] = a"
midpoint_distinct_2: fixes I :: "'p"   and A :: "'p"   and B :: "'p" assumes "I \<noteq> A"   and "I Midpoint A B" shows "A \<noteq> B \<and> I \<noteq> B"
atLeast0_lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "sum g {0..<Suc n} = g 0 + sum (g \<circ> Suc) {0..<n}"
fdg: shows "f21 = fst (snd (splitFace g v u f ws))"
assms_2: fixes y :: "'a" assumes "y \<in> S" shows "orthogonal x y"
bd_Cnotzero: shows "(F'bd, czero) \<notin> ordIso \<and> Card_order F'bd"
eucl_of_env_def: fixes "is" :: "nat list"   and vs :: "real list" shows "eucl_of_env is vs = eucl_of_list (map ((!) vs) is)"
ss2: shows "set (map ((*\<^sub>v) A) ss) = (*\<^sub>v) A ` S"
orbit_eqI_2: fixes z :: "'b"   and f :: "'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b" assumes "z = f y"   and "y \<in> orbit f x" shows "z \<in> orbit f x"
F6: shows "(lub H cl, f (lub H cl)) \<in> r"