hcju/isabelleps huggingface dataset:
prems: shows "B \<in> set (tr\<^sub>p\<^sub>c ((i, \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle>) # S) D)"
finvimage_frange: fixes r :: "V" shows "r -`\<^sub>\<bullet> \<R>\<^sub>\<bullet> r = \<D>\<^sub>\<bullet> r"
cycleE: fixes xs :: "'c list" assumes "T.cycle xs" shows "T.path xs \<and> xs \<noteq> [] \<and> 2 < length xs \<and> last xs \<rightarrow>\<^bsub>T\<^esub> hd xs"
Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e_inject: fixes x :: "\<langle>\<langle>'a set\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "\<langle>\<langle>'a set\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "x \<in> {X. X = UML_Types.bot_class.bot \<or> X = null \<or> (\<forall>x\<in>\<lceil>\<lceil>X\<rceil>\<rceil>. x \<noteq> UML_Types.bot_class.bot)}"   and "y \<in> {X. X = UML_Types.bot_class.bot \<or> X = null \<or> (\<forall>x\<in>\<lceil>\<lceil>X\<rceil>\<rceil>. x \<noteq> UML_Types.bot_class.bot)}" shows "(Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e x = Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e y) = (x = y)"
get_WIM_bit_low_equal2: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "get_WIM_bit (nat ((uint (fst (fst (get_curr_win () s1))) + 1) mod NWINDOWS)) (cpu_reg_val WIM (snd (fst (get_curr_win () s1)))) = get_WIM_bit (nat ((uint (fst (fst (get_curr_win () s2))) + 1) mod NWINDOWS)) (cpu_reg_val WIM (snd (fst (get_curr_win () s2))))"
inject_1: fixes x1 :: "label"   and y1 :: "label" shows "(Break x1 = Break y1) = (x1 = y1)"
strict_trans: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "b \<prec> a"   and "c \<prec> b" shows "c \<prec> a"
analz_insert_simps_3: fixes A :: "'a set"   and B :: "'a set"   and a :: "'a" assumes "A \<subseteq> B" shows "A \<subseteq> insert a B"
rewrite_negated_primitives_preserves_not_has_disc_negated: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and disc2 :: "'b \<Rightarrow> bool"   and negate_f :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "has_disc_negated disc False m \<Longrightarrow> \<forall>pts. \<not> has_disc_negated disc2 False (negate_f C pts)"   and "\<not> has_disc_negated disc2 False m" shows "\<not> has_disc_negated disc2 False (rewrite_negated_primitives (disc, sel) C negate_f m)"
inv_into_injective: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "inv_into A f x = inv_into A f y"   and "x \<in> f ` A"   and "y \<in> f ` A" shows "x = y"
env_restr_esing: fixes x :: "'c"   and S :: "'c set"   and v :: "'d" assumes "x \<in> S" shows "esing x\<cdot>v f|` S = esing x\<cdot>v"
N_2: fixes n :: "nat" assumes "N \<le> n" shows "Sigma_Algebra.measure (distr P borel (\<lambda>x. (g x + birkhoff_sum f n x) / B n)) {x. (L - 1) * a \<le> \<bar>x\<bar>} < alpha"
emv: shows "\<forall>i>M. emv (g i) (g (Suc i))"
sup_boolE: fixes P :: "bool"   and Q :: "bool" assumes "P \<squnion> Q" obtains "P"   | "Q"
vec_tensor_prod_2: fixes v :: "complex Matrix.vec"   and w :: "complex Matrix.vec" assumes "v \<in> state_qbit 1"   and "w \<in> state_qbit 1" shows "v \<otimes> w = vec_of_list [v $ 0 * w $ 0, v $ 0 * w $ 1, v $ 1 * w $ 0, v $ 1 * w $ 1]"
sum_upto_\<theta>_x_over_n_asymptotics: shows "(\<lambda>x. sum_upto (\<lambda>xa. \<theta> (x / real xa)) x - x * ln x) \<in> O(\<lambda>x. x)"
path_valid_node_1: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit.path n as n'" shows "CFGExit.valid_node n"
f1_def: shows "f1 \<equiv> \<lambda>A. THE x. \<exists>a b c d. A = {a, b, c, d} \<and> a < b \<and> b < c \<and> c < d \<and> x = f {(a, c), (b, d)}"
regular_mult_closed: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_p_algebra.regular x"   and "matrix_p_algebra.regular y" shows "matrix_p_algebra.regular (x \<odot> y)"
multiplicative_function'_axioms: shows "multiplicative_function' f f_prime_power f_prime"
can_in_hom: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.Ide t"   and "E.Ide u"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>" shows "\<guillemotleft>B.can u t : EVAL t \<rightarrow>\<^sub>B EVAL u\<guillemotright>"
order_trans: fixes x :: "'b list"   and y :: "'b list"   and z :: "'b list" assumes "subseq x y"   and "subseq y z" shows "subseq x z"
items_def: fixes f :: "'b \<Rightarrow>\<^sub>0 'c" shows "items f = list.map (\<lambda>k. (k, lookup f k)) (sorted_list_of_set (keys f))"
add_decreasing: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq a zero"   and "less_eq c b" shows "less_eq (plus a c) b"
a1': shows "\<forall>x i. (Functions I i has_derivative (THE f'. \<forall>x. (Functions I i has_derivative f' x) (at x)) x) (at x)"
Act_eq_iff_perm_renaming: fixes \<alpha>1 :: "'c"   and x1 :: "('a, 'b, 'c) formula"   and \<alpha>2 :: "'c"   and x2 :: "('a, 'b, 'c) formula" shows "(Act \<alpha>1 x1 = Act \<alpha>2 x2) = (\<exists>p. supp x1 - bn \<alpha>1 = supp x2 - bn \<alpha>2 \<and> (supp x1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> x1 = x2 \<and> supp \<alpha>1 - bn \<alpha>1 = supp \<alpha>2 - bn \<alpha>2 \<and> (supp \<alpha>1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> \<alpha>1 = \<alpha>2 \<and> supp p \<subseteq> bn \<alpha>1 \<union> p \<bullet> bn \<alpha>1)"
mmap_def: shows "mmap \<equiv> \<lambda>n. Inf (hd (F (Suc n)))"
traverse_cast_Dyn_to_norm: fixes xs :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and ys :: "'dyn list" assumes "traverse cast_Dyn xs = Some ys" shows "norm_stack xs = ys"
assms_4: shows "graph_homomorphism (LG E V) (chain_sup S) f"
zlcms_pos: fixes "is" :: "int list" assumes "\<forall>i\<in>set is. i \<noteq> 0" shows "0 < zlcms is"
mbisim_bisimulation: shows "bisimulation r1.redT r2.redT mbisim mta_bisim"
map2_val_compat'_def: fixes f :: "('a \<times> 'e) list \<times> 'o \<Rightarrow> ('a \<times> 'f) list \<times> 'o" shows "map2_val_compat' f = (\<forall>zs. oalist_inv zs \<longrightarrow> oalist_inv (f zs) \<and> snd (f zs) = snd zs \<and> fst ` set (fst (f zs)) \<subseteq> fst ` set (fst zs))"
cases: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "flow \<Delta> f" obtains "\<And>e. f e \<le> capacity \<Delta> e" and "\<And>x. \<lbrakk>x \<noteq> source \<Delta>; x \<noteq> sink \<Delta>\<rbrakk> \<Longrightarrow> KIR f x"
sinits_length: fixes xs :: "[:'a:]" shows "slength\<cdot>(sinits\<cdot>xs) = slength\<cdot>xs + 1"
martingaleAE: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure"   and X :: "'c \<Rightarrow> 'b \<Rightarrow> real"   and t :: "'c"   and s :: "'c" assumes "martingale M F X"   and "t \<le> s" shows "AEeq M (real_cond_exp M (F t) (X s)) (X t)"
narrow1: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "y \<le> x \<triangle> y"
ir_conseq: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_hoare P c c' Q"   and "\<And>s s'. P s s' \<Longrightarrow> P' s s'"   and "\<And>s s'. Q' s s' \<Longrightarrow> Q s s'" shows "ir_hoare P' c c' Q'"
Ax: shows "x n \<in> carrier (residue_ring (p ^ n))"
axioms_1: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f" assumes "vector_space_pair s1 s2" shows "vector_space s1"
closedin_closed_trans: fixes T :: "'a set"   and S :: "'a set" assumes "closedin (top_of_set T) S"   and "closed T" shows "closed S"
Inner: fixes x :: "'v"   and y :: "'v" assumes "edge \<Delta> x y" shows "edge' (Inner x) (Inner y)"
sep_plus_prod_conv: fixes ps :: "'a"   and n :: "'b"   and ps' :: "'a"   and n' :: "'b" shows "(ps, n) + (ps', n') = (ps + ps', n + n')"
homotopic_compose: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and Z :: "'d topology"   and g :: "'c \<Rightarrow> 'd"   and g' :: "'c \<Rightarrow> 'd" assumes "homotopic_with (\<lambda>x. True) X Y f f'"   and "homotopic_with (\<lambda>x. True) Y Z g g'" shows "homotopic_with (\<lambda>x. True) X Z (g \<circ> f) (g' \<circ> f')"
ptd_lookup_unchanged_1: fixes va :: "word32"   and ptp :: "word32"   and s1 :: "('b, 'c) sparc_state_scheme"   and addr :: "word36"   and val :: "word8" shows "ptd_lookup va ptp (mem s1) 1 = ptd_lookup va ptp ((mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))) 1"
nu_isotone: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "has_greatest_postfixpoint f"   and "has_greatest_postfixpoint g"   and "isotone f"   and "isotone g"   and "f \<le>\<le> g" shows "\<nu> f \<le> \<nu> g"
prems_2: shows "A\<^sub>F \<sharp>* P"
if_redT_non_speculative_vs_conf: fixes s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and t :: "'thread_id"   and ta :: "('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action"   and s' :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and vs :: "'addr \<times> addr_loc \<Rightarrow> 'addr val set"   and n :: "nat" assumes "h.mthr.if.redT_syntax1 s t ta s'"   and "ts_ok (init_fin_lift wfx) (thr s) (shr s)"   and "non_speculative P vs (llist_of (take n \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>))"   and "h.vs_conf P (shr s) vs" shows "h.vs_conf P (shr s') (w_values P vs (take n \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>))"
step: shows "|~ I \<and> _actrans (n1 \<or> n2) vars \<longrightarrow> \<circle>I"
assms_8: shows "yf = (\<lambda>x. (- n - k1 * x) / k2)"
distinct_161: shows "GreaterOrEqual \<noteq> ShiftRightSigned"
DBM_reset_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry"   and n :: "nat"   and k :: "nat"   and d :: "'b"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry" shows "DBM_reset M n k d M' \<equiv> (\<forall>j\<le>n. 0 < j \<and> k \<noteq> j \<longrightarrow> M' k j = \<infinity> \<and> M' j k = \<infinity>) \<and> M' k 0 = Le d \<and> M' 0 k = Le (- d) \<and> M' k k = M k k \<and> (\<forall>i\<le>n. \<forall>j\<le>n. i \<noteq> k \<and> j \<noteq> k \<longrightarrow> M' i j = min (dbm_add (M i k) (M k j)) (M i j))"
is_cnf_encode_problem_with_operator_interference_exclusion: fixes \<Pi> :: "'a strips_problem"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>" shows "is_cnf (\<Phi>\<^sub>\<forall> \<Pi> t)"
CHOnextConfig_def: fixes A :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and r :: "nat"   and cfg :: "'a \<Rightarrow> 'b"   and HO :: "'a \<Rightarrow> 'a set"   and coord :: "'a \<Rightarrow> 'a"   and cfg' :: "'a \<Rightarrow> 'b" shows "CHOnextConfig A r cfg HO coord cfg' \<equiv> CSHOnextConfig A r cfg HO HO coord cfg'"
finvimageD2: fixes a :: "V"   and r :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> r -`\<^sub>\<bullet> A" shows "a \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<bullet> `\<^sub>\<bullet> A"
validFrom_def: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and s :: "'a"   and tr :: "('a, 'b, 'c) trans list" shows "IO_Automaton.validFrom step s tr \<equiv> tr = [] \<or> IO_Automaton.valid step tr \<and> srcOf (hd tr) = s"
main: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "p \<in> F"   and "q \<in> F"   and "p \<noteq> 0"   and "q \<noteq> 0" shows "crit_pair_cbelow_on d m F p q"
proj_set_nonempty_of_proper: fixes A :: "'b set"   and x :: "'b" assumes "proper A"   and "A \<noteq> {}" shows "proj_set x A \<noteq> {}"
set_zero_plus: fixes C :: "'b set" shows "(0::'b) +o C = C"
ccsubspace_top_not_bot: shows "\<top> \<noteq> \<bottom>"
mset_right_cancel_elem: fixes a :: "'f"   and A :: "'f multiset"   and b :: "'f" assumes "a \<in># A + {#b#}"   and "a \<noteq> b" shows "a \<in># A"
Abs_lifted_distrib_lattice_top_inverse: fixes y :: "('c, 'd) phi \<Rightarrow> 'd" assumes "y \<in> {xf. True}" shows "Rep_lifted_distrib_lattice_top (Abs_lifted_distrib_lattice_top y) = y"
simps: fixes \<tau>move :: "('c, 'd, 'a, 'b, 'e, 'f) \<tau>moves"   and a1 :: "('c, 'd) locks \<times> (('d \<Rightarrow> ('a \<times> 'c \<Rightarrow>f nat) option) \<times> 'b) \<times> ('d \<Rightarrow> 'e wait_set_status option) \<times> 'd set"   and a2 :: "'d \<times> ('c, 'd, 'a, 'b, 'e, 'f) thread_action"   and a3 :: "('c, 'd) locks \<times> (('d \<Rightarrow> ('a \<times> 'c \<Rightarrow>f nat) option) \<times> 'b) \<times> ('d \<Rightarrow> 'e wait_set_status option) \<times> 'd set" shows "\<tau>multithreaded.m\<tau>move \<tau>move a1 a2 a3 = (\<exists>s t x s' x' ln' ta. a1 = s \<and> a2 = (t, ta) \<and> a3 = s' \<and> thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> thr s' t = \<lfloor>(x', ln')\<rfloor> \<and> \<tau>move (x, shr s) ta (x', shr s'))"
Basis_euclidean_space_def: shows "Basis \<equiv> range local.euclidean_space_basis_vector"
comp_converter_assoc: fixes conv1 :: "('h, 'i, 'n, 'o) converter"   and conv2 :: "('n, 'o, 'l, 'm) converter"   and conv3 :: "('l, 'm, 'j, 'k) converter" shows "(conv1 \<odot> conv2) \<odot> conv3 = conv1 \<odot> conv2 \<odot> conv3"
PrfP_imp_LstSeqP: fixes s :: "tm"   and k :: "tm"   and t :: "tm" shows "{PrfP s k t} \<turnstile> LstSeqP s k t"
ex_max_ultrafilter: fixes F :: "'b filter" assumes "F \<noteq> bot" shows "\<exists>U\<le>F. ultrafilter U"
single_valued_as_brE: fixes R :: "('e \<times> 'f) set" assumes "single_valued R" obtains \<alpha> :: "'e \<Rightarrow> 'f" and invar :: "'e \<Rightarrow> bool" where "R = br \<alpha> invar"
elims_1: fixes x :: "('c, 'd) graph"   and xa :: "'c"   and xb :: "('c \<times> 'd \<times> 'c) list"   and xc :: "'c"   and y :: "bool" assumes "is_path_undir x xa xb xc = y" obtains G :: "('c, 'd) graph" and v :: "'c" and v' :: "'c" where "x = Ga__" and "xa = v__" and "xb = []" and "xc = v'__" and "y = (v__ = v'__ \<and> v'__ \<in> nodes Ga__)"   | G :: "('c, 'd) graph" and v :: "'c" and v1 :: "'c" and w :: "'d" and v2 :: "'c" and p :: "('c \<times> 'd \<times> 'c) list" and v' :: "'c" where "x = G" and "xa = v" and "xb = (v1, w, v2) # p" and "xc = v'" and "y = (v = v1 \<and> ((v1, w, v2) \<in> edges G \<or> (v2, w, v1) \<in> edges G) \<and> is_path_undir G v2 p v')"
algebraicE: fixes x :: "'a" assumes "algebraic x" obtains p :: "'a poly" where "\<And>i. coeff p i \<in> \<int>" and "p \<noteq> 0" and "poly p x = (0::'a)"
alpha_lst: fixes bs :: "atom list"   and x :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom list"   and y :: "'a" shows "(bs, x) \<approx>lst R f p (cs, y) = (f x - set bs = f y - set cs \<and> (f x - set bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs)"
modified: shows "mem x \<noteq> mem' x \<or> dma mem x \<noteq> dma mem' x"
qSwapAbs_preserves_qGoodAbs2: fixes zs :: "'e"   and x :: "'d"   and y :: "'d"   and A :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGoodAbs (A $[[x \<and> y]]_zs)" shows "FixVars.qGoodAbs A"
Product_def: fixes U :: "ZF" shows "Product U = Sep (Fun U (HOLZF.Sum U)) (\<lambda>f. \<forall>u. Elem u U \<longrightarrow> Elem (f \<acute> u) u)"
induct: fixes x1 :: "('d, 'e) Term.term"   and x2 :: "('d, 'e) Term.term"   and P :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> bool" assumes "x1 \<rhd> x2"   and "\<And>s ss f. s \<in> set ss \<Longrightarrow> P (Fun f ss) s"   and "\<And>s ss t f. \<lbrakk>s \<in> set ss; s \<rhd> t; P s t\<rbrakk> \<Longrightarrow> P (Fun f ss) t" shows "P x1 x2"
converse_rtranclpE: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and z :: "'g" assumes "r\<^sup>*\<^sup>* x z" obtains "x = z"   | y :: "'g" where "r x y" and "r\<^sup>*\<^sup>* y z"
pp_np_rv_1: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "f \<in> rv M" shows "pp f \<in> rv M"
elims_1: fixes x :: "'a simple_match"   and xa :: "('a, 'b) simple_packet_scheme"   and y :: "bool" assumes "simple_matches x xa = y" obtains m :: "'a simple_match" and p :: "('a, 'b) simple_packet_scheme" where "x = m" and "xa = p" and "y = (match_iface (iiface m) (p_iiface p) \<and> match_iface (oiface m) (p_oiface p) \<and> simple_match_ip (src m) (p_src p) \<and> simple_match_ip (dst m) (p_dst p) \<and> match_proto (proto m) (p_proto p) \<and> simple_match_port (sports m) (p_sport p) \<and> simple_match_port (dports m) (p_dport p))"
merge_eq_Cons_imp: fixes xs :: "'a list"   and ys :: "'a list"   and z :: "'a"   and zs :: "'a list" assumes "local.merge xs ys = z # zs" shows "xs \<noteq> [] \<and> z = hd xs \<or> ys \<noteq> [] \<and> z = hd ys"
assms_2: shows "exec_step P h stk loc C M pc ics frs sh = (xp', h', frs', sh')"
simps: fixes R :: "('c, 'd) ring_scheme"   and p1 :: "'c list"   and p2 :: "'c list" assumes "ring R" shows "ring.poly_add R p1 p2 = (if length p2 \<le> length p1 then ring.normalize R (map2 (\<oplus>\<^bsub>R\<^esub>) p1 (replicate (length p1 - length p2) \<zero>\<^bsub>R\<^esub> @ p2)) else ring.poly_add R p2 p1)"
left_idem: fixes a :: "'c"   and b :: "'c" shows "inf a (inf a b) = inf a b"
Abs_discrete_market_inverse: fixes y :: "'c set \<times> ('c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real)" assumes "y \<in> {(s, a). stk_strict_subs s}" shows "Rep_discrete_market (Abs_discrete_market y) = y"
Rep_zimage_eq: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c zet" shows "Rep_zet (zimage f A) = f ` Rep_zet A"
walls_betw_def: fixes C :: "'a set"   and D :: "'a set" shows "walls_betw C D \<equiv> {H \<in> \<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}}. separated_by H C D}"
restrict_mem_restr: fixes A :: "'a list set"   and x :: "'a option list"   and V :: "nat set" assumes "mem_restr A x" shows "mem_restr A (restrict V x)"
A_2: shows "\<not> (p1 = [] \<or> p2 = [])"
resE_nonNegative: fixes e :: "nat \<times> nat" shows "(0::'capacity) \<le> cf e"
component_Join1: fixes F :: "'c program"   and G :: "'c program" shows "F \<le> F \<squnion> G"
le_nofailI: fixes M' :: "'b nres"   and M :: "'b nres" assumes "nofail M' \<Longrightarrow> M \<le> M'" shows "M \<le> M'"
r_code_const1_aux_def: shows "r_code_const1_aux \<equiv> Cn 3 r_prod_encode [r_constn 2 3, Cn 3 r_prod_encode [r_constn 2 1, Cn 3 r_prod_encode [r_constn 2 1, Cn 3 r_singleton_encode [recf.Id 3 1]]]]"
vv'_disjoint: shows "{v, v'} \<inter> odd_nodes_set (del_unEdge v w v' G) = {}"
transpose_conjugate: fixes A :: "'b mat" shows "(conjugate A)\<^sup>T = A\<^sup>H"
euler_maclaurin_strong_nat': fixes b :: "nat" assumes "1 \<le> b" shows "(\<Sum>k = 1..b. f (real k)) = F (real b) + z + (1 / 2) *\<^sub>R f (real b) + (\<Sum>i = 1..0. (bernoulli (2 * i) / fact (2 * i)) *\<^sub>R ([f, f'] ! (2 * i - 1)) (real b)) - EM_remainder (2 * 0 + 1) ([f, f'] ! (2 * 0 + 1)) (int b)"
simps_2: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and a :: "'Var ModeUpd"   and as :: "'Var ModeUpd list" shows "c \<otimes> (a # as) = c \<otimes> as@[a]"
S2: shows "\<forall>x\<in>S. \<forall>y\<in>S. {x, y} \<notin> E"
assms: assumes "I s" shows "M \<le> \<Down> R (WHILE\<^sub>T\<^bsup>I\<^esup> b f s)"
Abs: fixes xs :: "'varSort"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and x :: "'var" assumes "isInBar (xs, s)"   and "wls s X" shows "wlsAbs (xs, s) (Abs xs x X)"
OAlist_tc_insert_def: fixes x :: "'c \<times> 'd"   and xs :: "('c, 'd) oalist_tc" shows "OAlist_tc_insert x xs = OAlist_tc (update_by_pair_tc x (list_of_oalist_tc xs))"
weighted_majority_swap: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "weighted_majority R x y = - weighted_majority R y x"
has_density_emeasure_space: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "has_density M N f" shows "emeasure M (space M) = integral\<^sup>N N f"
div_gcd_coprime: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "a \<noteq> zero \<or> b \<noteq> zero" shows "algebraic_semidom.coprime times one (divide a (gcd a b)) (divide b (gcd a b))"
eq_iff: fixes x :: "'b"   and y :: "'b" shows "(x = y) = (x \<le> y \<and> y \<le> x)"
assms_2: fixes s :: "'a"   and a :: "'b"   and t :: "'a" assumes "reachable A s"   and "P s"   and "s \<midarrow>a\<midarrow>A\<longrightarrow> t" shows "P t"
simps_20: fixes x11 :: "'a"   and x12 :: "'b" shows "Stop \<noteq> x11 \<leftarrow> x12"
PC_ie: shows "D q t - D q s \<le> C p t - C p s"
sublist: fixes L :: "(real mpoly \<times> nat) list" shows "sublist (chop L) L"
fBall_alt_def: fixes S :: "'e fset"   and P :: "'e \<Rightarrow> bool" shows "fBall S P = (\<forall>x. x |\<in>| S \<longrightarrow> P x)"
oneStep_reachable: fixes b :: "'b"   and R :: "('b \<times> 'b) set"   and B :: "'b set" assumes "b \<in> R `` B" shows "b \<in> R\<^sup>* `` B"
\<G>\<^sub>F\<^sub>G_alt_def: fixes \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<^sub>F\<^sub>G \<phi> w = {G \<psi> |\<psi>. G \<psi> \<in> \<^bold>G \<phi> \<and> w \<Turnstile> F G \<psi>}"
cp_8: shows "contour_integral cp f = of_int (- n) * contour_integral p_circ f"
poincare_line_sym: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v" shows "poincare_line u v = poincare_line v u"
domI: fixes m :: "'d \<Rightarrow> 'c option"   and a :: "'d"   and b :: "'c" assumes "m a = Some b" shows "a \<in> dom m"
polysub_normh: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "isnpolyh (p -\<^sub>p q) (min n0 n1)"
fresh_list_cons: fixes a :: "'d"   and x :: "'e"   and xs :: "'e list" shows "a \<sharp> (x # xs) = (a \<sharp> x \<and> a \<sharp> xs)"
cr_stream_def: shows "stream_from_llist_setup.cr_stream \<equiv> \<lambda>x y. x = llist_of_stream y"
subst_sends_strand_step_fv_to_img: fixes x :: "('d, 'c) strand_step"   and \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>t\<^sub>p (x \<cdot>\<^sub>s\<^sub>t\<^sub>p \<delta>) \<subseteq> fv\<^sub>s\<^sub>t\<^sub>p x \<union> range_vars \<delta>"
coeff_finite_fourier_poly: fixes n :: "nat"   and ws :: "complex list" assumes "n < length ws" shows "coeff (finite_fourier_poly ws) n = 1 / of_nat (length ws) * (\<Sum>m<length ws. ws ! m * unity_root (length ws) (- int n * int m))"
simps_2: fixes id1 :: "nat list"   and id2 :: "nat list"   and t :: "(nat list \<times> nat list) list"   and e :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and s :: "nat list \<Rightarrow> nat list \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> nat" shows "step_score ((id1, id2) # t) e s = (let score = s id1 id2 e in if score = 0 then 0 else score + step_score t e s)"