hcju/isabelleps huggingface dataset:
neg_0_less_iff_less: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less zero (uminus a) = less a zero"
remove: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "Min A = (if A - {x} = {} then x else min x (Min (A - {x})))"
Subset_Eats_I: fixes H :: "fm set"   and x :: "tm"   and y :: "tm" shows "H \<turnstile> x SUBS Eats x y"
msort_remdups: fixes xs :: "'a list" shows "remdups (msort xs) = msort xs"
symmetric_mpoly_symmetric_sum: fixes A :: "nat set"   and g :: "(nat \<Rightarrow> nat) \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and f :: "'a \<Rightarrow> 'b mpoly" assumes "\<And>\<pi>. \<pi> permutes A \<Longrightarrow> g \<pi> permutes X"   and "\<And>x \<pi>. \<lbrakk>x \<in> X; \<pi> permutes A\<rbrakk> \<Longrightarrow> mpoly_map_vars \<pi> (f x) = f (g \<pi> x)" shows "symmetric_mpoly A (sum f X)"
A4: shows "A4 = mat nr2 nc2 (\<lambda>(i, j). four_block_mat A B C D $$ (i + dim_row A, j + dim_col A))"
sift_down_height: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, x, r\<rangle>" shows "height (sift_down l x r) \<le> height \<langle>l, x, r\<rangle>"
list_update_swap: fixes i :: "nat"   and i' :: "nat"   and xs :: "'a list"   and x :: "'a"   and x' :: "'a" assumes "i \<noteq> i'" shows "xs[i := x, i' := x'] = xs[i' := x', i := x]"
all_vec_append: fixes n :: "nat"   and m :: "nat"   and P :: "'a vec \<Rightarrow> bool" shows "(\<forall>x\<in>carrier_vec (n + m). P x) = (\<forall>x1\<in>carrier_vec n. \<forall>x2\<in>carrier_vec m. P (x1 @\<^sub>v x2))"
abs_eq: fixes x :: "int \<times> int" assumes "ratrel x x" shows "positive (Abs_Rat x) = (0 < fst x * snd x)"
span_Un: fixes S :: "'b set"   and T :: "'b set" shows "span (S \<union> T) = {x + y |x y. x \<in> span S \<and> y \<in> span T}"
msg: shows "matrix_seq.lowner_is_lub g (matrix_seq.lowner_lub g)"
Ide_Dom: fixes t :: "'a local.term" assumes "Arr t" shows "Ide (Dom t)"
ppp: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "uminus (uminus (uminus x)) = uminus x"
NFlow_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "NFlow c s t f \<equiv> NPreflow c s t f \<and> Flow c s t f"
lunit'_in_vhom: fixes f :: "'c"   and b :: "'c" assumes "C.ide f"   and "trg\<^sub>C f = b" shows "\<guillemotleft>\<l>\<^sub>C\<^sup>-\<^sup>1[f] : f \<Rightarrow>\<^sub>C b \<star>\<^sub>C f\<guillemotright>"
case_prod_conv: fixes f :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and a :: "'h"   and b :: "'i" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
PO_m1_inv0_fin_init: shows "init m1 \<subseteq> m1_inv0_fin"
emsapss_encode_help4_def: fixes PS :: "bit list"   and H :: "bit list"   and emBits :: "nat" shows "emsapss_encode_help4 PS H emBits = emsapss_encode_help5 (generate_DB PS) H emBits"
free_group_def: fixes gens :: "'d set" shows "\<F>\<^bsub>gens\<^esub> \<equiv> \<lparr>carrier = {l \<in> lists (UNIV \<times> gens). canceled l}, mult = \<lambda>x y. Cancelation.normalize (x @ y), one = []\<rparr>"
assms_1: shows "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"
morphism_locally_ringed_spaces_axioms_def: fixes X :: "'d set"   and is_open\<^sub>X :: "'d set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'d set \<Rightarrow> 'e set"   and \<rho>\<^sub>X :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and add_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and one_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and is_open\<^sub>Y :: "'f set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'f set \<Rightarrow> 'g set"   and \<rho>\<^sub>Y :: "'f set \<Rightarrow> 'f set \<Rightarrow> 'g \<Rightarrow> 'g"   and add_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and one_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and f :: "'d \<Rightarrow> 'f"   and \<phi>\<^sub>f :: "'f set \<Rightarrow> 'g \<Rightarrow> 'e" shows "morphism_locally_ringed_spaces_axioms X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f \<equiv> \<forall>x V. x \<in> X \<longrightarrow> is_open\<^sub>Y V \<longrightarrow> f x \<in> V \<longrightarrow> ind_mor_btw_stalks.is_local X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f x V \<phi>\<^bsub>X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y f \<phi>\<^sub>f x\<^esub>"
exact_decomp_def: fixes m :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "exact_decomp m ps = ((\<forall>(h, U)\<in>set ps. h \<in> P[X] \<and> U \<subseteq> X) \<and> (\<forall>(h, U)\<in>set ps. \<forall>(h', U')\<in>set ps. poly_deg h = poly_deg h' \<longrightarrow> m < card U \<longrightarrow> m < card U' \<longrightarrow> (h, U) = (h', U')))"
z_in_interval: fixes z :: "'a" assumes "z \<in> P"   and "\<forall>y\<in>Y. (y, z) \<in> induced P r" shows "z \<in> intY1"
cong_refl: fixes b :: "'a"   and a :: "'a" shows "[b = b] (mod a)"
Ord_odiff_less_odiff: fixes \<alpha> :: "V"   and x :: "V"   and y :: "V" assumes "\<alpha> \<le> x"   and "x < y"   and "Ord x"   and "Ord y"   and "Ord \<alpha>" shows "odiff x \<alpha> < odiff y \<alpha>"
IH: fixes xs :: "'a list" assumes "length xs = 2 ^ k" shows "C_msort xs \<le> k * 2 ^ k"
rel_G_Grp_weak: fixes l1 :: "'j \<Rightarrow> 'q"   and l2 :: "'k \<Rightarrow> 'r"   and co1 :: "'l \<Rightarrow> 's"   and co2 :: "'m \<Rightarrow> 't"   and contra1 :: "'u \<Rightarrow> 'n"   and contra2 :: "'v \<Rightarrow> 'o" shows "rel_G (Grp UNIV l1) (Grp UNIV l2) (Grp UNIV co1) (Grp UNIV co2) (Grp UNIV contra1)\<inverse>\<inverse> (Grp UNIV contra2)\<inverse>\<inverse> = Grp UNIV (map_G l1 l2 co1 co2 contra1 contra2)"
freshEnv_updEnv_E1: fixes xs :: "'varSort"   and y :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "freshEnv xs y (rho [x \<leftarrow> X]_xs)" shows "y \<noteq> x"
vconverse_mono: fixes r :: "V"   and s :: "V" assumes "vbrelation r"   and "vbrelation s" shows "(r\<inverse>\<^sub>\<circ> \<subseteq>\<^sub>\<circ> s\<inverse>\<^sub>\<circ>) = (r \<subseteq>\<^sub>\<circ> s)"
c: shows "\<forall>C\<in>Cn A .\<bottom>. \<phi>. AGM_relation C (Cn A) B"
eval_Lunit_Ide: fixes a :: "'c local.term" assumes "Ide a" shows "\<lbrace>\<^bold>\<l>\<^bold>[a\<^bold>]\<rbrace> = \<l>[\<lbrace>a\<rbrace>]"
uncurry_case_stmt: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x :: "'b \<times> 'c" shows "(case x of (a, b) \<Rightarrow> f a b) = Hs_Compat.uncurry f x"
D_not_diag_le': fixes xs :: "nat list"   and k :: "nat"   and i :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "set xs \<subseteq> {0..k}"   and "i \<notin> set xs"   and "j \<notin> set xs"   and "distinct xs" shows "D m i j k \<le> len m i j xs"
prod_eq_prob_T: fixes P1 :: "'s stream \<Rightarrow> bool"   and P2 :: "'s option stream \<Rightarrow> bool"   and x1 :: "'s"   and x2 :: "'s option" assumes "Measurable.pred (stream_space (count_space UNIV)) P1"   and "Measurable.pred (stream_space (count_space UNIV)) P2" shows "T.prob x1 {\<omega> \<in> space (T x1). P1 \<omega>} * K2.T.prob x2 {\<omega> \<in> space (K2.T x2). P2 \<omega>} = KN.T.prob (x1, x2) {\<omega> \<in> space (KN.T (x1, x2)). P1 (smap fst \<omega>) \<and> P2 (smap snd \<omega>)}"
taut_test_ifex: fixes t :: "'d ifex" shows "taut_test_ifex t = (\<forall>s. val_ifex t s)"
rev_append: fixes xs :: "'f list"   and ys :: "'f list" shows "rev (xs @ ys) = rev ys @ rev xs"
term_variants_pred_iff_in_term_variants: fixes P :: "'a \<Rightarrow> 'a list"   and t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term" shows "term_variants_pred P t s = (s \<in> set (term_variants P t))"
Exit_no_return_target: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" assumes "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "targetnode a = (_Exit_)" shows "False"
abs_eq: fixes x :: "('c, 'd) rbt" assumes "eq_onp (\<lambda>t. ord.is_rbt cless t \<or> ID ccompare = None) x x" shows "lookup (Mapping_RBT x) = rbt_comp_lookup ccomp x"
scale_zero_left: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "Modules.module scale" shows "scale (0::'b) x = (0::'c)"
sel: fixes x2 :: "'g" shows "the (Some x2) = x2"
wm_to_state_def: fixes M :: "object atom formula set" shows "wm_to_state M = (\<lambda>(p, xs). Atom (predAtm p xs) \<in> M)"
swap_swap: fixes i :: "'a"   and j :: "'a"   and x :: "'a" shows "swap i j (swap i j x) = x"
assms_1: fixes i :: "nat" shows "coeff p i \<in> R"
joinableI: fixes p :: "'a list"   and q :: "'a list" assumes "last p = hd q"   and "p \<noteq> []"   and "q \<noteq> []" shows "joinable p q"
pdg_Obj_vsubset_Vset: fixes J :: "V" assumes "J \<subseteq>\<^sub>\<circ> I" shows "dg_prod J \<AA>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
simps_4: fixes p :: "'c \<Rightarrow> bool"   and a :: "'c"   and g21 :: "'c \<Rightarrow> 'd"   and g22 :: "'c \<Rightarrow> 'c" assumes "\<not> p a" shows "ltl (unfold_llist p g21 g22 a) = unfold_llist p g21 g22 (g22 a)"
approx_mult_SReal_zero_cancel_iff: fixes a :: "real star"   and x :: "real star" assumes "a \<in> \<real>"   and "a \<noteq> 0" shows "(a * x \<approx> 0) = (x \<approx> 0)"
ib_f: shows "\<iota> \<star> f \<noteq> null"
InitConf_HAInitStates: fixes A :: "('d, 'e, 'f) hierauto" shows "InitConf A \<subseteq> HAInitStates A"
dist_nodes_def: shows "dist_nodes = (\<lambda>M n m p q. \<exists>w. length w = n \<and> list_all (is_alph m) w \<and> dfa_accepting M (dfa_steps M p w) \<noteq> dfa_accepting M (dfa_steps M q w))"
S'_def: shows "S' = filter (\<lambda>v. dim_vec v = dA) S"
inj_M_GmodH: shows "\<exists>f\<in>M \<rightarrow> rcosets H. inj_on f M"
L_C_def: fixes x :: "'a"   and y :: "'a" shows "L_C x y = seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x]"
call_return: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs\<^sub>1 :: "'c rule list"   and m' :: "'c match_expr"   and rs\<^sub>2 :: "'c rule list" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some (rs\<^sub>1 @ [Rule m' Return] @ rs\<^sub>2)"   and "matches \<gamma> m' p"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> Undecided"   and "no_matching_Goto \<gamma> p rs\<^sub>1" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m (Call chain)], Undecided\<rangle> \<Rightarrow> Undecided"
QTrans: shows "Q \<longmapsto> a[b] \<prec> Q'"
verit_la_disequality: fixes a :: "'e"   and b :: "'e" shows "a = b \<or> \<not> a \<le> b \<or> \<not> b \<le> a"
pred_mono: fixes m :: "nat"   and n :: "nat" assumes "m < n"   and "0 < m" shows "m - Suc 0 < n - Suc 0"
separation_t2: fixes "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.t2_space open" shows "(x \<noteq> y) = (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
at_pt_inst: assumes "at TYPE('a)" shows "pt TYPE('a) TYPE('a)"
ab_semigroup_add_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.ab_semigroup_add plus \<equiv> class.semigroup_add plus \<and> class.ab_semigroup_add_axioms plus"
infBranch_def: fixes subs :: "'b \<Rightarrow> 'b set"   and Gamma :: "'b"   and f :: "nat \<Rightarrow> 'b" shows "infBranch subs Gamma f = (f 0 = Gamma \<and> (\<forall>n. f (Suc n) \<in> subs (f n)))"
size_10: fixes x41 :: "'a formula"   and x42 :: "'a formula" shows "size (x41 \<^bold>\<and> x42) = size x41 + size x42 + Suc 0"
the_all_deriv: fixes f :: "'d \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'd \<Rightarrow> 'e" assumes "\<forall>x. (f has_derivative F x) (at x)" shows "(THE G. \<forall>x. (f has_derivative G x) (at x)) = F"
bbox_def: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "wp x y = rel_ad (x ; rel_ad y)"
rt_fresher_def': fixes i :: "nat"   and rt\<^sub>1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt\<^sub>2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" shows "(rt\<^sub>1 \<sqsubseteq>\<^bsub>i\<^esub> rt\<^sub>2) = (nsqn\<^sub>r (the (rt\<^sub>1 i)) < nsqn\<^sub>r (the (rt\<^sub>2 i)) \<or> nsqn\<^sub>r (the (rt\<^sub>1 i)) = nsqn\<^sub>r (the (rt\<^sub>2 i)) \<and> \<pi>\<^sub>5 (the (rt\<^sub>2 i)) \<le> \<pi>\<^sub>5 (the (rt\<^sub>1 i)))"
map_matrix_def: fixes f :: "'e \<Rightarrow> 'b"   and A :: "(('e, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "HMA_Connect.map_matrix f A \<equiv> \<chi>i. map_vector f (A $h i)"
sep_add_disjI1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x + y ## z"   and "x ## y" shows "x + z ## y"
subset: fixes c :: "'a" assumes "c \<in> G" shows "c \<in> M"
iT_Max_1: fixes n :: "nat" shows "Max [\<dots>n] = n"
\<pi>E_def: shows "macode.\<pi>E \<equiv> 1\<^sub>C"
cls: fixes c :: "real" shows "closed {x \<in> S. s x = c}"
soundnessForUNSAT: fixes state0 :: "State"   and F0 :: "Literal list list"   and state :: "State"   and decisionVars :: "nat set" assumes "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation decisionVars"   and "formulaFalse (getF state) (elements (getM state))"   and "decisions (getM state) = []" shows "\<not> satisfiable F0"
le_length_filter_update: fixes P :: "'a \<Rightarrow> bool"   and list :: "'a list"   and i :: "nat"   and j :: "'a" assumes "(\<not> P (list ! i) \<or> P j) \<and> i < length list" shows "length (filter P list) \<le> length (filter P (list[i := j]))"
add_point_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and p :: "'b" assumes "incidence_system point_set block_collection" shows "incidence_system.add_point point_set p \<equiv> insert p point_set"
t_in_Dags: shows "t \<in> Dags nodes low high"
R_def: shows "R \<equiv> measure (variant \<pi> b c)"
hcomp_isomorphic_ide: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "f \<cong> g"   and "ide h"   and "src f = trg h" shows "f \<star> h \<cong> g \<star> h"
ex_un_null: shows "\<exists>!n. \<forall>f. n \<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] f = n \<and> f \<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] n = n"
fps_Lcm: fixes A :: "'a fps set" assumes "A \<noteq> {}"   and "0 \<notin> A"   and "bdd_above (subdegree ` A)" shows "Lcm A = fps_X ^ Sup (subdegree ` A)"
extremum: fixes a :: "'c \<times> 'd" shows "less_eq_prod bot_prod a"
While_fun: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and f :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and c :: "com" assumes "\<And>n. \<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s \<and> n = f s} c {\<lambda>s. P s \<and> f s < n}" shows "\<turnstile>\<^sub>t {P} WHILE b DO c {\<lambda>s. P s \<and> \<not> bval b s}"
that: fixes u :: "'a \<Rightarrow> 'b"   and e :: "real" assumes "f (x, u x) = (0::'c)"   and "u x = y"   and "\<And>s. s \<in> cball x e \<Longrightarrow> f (s, u s) = (0::'c)"   and "continuous_on (cball x e) u"   and "(\<lambda>t. (t, u t)) ` cball x e \<subseteq> S"   and "0 < e"   and "(u has_derivative blinfun_apply (- T o\<^sub>L f' (x, y) o\<^sub>L embed1_blinfun)) (at x)"   and "\<And>s. s \<in> cball x e \<Longrightarrow> f' (s, u s) o\<^sub>L embed2_blinfun \<in> invertibles_blinfun"   and "\<And>U v s. \<lbrakk>\<And>s. s \<in> U \<Longrightarrow> f (s, v s) = (0::'c); u x = v x; continuous_on U v; s \<in> U; x \<in> U; U \<subseteq> cball x e; connected U; open U\<rbrakk> \<Longrightarrow> u s = v s" shows "thesis"
fps_to_fls_base_factor_to_fps: fixes f :: "'b fls" shows "fls_regpart_as_fls (fls_base_factor f) = fls_base_factor f"
assms: shows "(u, v) \<in> (edges g)\<^sup>*"
x: shows "(y \<longlongrightarrow> x) at_top"
CongA_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" shows "A B C CongA D E F \<equiv> A \<noteq> B \<and> C \<noteq> B \<and> D \<noteq> E \<and> F \<noteq> E \<and> (\<exists>A' C' D' F'. Bet B A A' \<and> Cong A A' E D \<and> Bet B C C' \<and> Cong C C' E F \<and> Bet E D D' \<and> Cong D D' B A \<and> Bet E F F' \<and> Cong F F' B C \<and> Cong A' C' D' F')"
intros_2: fixes \<sigma> :: "'a \<Rightarrow> 'b"   and k :: "nat"   and \<sigma>' :: "'a \<Rightarrow> 'b"   and k' :: "nat"   and prob :: "('c, 'a, 'b, 'd) ifc_problem"   and n :: "nat"   and v :: "'a"   and n' :: "nat" assumes "((\<sigma>, k), \<sigma>', k') \<in> IFC_def.cp prob"   and "IFC_def.is_ddi prob n (IFC_def.path prob \<sigma>) v k"   and "IFC_def.is_ddi prob n' (IFC_def.path prob \<sigma>') v k'"   and "IFC_def.cs prob (IFC_def.path prob \<sigma>) n = IFC_def.cs prob (IFC_def.path prob \<sigma>') n'"   and "IFC_def.kth_state prob \<sigma> n v \<noteq> IFC_def.kth_state prob \<sigma>' n' v" shows "((\<sigma>, n), \<sigma>', n') \<in> IFC_def.cp prob"
block_sizes: fixes bl :: "'a set" assumes "bl \<in># \<B> + mset_set {insert x g |g. g \<in> \<G>}" shows "int (card bl) \<in> {\<k>}"
Class_subset: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> kernel.Congruence" shows "kernel.Class a \<subseteq> kernel.Class b"
collapse: fixes stream :: "'a stream" shows "shd stream ## stl stream = stream"
k_reachable: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'g"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'h"   and jviewInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'g"   and jviewIncr :: "'b \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'g"   and simf :: "'e Trace \<Rightarrow> 'i"   and simRels :: "'b \<Rightarrow> ('i \<times> 'i) set"   and simVal :: "'i \<Rightarrow> 'c \<Rightarrow> bool"   and simAbs :: "'j \<Rightarrow> 'i set"   and simObs :: "'b \<Rightarrow> 'j \<Rightarrow> 'h"   and simInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'j"   and simTrans :: "'b \<Rightarrow> 'j \<Rightarrow> 'j list"   and simAction :: "'b \<Rightarrow> 'j \<Rightarrow> 'd list"   and aOps :: "('k, 'j, 'd list) MapOps"   and tOps :: "('l, 'j \<times> 'h, 'j) MapOps"   and a :: "'b" assumes "AlgorithmForAgent jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit simTrans simAction aOps tOps" shows "simAbs ` DFS.reachable (simTrans a) (set (Algorithm.k_frontier envInit envObs simInit a)) = (\<lambda>t. simf ` {t' \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview. jview a t' = jview a t}) ` PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview"
Sup_to_Inf_var: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and dual :: "'a \<Rightarrow> 'a"   and X :: "'a set" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "Sup X = dual (Inf (dual ` X))"
wf_J\<^sub>1_mdecl_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and C :: "char list" shows "wf_J\<^sub>1_mdecl P C \<equiv> \<lambda>(M, Ts, T, body). (\<exists>T'. P,Class C # Ts \<turnstile>\<^sub>1 body :: T' \<and> P \<turnstile> T' \<le> T) \<and> \<D> body \<lfloor>{..length Ts}\<rfloor> \<and> \<B> body (length Ts + 1)"
bigo_const_mult6: fixes c :: "'b"   and f :: "'a \<Rightarrow> 'b" shows "(\<lambda>x. c) *o O(f) \<subseteq> O(f)"
simps_1: fixes a :: "'b" shows "insert a \<langle>\<rangle> = \<langle>\<langle>\<rangle>, a, \<langle>\<rangle>\<rangle>"
assms_3: shows "h \<turnstile> get_host shadow_root_ptr \<rightarrow>\<^sub>r host"
basis_def: fixes A :: "'a vec set" shows "AB.basis A = (AB.lin_indpt A \<and> AB.span A = mat_kernel (A * B) \<and> A \<subseteq> mat_kernel (A * B))"
mrw_before: fixes E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and r :: "nat"   and w :: "nat" assumes "E \<in> \<E>"   and "P \<turnstile> (E, ws) \<surd>"   and "P,E \<turnstile> r \<leadsto>mrw w"   and "\<And>a. \<lbrakk>a < r; a \<in> read_actions E\<rbrakk> \<Longrightarrow> P,E \<turnstile> a \<leadsto>mrw ws a" shows "w < r"
openin_topology_base_unique: fixes X :: "'d topology"   and P :: "'d set \<Rightarrow> bool" shows "(openin X = arbitrary union_of P) = ((\<forall>V. P V \<longrightarrow> openin X V) \<and> (\<forall>U x. openin X U \<and> x \<in> U \<longrightarrow> (\<exists>V. P V \<and> x \<in> V \<and> V \<subseteq> U)))"
measure_measurable_subprob_algebra2: fixes M :: "'c measure"   and A :: "'c \<Rightarrow> 'd set"   and N :: "'d measure"   and L :: "'c \<Rightarrow> 'd measure" assumes "Sigma (space M) A \<in> sets (M \<Otimes>\<^sub>M N)"   and "L \<in> M \<rightarrow>\<^sub>M subprob_algebra N" shows "(\<lambda>x. Sigma_Algebra.measure (L x) (A x)) \<in> borel_measurable M"
ang_circ_def: fixes E :: "complex"   and \<mu>1 :: "complex"   and \<mu>2 :: "complex"   and p1 :: "bool"   and p2 :: "bool" shows "ang_circ E \<mu>1 \<mu>2 p1 p2 = \<angle> (circ_tang_vec \<mu>1 E p1) (circ_tang_vec \<mu>2 E p2)"
has_density_imp_is_subprob_density: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_density M N f"   and "integral\<^sup>N N f = 1" shows "is_subprob_density N f"
invertibleI: fixes u :: "'b"   and v :: "'b" assumes "u \<cdot>' v = \<one>'"   and "v \<cdot>' u = \<one>'"   and "u \<in> M'"   and "v \<in> M'" shows "invertible' u"
Inc: fixes U :: "('a \<Rightarrow> 'b) set" assumes "open U \<and> x \<in> U" shows "\<exists>k\<in>K. k \<subseteq> U"
linear_sum: fixes f :: "('d, 'e) vec \<Rightarrow> ('d, 'f) vec"   and g :: "'g \<Rightarrow> ('d, 'e) vec"   and S :: "'g set" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
spmf_of_pmf_def: shows "spmf_of_pmf = map_pmf Some"