hcju/isabelleps huggingface dataset:
WF_eventually: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> \<diamond>WF(A)_v = WF(A)_v"
assms: shows "is' \<in> it_paths (globalize is f t)"
simps_9: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "'b"   and va :: "value list" shows "GExp_Lexorder.less_gexp_aux (Eq e1 e2) (In v va) = True"
span_clauses_4: fixes x :: "'a"   and S :: "'a set"   and c :: "complex" assumes "x \<in> cspan S" shows "c *\<^sub>C x \<in> cspan S"
tuple_disj: fixes x :: "'a"   and z :: "'a" shows "{y. y = x \<or> y = z} = {x, z}"
eq': shows "U f = gfp (\<lambda>f. U (F (C f)))"
section_is_mono: fixes g :: "('a, 'a set) Maps.arr" assumes "Maps.section g" shows "Maps.mono g"
add_match_add_match_MatchAnd_foldr: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and ms :: "'c match_expr list"   and rs2 :: "'c rule list"   and s :: "state"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match m (add_match_MatchAnd_foldr ms rs2), s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match (MatchAnd_foldr (m # ms)) rs2, s\<rangle> \<Rightarrow> t"
induct: fixes x1 :: "nat"   and x2 :: "'b \<Rightarrow> 'b"   and P :: "nat \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> bool" assumes "swapidseq x1 x2"   and "P 0 id"   and "\<And>n p a b. \<lbrakk>swapidseq n p; P n p; a \<noteq> b\<rbrakk> \<Longrightarrow> P (Suc n) (Fun.swap a b id \<circ> p)" shows "P x1 x2"
abstract_correct: fixes \<eta> :: "nat" shows "rivest_commit.correct \<eta>"
"1_4": shows "vars_change as2 vs (state_succ (exec_plan s as1) act) = ss"
res_of_integer_eq: fixes i :: "int" shows "\<guillemotleft>i\<guillemotright> = i mod int p"
remdups_def: shows "remdups \<equiv> rec_list [] (\<lambda>x xs xsa. if x \<in> set xs then xsa else x # xsa)"
satPB_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi> :: "('fsym, 'psym) lit list set" shows "M.satPB \<xi> \<Phi> \<equiv> \<forall>c\<in>\<Phi>. M.satC \<xi> c"
abs_eq: fixes x :: "int" shows "of_int_mod_ring x = Abs_mod_ring (x mod int CARD('b))"
star_unfoldr_eq: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<star> \<cdot> x = x\<^sup>\<star>"
finite_Collect_conjI: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "finite {x. P x} \<or> finite {x. Q x}" shows "finite {x. P x \<and> Q x}"
step_suc_sem: fixes n :: "'n"   and \<sigma> :: "'var \<Rightarrow> 'val" shows "local.step (n, \<sigma>) = (suc n \<sigma>, sem n \<sigma>)"
cnj_0_eq: fixes a :: "'a" shows "(cnj_class.cnj a = (0::'a)) = (a = (0::'a))"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "vc (C\<^sub>1;; C\<^sub>2) Q = (vc C\<^sub>1 (pre C\<^sub>2 Q) \<and> vc C\<^sub>2 Q)"
row_def: shows "Square_Matrix.row \<equiv> map_fun to_fun id (\<lambda>M i. \<chi>j. M i j)"
all_zero: shows "\<forall>n. is_zero_row_upt_k n k A"
a_subset_d: shows "ideal_generated {a} \<subseteq> ideal_generated {d}"
DAAux: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" shows "x \<in> dom (C DenyAll)"
all_ideals: fixes R :: "('a, 'b) ring_scheme" assumes "field R" shows "{I. ideal I R} = {{\<zero>\<^bsub>R\<^esub>}, carrier R}"
N_bot: shows "- (- bot \<sqinter> top) \<sqinter> top = bot"
prj_simps_12: shows "C.seq \<pi>\<rho>.prj\<^sub>0 \<p>\<^sub>0[\<mu>\<nu>.leg0, \<pi>.leg1 \<cdot> \<pi>\<rho>.prj\<^sub>1]"
exhaust: fixes y :: "'a array" obtains x :: "'a list" where "y = Array x"
new_asms_only_for_priv_def: fixes mdss\<^sub>C :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list" shows "new_asms_only_for_priv mdss\<^sub>C \<equiv> \<forall>i<length cms. ((mdss\<^sub>C ! i) AsmNoReadOrWrite \<union> (mdss\<^sub>C ! i) AsmNoWrite) \<inter> - range var\<^sub>C_of \<subseteq> priv_mem\<^sub>C ! i"
validE: fixes X :: "msg" assumes "X \<in> asym.valid" obtains x :: "tag" and A :: "agent" and B :: "agent" and M :: "msg" where "X = implem_asym (Chan x A B M)" and "M \<in> payload"
CFG_wf_def: fixes \<alpha>e :: "'f \<Rightarrow> ('g \<times> 'h \<times> 'g) set"   and \<alpha>n :: "'f \<Rightarrow> 'g list"   and invar :: "'f \<Rightarrow> bool"   and inEdges' :: "'f \<Rightarrow> 'g \<Rightarrow> ('g \<times> 'h) list"   and Entry :: "'f \<Rightarrow> 'g"   and defs :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set"   and "uses" :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set" shows "CFG_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<equiv> CFG \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<and> CFG_wf_axioms \<alpha>n invar inEdges' Entry defs uses"
acomplete_bal: fixes n :: "nat"   and xs :: "'a list"   and t :: "'a tree"   and ys :: "'a list" assumes "n \<le> length xs"   and "bal n xs = (t, ys)" shows "acomplete t"
inv_solve_left: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<in> carrier (G i)"   and "b \<in> carrier (G i)"   and "c \<in> carrier (G i)" shows "(a = inv\<^bsub>G i\<^esub> b \<otimes>\<^bsub>G i\<^esub> c) = (c = b \<otimes>\<^bsub>G i\<^esub> a)"
subset_hull: fixes S :: "'d set \<Rightarrow> bool"   and t :: "'d set"   and s :: "'d set" assumes "S t" shows "(S hull s \<subseteq> t) = (s \<subseteq> t)"
dom_comp: fixes g :: "'a"   and f :: "'a" assumes "A.seq g f" shows "A.dom (A g f) = A.dom f"
n_rules_def: shows "task2_11.n_rules = flat (smap (\<lambda>r. map (Pair r) (consequent r)) prop_rules)"
simps_2: fixes x :: "'d"   and t :: "('e, 'f) Term.term"   and P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) list"   and \<sigma> :: "'d \<Rightarrow> ('e, 'f) Term.term option" shows "match_term_list ((Var x, t) # P) \<sigma> = (if \<sigma> x = None \<or> \<sigma> x = Some t then match_term_list P (\<sigma>(x \<mapsto> t)) else None)"
has_prod_unique2: fixes f :: "nat \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "f has_prod a"   and "f has_prod b" shows "a = b"
prv_imp_neg_encF_N_aux: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "prv (imp (PP \<langle>neg \<phi>\<rangle>) (PP (encF N \<langle>\<phi>\<rangle>)))"
Complex_eq: fixes a :: "real"   and b :: "real" shows "Complex a b = cor a + \<i> * cor b"
E_state_fix: shows "flows_state \<T> \<subseteq> flows_fix \<T>"
prime_gcd: fixes m :: "nat"   and n :: "nat" assumes "prime m"   and "prime n"   and "m \<noteq> n" shows "gcd m n = 1"
fun_upd_idem_iff: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'e" shows "(f(x := y) = f) = (f x = y)"
assms: shows "v \<in> fmdom' (action_needed_asses a f)"
locally_respects_next_state: shows "\<forall>s u execs. \<not> ifp (current s) u \<longrightarrow> vpeq u s (int.next_state s execs)"
smc_10_components_1: fixes \<aa> :: "V" shows "smc_10 \<aa>\<lparr>Obj\<rparr> = ZFC_in_HOL.set {\<aa>}"
inject_2: fixes x3 :: "nat"   and y3 :: "nat" shows "(HOLD x3 = HOLD y3) = (x3 = y3)"
non_overlapping_without_is_non_overlapping: fixes P :: "'b set set"   and elem :: "'b" assumes "is_non_overlapping P" shows "is_non_overlapping (partition_without elem P)"
zero_eq_gauss2complex_iff: fixes x :: "gauss_int" shows "(0 = gauss2complex x) = (x = 0)"
uv': shows "(u, v) \<in> (\<Union> (range (relations M)))\<^sup>*"
abs_eq: shows "1 = of_fun (\<lambda>i j. if i = j then 1::'c else (0::'c))"
subst_cls_lists_def: fixes Cs :: "('c, 'd) Term.term literal multiset list"   and \<sigma>s :: "('d \<Rightarrow> ('c, 'd) Term.term) list" shows "subst_cls_lists Cs \<sigma>s = Map2.map2 subst_cls Cs \<sigma>s"
acc_listrel: fixes r :: "('b \<times> 'b) set" assumes "refl r"   and "trans r"   and "acc r" shows "acc (listrel r - {([], [])})"
xbeta: fixes E :: "ctx"   and x :: "name"   and t :: "lam"   and t' :: "lam" shows "E\<lbrakk>App (Lam [x].t) t'\<rbrakk> \<longrightarrow>x E\<lbrakk>t[x::=t']\<rbrakk>"
map_id_dom: fixes xs :: "'c set" shows "dom (map_id_on xs) = xs"
sum_cong_aux: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "sum f A = sum g A"
map_disj_update_left: fixes p :: "'c"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and v :: "'d" assumes "p \<in> dom h\<^sub>1" shows "h\<^sub>0 \<bottom> h\<^sub>1(p \<mapsto> v) = h\<^sub>0 \<bottom> h\<^sub>1"
1: shows "open {w. - pi < Im w \<and> Im w < pi}"
start_heap_write_typeable: fixes ad :: "'addr"   and al :: "addr_loc"   and v :: "'addr val" assumes "WriteMem ad al v \<in> set h.start_heap_obs" shows "\<exists>T. h.addr_loc_type P h.start_heap ad al T \<and> h.conf P h.start_heap v T"
path_image_join: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "pathfinish g1 = pathstart g2" shows "path_image (g1 +++ g2) = path_image g1 \<union> path_image g2"
surj_pair: fixes p :: "'r \<times> 's" shows "\<exists>x y. p = (x, y)"
cases: fixes a1 :: "(nat \<times> nat) multiset"   and a2 :: "(nat \<times> nat) multiset" assumes "pw_leq a1 a2" obtains "a1 = {#}" and "a2 = {#}"   | x :: "nat \<times> nat" and y :: "nat \<times> nat" and X :: "(nat \<times> nat) multiset" and Y :: "(nat \<times> nat) multiset" where "a1 = {#x#} + X" and "a2 = {#y#} + Y" and "(x, y) \<in> pair_leq" and "pw_leq X Y"
replace5: fixes f :: "'c"   and oldF :: "'c"   and newfs :: "'c list"   and fs :: "'c list" assumes "f \<in> set (replace oldF newfs fs)" shows "f \<in> set fs \<or> f \<in> set newfs"
principal_ideal_def: fixes S :: "'a set" shows "principal_ideal S = (\<exists>a. ideal_generated {a} = S)"
regset_of_DA_def: fixes A :: "nat \<times> ('b \<Rightarrow> nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool)"   and k :: "nat" shows "regset_of_DA A k = (\<Union>j\<in>{j. j < k \<and> fin A j}. regset (next A) (start A) j k)"
prems: shows "v \<in> vars\<^sub>s\<^sub>t (x # S)"
get_child_nodes_new_shadow_root: fixes ptr' :: "(_) object_ptr"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr' \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "r \<in> get_child_nodes_locs ptr'" shows "r h h'"
resultant_const_1: fixes a :: "'c"   and q :: "'c poly" shows "resultant [:a:] q = a ^ degree q"
a_in_range_of_v: shows "a \<in> \<R>\<^sub>+ \<Psi> v"
proper_designII: fixes V :: "'b set"   and B :: "'b set multiset" assumes "design V B"   and "B \<noteq> {#}" shows "proper_design V B"
le_rexp_antisym: fixes r :: "'b rexp"   and s :: "'b rexp" assumes "le_rexp r s"   and "le_rexp s r" shows "r = s"
path_Append: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and n'' :: "'b"   and as' :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n''"   and "CFG.path sourcenode targetnode valid_edge n'' as' n'" shows "CFG.path sourcenode targetnode valid_edge n (as @ as') n'"
simps_13: fixes f :: "'b \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b llist" shows "lmap f (LCons x21 x22) = LCons (f x21) (lmap f x22)"
Delta_target_States: fixes T :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd"   and A :: "('d, 'e, 'f) seqauto" assumes "T \<in> Delta A" shows "target T \<in> States A"
preserves_ide: fixes a :: "'b" assumes "B.ide a" shows "D.ide (HoF.map a)"
Sim: fixes R :: "pi"   and S :: "pi" assumes "(R, S) \<in> Rel" shows "R \<leadsto><Rel> S"
span_induct': fixes p :: "'a"   and B :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "p \<in> ideal B"   and "P (0::'a)"   and "\<And>a q p. \<lbrakk>a \<in> ideal B; P a; p \<in> B; q \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> P (a + q * p)" shows "P p"
Roots_def: shows "Roots = {x. ipoly P x = 0}"
subset_psubset_trans: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "A \<subseteq> B"   and "B \<subset> C" shows "A \<subset> C"
seqSubs''_def: fixes C :: "('a, 'b, 'c) psiCase"   and \<sigma> :: "(name list \<times> 'a list) list" shows "seqSubs'' C \<sigma> \<equiv> foldl (\<lambda>Q (xvec, Tvec). subs'' Q xvec Tvec) C \<sigma>"
rep_eq: fixes x :: "uint16" shows "int_of_uint16 x = uint (Rep_uint16 x)"
linE: fixes x :: "'b"   and v :: "'b" assumes "x \<in> lin v" obtains a :: "real" where "x = a \<cdot> v"
correctCompositionKeys_subcomp1: fixes C :: "specID"   and x :: "specID"   and xb :: "Keys" assumes "correctCompositionKeys C"   and "x \<in> subcomponents C"   and "xb \<in> specKeys C" shows "\<exists>x\<in>subcomponents C. xb \<in> specKeys x"
dord_pm_lin: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" assumes "class.linorder less_eq less"   and "ord s t \<or> ord t s" shows "linorder.dord_pm ord s t \<or> linorder.dord_pm ord t s"
assms_2: shows "set (mat_leqb_eqc A b c) \<subseteq> set S"
dim_component_mult: fixes v :: "'a vec"   and w :: "'a vec" shows "dim_vec (component_mult v w) = min (dim_vec v) (dim_vec w)"
rebase_poly_smult: fixes a :: "'pq mod_ring"   and p :: "'pq mod_ring poly" shows "#Polynomial.smult a p = Polynomial.smult (@a) (#p)"
simps_1: fixes A :: "'b set"   and B :: "'b set"   and t2 :: "'b list" shows "merge A B [] t2 = t2"
pref_share_take: fixes u :: "'b list"   and v :: "'b list"   and q :: "nat" assumes "u \<le>p v"   and "q \<le> \<^bold>|u\<^bold>|" shows "take q u = take q v"
natural_isomorphism_axioms: shows "natural_isomorphism A B G F \<psi>.map"
prod_simps_1: fixes f0 :: "'a"   and f1 :: "'a" assumes "arr f0"   and "arr f1" shows "arr (f0 \<otimes> f1)"
del_has_vector_derivative: fixes t :: "real"   and u :: "real set" shows "(\<tau> has_vector_derivative \<tau>' t) (at t within u)"
sum_disj_Union: fixes V :: "'a set"   and F :: "'a \<Rightarrow> 'b set"   and w :: "'b \<Rightarrow> nat" assumes "finite V"   and "\<And>f. finite (F f)"   and "separating V F" shows "(\<Sum>v\<in>V. sum w (F v)) = sum w (\<Union> (F ` V))"
invE2: fixes G :: "graph"   and S :: "state"   and i :: "nat" assumes "Dijkstra.inv G S"   and "i \<in> known_set S" shows "est S ! i = dist G 0 i"
diff_right_commute: fixes a :: "'i"   and c :: "'i"   and b :: "'i" shows "a - c - b = a - b - c"
pwd_spy_id_pwd_def: fixes n :: "nat" shows "pwd_spy_id_pwd n \<equiv> case pwd_spy_id_sesk n of (s, S, A, B, C, D) \<Rightarrow> (insert (Spy, \<langle>n, Pwd n\<rangle>) s, S, A, B, C, D)"
rewrite_X_sound: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n rewrite_X \<phi> = w \<Turnstile>\<^sub>n \<phi>"
Suc_1: assumes "k < n" shows "ket_zero_k k = Matrix.vec (2 ^ Suc k) (\<lambda>k. if k = 0 then 1 else 0)"
bij: fixes f :: "'b \<Rightarrow> 'b" assumes "bijection f" shows "bij f"
list_all_iff: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c list" shows "list_all P x = Ball (set x) P"
simps_8: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi> = (\<exists>i. suffix i \<xi> \<Turnstile>\<^sub>n \<psi> \<and> (\<forall>j<i. suffix j \<xi> \<Turnstile>\<^sub>n \<phi>))"
sum_list_map_times: fixes x :: "'b"   and xs :: "'b list" shows "sum_list (map ((*) x) xs) = x * sum_list xs"
empty_sb: 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"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "empty_store_buffers ts"   and "i < length ts"   and "ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)" shows "sb = []"
diff_fun_charts_euclI: fixes k :: "enat"   and g :: "'c \<Rightarrow> 'd" assumes "k-smooth_on UNIV g" shows "diff_fun k charts_eucl g"
distinct_21: fixes x5 :: "String.literal" shows "Integer \<noteq> Class x5"
rtrancl_eq_or_trancl: fixes x :: "'c"   and y :: "'c"   and R :: "('c \<times> 'c) set" shows "((x, y) \<in> R\<^sup>*) = (x = y \<or> x \<noteq> y \<and> (x, y) \<in> R\<^sup>+)"
assm_4: fixes x :: "'b"   and K :: "'b set" assumes "(x, K) \<in> p" shows "\<exists>a b. K = cbox a b"
ForallE: fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form"   and t :: "'a FOL_Fitting.term" assumes "G \<turnstile> Forall a" shows "G \<turnstile> a[t/0]"
f_Int_fin: shows "f (\<Inter> (range A)) \<noteq> \<infinity>"
replicate_mset_0: fixes x :: "'b" shows "replicate_mset 0 x = {#}"
synth_increasing: fixes H :: "msg set" shows "H \<subseteq> synth H"
IH_2: assumes "x \<in> vars_of t2" shows "t2 \<lhd> \<sigma> \<noteq> t2 \<lhd> \<eta>"
altc_sample_option_def: fixes altc_sample :: "'b \<Rightarrow> ('c \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'b"   and f :: "'c \<Rightarrow> ('a, 'm) optionT" shows "local.altc_sample_option altc_sample p f = OptionT (altc_sample p (\<lambda>x. run_option (f x)))"
mlex_iff: fixes x :: "'b"   and y :: "'b"   and f :: "'b \<Rightarrow> nat"   and R :: "('b \<times> 'b) set" shows "((x, y) \<in> f <*mlex*> R) = (f x < f y \<or> f x = f y \<and> (x, y) \<in> R)"
path_entry_cons_conv: fixes E :: "(nat \<times> nat) set"   and u' :: "nat"   and l :: "nat list"   and w :: "nat" shows "path_entry E (u' # l) w = ((u', w) \<in> E \<and> path_entry E l u')"
his_ex: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a"   and n' :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "\<not> Option.is_none (Blockchain.devBC active cmp bc mining t n nid n')" shows "\<exists>nid'. (n', nid') \<in> Blockchain.his active cmp bc mining t n nid"
closure_bounded_linear_image_subset: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "bounded_linear f" shows "f ` closure S \<subseteq> closure (f ` S)"
elims: fixes h :: "nat"   and n_ops :: "nat"   and x :: "sat_plan_variable"   and y :: "nat" assumes "cnf_to_dimacs.var_to_dimacs h n_ops x = y" obtains t :: "nat" and k :: "nat" where "x = Operator t__ k__" and "y = 1 + t__ + k__ * h"   | t :: "nat" and k :: "nat" where "x = State t k" and "y = 1 + n_ops * h + t + k * h"
mulex_on_self_add_right: fixes M :: "'b multiset"   and A :: "'b set"   and K :: "'b multiset"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "M \<in> multisets A"   and "K \<in> multisets A"   and "K \<noteq> {#}" shows "mulex_on P A M (M + K)"
measurable_def: fixes A :: "'b measure"   and B :: "'c measure" shows "A \<rightarrow>\<^sub>M B = {f \<in> space A \<rightarrow> space B. \<forall>y\<in>sets B. f -` y \<inter> space A \<in> sets A}"
K_def: shows "K = {n \<in> {1..}. (T ^^ n) x \<in> A}"
simps_2: fixes n :: "nat"   and s :: "'c stream" shows "stake (Suc n) s = shd s # stake n (stl s)"
kcomp_assoc: fixes f :: "'b \<Rightarrow> 'e set"   and g :: "'e \<Rightarrow> 'd set"   and h :: "'d \<Rightarrow> 'c set" shows "f \<circ>\<^sub>K g \<circ>\<^sub>K h = f \<circ>\<^sub>K (g \<circ>\<^sub>K h)"
resolvents_do_not_contain_atom: fixes P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set"   and E2 :: "'at" assumes "\<not> tautology P1"   and "\<not> tautology P2"   and "C = resolvent_upon P1 P2 E2"   and "\<not> subsumes P1 C"   and "\<not> subsumes P2 C" shows "Neg E2 \<notin> C \<and> Pos E2 \<notin> C"
wtE_def: fixes \<xi> :: "'a \<Rightarrow> 'univ" shows "wtE \<xi> \<equiv> \<forall>x. D (\<xi> x)"
preserve_min_P_Sts: fixes k :: "nat"   and C :: "'a literal multiset"   and i :: "nat"   and j :: "nat" assumes "enat (Suc k) < llength Sts"   and "(C, i) \<in># wP_of_wstate (lnth Sts k)"   and "(C, j) \<in># wP_of_wstate (lnth Sts (Suc k))"   and "\<forall>j. (C, j) \<in># wP_of_wstate (lnth Sts k) \<longrightarrow> i \<le> j" shows "(C, i) \<in># wP_of_wstate (lnth Sts (Suc k))"
assms: shows "(([], c), w, s', c') \<in> trcl (trss fg)"
rtranclp_into_tranclp1: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>+\<^sup>+ a c"
distr_pair_swap: shows "M x \<Otimes>\<^sub>M M y = distr (M y \<Otimes>\<^sub>M M x) (M x \<Otimes>\<^sub>M M y) (\<lambda>(x, y). (y, x))"
exhaust_sel: fixes fundef :: "'a fundef" obtains "fundef = Fundef (body fundef) (arity fundef)"
listsetD_2: fixes ys :: "'a list"   and xs :: "'a set list"   and i :: "nat" assumes "ys \<in> listset xs"   and "i < length xs" shows "ys ! i \<in> xs ! i"
preserves_comp: fixes g :: "'b"   and f :: "'b" assumes "B.seq g f" shows "local.inv (g \<cdot>\<^sub>B f) = local.inv g \<cdot>\<^sub>A local.inv f"
points_index_on_set_def: fixes B :: "'b set set"   and ps :: "'b set" assumes "finite B" shows "mset_set B index ps = card {b \<in> B. ps \<subseteq> b}"
simps: fixes a1 :: "'d"   and a2 :: "'d"   and r :: "('d \<times> 'd) set" shows "((a1, a2) \<in> r\<^sup>*) = ((\<exists>a. a1 = a \<and> a2 = a) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r\<^sup>* \<and> (b, c) \<in> r))"
abstract_primitive_in_doubt_deny_Deny: fixes \<beta> :: "'c common_primitive \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and m :: "'c common_primitive match_expr"   and p :: "('c, 'd) tagged_packet_scheme"   and disc :: "'c common_primitive negation_type \<Rightarrow> bool" assumes "primitive_matcher_generic \<beta>"   and "normalized_nnf_match m"   and "matches (\<beta>, in_doubt_deny) m Drop p" shows "matches (\<beta>, in_doubt_deny) (abstract_primitive disc m) Drop p"
alpha_Tree_rel_def: shows "FL_Validity.alpha_Tree_rel \<equiv> {(x, y). x =\<^sub>\<alpha> y}"
rv2_Nil: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and x :: "'e" shows "rv2 s1 s2 [] x = Inr x"
sgn_0_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(sgn a = zero) = (a = zero)"
get_fin_long_ch_bounds: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set" assumes "long_ch_by_ord f X"   and "finite X" shows "\<exists>x\<in>X. \<exists>y\<in>X. \<exists>z\<in>X. [f[x .. y ..  z]X]"
top_apply: fixes x :: "'c" shows "\<top> x = \<top>"
v11_vconverse_app_in_vdomain: fixes y :: "V" assumes "y \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_inverse" shows "vrat_inverse\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vrat_inverse"
steps_ConsD: fixes x :: "'a"   and xs :: "'a list" assumes "steps (x # xs)"   and "xs \<noteq> []" shows "steps xs"