hcju/isabelleps huggingface dataset:
elims: fixes x :: "label"   and xa :: "nat"   and y :: "label" assumes "x \<oplus> xa = y" obtains l :: "nat" and i :: "nat" where "x = Label l" and "xa = i__" and "y = Label (l + i__)"   | i :: "nat" where "x = Entry" and "xa = ia__" and "y = Entry"   | i :: "nat" where "x = Exit" and "xa = i" and "y = Exit"
a_gla2: fixes x :: "'a"   and y :: "'a" assumes "ad x \<le> ad y" shows "ad x \<cdot> y = (0::'a)"
length_rotate1: fixes xs :: "'g list" shows "length (rotate1 xs) = length xs"
hfref_weaken_pre: fixes P :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c Heap"   and h :: "'a \<Rightarrow> 'd nres"   and R :: "('a \<Rightarrow> 'b \<Rightarrow> assn) \<times> ('a \<Rightarrow> 'b \<Rightarrow> assn)"   and S :: "'d \<Rightarrow> 'c \<Rightarrow> assn" assumes "\<And>x. P x \<longrightarrow> P' x"   and "(f, h) \<in> [P']\<^sub>a R \<rightarrow> S" shows "(f, h) \<in> [P]\<^sub>a R \<rightarrow> S"
extremum_unique: fixes a :: "'b list" shows "??.Sublist.prefix a [] = (a = [])"
ShiftP_Zero: fixes k :: "tm"   and d :: "tm" shows "{} \<turnstile> ShiftP Zero k d Zero"
trg_src: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "trg\<^sub>B (src\<^sub>B \<mu>) = src\<^sub>B \<mu>"
vector_space_with_def: fixes plus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S :: "'d"   and minus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" shows "vector_space_with plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S scale \<equiv> class.ab_group_add plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S \<and> vector_space_with_axioms plus\<^sub>V\<^sub>S scale"
mono_hide_ref: fixes P :: "'b process"   and Q :: "'b process"   and A :: "'b set" assumes "P \<sqsubseteq> Q" shows "P \ A \<sqsubseteq> Q \ A"
put_get: fixes s :: "'s"   and f :: "'s \<Rightarrow> 'm" shows "put s (get f) = put s (f s)"
par_strict_neq1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "A \<noteq> B"
simps_6: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and x21 :: "nat"   and x22 :: "nat" shows "(case Operator x21 x22 of State x xa \<Rightarrow> f1 x xa | Operator x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
sym_preproc_aux_term1_def: fixes d :: "'a \<Rightarrow> nat" shows "sym_preproc_aux_term1 d = {((gs1, ks1, ts1, fs1), gs2, ks2, ts2, fs2). \<exists>t2\<in>set ts2. \<forall>t1\<in>set ts1. t1 \<prec>\<^sub>t t2}"
connected_in_V_2: fixes v :: "'c"   and w :: "'c" assumes "T.connected v w" shows "w \<in> V\<^bsub>T\<^esub>"
sublist_SING_MEM: fixes h :: "'a"   and l :: "'a list" shows "subseq [h] l = ListMem h l"
z1_eq: shows "[(r1 + c * x1) mod order \<G> + order \<G> * c * x1 = r1 + c * x1] (mod order \<G>)"
simps_2: fixes r :: "'b rexp"   and X :: "'b list set"   and rexp :: "'b rexp" shows "Append_rexp r (Trn X rexp) = Trn X (Times rexp r)"
stable_img: fixes t :: "'s" assumes "terminal t" shows "t \<in> img ` hom unity t"
map_fixed_eqvt: fixes p :: "perm"   and l :: "Syntax.var list"   and v :: "Syntax.var"   and t :: "Syntax.term" assumes "p \<bullet> l = l" shows "map_fixed v l (p \<bullet> t) = map_fixed v l t"
D_def: shows "D = mat_of_cols (dim_row A) [col C 0, col C 1]"
pdo2: shows "partial_density_operator \<rho>\<^sub>2"
class_add_norm_eff_sup_state_opt: fixes i :: "instr"   and P :: "jvm_prog"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list"   and \<tau>s :: "(ty list \<times> ty err list) option list"   and C :: "char list"   and mxs :: "nat"   and T :: "ty"   and cdec :: "jvm_method class" assumes "\<forall>(pc', \<tau>')\<in>set (norm_eff i P pc \<tau>). P \<turnstile> \<tau>' \<le>' \<tau>s ! pc'"   and "\<not> is_class P C"   and "app\<^sub>i (i, P, pc, mxs, T, \<tau>)" shows "\<forall>(pc', \<tau>')\<in>set (norm_eff i (class_add P (C, cdec)) pc \<tau>). class_add P (C, cdec) \<turnstile> \<tau>' \<le>' \<tau>s ! pc'"
bernoulli_stream_stake_prob: fixes M :: "bool stream measure"   and p :: "real"   and n :: "nat"   and x :: "bool stream" assumes "M = bernoulli_stream p"   and "p \<le> 1"   and "0 \<le> p" shows "\<P>(w in M. stake n w = stake n x) = prod (prob_component p x) {0..<n}"
nat_Suc_predicate_change_exists2: fixes a :: "nat"   and b :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "a \<le> b"   and "\<not> P a"   and "P b" shows "\<exists>n\<ge>a. n < b \<and> \<not> P n \<and> (\<forall>k\<le>b. n < k \<longrightarrow> P k)"
F_def: shows "F \<equiv> \<lambda>lzip (xs, ys). case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> case ys of LNil \<Rightarrow> LNil | LCons y ys' \<Rightarrow> LCons (x, y) (curry lzip xs' ys')"
simps_3: fixes x2 :: "'m" shows "Some x2 \<noteq> None"
sup_commute: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
poly2_ext: fixes p :: "'a poly poly"   and q :: "'a poly poly" assumes "\<And>x y. poly2 p x y = poly2 q x y" shows "p = q"
ln_ln_primorial'_asymp_equiv: shows "(\<lambda>k. ln (ln (real (primorial' k)))) \<sim>[sequentially] (\<lambda>x. ln (real x))"
alw_eq_sconst: fixes h :: "'a"   and t :: "'a stream" shows "alw (HLD {h}) t = (t = sconst h)"
that: fixes f :: "'a" assumes "ide (g \<cdot> f)" shows "thesis"
simulation_reaches1: fixes a :: "'a"   and a' :: "'a"   and b :: "'b" assumes "A.reaches1 a a'"   and "Pre.equiv' a b" shows "\<exists>b'. B.reaches1 b b' \<and> Pre.equiv' a' b'"
all_regs_def: fixes e :: "((nat \<times> nat) \<times> transition) fset" shows "all_regs e = (\<Union>(uu_, t)\<in>fset e. Transition.enumerate_regs t)"
map_simp: fixes f :: "'c local.term set \<times> 'c local.term set \<times> 'c local.term set" assumes "CCC.arr f" shows "\<alpha>' f = local.inv (\<alpha>\<^sub>F\<^sub>M\<^sub>C (CCC.cod f)) \<cdot> T.ToCT f"
new2_finite: fixes \<psi> :: "'d ltlr" shows "finite (new2 \<psi>)"
S: shows "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x"
assms_2: fixes x :: "'a" assumes "x \<in># A" shows "B x \<subseteq># D x"
cases: fixes x :: "('v \<times> 'v \<Rightarrow> ennreal) \<times> 'v vertex \<times> 'v vertex" obtains f :: "'v \<times> 'v \<Rightarrow> ennreal" and xa :: "'v" and x' :: "'v" and y :: "'v" where "x = (f__, Vertex xa__, Edge x' y__)"   | f :: "'v \<times> 'v \<Rightarrow> ennreal" and xa :: "'v" and y' :: "'v" and y :: "'v" where "x = (fa__, Edge xa y', Vertex y)"   | f :: "'v \<times> 'v \<Rightarrow> ennreal" and vb :: "'v" and vc :: "'v" and v :: "'v" and vd :: "'v" where "x = (fb__, Edge vb__ vc, Edge v vd)"   | f :: "'v \<times> 'v \<Rightarrow> ennreal" and va :: "'v" and vb :: "'v" where "x = (f, Vertex va, Vertex vb)"
word_less_iff_unsigned: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'c word"   and b :: "'c word" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "(a < b) = less (semiring_1.unsigned one plus zero a) (semiring_1.unsigned one plus zero b)"
space_stream_space: fixes M :: "'b measure" shows "space (stream_space M) = streams (space M)"
finite_set_of_finite_funs: fixes A :: "'d set"   and B :: "'e set"   and d :: "'e" assumes "finite A"   and "finite B" shows "finite {f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}"
relhoare_conseq_left: fixes pre' :: "('a \<times> 'b) set"   and Ra :: "('a \<times> 'a) set"   and Rc :: "('b \<times> 'b) set"   and post :: "('a \<times> 'b) set"   and pre :: "('a \<times> 'b) set" assumes "{pre'} Ra, Rc {> post}"   and "pre \<subseteq> pre'" shows "{pre} Ra, Rc {> post}"
leq_add_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.add_linorder_min plus minus zero less_eq less" shows "less_eq x (plus x y)"
cspan_raw_def: shows "cspan \<equiv> module.span (*\<^sub>C)"
LstSeqP_SeqAppendP: fixes H :: "fm set"   and f1 :: "tm"   and k1 :: "tm"   and f2 :: "tm"   and k2 :: "tm"   and g :: "tm"   and y1 :: "tm"   and y2 :: "tm"   and k :: "tm" assumes "H \<turnstile> SeqAppendP f1 (SUCC k1) f2 (SUCC k2) g"   and "H \<turnstile> LstSeqP f1 k1 y1"   and "H \<turnstile> LstSeqP f2 k2 y2"   and "H \<turnstile> HaddP k1 k2 k" shows "H \<turnstile> LstSeqP g (SUCC k) y2"
V_equalityI: fixes a :: "V"   and b :: "V" assumes "\<And>x. (x \<in> elts a) = (x \<in> elts b)" shows "a = b"
prod_fun_imageE: fixes c :: "'a \<times> 'b"   and f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and R :: "('c \<times> 'd) set" assumes "c \<in> map_prod f g ` R" obtains x :: "'c" and y :: "'d" where "c = (f x, g y)" and "(x, y) \<in> R"
A: shows "A = four_block_mat B C (0\<^sub>m m n) D"
round_down_nonneg: fixes s :: "real"   and p :: "int" assumes "0 \<le> s" shows "0 \<le> round_down p s"
poly_deg_one: shows "poly_deg 1 = (0::'a)"
ASSERT_simps_1: shows "cASSERT True = creturn ()"
tllist_all2_thdD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist" assumes "tllist_all2 P Q xs ys"   and "\<not> is_TNil xs \<or> \<not> is_TNil ys" shows "P (thd xs) (thd ys)"
eq: shows "take i xs @ xs ! i # drop (Suc i) xs = fst (splitAt x xs) @ x # snd (splitAt x xs)"
cong: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> real" shows "transition_rates.J R \<equiv> transition_rates.J R"
eq_iff: fixes a :: "'b"   and b :: "'b" shows "(a = b) = (c1.le b a \<and> c1.le a b)"
is_projection_on_is_cadjoint: fixes \<pi> :: "'a \<Rightarrow> 'a"   and M :: "'a set" assumes "is_projection_on \<pi> M"   and "closed_csubspace M" shows "is_cadjoint \<pi> \<pi>"
assms: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>)"
assms_1: shows "s\<^sub>o \<le> u_max"
imp_graph_nodes: fixes cnf :: "lit uprod set" shows "imp_graph cnf \<subseteq> lits_of_cnf cnf \<times> lits_of_cnf cnf"
emb_inverse: fixes x :: "'e" shows "PRJ('e)\<cdot>(EMB('e)\<cdot>x) = x"
B: assumes "P \<sigma>"   and "\<sigma> \<Turnstile> B\<^sub>1" shows "Q"
valid_extractable: fixes implem :: "chan \<Rightarrow> msg"   and I :: "msg set"   and K :: "msg set" assumes "semivalid_implem implem"   and "I \<subseteq> basic_implem.valid implem" shows "semivalid_implem.extractable implem K I \<subseteq> basic_implem.valid implem"
inf_compl_bot: fixes x :: "'a" shows "x \<sqinter> - x = \<bottom>"
wt_elim_cases_18: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and Ts :: "tys" assumes "E,dt\<Turnstile>In3 []\<Colon>Ts" obtains "Ts = Inr []"
if_split_fmem1: fixes Q :: "bool"   and x :: "'a"   and y :: "'a"   and b :: "'a fset" shows "((if Q then x else y) |\<in>| b) = ((Q \<longrightarrow> x |\<in>| b) \<and> (\<not> Q \<longrightarrow> y |\<in>| b))"
cexpr_typing_cexpr_comp_aux: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and x :: "nat"   and t1 :: "pdf_type"   and e' :: "cexpr"   and t2 :: "pdf_type"   and e :: "cexpr" assumes "\<Gamma>(x := t1) \<turnstile>\<^sub>c e' : t2"   and "\<Gamma> \<turnstile>\<^sub>c e : t1" shows "\<Gamma> \<turnstile>\<^sub>c cexpr_comp_aux x e e' : t2"
carrier_relcycle_group: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "carrier (relcycle_group p X S) = singular_relcycle_set p X S"
pth_d: fixes x :: "'a" shows "x + (0::'a) \<equiv> x"
inj_double_dual_iso: shows "inj_on G''.double_dual_iso (carrier (Characters (Characters G)))"
le_inf_transp: fixes w\<^sub>1 :: "'item stream"   and w\<^sub>2 :: "'item stream"   and w\<^sub>3 :: "'item stream" assumes "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>3"
Rep_formula': fixes x :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "FL_Formula.hereditarily_fs (FL_Formula.formula.Rep_formula x)"
opp_mult_def: fixes x :: "'b"   and y :: "'b" shows "x \<odot> y \<equiv> y \<cdot> x"
finitesum_sprodTr: fixes f :: "'f \<Rightarrow> 'a set"   and I :: "'f set"   and r :: "'b"   and g :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> I \<rightarrow> {X. submodule R M X}"   and "I \<noteq> {}"   and "r \<in> carrier R" shows "g \<in> {j. j \<le> n} \<rightarrow> finitesum_base M I f \<longrightarrow> r \<cdot>\<^sub>s \<Sigma>\<^sub>e M g n = \<Sigma>\<^sub>e M (\<lambda>x. r \<cdot>\<^sub>s g x) n"
valof_alt: fixes x :: "('c, 'd) IEEE.float" shows "valof x = (if IEEE.exponent x = 0 then if sign x = 0 then valofd x else - valofd x else if sign x = 0 then valofn x else - valofn x)"
test_axiom_def: shows "test_axiom \<equiv> \<langle> ? (Pred pid2 nothing) \<rangle> Pred pid1 nothing \<leftrightarrow> (Pred pid2 nothing && Pred pid1 nothing)"
regular1: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" shows "regular (mk_regular1 P F)"
cos_def: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event llist"   and i :: "nat" shows "local.cos c t i = s c (Co_Snapshot.ltake i t) i"
primitive_imp_no_constant_factor: fixes p :: "'a poly"   and F :: "'a poly multiset"   and f :: "'a poly" assumes "primitive p"   and "mset_factors F p"   and "f \<in># F" shows "degree f \<noteq> 0"
b43: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "b ((4 * real m + 3) / 2 ^ Suc n) = b ((2 * real m + 1) / 2 ^ n)"
is_functor: shows "functor (\<cdot>) (\<cdot>) \<Omega>.map"
fmrel_iff: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "('f, 'd) fmap"   and n :: "('f, 'e) fmap" shows "fmrel R m n = (\<forall>x. rel_option R (fmlookup m x) (fmlookup n x))"
RU_call: fixes u :: "'a"   and p :: "'b"   and u' :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and v :: "'a"   and M :: "'d set"   and P :: "'b multiset"   and Ml :: "'d set"   and Me :: "'d set"   and h :: "'d \<Rightarrow> 'd set"   and U :: "'a set" assumes "(u, Call p, u') \<in> edges fg"   and "proc_of fg v = p"   and "(v, M, P) \<in> S_cs fg 0"   and "(v, Ml, Me, h) \<in> RU_cs fg U"   and "mon_n fg u \<inter> Me = {}" shows "(u, mon fg p \<union> M \<union> Ml, Me, ah_update h (mon fg p, M) (Ml \<union> Me)) \<in> RU_cs fg U"
less: fixes y :: "'a" assumes "y \<in> S" shows "x < f y"
simps_2: fixes x :: "nat"   and v :: "exp"   and n :: "nat" shows "subst x v (ENat n) = ENat n"
PSimQ: shows "\<Psi> \<rhd> P \<leadsto><Rel> Q"
a_set_val_def: fixes ptr :: "(_) character_data_ptr"   and v :: "char list" shows "l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_val ptr v = Heap_Error_Monad.bind (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr val) (\<lambda>m. put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr val_update v)"
add: fixes b1 :: "'b"   and b2 :: "'b" shows "(b1 + b2, 0::'c) = (b1, 0::'c) + (b2, 0::'c)"
\<phi>_cancel_left: fixes d :: "V" assumes "d \<in> D" shows "\<phi> (ordermap D VWF d) = d"
pw: shows "pairwise (\<lambda>i j. disjnt (singular_simplex_set p (subtopology X i)) (singular_simplex_set p (subtopology X j))) \<U>"
prems_1: shows "mset (d # ds) \<subseteq># mset bs + as"
homotopic_paths_imp_pathstart: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "homotopic_paths s p q" shows "pathstart p = pathstart q"
aff_dim_zero: fixes S :: "'b set" assumes "(0::'b) \<in> affine hull S" shows "aff_dim S = int (dim S)"
execn_Abrupt_end: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate"   and s' :: "'e" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "s = Abrupt s'" shows "t = Abrupt s'"
uadmit_sterm_adjoint': fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<theta> :: "('sf, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp" assumes "\<And>f f'. SFunctions \<sigma> f = Some f' \<Longrightarrow> dsafe f'"   and "\<And>f f'. SPredicates \<sigma> f = Some f' \<Longrightarrow> fsafe f'"   and "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT \<theta>. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)" shows "sterm_sem (local.adjoint I \<sigma> \<nu>) \<theta> = sterm_sem (local.adjoint I \<sigma> \<omega>) \<theta>"
not_refTE: fixes T :: "ty" assumes "\<not> is_refT T" obtains "T = Void"   | "T = Boolean"   | "T = Integer"
intro: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and T :: "'c \<Rightarrow> 'c" assumes "FGModuleHom G smult V smult T"   and "FGModuleEnd_axioms V T" shows "FGModuleEnd G smult V T"
RTrans: shows "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> R \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> R'"
case_prod_beta': fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
vreal_minus_def: shows "vreal_minus = (\<lambda>x\<in>\<^sub>\<circ>\<real>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>. (real_of_vreal (x\<lparr>0\<^sub>\<nat>\<rparr>) - real_of_vreal (x\<lparr>1\<^sub>\<nat>\<rparr>))\<^sub>\<real>)"
assms: shows "L\<lbrakk>Vx : A \<turnstile> e1\<equiv>e2 : B\<rbrakk> \<rightarrow> IBool True"
essential_classI: fixes C :: "'a set"   and K :: "'a \<Rightarrow> 'a pmf" assumes "C \<in> UNIV // MC_syntax.communicating K"   and "\<And>x y. \<lbrakk>x \<in> C; (x, y) \<in> (SIGMA x:UNIV. set_pmf (K x))\<^sup>*\<rbrakk> \<Longrightarrow> y \<in> C" shows "MC_syntax.essential_class K C"
has_locks_acquire_locks_conv': fixes l :: "('c \<times> nat) option"   and t :: "'c"   and t' :: "'c"   and n :: "nat" assumes "may_lock l t"   and "t \<noteq> t'" shows "has_locks (acquire_locks l t n) t' = has_locks l t'"
transfer_bounded_bilinear_bounded_linearI: fixes g :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and f :: "'d \<Rightarrow> 'e \<Rightarrow>\<^sub>L 'f" assumes "g = (\<lambda>i. blinfun_apply (f i))" shows "bounded_bilinear g = bounded_linear f"
norm_cvec_gt_0: fixes z :: "complex \<times> complex" assumes "z \<noteq> vec_zero" shows "0 < \<langle>z\<rangle>"
predict_imp_adapt: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'b"   and N :: "'b measure" assumes "predict_stoch_proc G X N" shows "adapt_stoch_proc G X N"
traverse_cast_Dyn_replicate: fixes xs :: "('a, 'b, 'c) unboxed list"   and ys :: "'a list" assumes "traverse cast_Dyn xs = Some ys" shows "map typeof xs = replicate (length xs) None"
adv_P1_def: fixes \<eta> :: "nat"   and m1 :: "'grp \<times> 'grp"   and m2 :: "bool"   and D :: "('grp \<times> 'grp) \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp \<Rightarrow> bool spmf" shows "sim_def.adv_P1 \<eta> m1 m2 D \<equiv> \<bar>spmf (R1 \<eta> m1 m2 \<bind> D) True - spmf (funct_OT_12 m1 m2 \<bind> (\<lambda>(o1, o2). S1 \<eta> m1 o1 \<bind> D)) True\<bar>"
ans_d_def: fixes x :: "'a" shows "ans_d x = ad (ad x)"
simps_9: fixes \<sigma> :: "(char list \<times> 'b list) Trace.trace"   and v :: "'b list"   and i :: "nat"   and \<phi> :: "'b formula"   and I :: "\<I>"   and \<psi> :: "'b formula" shows "sat \<sigma> v i (Until \<phi> I \<psi>) = (\<exists>j\<ge>i. mem (\<tau> \<sigma> j - \<tau> \<sigma> i) I \<and> sat \<sigma> v j \<psi> \<and> (\<forall>k\<in>{i..<j}. sat \<sigma> v k \<phi>))"
disjuncts_list_nFOR: fixes \<phi>s :: "('a, 'b) aformula list" assumes "list_all (\<lambda>x. \<not> is_FOr x) \<phi>s"   and "\<phi>s \<noteq> []" shows "disjuncts_list (nFOR \<phi>s) = \<phi>s"
hoare_assume: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and p :: "'c Assertion"   and b :: "'c Assertion"   and q :: "'c Assertion" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "hoare p [\<cdot> b ] q = (p \<sqinter> b \<le> q)"
F_1: fixes i :: "'i" shows "range (F i) \<subseteq> sets (M i)"
Fubini': fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> ennreal" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (lborel \<Otimes>\<^sub>M lborel)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f x y \<partial>lborel \<partial>lborel = \<integral>\<^sup>+ x. integral\<^sup>N lborel (f x) \<partial>lborel"
r_neg1: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "\<ominus>\<^bsub>P\<^esub> x \<oplus>\<^bsub>P\<^esub> (x \<oplus>\<^bsub>P\<^esub> y) = y"
blinop_apply_one_blinop: fixes x :: "'a" shows "1 $ x = x"
minus_add: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "\<ominus> (x \<oplus> y) = \<ominus> x \<oplus> \<ominus> y"
SN_s_mul_ext_strong: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set"   and M :: "'a multiset" assumes "order_pair s ns"   and "\<forall>y. y \<in># M \<longrightarrow> SN_on s {y}" shows "SN_on (s_mul_ext ns s) {M}"
simps_3: fixes var :: "nat"   and uv :: "real list" shows "convert_fm var FalseF uv = Some FalseFUni"
PPs_Un: fixes X :: "'b set"   and Y :: "'b set" shows ".[X \<union> Y] = (\<Union>t\<in>.[X]. (+) t ` .[Y])"
map_comp: fixes g :: "'f \<Rightarrow> 'e"   and f :: "'g \<Rightarrow> 'f"   and v :: "'g stream" shows "smap g (smap f v) = smap (g \<circ> f) v"
H_range_1: fixes x :: "'c" shows "0 \<le> H x"
quotient_of_def: fixes x :: "rat" shows "quotient_of x = (THE pair. x = Fract (fst pair) (snd pair) \<and> 0 < snd pair \<and> coprime (fst pair) (snd pair))"
cIP: shows "IDirProds G S I \<subseteq> carrier G"
kh': fixes x :: "real \<times> 'a" assumes "x \<in> {0, 1} \<times> S \<union> {0..1} \<times> T" shows "k x = h' x"
pos: fixes n :: "nat" shows "0 < inverse (real (Suc n))"
2: shows "\<forall>x y. d (d y \<sqinter> x) \<le> d y"
list_update_swap: fixes i :: "nat"   and i' :: "nat"   and xs :: "'b list"   and x :: "'b"   and x' :: "'b" assumes "i \<noteq> i'" shows "xs[i := x, i' := x'] = xs[i' := x', i := x]"
vcard_disjoint_sup: fixes x :: "V"   and y :: "V" assumes "vdisjnt x y" shows "vcard (x \<union>\<^sub>\<circ> y) = vcard x \<oplus> vcard y"
coeff_pcompose_linear: fixes p :: "'a poly"   and c :: "'a"   and i :: "nat" shows "poly.coeff (p \<circ>\<^sub>p [:0::'a, c:]) i = c ^ i * poly.coeff p i"
det_row_add': fixes i :: "'b"   and j :: "'b"   and A :: "(('c, 'b) vec, 'b) vec"   and q :: "'c" assumes "i \<noteq> j" shows "det (row_add A i j q) = det A"
B_FMT: shows "it' s (case_dres False False c) (\<lambda>(k, v) s. s \<bind> f' k v) \<sigma> = foldli l (case_dres False False c) (\<lambda>(k, v) s. s \<bind> f' k v) \<sigma>"
option_while'_term: fixes I :: "'d \<Rightarrow> bool"   and r :: "'d"   and M :: "('d \<times> 'd) set"   and C :: "'d \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd option" assumes "I r"   and "wf M"   and "\<And>r r'. \<lbrakk>I r; C r; B r = Some r'\<rbrakk> \<Longrightarrow> (r', r) \<in> M"   and "\<And>r r'. \<lbrakk>I r; C r; B r = Some r'\<rbrakk> \<Longrightarrow> I r'" obtains sr' :: "'d option" where "(Some r, sr') \<in> option_while' C B"
rs_Union_def: shows "rs_Union = foldl rs.union (rs.empty ())"