hcju/isabelleps huggingface dataset:
inverse_zero: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse zero = zero"
atLeastLessThan_iff: fixes i :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and u :: "'b" shows "(i \<in> ord.atLeastLessThan less_eq less l u) = (less_eq l i \<and> less i u)"
lift_gi_closed: fixes f :: "'d \<Rightarrow> 'a"   and gens :: "'d set"   and gi :: "bool \<times> 'd" assumes "f \<in> gens \<rightarrow> carrier G"   and "snd gi \<in> gens" shows "lift_gi f gi \<in> carrier G"
ordered_map_remove: fixes S :: "('a \<times> 'b) set"   and x :: "'a \<times> 'b" assumes "is_map S"   and "finite S"   and "x \<in> S" shows "ordered_map (S - {x}) = remove1 x (ordered_map S)"
analz_Un_Tag: fixes H :: "msg set" assumes "Enc_keys_clean H" shows "analz (Tags \<union> H) \<subseteq> Tags \<union> analz H"
field_cont_on_left_edge_image: shows "continuous_on {p. \<exists>y. y \<in> cbox (g2 b) (g1 b) \<and> p = (b, y)} F"
usubstappf_antimon: fixes V :: "variable set"   and U :: "variable set"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and \<phi> :: "fml" assumes "V \<subseteq> U"   and "usubstappf \<sigma> U \<phi> \<noteq> undeff" shows "usubstappf \<sigma> U \<phi> = usubstappf \<sigma> V \<phi>"
hom_poly_lift: fixes q :: "'a poly" shows "map_poly (map_poly hom) (poly_lift q) = poly_lift (map_poly hom q)"
SeqHaddP_graph_aux_def: shows "SeqHaddP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SeqHaddP_aux l sl s k j y. x1 = (s, j, k, y) \<and> x2 = LstSeqP s k y AND HPair Zero j IN s AND All2 l k (SyntaxN.Ex sl (HPair (Var l) (Var sl) IN s AND HPair (SUCC (Var l)) (SUCC (Var sl)) IN s)) \<and> atom l \<sharp> (sl, s, k, j) \<and> atom sl \<sharp> (s, j))"
trcl_single: fixes c :: "'c"   and a :: "'d"   and c' :: "'c"   and t :: "('c \<times> 'd \<times> 'c) set" shows "((c, [a], c') \<in> trcl t) = ((c, a, c') \<in> t)"
Release_ltlp_def: fixes \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "\<phi> R\<^sub>p \<psi> \<equiv> not\<^sub>p (not\<^sub>p \<phi> U\<^sub>p not\<^sub>p \<psi>)"
OV_inv2D: fixes x :: "'c obsv_state_scheme"   and r :: "nat" assumes "x \<in> OV_inv2"   and "v_state.next_round x \<le> r" shows "obsv_state.obs x r = Map.empty"
stream_all_iff: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a stream" shows "stream_all P s = Ball (sset s) P"
0: shows "igSwapInpIPresIGWlsInp MOD \<and> igSwapBinpIPresIGWlsBinp MOD"
assms_2: fixes v :: "'a vec"   and a :: "'a" assumes "P v" shows "P (v @\<^sub>v [a]\<^sub>v)"
carrier_eliminate_entries_1: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and v :: "nat \<Rightarrow> 'c"   and i :: "nat"   and bs :: "nat" assumes "A \<in> carrier_mat nr nc" shows "eliminate_entries_gen minus times v A i bs \<in> carrier_mat nr nc"
partial_state2_axioms: shows "partial_state2 {0..<Suc k} {Suc k..<n}"
layout_id_cons: fixes ap :: "abc_inst list"   and p :: "abc_inst" shows "layout_of (ap @ [p]) = layout_of ap @ [length_of p]"
redT_updW_Woken_Up_same_no_Notified_Interrupted: fixes t :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and wa :: "('b, 'c) wait_set_action"   and ws' :: "'b \<Rightarrow> 'c wait_set_status option"   and w :: "wake_up_status" assumes "redT_updW t ws wa ws'"   and "ws' t = \<lfloor>PostWS w\<rfloor>"   and "ws t = \<lfloor>PostWS w\<rfloor>"   and "\<And>w. wa \<noteq> Suspend w" shows "wa \<noteq> Notified \<and> wa \<noteq> WokenUp"
perm_is_ub_eqvt: fixes S :: "'c set"   and x :: "'c"   and \<pi> :: "perm" assumes "S <| x" shows "\<pi> \<bullet> S <| \<pi> \<bullet> x"
trimmed_neg_imp_trimmed: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and x :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "multiseries.trimmed_neg trimmed dominant_term x" shows "trimmed x"
assms: shows "inter_aform_plane_ortho p Z n g = Some X"
"case": fixes f :: "real interval \<Rightarrow> real interval \<Rightarrow> 'a"   and x1 :: "real interval"   and x2 :: "real interval" shows "(case Complex_Interval x1 x2 of Complex_Interval x xa \<Rightarrow> f x xa) = f x1 x2"
wcode_fourtimes_case: fixes m :: "nat"   and ires :: "cell list"   and rs :: "nat"   and n :: "nat" shows "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk \<up> m @ Oc # Bk # Oc # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main stp = (Suc 0, Bk # Bk \<up> ln @ Oc # ires, Bk # Oc \<up> Suc (4 * rs + 4) @ Bk \<up> rn)"
50: shows "bprv (all yy (imp (eql \<langle>\<psi>\<rangle> (Var yy)) (imp (subst \<phi> (Var yy) xx) (subst \<phi> \<langle>\<psi>\<rangle> xx))))"
verdicts_def: fixes \<sigma> :: "'a trace" shows "verdicts \<sigma> = {(i, v). wf_tuple nfv fv v \<and> sat \<sigma> (map the v) i}"
rep_eq: fixes x :: "('a, 'b) alist" shows "keys x = list.set (map fst (alist.impl_of x))"
a1: shows "A \<Longrightarrow>\<^sub>A B * true"
a0i_defs_3: shows "a0n_trans \<equiv> ((\<Union>A B. \<Union> (range (a0i_running A B))) \<union> (\<Union>A B. \<Union> (range (a0n_commit A B))))\<^sup>="
Max_less_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less (linorder.Max less_eq A) x = (\<forall>a\<in>A. less a x)"
compare_lift_def: shows "compare \<equiv> \<Lambda> (Def x) (Def y). if x < y then LT else if y < x then GT else EQ"
subst_compose1: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y1 :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y2 :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y1"   and "good Y2" shows "X #[Y1 / y]_ys #[Y2 / y]_ys = X #[Y1 #[Y2 / y]_ys / y]_ys"
emeasure_lub_spmf: fixes Y :: "'c spmf set"   and A :: "'c set" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y"   and "Y \<noteq> {}" shows "emeasure (measure_spmf (lub_spmf Y)) A = (\<Squnion>y\<in>Y. emeasure (measure_spmf y) A)"
elims_3: fixes x :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xa :: "'g list" assumes "\<not> successively x xa" obtains P :: "'g \<Rightarrow> 'g \<Rightarrow> bool" and xb :: "'g" and y :: "'g" and xs :: "'g list" where "x = P" and "xa = xb # y # xs" and "\<not> (P xb y \<and> successively P (y # xs))"
assms_3: shows "X = Re (2 * z / (1 + z * cnj z))"
case_unit_unfold: fixes x :: "'l"   and u :: "unit" shows "case u of () \<Rightarrow> x \<equiv> x"
inj_on_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and A :: "'a set" assumes "class.cancel_semigroup_add plus" shows "inj_on (plus a) A"
inv_comm: fixes x :: "'c"   and y :: "'c" assumes "x \<otimes>\<^bsub>H\<^esub> y = \<one>\<^bsub>H\<^esub>"   and "x \<in> carrier H"   and "y \<in> carrier H" shows "y \<otimes>\<^bsub>H\<^esub> x = \<one>\<^bsub>H\<^esub>"
card_of_Plus_empty2: fixes A :: "'c set" shows "|A| =o |{} <+> A|"
prepend_eqI: fixes vs :: "nat list"   and g :: "nat \<Rightarrow> nat option"   and f :: "nat \<Rightarrow> nat option" assumes "\<And>x. x < length vs \<Longrightarrow> g x \<down>= vs ! x"   and "\<And>x. g (length vs + x) = f x" shows "g = vs \<odot> f"
add: fixes b1 :: "'a"   and b2 :: "'a" shows "T (b1 + b2) = T b1 + T b2"
eqv_class_eqD': fixes a :: "'b"   and b :: "'b" assumes "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"   and "a \<in> domain" shows "a \<sim> b"
cblinfun_eqI: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and y :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "\<And>i. cblinfun_apply x i = cblinfun_apply y i" shows "x = y"
path_Entry_target_1: fixes prog :: "cmd"   and n :: "w_node"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list" assumes "While_CFG.path prog n as (_Entry_)" shows "n = (_Entry_)"
pmf_embed_pmf: fixes f :: "'c \<Rightarrow> real"   and x :: "'c" assumes "\<And>x. 0 \<le> f x"   and "(\<Sum>\<^sup>+ x. ennreal (f x)) = 1" shows "pmf (embed_pmf f) x = f x"
impl_red_dfs_def: shows "impl_red_dfs \<equiv> \<lambda>x xa xb xc. REC\<^sub>T (\<lambda>xd (xe, xf). let y' = map2set_insert rbt_insert xf xe in NDFS_SI_Statistics.vis_red_nres \<bind> (\<lambda>xg. LIST_FOREACH' (RETURN (id (x xf))) is_None (\<lambda>xh xi. if map2set_memb (\<lambda>k t. rbt_lookup t k) xh xa then RETURN (red_init_witness xf xh) else RETURN None) None \<bind> (\<lambda>xh. case xh of None \<Rightarrow> LIST_FOREACH' (RETURN (id (x xf))) (\<lambda>(xi, y). is_None y) (\<lambda>xi (xj, xk). if \<not> map2set_memb (\<lambda>k t. rbt_lookup t k) xi xj then xd (xj, xi) \<bind> (\<lambda>(xl, xm). RETURN (xl, prep_wit_red xf xm)) else RETURN (xj, None)) (y', None) | Some xi \<Rightarrow> RETURN (y', xh)))) (xb, xc)"
sat_Since_pointD: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and \<phi> :: "Formula.formula"   and t :: "nat"   and \<psi> :: "Formula.formula"   and I :: "\<I>" assumes "Formula.sat \<sigma> V v i (formula.Since \<phi> (point t) \<psi>)"   and "mem t I" shows "Formula.sat \<sigma> V v i (formula.Since \<phi> I \<psi>)"
index_of_const_def: shows "index_of_const = PrimRecOp1 0 (\<lambda>x y. c_pair 4 (c_pair 2 y))"
assms_6: fixes s :: "'a" assumes "s \<in> S"   and "compo s" shows "nF s = 1"
card_of_Times_singl1: fixes A :: "'a set"   and b :: "'b" shows "|A| =o |A \<times> {b}|"
assms_1: shows "\<forall>\<^sub>F x in at_top. l x \<le> f x"
set_swap: fixes i :: "nat"   and xs :: "'b list"   and j :: "nat" assumes "i < length xs"   and "j < length xs" shows "set (xs[i := xs ! j, j := xs ! i]) = set xs"
finite_subset: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "finite B" shows "finite A"
type_aexpr_Load': fixes v :: "Eg1.var"   and \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and t :: "bexp set" assumes "v \<in> dom \<Gamma>"   and "t = the (\<Gamma> v)" shows "type_aexpr \<Gamma> (Load v) t"
long_inf_chain_is_semifin: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set" assumes "long_ch_by_ord f X \<and> infinite X" shows "\<exists>x. [f[x ..]X]"
q_2: shows "\<forall>i<n. \<exists>r s. (\<forall>j<n. q j \<le> r j \<and> r j \<le> q j + 1) \<and> (\<forall>j<n. q j \<le> s j \<and> s j \<le> q j + 1) \<and> (label (\<Sum>i\<in>Basis. (real (r (b' i)) / real p) *\<^sub>R i) \<circ> b) i \<noteq> (label (\<Sum>i\<in>Basis. (real (s (b' i)) / real p) *\<^sub>R i) \<circ> b) i"
assms: shows "P \<subseteq> {s. \<exists>P' Q' A'. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P' c s Q',A' \<and> P \<subseteq> P' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A}"
filterlim_at_bot: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "filterlim f at_bot F = (\<forall>Z. \<forall>\<^sub>F x in F. f x \<le> Z)"
run_sim1: fixes G :: "('a, 'b) graph_rec_scheme"   and f :: "'a \<Rightarrow> 'c"   and r :: "nat \<Rightarrow> 'a"   and ecnv :: "('a, 'b) graph_rec_scheme \<Rightarrow> 'd" assumes "g_rename_precond G f"   and "graph_defs.is_run G r" shows "graph_defs.is_run (fr_rename_ext ecnv f G) (f \<circ> r)"
seq_split: fixes mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" assumes "\<langle>Stop, mds', mem'\<rangle> \<in> loc_reach \<langle>c\<^sub>1 ;; c\<^sub>2, mds, mem\<rangle>" shows "\<exists>mds'' mem''. \<langle>Stop, mds'', mem''\<rangle> \<in> loc_reach \<langle>c\<^sub>1, mds, mem\<rangle> \<and> \<langle>Stop, mds', mem'\<rangle> \<in> loc_reach \<langle>c\<^sub>2, mds'', mem''\<rangle>"
AE_2_2: shows "AE x in M. (\<lambda>n. u n x / real n) \<longlonglongrightarrow> subcocycle_lim u x"
jump_def: fixes f :: "real \<Rightarrow> real"   and a :: "real" shows "jump f a = (if filterlim f at_bot (at_left a) \<and> filterlim f at_top (at_right a) then 1 else if filterlim f at_top (at_left a) \<and> filterlim f at_bot (at_right a) then - 1 else 0)"
List_def: fixes h :: "'c \<Rightarrow> 'c"   and x :: "'c"   and as :: "'c list" shows "List h x as = Path h x as Null"
mu_help_1: fixes u :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and h :: "'b \<Rightarrow> real" assumes "\<And>n. (\<lambda>m. u m n)\<up>f n"   and "f\<up>h" shows "mu u\<up>h"
assms_3: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "A i \<in>\<^sub>\<circ> Vset \<alpha>"
prems_2: shows "are_all_operators_non_interfering (op # \<pi>\<^sub>1)"
infiter_annil: fixes c :: "'a::refinement_lattice"   and x :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<infinity> ; (x::'a::refinement_lattice) = c\<^sup>\<infinity>"
possible_singles: fixes S :: "'b set"   and x :: "'b" shows "possible (singles S) x"
set_finsert: fixes x :: "'a"   and A :: "'a fset" assumes "x |\<in>| A" obtains B :: "'a fset" where "A = finsert x B" and "x |\<notin>| B"
fresh_star_set_subset: fixes x :: "atom set"   and B :: "'b list"   and A :: "'b list" assumes "x \<sharp>* B"   and "set A \<subseteq> set B" shows "x \<sharp>* A"
to_Gromov_completion: shows "eta = to_Gromov_completion y"
Utrans_P_def: fixes dims :: "nat list"   and vars :: "nat set"   and U :: "complex mat" shows "state_sig.Utrans_P dims vars U = Utrans (mat_extension dims vars U)"
jw_Sum_inv: fixes e :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "{$$} \<turnstile>\<^sub>W e : Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2" obtains e :: "Syntax.term" and \<tau>\<^sub>1' :: "ty" and e' :: "Syntax.term" where "ea__ = App eb__ e'" and "{$$} \<turnstile>\<^sub>W eb__ : Fun \<tau>\<^sub>1'__ (Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2)" and "{$$} \<turnstile>\<^sub>W e' : \<tau>\<^sub>1'__"   | x :: "Syntax.var" and e\<^sub>1 :: "Syntax.term" and \<tau>\<^sub>1' :: "ty" and e\<^sub>2 :: "Syntax.term" where "ea__ = Syntax.Let e\<^sub>1__ x e\<^sub>2__" and "atom x \<sharp> ({$$}, e\<^sub>1__)" and "{$$} \<turnstile>\<^sub>W e\<^sub>1__ : \<tau>\<^sub>1''__" and "{$$}(x $$:= \<tau>\<^sub>1''__) \<turnstile>\<^sub>W e\<^sub>2__ : Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2"   | e :: "Syntax.term" where "ea__ = Inj1 ec__" and "{$$} \<turnstile>\<^sub>W ec__ : \<tau>\<^sub>1"   | e :: "Syntax.term" where "ea__ = Inj2 ed__" and "{$$} \<turnstile>\<^sub>W ed__ : \<tau>\<^sub>2"   | e :: "Syntax.term" and \<tau>\<^sub>1' :: "ty" and \<tau>\<^sub>2' :: "ty" and e\<^sub>1 :: "Syntax.term" and e\<^sub>2 :: "Syntax.term" where "ea__ = Case ee__ e\<^sub>1 e\<^sub>2" and "{$$} \<turnstile>\<^sub>W ee__ : Syntax.Sum \<tau>\<^sub>1'''__ \<tau>\<^sub>2'__" and "{$$} \<turnstile>\<^sub>W e\<^sub>1 : Fun \<tau>\<^sub>1'''__ (Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2)" and "{$$} \<turnstile>\<^sub>W e\<^sub>2 : Fun \<tau>\<^sub>2'__ (Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2)"   | e :: "Syntax.term" and \<tau>\<^sub>2' :: "ty" where "ea__ = Prj1 ef__" and "{$$} \<turnstile>\<^sub>W ef__ : Syntax.Prod (Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2) \<tau>\<^sub>2'"   | e :: "Syntax.term" and \<tau>\<^sub>1' :: "ty" where "ea__ = Prj2 eg__" and "{$$} \<turnstile>\<^sub>W eg__ : Syntax.Prod \<tau>\<^sub>1' (Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2)"   | \<alpha> :: "Syntax.tvar" and e :: "Syntax.term" and \<tau> :: "ty" where "ea__ = Unroll eh__" and "Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2 = subst_type \<tau> (Mu \<alpha> \<tau>) \<alpha>" and "{$$} \<turnstile>\<^sub>W eh__ : Mu \<alpha> \<tau>"   | e :: "Syntax.term" where "ea__ = Auth ei__" and "{$$} \<turnstile>\<^sub>W ei__ : Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2"   | e :: "Syntax.term" where "ea__ = Unauth e" and "{$$} \<turnstile>\<^sub>W e : Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2"
set_map: fixes f :: "'c \<Rightarrow> 'b"   and v :: "'c pmf" shows "set_pmf (map_pmf f v) = f ` set_pmf v"
assms_2: shows "p2 tagged_division_of {a..b} \<inter> {x. c \<le> x \<bullet> k}"
compositionality: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f"   and llist :: "'g llist" shows "lmap f (lmap g llist) = lmap (f \<circ> g) llist"
subsumE_fringe: fixes sub :: "('d \<times> nat) \<times> 'd \<times> nat"   and prb :: "('d, 'e, 'f) pre_RedBlack"   and prb' :: "('d, 'e, 'f) pre_RedBlack" assumes "(subsumee sub \<noteq> subsumer sub \<and> fst (subsumee sub) = fst (subsumer sub) \<and> subsumee sub \<in> red_vertices prb \<and> subsumee sub \<notin> subsumers (subs prb) \<and> subsumee sub \<notin> subsumees (subs prb) \<and> subsumer sub \<in> red_vertices prb \<and> subsumer sub \<notin> subsumees (subs prb) \<and> out_edges (red prb) (subsumee sub) = {} \<and> subs prb' = subs prb \<union> {sub}) \<and> \<not> marked prb (subsumer sub) \<and> \<not> marked prb (subsumee sub) \<and> confs prb (subsumee sub) \<sqsubseteq> confs prb (subsumer sub) \<and> prb' = \<lparr>red = red prb, black = black prb, subs = insert sub (subs prb), init_conf = init_conf prb, confs = confs prb, marked = marked prb, strengthenings = strengthenings prb, \<dots> = pre_RedBlack.more prb\<rparr>" shows "fringe prb' = fringe prb - {subsumee sub}"
gen_succ: fixes set_of :: "'c \<Rightarrow> 'd set"   and memb :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and empty :: "'c"   and un :: "'d list \<Rightarrow> 'c \<Rightarrow> 'c"   and gen_succ :: "('d \<times> 'd) list \<Rightarrow> 'd list \<Rightarrow> 'd list"   and rel :: "('d \<times> 'd) list"   and as :: "'d list" assumes "set_access_gen set_of memb empty un gen_succ" shows "set (gen_succ rel as) = {b. \<exists>a\<in>set as. (a, b) \<in> set rel}"
fun_upd_FINFUNC: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'a" assumes "f \<in> FINFUNC"   and "x \<in> Field s"   and "y \<in> Field r" shows "f(x := y) \<in> FINFUNC"
snoc_3: fixes x :: "variable \<times> typ" assumes "x \<in> \<Union> (fv ` snd ` set (xs @ [x__]))" shows "x \<notin> fst ` set (xs @ [x__])"
int_inc_closed: fixes n :: "int" shows "[n] \<cdot> \<one> \<in> carrier Zp"
tendsto_Lim: fixes net :: "'c filter"   and f :: "'c \<Rightarrow> 'd"   and l :: "'d" assumes "net \<noteq> bot"   and "(f \<longlongrightarrow> l) net" shows "Lim net f = l"
permutation_insert_column_step: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and j :: "nat"   and i :: "nat" assumes "p permutes {0..<n}"   and "j < n" shows "Fun.swap j (Suc j) id \<circ> permutation_insert i (Suc j) p = permutation_insert i j p"
a1: shows "s' = snd (fst (store_sub1 instr rd s_val s)) \<and> get_S (cpu_reg_val PSR s) = 0"
map: fixes f :: "'b \<Rightarrow> 'a"   and x1 :: "'b"   and x2 :: "'b stream" shows "smap f (x1 ## x2) = f x1 ## smap f x2"
assms_1: shows "b2 \<noteq> B2 \<or> b1 \<noteq> B1"
scopeExtPar: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<parallel> Q) \<approx> P \<parallel> <\<nu>x>Q"
ord_eq_le_subst: fixes a :: "'a::ord"   and f :: "'c::ord \<Rightarrow> 'a::ord"   and b :: "'c::ord"   and c :: "'c::ord" assumes "(a::'a::ord) = (f::'c::ord \<Rightarrow> 'a::ord) (b::'c::ord)"   and "(b::'c::ord) \<le> (c::'c::ord)"   and "\<And>(x::'c::ord) y::'c::ord. x \<le> y \<Longrightarrow> (f::'c::ord \<Rightarrow> 'a::ord) x \<le> f y" shows "(a::'a::ord) \<le> (f::'c::ord \<Rightarrow> 'a::ord) (c::'c::ord)"
rel_simps_27: fixes n :: "num" shows "(0::'a) \<le> numeral n"
completion_ex_simple_function: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "simple_function (completion M) f" shows "\<exists>f'. simple_function M f' \<and> (AE x in M. f x = f' x)"
inject_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(Eff x2 = Eff y2) = (x2 = y2)"
sem_def: fixes c :: "('a, 'b) conf" shows "sem c = (\<lambda>\<sigma>. \<sigma> \<in> states c)"
s_nc_par_U: fixes x :: "'a" shows "1\<^sub>\<sigma> \<sqinter> x \<parallel> nc = 1\<^sub>\<sigma> \<sqinter> x \<parallel> U"
IH: shows "\<forall>n'>n'. n' \<le> n \<longrightarrow> (\<exists>nid'. (n', nid') \<in> his t n nid) \<longrightarrow> (\<exists>!nid'. (n', nid') \<in> his t n nid) \<and> ((\<exists>n''<n'. \<exists>nid'. (n'', nid') \<in> his t n nid) \<longrightarrow> (\<exists>x. (\<exists>na. latestAct_cond (THE nid'. (n', nid') \<in> his t n nid) t n' na) \<and> \<parallel>snd x\<parallel>\<^bsub>t (fst x)\<^esub> \<and> fst x = \<langle>THE nid'. (n', nid') \<in> his t n nid \<leftarrow> t\<rangle>\<^bsub>n'\<^esub> \<and> (prefix (bc (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n')) (bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x))) \<or> (\<exists>b. bc (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n') = bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x)) @ [b] \<and> mining (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n')))) \<and> (hisPred t n nid n', SOME nid'. (hisPred t n nid n', nid') \<in> his t n nid) = (SOME x. (\<exists>na. latestAct_cond (THE nid'. (n', nid') \<in> his t n nid) t n' na) \<and> \<parallel>snd x\<parallel>\<^bsub>t (fst x)\<^esub> \<and> fst x = \<langle>THE nid'. (n', nid') \<in> his t n nid \<leftarrow> t\<rangle>\<^bsub>n'\<^esub> \<and> (prefix (bc (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n')) (bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x))) \<or> (\<exists>b. bc (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n') = bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x)) @ [b] \<and> mining (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n')))))"
hcomplex_hIm_one: shows "hIm 1 = 0"
Cons_5: fixes a :: "'a"   and b :: "'a" assumes "a \<in> set (x0 # xs) \<union> set ys"   and "b \<in> set ys"   and "rel a b" shows "a = b"
gx1: fixes x :: "nat"   and i :: "nat" assumes "gx1 \<le> x"   and "i < k" shows "gx0 \<le> (ts ! i) x"
inject_8: fixes x1 :: "name"   and x2 :: "pi"   and y1 :: "name"   and y2 :: "pi" shows "(<\<nu>x1>x2 = <\<nu>y1>y2) = ([x1].x2 = [y1].y2)"
GmultD: fixes g :: "'g"   and v :: "('f, 'g) aezfun \<Rightarrow> 'v" shows "g *\<currency> v = (1::'f) \<delta>\<delta> g \<currency> v"
subgroupE_3: fixes H :: "'c set"   and a :: "'c" assumes "subgroup H H"   and "a \<in> H" shows "inv\<^bsub>H\<^esub> a \<in> H"
not_neg_one_less_neg_numeral: 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less (uminus one) (uminus (numeral.numeral one plus m))"