hcju/isabelleps huggingface dataset:
C: fixes s :: "'a"   and s' :: "'b" assumes "(s, s') \<in> R"   and "rwof m c f s"   and "rwof m' c' f' s'" shows "c s = c' s'"
iuntilE: fixes I :: "nat set"   and P' :: "nat \<Rightarrow> bool"   and P :: "nat \<Rightarrow> bool" assumes "P' t'. t' \<U> t I. P t" obtains t :: "nat" where "t \<in> I" and "P t"
measure_interior: fixes S :: "'c set" assumes "bounded S"   and "negligible (frontier S)" shows "Sigma_Algebra.measure lebesgue (interior S) = Sigma_Algebra.measure lebesgue S"
pred_prod_beta: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> bool"   and xy :: "'e \<times> 'f" shows "pred_prod P Q xy = (P (fst xy) \<and> Q (snd xy))"
bind_return_val_distr: fixes M :: "val measure"   and t :: "pdf_type"   and f :: "val \<Rightarrow> val"   and t' :: "pdf_type" assumes "sets M = sets (stock_measure t)"   and "f \<in> stock_measure t \<rightarrow>\<^sub>M stock_measure t'" shows "M \<bind> return_val \<circ> f = distr M (stock_measure t') f"
is_heap_max: fixes v :: "'c"   and t :: "'c Tree" assumes "in_tree v t"   and "is_heap t" shows "v \<le> val t"
currentD_outside_IN: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "current \<Gamma> f"   and "x \<notin> \<^bold>V\<^bsub>\<Gamma>\<^esub>" shows "d_IN f x = 0"
r_minus: fixes x :: "'a"   and y :: "'a" shows "x * - y = - (x * y)"
4: shows "((\<box>a)\<box> \<cong> \<box>a \<^bold>\<leftarrow> E ((\<box>a) \<^bold>\<cdot> a)) \<^bold>\<and> (E ((\<box>a) \<^bold>\<cdot> a) \<^bold>\<leftarrow> (\<box>a)\<box> \<cong> \<box>a)"
override_on_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "f ++\<^bsub>A\<^esub> g = (\<lambda>a. if a \<in> A then g a else f a)"
interleaves_nil_all: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list" shows "xs \<simeq> {[], xs, P}"
isometryI: fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>x y. dist (f x) (f y) = dist x y"   and "surj f" shows "isometry f"
homotopy_eqv_homotopic_triviality_null_imp: fixes S :: "'b set"   and T :: "'c set"   and U :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "S homotopy_eqv T"   and "continuous_on U f"   and "f ` U \<subseteq> T"   and "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>c. homotopic_with_canon (\<lambda>x. True) U S f (\<lambda>x. c)" shows "\<exists>c. homotopic_with_canon (\<lambda>x. True) U T f (\<lambda>x. c)"
HEndPhase1_HInv4a1: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv4a1 s p" shows "HInv4a1 s' p"
is_constant_constructors_simps2_5: fixes y :: "'a ltln"   and x :: "'a ltln" shows "is_constant (mk_until y x) = is_constant x"
rank_Some_sink: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and j :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = Some j" shows "\<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n)"
conth: shows "continuous_on ({0..1} \<times> S) h"
mod: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and a :: "nat" assumes "dcharacter n \<chi>" shows "\<chi> (a mod n) = \<chi> a"
bounded_iff: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le> Min A) = (\<forall>a\<in>A. x \<le> a)"
subgraph_induce_subgraphI2: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "subgraph H G" shows "subgraph H (G \<restriction> verts H)"
path_entry_reachable: fixes p :: "nat list"   and n :: "nat" assumes "path_entry (g_E G) p n" shows "reachable n"
equalityCE: fixes A :: "'b set"   and B :: "'b set"   and c :: "'b" assumes "(A::'b set) = (B::'b set)" obtains "(c::'b) \<in> (A::'b set)" and "(c::'b) \<in> (B::'b set)"   | "(c::'b) \<notin> (A::'b set)" and "(c::'b) \<notin> (B::'b set)"
berlekamp_basis_def: fixes u :: "'b mod_ring poly" shows "berlekamp_basis u = map (Poly \<circ> list_of_vec) (find_base_vectors (berlekamp_resulting_mat u))"
liftM_ctxt_struct_subst: fixes E :: "ctxt"   and i :: "nat"   and t :: "trm"   and F :: "ctxt" shows "ctxt_subst (liftM_ctxt E i) t[i=i F]\<^sup>T = ctxt_subst (liftM_ctxt E i) (t[i=i F]\<^sup>T)"
adopt_node_pointers_preserved: fixes w :: "((_) heap, exception, unit) prog"   and parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent owner_document document_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
append_butlast_last_id: fixes xs :: "'c list" assumes "xs \<noteq> \<epsilon>" shows "butlast xs \<cdot> [last xs] = xs"
countable_Un_Int_1: fixes I :: "'c set"   and N :: "'c \<Rightarrow> 'd set"   and M :: "'d measure" assumes "\<And>i. i \<in> I \<Longrightarrow> N i \<in> sets M" shows "\<Union> (N ` I) \<in> sets M"
assms: shows "y \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
Suc_1: fixes v :: "'a" assumes "n' = num v - 1"   and "v \<in> verts G"   and "v \<noteq> s"   and "dist v \<noteq> \<infinity>" shows "\<exists>e. s \<rightarrow>\<^sup>* tail G e \<and> e \<in> arcs G \<and> head G e = v \<and> dist (tail G e) \<noteq> \<infinity> \<and> parent_edge v = Some e \<and> num v = num (tail G e) + 1"
that: fixes g :: "'a \<Rightarrow> 'b" assumes "continuous_on (\<Union> \<F>) g"   and "g ` \<Union> \<F> \<subseteq> rel_frontier T"   and "\<And>x. x \<in> S \<Longrightarrow> g x = f x" shows "thesis"
invariant_def: fixes P :: "'c \<Rightarrow> bool"   and succs :: "'c \<Rightarrow> 'c list" shows "invariant P succs \<equiv> \<forall>g g'. g' \<in> set (succs g) \<longrightarrow> P g \<longrightarrow> P g'"
Equality_sf: fixes t :: "tm"   and u :: "tm" shows "Sigma_fm (t EQ u)"
prod_emb_Diff: fixes M :: "'b set"   and L :: "'b \<Rightarrow> 'c measure"   and K :: "'b set"   and A :: "('b \<Rightarrow> 'c) set"   and B :: "('b \<Rightarrow> 'c) set" shows "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B"
subst_of_append: fixes ss :: "('c \<times> ('d, 'c) Term.term) list"   and ts :: "('c \<times> ('d, 'c) Term.term) list" shows "subst_of (ss @ ts) = subst_of ts \<circ>\<^sub>s subst_of ss"
abs_eq: fixes x :: "ltkey" shows "LtK x = Ab (cLtK x)"
base: shows "P (G\<lparr>carrier := H0\<rparr>)"
assms_1: fixes x :: "constraint" assumes "x \<in> set cs" shows "\<nexists>a b. LT a b = x"
sum_mult_pOp_closedn: fixes A :: "'a set"   and B :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "A \<subseteq> carrier R"   and "ideal R B" shows "(\<forall>j\<le>n. f j \<in> set_mult R A B) \<longrightarrow> \<Sigma>\<^sub>e R f n \<in> sum_mult R A B"
normalize_rules_property: fixes rs :: "'b rule list"   and P :: "'b match_expr \<Rightarrow> bool"   and f :: "'b match_expr \<Rightarrow> 'b match_expr list"   and Q :: "'b match_expr \<Rightarrow> bool" assumes "\<forall>r\<in>set rs. P (get_match r)"   and "\<forall>m. P m \<longrightarrow> (\<forall>m'\<in>set (f m). Q m')" shows "\<forall>r\<in>set (normalize_rules f rs). Q (get_match r)"
basic_div_exp_def: fixes x :: "'b"   and y :: "'b"   and n :: "nat" shows "basic_div_exp x y n = x ^ n div y ^ (n - 1)"
rsp: shows "rel_fun (=) (eq_onp (\<lambda>x. constant_term (denominator x) \<noteq> (0::'b))) (\<lambda>(x, y). let (x', y') = normalize_quot (x, y) in if constant_term y' = (0::'b) then 0 else quot_to_fract (x', y')) (\<lambda>(x, y). let (x', y') = normalize_quot (x, y) in if constant_term y' = (0::'b) then 0 else quot_to_fract (x', y'))"
vsubset_vsingleton_leftI: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "set {a} \<subseteq>\<^sub>\<circ> A"
rep_eq: fixes x :: "('a, 'b) IEEE.float" shows "IEEE.exponent x = (case Rep_float x of (uu_, e, uua_) \<Rightarrow> unat e)"
disj_sets_disj_Edges: fixes xs :: "nat list"   and ys :: "nat list" assumes "set xs \<inter> set ys = {}" shows "Edges xs \<inter> Edges ys = {}"
expandSet_def: fixes P :: "pi"   and Q :: "pi" shows "expandSet P Q \<equiv> {\<tau>.(P' \<parallel> Q) |P'. \<tau>.P' \<in> summands P} \<union> {\<tau>.(P \<parallel> Q') |Q'. \<tau>.Q' \<in> summands Q} \<union> {a{b}.(P' \<parallel> Q) |a b P'. a{b}.P' \<in> summands P} \<union> {a{b}.(P \<parallel> Q') |a b Q'. a{b}.Q' \<in> summands Q} \<union> {a<x>.(P' \<parallel> Q) |a x P'. a<x>.P' \<in> summands P \<and> x \<sharp> Q} \<union> {a<x>.(P \<parallel> Q') |a x Q'. a<x>.Q' \<in> summands Q \<and> x \<sharp> P} \<union> {<\<nu>x>a{x}.(P' \<parallel> Q) |a x P'. <\<nu>x>a{x}.P' \<in> summands P \<and> x \<sharp> Q} \<union> {<\<nu>x>a{x}.(P \<parallel> Q') |a x Q'. <\<nu>x>a{x}.Q' \<in> summands Q \<and> x \<sharp> P} \<union> {\<tau>.(P'[x::=b] \<parallel> Q') |x P' b Q'. \<exists>a. a<x>.P' \<in> summands P \<and> a{b}.Q' \<in> summands Q} \<union> {\<tau>.(P' \<parallel> Q'[x::=b]) |b P' x Q'. \<exists>a. a{b}.P' \<in> summands P \<and> a<x>.Q' \<in> summands Q} \<union> {\<tau>.(<\<nu>y>(P'[x::=y] \<parallel> Q')) |x P' y Q'. \<exists>a. a<x>.P' \<in> summands P \<and> <\<nu>y>a{y}.Q' \<in> summands Q \<and> y \<sharp> P} \<union> {\<tau>.(<\<nu>y>(P' \<parallel> Q'[x::=y])) |y P' x Q'. \<exists>a. <\<nu>y>a{y}.P' \<in> summands P \<and> a<x>.Q' \<in> summands Q \<and> y \<sharp> Q}"
fls_residue_deriv: fixes f :: "'b fls" shows "fls_residue (fls_deriv f) = (0::'b)"
map_dual_def_var: fixes f :: "'b \<Rightarrow> 'c" shows "map_dual f = \<partial> \<circ> f \<circ> \<partial>\<^sup>-"
lipschitz_on_cmult: fixes C :: "real"   and U :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and a :: "real" assumes "C-lipschitz_on U f" shows "(\<bar>a\<bar> * C)-lipschitz_on U (\<lambda>x. a *\<^sub>R f x)"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "dec_left_move (as, lm) (s, l, r) ires = (\<exists>lm1 m rn. lm = lm1 @ [m] \<and> 0 < rn \<and> (if lm1 = [] then l = Bk # Oc \<up> Suc m @ Bk # Bk # ires else l = Bk # Oc \<up> Suc m @ Bk # <rev lm1> @ Bk # Bk # ires) \<and> r = Bk \<up> rn)"
smallI: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<And>c. 0 < c \<Longrightarrow> \<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)" shows "f \<in> o[F](g)"
tame11b_def: fixes g :: "graph" shows "tame11b g = (\<forall>v\<in>\<V> g. degree g v \<le> (if except g v = 0 then 7 else 6))"
map_default_def: fixes k :: "'a"   and v :: "'b"   and f :: "'b \<Rightarrow> 'b"   and m :: "('a, 'b) mapping" shows "Mapping.map_default k v f m = Mapping.map_entry k f (Mapping.default k v m)"
igSubstCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) Iteration.model_scheme" shows "igSubstCls MOD \<equiv> igSubstIGVar1 MOD \<and> igSubstIGVar2 MOD \<and> igSubstIGAbs MOD \<and> igSubstIGOp MOD"
add_mono: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd"   and d :: "'c \<times> 'c \<Rightarrow> 'd" assumes "a \<preceq> b"   and "c \<preceq> d" shows "a \<oplus> c \<preceq> b \<oplus> d"
split_asm: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'d \<Rightarrow> 'b"   and option :: "'d option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> (\<exists>x2. option = Some x2 \<and> \<not> P (f2 x2))))"
upd_def: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and x :: "nat"   and y :: "nat"   and v :: "'b" shows "upd m x y v = m(x := (m x)(y := v))"
Lcm_1_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "(Lcm A = one) = (\<forall>a\<in>A. dvd.dvd times a one)"
rr: shows "(y, v') # \<rho> \<approx> (y, v') # \<rho>"
space_PiF: fixes I :: "'d set set"   and M :: "'d \<Rightarrow> 'e measure" shows "space (Pi\<^sub>F I M) = (\<Union>J\<in>I. \<Pi>' j\<in>J. space (M j))"
times_enat_simps_2: shows "\<infinity> * \<infinity> = \<infinity>"
simps_3: fixes e :: "(name, nat) finfun"   and A :: "fmla" shows "eval_fmla e (neg A) = (\<not> eval_fmla e A)"
s: shows "tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p s"
lasso_rel_ext_def: fixes Re :: "('f \<times> 'h) set"   and R :: "('e \<times> 'g) set" shows "\<langle>Re, R\<rangle>lasso_rel_ext = {(\<lparr>lasso_reach = r', lasso_va = va', lasso_cysfx = cysfx', \<dots> = m'\<rparr>, \<lparr>lasso_reach = r, lasso_va = va, lasso_cysfx = cysfx, \<dots> = m\<rparr>) |r' r va' va cysfx' cysfx m' m. (r', r) \<in> \<langle>R\<rangle>list_rel \<and> (va', va) \<in> R \<and> (cysfx', cysfx) \<in> \<langle>R\<rangle>list_rel \<and> (m', m) \<in> Re}"
Next: fixes valid_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> event_data option list set) list \<Rightarrow> bool"   and init_msaux :: "Monitor.args \<Rightarrow> 'a"   and add_new_ts_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and join_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and add_new_table_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and result_msaux :: "Monitor.args \<Rightarrow> 'a \<Rightarrow> event_data option list set"   and valid_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> event_data option list set \<times> event_data option list set) list \<Rightarrow> bool"   and init_muaux :: "Monitor.args \<Rightarrow> 'b"   and add_new_muaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b"   and length_muaux :: "Monitor.args \<Rightarrow> 'b \<Rightarrow> nat"   and eval_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> event_data option list set list \<times> 'b"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('a, 'b) mformula"   and \<phi>' :: "Formula.formula"   and first :: "bool"   and nts :: "nat list"   and I :: "\<I>" assumes "maux valid_msaux init_msaux add_new_ts_msaux join_msaux add_new_table_msaux result_msaux valid_muaux init_muaux add_new_muaux length_muaux eval_muaux"   and "maux.wf_mformula valid_msaux valid_muaux length_muaux \<sigma> j P V n R \<phi> \<phi>'"   and "first = (Monitor.progress \<sigma> P \<phi>' j = 0)"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [Monitor.progress \<sigma> P \<phi>' j - 1..<j] nts" shows "maux.wf_mformula valid_msaux valid_muaux length_muaux \<sigma> j P V n R (MNext I \<phi> first nts) (formula.Next I \<phi>')"
division_segment_not_0: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "division_segment a \<noteq> zero"
retract_of_space_retraction_maps: fixes S :: "'b set"   and X :: "'b topology" shows "(S retract_of_space X) = (S \<subseteq> topspace X \<and> (\<exists>r. retraction_maps X (subtopology X S) r id))"
simplex_likeD_iso: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set" assumes "ordering less_eq less"   and "ordering.simplex_like less_eq less P" shows "\<exists>f A. OrderingSetIso less_eq less dual_order.greater_eq dual_order.greater P f \<and> f ` P = Pow A"
Rep_inject: fixes Rep :: "'d \<Rightarrow> 'e"   and Abs :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and x :: "'d"   and y :: "'d" assumes "type_definition Rep Abs A" shows "(Rep x = Rep y) = (x = y)"
obligation2: fixes s :: "(nat list \<times> (bool list \<times> nat list + nat list)) pmf"   and Da :: "(nat list \<times> bool list \<times> nat list) pmf"   and Db :: "(nat list \<times> nat list) pmf"   and qs :: "nat list" assumes "map_pmf con s = Sum_pmf (8 / 10) Da Db"   and "finite (set_pmf Da)"   and "finite (set_pmf Db)" shows "T\<^sub>p_on_rand' (COMB []) s qs = 2 / 10 * T\<^sub>p_on_rand' (Partial_Cost_Model.embed (rTS [])) Db qs + 8 / 10 * T\<^sub>p_on_rand' BIT Da qs"
finite_open_segment: fixes a :: "'b"   and b :: "'b" shows "finite (open_segment a b) = (a = b)"
max_ev_def: shows "local.max_ev = snd local.max_v_ev"
maxim_insert: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}"   and "A \<subseteq> Field r"   and "x \<in> Field r" shows "maxim (insert x A) = local.max2 x (maxim A)"
assms: fixes x :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "x \<in> A"   and "\<And>y. \<lbrakk>y \<in> A; \<And>x. \<lbrakk>x \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x\<rbrakk> \<Longrightarrow> Q y" shows "Q x"
Complexs_of_real: fixes r :: "real" shows "of_real r \<in> \<complex>"
bind_spmf_const: fixes p :: "'c spmf"   and q :: "'b spmf" shows "p \<bind> (\<lambda>x. q) = scale_spmf (weight_spmf p) q"
sumhr_app: fixes M :: "nat star"   and N :: "nat star"   and f :: "nat \<Rightarrow> real" shows "sumhr (M, N, f) = (*f2* (\<lambda>m n. sum f {m..<n})) M N"
fSum_Suc: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<in>nset n (n + Suc m). f j \<in> carrier A" shows "\<Sigma>\<^sub>f A f n (n + Suc m) = \<Sigma>\<^sub>f A f n (n + m) \<plusminus> f (n + Suc m)"
open_Pi'I: fixes I :: "'d set"   and A :: "'d \<Rightarrow> 'e set" assumes "\<And>i. i \<in> I \<Longrightarrow> open (A i)" shows "open (Pi' I A)"
trace_simps_5: fixes A :: "('b, 'c) ioa"   and e :: "'b \<times> ('c \<times> 'b) list" shows "is_exec_of A e \<equiv> fst e \<in> start A \<and> is_exec_frag_of A e"
not_eq_order_implies_strict: fixes a :: "'a"   and b :: "'a" assumes "a \<in> U"   and "b \<in> U"   and "a \<noteq> b"   and "b \<le>\<^sub>o\<^sub>w a" shows "b <\<^sub>o\<^sub>w a"
intermediate_2: shows "R\<^sub>d \<subseteq> R\<^sup>*"
run_bind_nondet: fixes m :: "('a, 'm) nondetT"   and f :: "'a \<Rightarrow> ('a, 'm) nondetT" shows "run_nondet (bind_nondet m f) = bind (run_nondet m) (\<lambda>A. merge A (run_nondet \<circ> f))"
vimage_VLambda_vrange_rep: fixes A :: "V"   and f :: "V \<Rightarrow> V" shows "VLambda A f `\<^sub>\<circ> A = \<R>\<^sub>\<circ> (VLambda A f)"
lt_minus_distinct_eq_max: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lt p \<noteq> lt q" shows "lt (p - q) = ord_term_lin.max (lt p) (lt q)"
functor_abbrev_def: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme"   and A :: "('g, 'i, 'k) Category_scheme"   and B :: "('h, 'j, 'l) Category_scheme" shows "Ftor F : A \<longrightarrow> B \<equiv> Functor F \<and> CatDom F = A \<and> CatCod F = B"
not_less_zero: fixes n :: "'b multiset" shows "\<not> n \<subset># {#}"
axioms_1: fixes G :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "subgroup G M composition unit" shows "submonoid G M composition unit"
simps_2: fixes f :: "(val \<times> val) fset" shows "good (VFun f) = {BClos x e \<rho> |x e \<rho>. ffold (good_entry x e \<rho>) True (good_prod |`| f)}"
maybe_not_eq: fixes x :: "trilean"   and y :: "trilean" shows "(\<not>? x = \<not>? y) = (x = y)"
set_dfs_rev: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and xs :: "('d, 'e) bintree option list" shows "set (dfs f (rev xs)) = set (dfs f xs)"
rel_nebset_aux_infinite: fixes a :: "'d set!['e]"   and b :: "'f set!['e]"   and R :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "((\<forall>t\<in>set_nebset a. \<exists>u\<in>set_nebset b. R t u) \<and> (\<forall>u\<in>set_nebset b. \<exists>t\<in>set_nebset a. R t u)) = ((BNF_Def.Grp {a. set_nebset a \<subseteq> {(a, b). R a b}} (map_nebset fst))\<inverse>\<inverse> OO BNF_Def.Grp {a. set_nebset a \<subseteq> {(a, b). R a b}} (map_nebset snd)) a b"
cases: fixes x :: "(bool list cstate \<times> (bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)) \<times> unit" obtains m :: "bool list" and as :: "bool list \<Rightarrow> bool list option" where "x = ((cstate.Collect m, None, as__), ())"   | ms :: "bool list cstate" and a' :: "bool list" and m' :: "bool list" and as :: "bool list \<Rightarrow> bool list option" where "x = ((ms, Some (a', m'), as), ())"   | vb :: "bool list \<Rightarrow> bool list option" where "x = ((Void, None, vb__), ())"   | vb :: "bool list \<Rightarrow> bool list option" where "x = ((cstate.Fail, None, vba__), ())"   | vb :: "bool list" and vc :: "bool list \<Rightarrow> bool list option" where "x = ((Store vb, None, vc), ())"
lin_indpt_list_def: fixes fs :: "rat vec list" shows "gs.lin_indpt_list fs = (set fs \<subseteq> carrier_vec n \<and> distinct fs \<and> gs.lin_indpt (set fs))"
hb: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) b = Bv"
someI2: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and Q :: "'b \<Rightarrow> bool" assumes "P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (SOME x. P x)"
path_append: fixes r :: "'action list"   and p :: "'state"   and s :: "'action list" assumes "R.path r p"   and "R.path s (target r p)" shows "R.path (r @ s) p"
abs_eq: fixes x :: "int \<times> int" assumes "ratrel x x" shows "inverse (Abs_Rat x) = Abs_Rat (if fst x = 0 then (0, 1) else (snd x, fst x))"
triple_valid2_eq: fixes G :: "prog" assumes "wf_prog G" shows "triple_valid2 G = triple_valid G"
minimal_unsat_core_tabl_atoms_mono: fixes as :: "('a \<times> 'b atom) set"   and bs :: "('a \<times> 'b atom) set"   and I :: "'a set"   and t :: "(nat \<times> linear_poly) list" assumes "as \<subseteq> bs"   and "minimal_unsat_core_tabl_atoms I t as" shows "minimal_unsat_core_tabl_atoms I t bs"
mult_botl: fixes x :: "'a" shows "\<bottom> \<cdot> x = \<bottom>"
make_F_def: fixes Vs :: "name set"   and p :: "perm" shows "make_F Vs p \<equiv> \<lambda>i. if i \<in> Vs then Var (p \<bullet> i) else Var i"
properSeparationOfViews_def: fixes ES1 :: "'b ES_rec"   and ES2 :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>1 :: "'b V_rec"   and \<V>2 :: "'b V_rec" shows "properSeparationOfViews ES1 ES2 \<V> \<V>1 \<V>2 \<equiv> V\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ES1\<^esub> = V\<^bsub>\<V>1\<^esub> \<and> V\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ES2\<^esub> = V\<^bsub>\<V>2\<^esub> \<and> C\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ES1\<^esub> \<subseteq> C\<^bsub>\<V>1\<^esub> \<and> C\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ES2\<^esub> \<subseteq> C\<^bsub>\<V>2\<^esub> \<and> N\<^bsub>\<V>1\<^esub> \<inter> N\<^bsub>\<V>2\<^esub> = {}"
Phi_def: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and n :: "nat" assumes "BIT_Off acts qs init" shows "BIT_Off.Phi acts qs init n = E (map_pmf (BIT_Off.phi acts qs init n) (config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) (take n qs)))"
sorted_drop: fixes xs :: "'b list"   and n :: "nat" assumes "sorted xs" shows "sorted (drop n xs)"