hcju/isabelleps huggingface dataset:
any_sdominate_unreachable: fixes G :: "nat graph_rec"   and n :: "nat"   and d :: "nat" assumes "cfg_doms G"   and "\<not> cfg_doms.reachable G n" shows "cfg_doms.strict_dominate G d n"
stepsST_STCal_steps: fixes P :: "('procS, 'procT) Proc"   and P' :: "('procS, 'procT) Proc" shows "P \<longmapsto>STCal Source Target* P' = P \<longmapsto>ST* P'"
blocking_onI: fixes X'' :: "'a set"   and h :: "'c"   and X :: "'a set"   and ds :: "'b set" assumes "X'' \<noteq> Ch h X"   and "X'' = Ch h (X \<union> X'')"   and "\<And>x. x \<in> X'' \<Longrightarrow> x \<in> PdXXX.CD_on ds (X \<union> X'')" shows "PdXXX.blocking_on ds X h X''"
simps_17: fixes f :: "'b \<Rightarrow> 'a"   and x3 :: "'b" shows "map_behaviour f (Goes_wrong x3) = Goes_wrong (f x3)"
assms_2: shows "v\<^sub>e \<le> other.s' \<delta>"
subseqs_axioms: shows "Diagonal_Subsequence.subseqs P"
Eq\<kappa>S: fixes v :: "i"   and x :: "\<kappa>"   and y :: "\<kappa>" shows "[x \<^bold>=\<^sub>\<kappa> y in v] = (\<exists>o\<^sub>1 o\<^sub>2. Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = Semantics.d\<^sub>\<kappa> y \<and> o\<^sub>1 = o\<^sub>2)"
nsqn_invalidate_eq: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dests :: "nat \<Rightarrow> nat option"   and rsn :: "nat" assumes "dip \<in> kD rt"   and "dests dip = Some rsn" shows "nsqn (invalidate rt dests) dip = rsn - 1"
Conjunctive_def: fixes H :: "'b upred \<Rightarrow> 'b upred" shows "Conjunctive H = (\<exists>Q. \<forall>P. H P = (P \<and> Q))"
nt: fixes P :: "'a \<Rightarrow> real" assumes "sound P" shows "sound (t P)"
spr_jview_def: fixes a :: "'a" shows "spr_jview a = tMap (envObs a)"
elims_2: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and x :: "'a option"   and xa :: "'a list"   and xb :: "'a list" assumes "proper_intrvl.Compl_set_less_aux less proper_interval x xa xb" obtains ao :: "'a option" and xc :: "'a" and xs :: "'a list" and y :: "'a" and ys :: "'a list" where "x = ao" and "xa = xc # xs" and "xb = y # ys" and "if less xc y then \<not> proper_interval ao (Some xc) \<and> proper_intrvl.Compl_set_less_aux less proper_interval (Some xc) xs (y # ys) else if less y xc then \<not> proper_interval ao (Some y) \<and> proper_intrvl.Compl_set_less_aux less proper_interval (Some y) (xc # xs) ys else \<not> proper_interval ao (Some y)"
intro: 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" assumes "\<And>i j p\<^sub>i is\<^sub>i \<O>\<^sub>i \<R>\<^sub>i \<D>\<^sub>i \<theta>\<^sub>i sb\<^sub>i p\<^sub>j is\<^sub>j \<O>\<^sub>j \<R>\<^sub>j \<D>\<^sub>j \<theta>\<^sub>j sb\<^sub>j. \<lbrakk>i < length ts; j < length ts; i \<noteq> j; ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i); ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)\<rbrakk> \<Longrightarrow> (\<O>\<^sub>i \<union> all_acquired sb\<^sub>i) \<inter> (\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) = {}" shows "ownership_distinct ts"
measure_eq_emeasure_eq_ennreal: fixes x :: "real"   and M :: "'a measure"   and A :: "'a set" assumes "0 \<le> x"   and "emeasure M A = ennreal x" shows "Sigma_Algebra.measure M A = x"
S_Min_Le: shows "S_Min_Le = {- d - u c |c d. 0 < v c \<and> v c \<le> n \<and> M 0 (v c) = Le d}"
mult_scaleR_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "real"   and x :: "'b"   and y :: "'b" assumes "class.real_algebra plus zero minus uminus scaleR times" shows "times (scaleR a x) y = scaleR a (times x y)"
convex_hull_explicit: fixes p :: "'c set" shows "convex hull p = {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y}"
viewRn_empty: fixes t :: "('e, 'f) FingerTreeStruc" shows "(t \<noteq> Empty) = (viewRn t \<noteq> None)"
of_int_less_0_iff: 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 z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus z) zero = (z < 0)"
is_const_list1: fixes ves :: "e list"   and vs :: "v list" assumes "ves = map (Basic \<circ> EConst) vs" shows "const_list ves"
sum_subspace: fixes U :: "'b set"   and E :: "'b set"   and V :: "'b set" assumes "Subspace.subspace U E"   and "vectorspace E"   and "Subspace.subspace V E" shows "Subspace.subspace (U + V) E"
csmall_to_csmall_nstep_prop: fixes R :: "'prog \<Rightarrow> 'coll \<Rightarrow> bool"   and Q :: "'prog \<Rightarrow> 'state \<Rightarrow> bool"   and R' :: "'prog \<Rightarrow> 'state \<Rightarrow> 'state \<Rightarrow> 'coll \<Rightarrow> bool"   and P :: "'prog"   and \<sigma>' :: "'state"   and coll :: "'coll"   and \<sigma> :: "'state"   and n :: "nat" assumes "\<And>P \<sigma> \<sigma>' coll. \<lbrakk>(\<sigma>', coll) \<in> csmall P \<sigma>; R P coll; Q P \<sigma>\<rbrakk> \<Longrightarrow> R' P \<sigma> \<sigma>' coll"   and "\<And>P coll1 coll2. R P (combine coll1 coll2) = (R P coll1 \<and> R P coll2)"   and "\<And>P \<sigma> \<sigma>'. \<lbrakk>Q P \<sigma>; \<sigma>' \<in> small P \<sigma>\<rbrakk> \<Longrightarrow> Q P \<sigma>'"   and "\<And>P \<sigma> \<sigma>1 \<sigma>' coll1 coll2. R' P \<sigma> \<sigma>1 coll1 \<and> R' P \<sigma>1 \<sigma>' coll2 \<Longrightarrow> R' P \<sigma> \<sigma>' (combine coll1 coll2)"   and "\<And>\<sigma>. R' P \<sigma> \<sigma> collect_id"   and "(\<sigma>', coll) \<in> csmall_nstep P \<sigma> n"   and "R P coll"   and "Q P \<sigma>" shows "R' P \<sigma> \<sigma>' coll"
l2_dy_fake_chan_def: fixes M :: "chan" shows "l2_dy_fake_chan M \<equiv> {(s, s'). M \<in> dy_fake_chan (l2_state.bad s) (ik s) (chan s) \<and> s' = s\<lparr>chan := {M} \<union> chan s\<rparr>}"
pure_def: fixes f :: "('l, 'm, 'n) prog"   and h :: "'l" shows "pure f h = (h \<turnstile> ok f \<longrightarrow> h \<turnstile> f \<rightarrow>\<^sub>h h)"
representation_neg: fixes basis :: "'b set"   and v :: "'b" assumes "independent basis"   and "v \<in> span basis" shows "representation basis (- v) = (\<lambda>b. - representation basis v b)"
children_eq2_h2: fixes ptr' :: "(_) object_ptr" 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 shadow_root_ptr" shows "|h2 \<turnstile> get_child_nodes ptr'|\<^sub>r = |h' \<turnstile> get_child_nodes ptr'|\<^sub>r"
CFG_axioms_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'c set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'c set" shows "CFG_axioms \<alpha>n invar defs uses \<equiv> ((\<forall>n g. n \<in> set (\<alpha>n g) \<longrightarrow> defs g n \<inter> uses g n = {}) \<and> (\<forall>g n. finite (defs g n))) \<and> (\<forall>v g n. v \<in> uses g n \<longrightarrow> n \<in> set (\<alpha>n g)) \<and> (\<forall>g n. finite (uses g n)) \<and> (\<forall>g. invar g)"
exec_Normal_elim_cases_11: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p :: "'d"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow> t" obtains bdy :: "('e, 'd, 'f) com" where "\<Gamma> p = Some bdy" and "\<Gamma>\<turnstile> \<langle>bdy,Normal s\<rangle> \<Rightarrow> t"   | "t = Stuck" and "\<Gamma> p = None"
scale_zero_left: fixes x :: "'d" shows "0 *\<^sub>C x = (0::'d)"
assms_3: shows "B = [b, b', b'']\<^sub>\<circ>"
entails_member: fixes C :: "'at Literal set"   and S :: "'at Literal set set" assumes "C \<in> S" shows "entails S C"
flow_def: fixes t0 :: "real"   and x0 :: "'a" shows "flow t0 x0 = (\<lambda>t. if t \<in> existence_ivl t0 x0 then csol t0 x0 t t else (0::'a))"
w_weakly_prefers_def: fixes w :: "W"   and \<mu> :: "(M \<times> W) set" shows "Non_Strict.w_weakly_prefers w \<mu> = {m' \<in> Field (Pw w). \<forall>m\<in>{m. (m, w) \<in> \<mu>}. (m, m') \<in> Pw w}"
2: shows "X_poly_plus (R\<lparr>carrier := S\<rparr>) a = X_poly_plus R a"
cond_actions_ok_\<tau>mRed1_inv: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s1' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ct :: "'t conditional_action" assumes "\<tau>mRed1 s1 s1'"   and "r1.cond_action_ok s1 t ct" shows "r1.cond_action_ok s1' t ct"
rtranclE: fixes a :: "'e"   and b :: "'e"   and r :: "('e \<times> 'e) set" assumes "(a, b) \<in> r\<^sup>*" obtains "a = b"   | y :: "'e" where "(a, y) \<in> r\<^sup>*" and "(y, b) \<in> r"
assms_1: shows "homotopic_with (\<lambda>x. True) X Y f f'"
unit_disc_iff_cmod_lt_1: fixes c :: "complex" shows "(of_complex c \<in> unit_disc) = (cmod c < 1)"
crename_AndR': fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname"   and M :: "trm"   and d :: "coname"   and N :: "trm"   and e :: "coname" assumes "R[a\<turnstile>c>b] = AndR <c>.M <d>.N e"   and "c \<sharp> (a, b, d, e, N, R)"   and "d \<sharp> (a, b, c, e, M, R)"   and "e \<sharp> a" shows "(\<exists>M' N'. R = AndR <c>.M' <d>.N' e \<and> M'[a\<turnstile>c>b] = M \<and> N'[a\<turnstile>c>b] = N \<and> c \<sharp> N' \<and> d \<sharp> M') \<or> (\<exists>M' N'. R = AndR <c>.M' <d>.N' a \<and> b = e \<and> M'[a\<turnstile>c>b] = M \<and> N'[a\<turnstile>c>b] = N \<and> c \<sharp> N' \<and> d \<sharp> M')"
convex_hull: fixes C :: "(real \<times> real) set"   and a :: "real \<times> real"   and b :: "real \<times> real"   and x :: "real \<times> real" assumes "finite C"   and "\<And>c. c \<in> C \<Longrightarrow> ccw a b c"   and "x \<in> convex hull C"   and "Counterclockwise_2D_Arbitrary.lex b a" shows "ccw a b x"
constant_function_comp_is_closed_seq: fixes a :: "'a"   and s :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "s \<in> carrier R\<^bsup>\<omega>\<^esup>" shows "constant_function (carrier R) a \<circ> s \<in> carrier R\<^bsup>\<omega>\<^esup>"
permutations_Cons: fixes v :: "'b"   and va :: "'b list" shows "permutations (v # va) = concat (map (\<lambda>y. map ((#) y) (permutations (remove1 y (v # va)))) (v # va))"
simps_19: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "rec_finite_4 f1 f2 f3 f4 a\<^sub>3 = f3"
ab_group_add_ow_axioms_def: fixes U :: "'b set"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" shows "ab_group_add_ow_axioms U plus zero minus uminus \<equiv> (\<forall>a. a \<in> U \<longrightarrow> plus (uminus a) a = zero) \<and> (\<forall>a b. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> minus a b = plus a (uminus b))"
almost_full_on_Plus: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set" assumes "almost_full_on P A"   and "almost_full_on Q B" shows "almost_full_on (sum_le P Q) (A <+> B)"
elims_2: fixes x :: "'b gexp" assumes "contains_bool x" obtains uu_ :: "bool" where "x = Bc uu_"   | g1 :: "'b gexp" and g2 :: "'b gexp" where "x = Nor g1 g2" and "contains_bool g1 \<or> contains_bool g2"
product_nn_integral_fold: fixes I :: "'i set"   and J :: "'i set"   and f :: "('i \<Rightarrow> 'a) \<Rightarrow> ennreal" assumes "I \<inter> J = {}"   and "finite I"   and "finite J"   and "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)" shows "integral\<^sup>N (Pi\<^sub>M (I \<union> J) M) f = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (merge I J (x, y)) \<partial>Pi\<^sub>M J M \<partial>Pi\<^sub>M I M"
B_def: shows "B \<equiv> \<Union> (trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s ` set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F' (map snd (dbproj i D))))"
is_span: shows "C.span trg_f.dom.leg0 trg_f.dom.leg1"
Some_image_these_eq: fixes A :: "'a option set" shows "Some ` Option.these A = {x \<in> A. x \<noteq> None}"
in_chain_finite: fixes A :: "'b set" assumes "Complete_Partial_Order.chain (\<le>) A"   and "finite A"   and "A \<noteq> {}" shows "Sup A \<in> A"
o_assoc: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and h :: "'c \<Rightarrow> 'f" shows "f \<circ> (g \<circ> h) = f \<circ> g \<circ> h"
le_inf_iff: fixes x :: "'g multiset"   and y :: "'g multiset"   and z :: "'g multiset" shows "(x \<subseteq># y \<inter># z) = (x \<subseteq># y \<and> x \<subseteq># z)"
assms: shows "set (coeffs p) \<subseteq> range to_fract"
ereal_uminus_less_reorder: fixes a :: "ereal"   and b :: "ereal" shows "(- a < b) = (- b < a)"
class_of_def: fixes U :: "'c set"   and s :: "'d" shows "stfx.class_of U s \<equiv> eqY.Class (U, s)"
GSMP_intersection_is_homogeneous: fixes \<A> :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and l :: "'b"   and l' :: "'b" assumes "\<forall>l l'. l \<noteq> l' \<longrightarrow> GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) Sec"   and "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) \<inter> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>))"   and "l \<noteq> l'" shows "homogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec"
real_interval_max_interval: fixes a :: "float interval"   and b :: "float interval" shows "real_interval (max_interval a b) = max_interval (real_interval a) (real_interval b)"
simps_9: fixes p :: "('a, 'b) form" shows "FOL_Fitting.size_form (Exists p) = FOL_Fitting.size_form p + 1"
prems: fixes \<alpha>s :: "V list"   and ms :: "nat list" assumes "list.set \<alpha>s \<subseteq> ON"   and "list.set ms \<subseteq> {0<..}"   and "length \<alpha>s = length ms"   and "sorted_wrt (\<lambda>x y. y < x) \<alpha>s"   and "\<alpha> = Cantor_sum \<alpha>s ms" shows "thesis"
dvd_cong_not_eq_mod: fixes xa :: "'a"   and c :: "'a"   and xb :: "'a" assumes "xa mod c \<noteq> xb"   and "c dvd xa mod c - xb"   and "c \<noteq> (0::'a)" shows "xb mod c \<noteq> xb"
irreducible_cong: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalize a = normalize b" shows "comm_semiring_1.irreducible times one zero a = comm_semiring_1.irreducible times one zero b"
rel_setD2: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and A :: "'e set"   and B :: "'f set"   and y :: "'f" assumes "rel_set R A B"   and "y \<in> B" shows "\<exists>x\<in>A. R x y"
increasing_Bseq_subseq_iff: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> nat" assumes "\<And>x y. x \<le> y \<Longrightarrow> norm (f x) \<le> norm (f y)"   and "strict_mono g" shows "Bseq (\<lambda>x. f (g x)) = Bseq f"
less_eq_set_def: fixes A :: "'a set"   and B :: "'a set" shows "(A \<subseteq> B) = ((\<lambda>x. x \<in> A) \<le> (\<lambda>x. x \<in> B))"
plus_Box_def: fixes x :: "'c Box"   and y :: "'c Box" shows "x + y \<equiv> bliftM2 (\<Lambda> a b. a + b)\<cdot>x\<cdot>y"
gauss_jordan_single_def: fixes A :: "'c mat" shows "gauss_jordan_single A = fst (gauss_jordan A (0\<^sub>m (dim_row A) 0))"
assms_2: shows "a \<notin>\<^sub>\<circ> A"
iLast_iff_singleton: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iLast t0 I P = (\<box> t {iprev t0 I}. P t)"
finite_density_unique: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and g :: "'b \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "AE x in M. 0 \<le> g x"   and "integral\<^sup>N M f \<noteq> \<infinity>" shows "(density M f = density M g) = (AE x in M. f x = g x)"
a_transpose_inv: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int"   and z :: "nat \<Rightarrow> int" assumes "x \<oplus> y = z"   and "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "z \<in> carrier Zp" shows "\<ominus> x \<oplus> z = y"
assms_4: shows "x0' \<equiv> einterpret x0 vs"
assms_1: shows "i \<equiv> if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1"
iT_Div_cut_less: fixes k :: "nat"   and I :: "nat set"   and t :: "nat" assumes "0 < k" shows "(I \<oslash> k) \<down>< t = I \<down>< (t * k) \<oslash> k"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "monoid_add_isom hom" shows "comm_monoid_add_isom hom"
sigma_sets_sigma_sets_eq: fixes M :: "'b set set"   and S :: "'b set" assumes "M \<subseteq> Pow S" shows "sigma_sets S (sigma_sets S M) = sigma_sets S M"
finally_until: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n F\<^sub>n (\<phi> U\<^sub>n \<psi>) = w \<Turnstile>\<^sub>n F\<^sub>n \<psi>"
gcd_diff_dvd_left1: 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 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 uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus"   and "dvd.dvd times a b" shows "gcd (minus b c) a = gcd c a"
sd_chain_preferred: fixes m :: "nat"   and n :: "nat"   and q :: "'alt pmf"   and s :: "'alt pmf" assumes "m \<le> n"   and "sd_chain m = Some q"   and "sd_chain n = Some s" shows "Pareto (SD \<circ> R) q s"
rk2_fas_def: fixes rkp :: "floatarith"   and x0 :: "floatarith list"   and h :: "floatarith"   and cx :: "floatarith list"   and s2 :: "floatarith" shows "rk2_fas rkp x0 h cx s2 = map (\<lambda>i. x0 ! i + h * ((1 - 1 / (rkp * R\<^sub>e 2)) * ode_fa_nth x0 i + 1 / (rkp * R\<^sub>e 2) * ode_fa_nth (euler_incr_fas x0 (h * rkp) x0) i) + rk2_fas_err_nth rkp x0 h cx s2 i) [0..<length x0] @ rk2_fas_err rkp x0 h cx s2"
mapping_empty_code_1: shows "mapping_empty mapping_Choose = mapping_empty_choose"
obj_at_cong: fixes s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and r :: "'g"   and P :: "('e, 'f, 'g) object \<Rightarrow> bool"   and P' :: "('e, 'f, 'g) object \<Rightarrow> bool"   and r' :: "'g"   and s' :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state" assumes "\<And>obj. sys_heap s r = Some obj \<Longrightarrow> P obj = P' obj"   and "r = r'"   and "s = s'" shows "obj_at P r s = obj_at P' r' s'"
elims_2: fixes x :: "(String.literal \<times> String.literal) set \<times> ('a \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" assumes "wf_osig x" obtains sub :: "(String.literal \<times> String.literal) set" and tcs :: "'a \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option" where "x = (sub, tcs)" and "wf_subclass sub \<and> wf_tcsigs sub tcs"
sum_subset_le_1: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and c :: "('c, 'd, 'a) cmd"   and I :: "nat set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b" assumes "PL cval"   and "proper c"   and "I \<subseteq> {..<PL.brn c}" shows "sum (PL.wt aval tval cval c s) I \<le> 1"
\<psi>_\<phi>: fixes t :: "'s" assumes "t \<in> S.Univ" shows "\<psi> (\<phi> t) = t"
mA_def: shows "mA = map_mat (\<lambda>a. [:- a:]) A"
max2_underS'_2: fixes x :: "'a Bonk_Schramm_extension_unfolded"   and y :: "'a Bonk_Schramm_extension_unfolded" assumes "x \<in> wo.underS y" shows "wo.max2 y x = y"
left_commute: fixes b :: "'b::semilattice_inf"   and a :: "'b::semilattice_inf"   and c :: "'b::semilattice_inf" shows "(b::'b::semilattice_inf) \<sqinter> ((a::'b::semilattice_inf) \<sqinter> (c::'b::semilattice_inf)) = a \<sqinter> (b \<sqinter> c)"
cong: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and a :: "nat"   and b :: "nat" assumes "dcharacter n \<chi>"   and "[a = b] (mod n)" shows "\<chi> a = \<chi> b"
right_commute: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b * c = a * c * b"
lunit'_ide_eq: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "\<l>\<^sup>-\<^sup>1[f] = \<lparr>Chn = \<langle>Leg1 (Dom f) \<lbrakk>C.cod (Leg1 (Dom f)), Leg1 (Dom f)\<rbrakk> Chn f\<rangle>, Dom = Cod f, Cod = \<lparr>Leg0 = Leg0 (Dom f) \<cdot> \<p>\<^sub>0[C.cod (Leg1 (Dom f)), Leg1 (Dom f)], Leg1 = \<p>\<^sub>1[C.cod (Leg1 (Dom f)), Leg1 (Dom f)]\<rparr>\<rparr>"
fresh_subst_term: fixes x :: "Syntax.var"   and t :: "Syntax.term"   and t' :: "Syntax.term"   and x' :: "Syntax.var" shows "atom x \<sharp> t[t' / x'] = ((x = x' \<or> atom x \<sharp> t) \<and> (atom x' \<sharp> t \<or> atom x \<sharp> t'))"
map_simp: fixes f :: "'b" assumes "B.arr f" shows "B.map f = f"
SAT_def: fixes \<Phi> :: "('a, 'b) lit list set" shows "U.SAT \<Phi> \<equiv> \<forall>\<xi>. U.wtE \<xi> \<longrightarrow> U.satPB \<xi> \<Phi>"
hyps_free_def: fixes pth :: "(('d \<times> 'c) \<times> 'e \<times> 'f) list" shows "hyps_free pth = (\<forall>v\<^sub>1 p\<^sub>1 v\<^sub>2 p\<^sub>2. ((v\<^sub>1, p\<^sub>1), v\<^sub>2, p\<^sub>2) \<in> set pth \<longrightarrow> hyps (nodeOf v\<^sub>1) p\<^sub>1 = None)"
H_iso_fInf: fixes F :: "('b \<Rightarrow> 'c) set"   and x :: "'c"   and y :: "'b" assumes "\<forall>f\<in>F. x \<le> f y" shows "x \<le> (\<Sqinter> F) y"
x_lin_x: fixes x :: "'a" assumes "x \<in> E" shows "x \<in> lin x"
measure_nonneg: fixes M :: "'c measure"   and A :: "'c set" shows "0 \<le> Sigma_Algebra.measure M A"
atom_fresh_star_atom_set_conv: fixes i :: "'a"   and bs :: "atom set" assumes "atom i \<sharp> bs"   and "finite bs" shows "bs \<sharp>* i"
monomial_mem: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c" shows "\<forall>j\<le>fst c. snd c j \<cdot>\<^sub>r X^\<^bsup>R j\<^esup> \<in> carrier R"