hcju/isabelleps huggingface dataset:
mat_kernel_carrier: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "mat_kernel A \<subseteq> carrier_vec nc"
bounded_bilinear_inner: shows "bounded_bilinear (\<bullet>)"
mpoly_base_conv_2: fixes c :: "'c"   and x :: "'c" shows "c = poly [:c:] x"
Ostrowski: fixes n :: "nat" shows "\<forall>vv. vals_nonequiv K (Suc n) vv \<longrightarrow> (\<exists>x\<in>carrier K. Ostrowski_elem K (Suc n) vv x)"
par_strict_col2_par_strict: fixes E :: "'p"   and F :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "E \<noteq> F"   and "A B ParStrict C D"   and "Col C D E"   and "Col C D F" shows "A B ParStrict E F"
simps_1: fixes s :: "'c \<Rightarrow> 'd option" shows "execute_parallel_plan_sas_plus s [] = s"
Max_insert: 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 "linorder.Max less_eq (insert x A) = ord.max less_eq x (linorder.Max less_eq A)"
1: shows "point p \<and> point q \<and> p ; q\<^sup>T \<le> x"
has_integral_reflect_lemma: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and a :: "'a"   and b :: "'a" assumes "(f has_integral i) (cbox a b)" shows "((\<lambda>x. f (- x)) has_integral i) (cbox (- b) (- a))"
prob_space_distr: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and M' :: "'c measure" assumes "prob_space M"   and "f \<in> M \<rightarrow>\<^sub>M M'" shows "prob_space (distr M M' f)"
iT_infinite_1: fixes n :: "nat" shows "infinite [n\<dots>]"
implies_inf_absorb: fixes x :: "'c"   and y :: "'c" shows "- x \<squnion> x \<sqinter> y = - x \<squnion> y"
inject: fixes x :: "'g"   and ya :: "'g" shows "(object_ptr.Ext x = object_ptr.Ext ya) = (x = ya)"
cl_ord_eq_def: shows "cl_ord_eq = cl_ord \<union> {(x, y). mset_cl x = mset_cl y}"
nat_times_as_int: shows "(*) = (\<lambda>a b. nat (int a * int b))"
finite_sorted_on_distinct_unique: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "Linear_order r"   and "A \<subseteq> Field r"   and "finite A" shows "\<exists>!xs. set xs = A \<and> sorted_on r xs \<and> distinct xs"
hcomp_reassoc_2: fixes \<tau> :: "'a"   and \<mu> :: "'a"   and \<nu> :: "'a" assumes "arr \<tau>"   and "arr \<mu>"   and "arr \<nu>"   and "src \<tau> = trg \<mu>"   and "src \<mu> = trg \<nu>" shows "\<tau> \<star> \<mu> \<star> \<nu> = \<a>[cod \<tau>, cod \<mu>, cod \<nu>] \<cdot> ((\<tau> \<star> \<mu>) \<star> \<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom \<tau>, local.dom \<mu>, local.dom \<nu>]"
check_eqv_soundness: fixes \<AA> :: "interp"   and idx :: "nat"   and \<phi> :: "(presb, unit) aformula"   and \<psi> :: "(presb, unit) aformula" assumes "#\<^sub>V \<AA> = idx"   and "Presb.check_eqv idx \<phi> \<psi>" shows "Presb.sat \<AA> \<phi> = Presb.sat \<AA> \<psi>"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_in_set_prefix_obtain: fixes s :: "('d, 'e) stateful_strand_step"   and A :: "('f strand_label \<times> ('d, 'e) stateful_strand_step) list" assumes "s \<in> set (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))" shows "\<exists>l B s'. (l, s) = dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (l, s') \<and> prefix (B @ [(l, s')]) A"
assms_1: shows "\<not> hb (i, Store e1 mo i) (ir, Delete is e2)"
show_law_float10_aux: fixes m :: "int"   and e :: "int" shows "show_law showsp_float10 (m \<e> e)"
set_2: fixes x21 :: "'a"   and x22 :: "'a list" shows "set (x21 # x22) = {x21} \<union> set x22"
dPk: fixes k :: "nat" assumes "k < n" shows "P k \<in> carrier_mat d d"
ls_galois: shows "\<nu> \<stileturn> \<nu>\<^sup>\<natural>"
left_part_def: fixes s :: "'c"   and t :: "'c" shows "left_part s t \<equiv> \<Union> {bag u |u. u \<in> T.left_tree s t}"
isReason_def: fixes clause :: "Literal list"   and literal :: "Literal"   and valuation :: "Literal list" shows "isReason clause literal valuation \<equiv> literal el clause \<and> clauseFalse (removeAll literal clause) valuation \<and> (\<forall>literal'. literal' el removeAll literal clause \<longrightarrow> precedes (opposite literal') literal valuation \<and> opposite literal' \<noteq> literal)"
simps_1: fixes y :: "'b" shows "count_list [] y = 0"
square_free_m_cong: fixes f :: "int poly"   and g :: "int poly" assumes "q.square_free_m f"   and "q.eq_m f g" shows "q.square_free_m g"
eq_monom_sum_var_list: fixes m :: "('a \<times> nat) list"   and n :: "('a \<times> nat) list" assumes "monom_inv m"   and "monom_inv n" shows "(m = n) = (\<forall>x. sum_var_list m x = sum_var_list n x)"
simps_2: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "sum_le P Q (Inr x) (Inr y) = Q x y"
scalar_prod_right_sum_distrib: fixes vvs :: "'a vec list"   and w :: "'a vec" assumes "\<And>v. v \<in> set vvs \<Longrightarrow> v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "w \<bullet> M.sumlist vvs = sum_list (map ((\<bullet>) w) vvs)"
type_conf_Class: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and C :: "char list" shows "(P,E,h \<turnstile> e :\<^bsub>NT\<^esub> Class C) = (P,E,h \<turnstile> e : Class C \<or> P,E,h \<turnstile> e : NT)"
in_scc_of_self: fixes u :: "'a" assumes "u \<in> verts G" shows "u \<in> scc_of u"
cc_lub_below_iff: fixes X :: "CoCalls set"   and G :: "CoCalls" shows "(lub X \<sqsubseteq> G) = (\<forall>G'\<in>X. G' \<sqsubseteq> G)"
AbstAtomic_def: fixes v :: "hf"   and i :: "hf"   and y :: "hf"   and y' :: "hf" shows "AbstAtomic v i y y' \<equiv> \<exists>t u t' u'. AbstTerm v i t t' \<and> AbstTerm v i u u' \<and> (y = q_Eq t u \<and> y' = q_Eq t' u' \<or> y = q_Mem t u \<and> y' = q_Mem t' u')"
mono: fixes a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" assumes "a \<le> c"   and "b \<le> d" shows "inf a b \<le> inf c d"
fps_integral0_neg: fixes a :: "'a fps" shows "fps_integral0 (- a) = - fps_integral0 a"
chain_def: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'a" assumes "class.po below" shows "po.chain below Y = (\<forall>i. below (Y i) (Y (Suc i)))"
collinear_3_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "collinear {a, b, c}"   and "collinear {b, c, d}"   and "b \<noteq> c" shows "collinear {a, b, d}"
norm_axis_1: fixes m :: "'a" shows "norm (axis m 1) = 1"
less: shows "degree (y mod prod m A) < (\<Sum>i\<in>A. degree (m i))"
return_time_step: fixes P :: "'a set"   and x :: "'a"   and t :: "real" assumes "returns_to P (flow0 x t)"   and "closed P"   and "\<And>s. \<lbrakk>0 < s; s \<le> t\<rbrakk> \<Longrightarrow> flow0 x s \<notin> P"   and "0 < t"   and "t \<in> existence_ivl0 x" shows "return_time P (flow0 x t) = return_time P x - t"
mangoldt_le: fixes n :: "nat" assumes "0 < n" shows "mangoldt n \<le> ln (real n)"
expl_cond_eq_sets: fixes N :: "bool stream measure"   and q :: "real"   and n :: "nat"   and X :: "bool stream \<Rightarrow> real" assumes "N = bernoulli_stream q" shows "expl_cond_expect N (proj_stoch_proc geom_proc n) X \<in> borel_measurable (fct_gen_subalgebra N (stream_space borel) (proj_stoch_proc geom_proc n))"
CatCod: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" assumes "PreFunctor F" shows "Category (CatCod F)"
assms_2: shows "negligible {x \<in> T - S. f x \<noteq> (0::'b)}"
st: fixes P :: "'s \<Rightarrow> real" assumes "sound P" shows "sound (t P)"
comm_monoid_fun_def: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" shows "comm_monoid_fun f z \<equiv> comm_monoid f z"
o: shows "\<forall>a\<in>B. orthogonal a v"
leg1_composite: shows "fg.leg1 = f.leg1 \<cdot> fg.prj\<^sub>1"
iprev_mono2: fixes n :: "nat"   and I :: "nat set" assumes "n \<in> I"   and "\<exists>x\<in>I. x < n" shows "iprev n I < n"
z2: shows "\<forall>x. z2 \<sqsubset> x \<longrightarrow> P2 x = P2'"
quasinorm_subset_space: fixes N1 :: "'c quasinorm"   and N2 :: "'c quasinorm" assumes "N1 \<subseteq>\<^sub>N N2" shows "space\<^sub>N N1 \<subseteq> space\<^sub>N N2"
cexpr_sem_integrate_vars': fixes \<rho> :: "nat \<Rightarrow> val"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and vs :: "nat list"   and e :: "cexpr" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "distinct vs"   and "set vs \<inter> V' = {}"   and "nonneg_cexpr (set vs \<union> V') \<Gamma> e"   and "integrable (state_measure (set vs) \<Gamma>) (\<lambda>\<sigma>. extract_real (cexpr_sem (merge (set vs) V' (\<sigma>, \<rho>)) e))"   and "\<Gamma> \<turnstile>\<^sub>c e : REAL"   and "free_vars e \<subseteq> set vs \<union> V'" shows "ennreal (extract_real (cexpr_sem \<rho> (integrate_vars \<Gamma> vs e))) = \<integral>\<^sup>+ x. ennreal (extract_real (cexpr_sem (merge (set vs) V' (x, \<rho>)) e)) \<partial>state_measure (set vs) \<Gamma>"
max_fun_diff_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "s.max_fun_diff f g \<equiv> s.maxim {a \<in> Field s. f a \<noteq> g a}"
g1_def: shows "PROP g1 \<equiv> (\<And>kvs. Q (Suc 0) kvs)"
sterms_not_call: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and q :: "('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>"   and "q \<in> sterms \<Gamma> p" shows "not_call q"
finprod_one_eqI: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'a" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = \<one>" shows "finprod G f A = \<one>"
P5: fixes P :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd"   and w :: "'c \<Rightarrow> 'e" shows "\<turnstile> \<box>[P]_v \<longrightarrow> \<box>[\<box>[P]_v]_w"
always_conj_distrib: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<box>(\<lambda>s. P s \<and> Q s) = (\<lambda>s. (\<box>P) s \<and> (\<box>Q) s)"
cod_dom: fixes f :: "'a1" assumes "C1.arr f" shows "C1.cod (C1.dom f) = C1.dom f"
is_Eulerian_trail_def: fixes G :: "('a, 'b) graph"   and v :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_unMultigraph G" shows "valid_unMultigraph.is_Eulerian_trail G v ps v' \<equiv> valid_unMultigraph.is_trail G v ps v' \<and> edges (rem_unPath ps G) = {}"
finished_ss_sccs_tj_stack: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "'c tarjan_state" assumes "Tarjan_invar G s" shows "dom (finished s) \<subseteq> \<Union> (sccs s) \<union> set (tj_stack s)"
map_gpv'_id: shows "map_gpv' id id id = id"
\<pi>_step_regular: fixes T :: "(symbol \<times> character list) set"   and k :: "nat" shows "regular (Scan T k \<circ> Complete k \<circ> Predict k)"
poly_eval_plus: fixes a :: "'c \<Rightarrow> 'b"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "poly_eval a (p + q) = poly_eval a p + poly_eval a q"
elims_1: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a list"   and y :: "bool" assumes "distinct_on x xa = y" obtains uu_ :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "x = uu_" and "xa = []" and "y"   | cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and xb :: "'a" and xs :: "'a list" where "x = cmp" and "xa = xb # xs" and "y = (\<not> Bex (set xs) (cmp xb) \<and> distinct_on cmp xs)"
funcong: fixes l :: "tm list"   and l' :: "tm list"   and e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool"   and i :: "String.literal" assumes "equal_length l l'" shows "semantics e f g (foldr Imp (zip_eq l l') (Atom (Rl STR ''='' [Fn i l, Fn i l'])))"
ps_1: shows "\<pi>\<^sub>2 = \<pi>\<^sub>1 @ \<pi>'"
pick_NE: fixes X :: "'a set" assumes "X \<noteq> {}" shows "pick X \<in> X"
cnj_x_x_geq0: fixes x :: "complex" shows "0 \<le> cnj x * x"
resolution_classI: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and S :: "'b set set" assumes "incidence_system point_set block_collection"   and "partition_on point_set S"   and "\<And>bl. bl \<in> S \<Longrightarrow> bl \<in># block_collection" shows "incidence_system.resolution_class point_set block_collection S"
f2: shows "cpu_reg_val PC (snd (fst (get_curr_win () s1))) = cpu_reg_val PC (snd (fst (get_curr_win () s2)))"
ballE: fixes A :: "'g set"   and P :: "'g \<Rightarrow> bool"   and x :: "'g" assumes "\<forall>x\<in>A. P x" obtains "P x"   | "x \<notin> A"
measurable_space: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and A :: "'c measure"   and x :: "'b" assumes "f \<in> M \<rightarrow>\<^sub>M A"   and "x \<in> space M" shows "f x \<in> space A"
continuous_at_dist: fixes b :: "'b"   and a :: "'b" shows "isCont (dist a) b"
set_snd_apply_icrit: fixes crit :: "nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and data :: "nat \<times> 'g"   and gs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "(bool \<times> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "snd ` set (punit.apply_icrit crit data gs bs hs ps) = set ps"
member_delete_list_eq2: fixes r :: "'b list"   and e :: "'b" assumes "List.member r e" shows "e # removeAll e r \<approx>2 r"
M_index: fixes i :: "nat"   and j :: "nat" assumes "i < length (map of_int_hom.vec_hom fs')"   and "j < length (map of_int_hom.vec_hom fs')" shows "gs2.M (length (map of_int_hom.vec_hom fs')) $$ (i, j) = gs2.\<mu> i j"
fds_truncate_one: fixes m :: "nat" assumes "0 < m" shows "fds_truncate m 1 = 1"
LT_def: shows "LT = {y. \<exists>x\<in>T. - a + x = y}"
R1_R3_commute: shows "R1 \<circ> R3 = R3 \<circ> R1"
not_call_def: fixes p :: "('ap, 'aq, 'ar, 'as) seqp" shows "not_call p \<equiv> \<forall>pn. p \<noteq> call(pn)"
card_low_digits: fixes m :: "nat"   and ds :: "nat list" assumes "0 < m"   and "\<And>d. d \<in> set ds \<Longrightarrow> m \<le> d" shows "card {i. i < prod_list ds \<and> (\<forall>i0\<in>set (digit_encode ds i). i0 < m)} = m ^ length ds"
node_rel_def_internal: fixes Re :: "('b \<times> 'd) set"   and R :: "('a \<times> 'c) set" shows "node_rel Re R \<equiv> {(\<lparr>name_impl = namei, incoming_impl = inci, new_impl = newi, old_impl = oldi, next_impl = nexti, \<dots> = morei\<rparr>, \<lparr>name = name, incoming = inc, new = new, old = old, next = next, \<dots> = more\<rparr>) |namei name inci inc newi new oldi old nexti next morei more. (namei, name) \<in> nat_rel \<and> (inci, inc) \<in> \<langle>nat_rel\<rangle>dflt_rs_rel \<and> (newi, new) \<in> \<langle>\<langle>R\<rangle>ltlr_rel\<rangle>lss.rel \<and> (oldi, old) \<in> \<langle>\<langle>R\<rangle>ltlr_rel\<rangle>lss.rel \<and> (nexti, next) \<in> \<langle>\<langle>R\<rangle>ltlr_rel\<rangle>lss.rel \<and> (morei, more) \<in> Re}"
induct: fixes P :: "'b acom \<Rightarrow> 'b acom \<Rightarrow> bool"   and a0 :: "'b acom"   and a1 :: "'b acom" assumes "\<And>S S'. P (SKIP {S}) (SKIP {S'})"   and "\<And>x e S x' e' S'. P (x ::= e {S}) (x' ::= e' {S'})"   and "\<And>c1 c2 c1' c2'. \<lbrakk>P c1 c1'; P c2 c2'\<rbrakk> \<Longrightarrow> P (c1;; c2) (c1';; c2')"   and "\<And>b c1 c2 S b' c1' c2' S'. \<lbrakk>P c1 c1'; P c2 c2'\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2 {S}) (IF b' THEN c1' ELSE c2' {S'})"   and "\<And>Inv b c Pa Inv' b' c' P'. P c c' \<Longrightarrow> P ({Inv} WHILE b DO c {Pa}) ({Inv'} WHILE b' DO c' {P'})"   and "\<And>v va vb vc. P (v ::= va {vb}) (SKIP {vc})"   and "\<And>v va vb vc vd. P (v ::= va {vb}) (vc;; vd)"   and "\<And>v va vb vc vd ve vf. P (v ::= va {vb}) (IF vc THEN vd ELSE ve {vf})"   and "\<And>v va vb vc vd ve vf. P (v ::= va {vb}) ({vc} WHILE vd DO ve {vf})"   and "\<And>v va vb. P (v;; va) (SKIP {vb})"   and "\<And>v va vb vc vd. P (v;; va) (vb ::= vc {vd})"   and "\<And>v va vb vc vd ve. P (v;; va) (IF vb THEN vc ELSE vd {ve})"   and "\<And>v va vb vc vd ve. P (v;; va) ({vb} WHILE vc DO vd {ve})"   and "\<And>v va vb vc vd. P (IF v THEN va ELSE vb {vc}) (SKIP {vd})"   and "\<And>v va vb vc vd ve vf. P (IF v THEN va ELSE vb {vc}) (vd ::= ve {vf})"   and "\<And>v va vb vc vd ve. P (IF v THEN va ELSE vb {vc}) (vd;; ve)"   and "\<And>v va vb vc vd ve vf vg. P (IF v THEN va ELSE vb {vc}) ({vd} WHILE ve DO vf {vg})"   and "\<And>v va vb vc vd. P ({v} WHILE va DO vb {vc}) (SKIP {vd})"   and "\<And>v va vb vc vd ve vf. P ({v} WHILE va DO vb {vc}) (vd ::= ve {vf})"   and "\<And>v va vb vc vd ve. P ({v} WHILE va DO vb {vc}) (vd;; ve)"   and "\<And>v va vb vc vd ve vf vg. P ({v} WHILE va DO vb {vc}) (IF vd THEN ve ELSE vf {vg})"   and "\<And>vc v va vb. P (SKIP {vc}) (v ::= va {vb})"   and "\<And>vc vd v va vb. P (vc;; vd) (v ::= va {vb})"   and "\<And>vc vd ve vf v va vb. P (IF vc THEN vd ELSE ve {vf}) (v ::= va {vb})"   and "\<And>vc vd ve vf v va vb. P ({vc} WHILE vd DO ve {vf}) (v ::= va {vb})"   and "\<And>vb v va. P (SKIP {vb}) (v;; va)"   and "\<And>vb vc vd v va. P (vb ::= vc {vd}) (v;; va)"   and "\<And>vb vc vd ve v va. P (IF vb THEN vc ELSE vd {ve}) (v;; va)"   and "\<And>vb vc vd ve v va. P ({vb} WHILE vc DO vd {ve}) (v;; va)"   and "\<And>vd v va vb vc. P (SKIP {vd}) (IF v THEN va ELSE vb {vc})"   and "\<And>vd ve vf v va vb vc. P (vd ::= ve {vf}) (IF v THEN va ELSE vb {vc})"   and "\<And>vd ve v va vb vc. P (vd;; ve) (IF v THEN va ELSE vb {vc})"   and "\<And>vd ve vf vg v va vb vc. P ({vd} WHILE ve DO vf {vg}) (IF v THEN va ELSE vb {vc})"   and "\<And>vd v va vb vc. P (SKIP {vd}) ({v} WHILE va DO vb {vc})"   and "\<And>vd ve vf v va vb vc. P (vd ::= ve {vf}) ({v} WHILE va DO vb {vc})"   and "\<And>vd ve v va vb vc. P (vd;; ve) ({v} WHILE va DO vb {vc})"   and "\<And>vd ve vf vg v va vb vc. P (IF vd THEN ve ELSE vf {vg}) ({v} WHILE va DO vb {vc})" shows "P a0 a1"
mk_minsky_mono: fixes P :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool" assumes "mk_minsky P"   and "\<And>vs vs'. P vs vs' \<Longrightarrow> Q vs vs'" shows "mk_minsky Q"
op_map_lookup_def: fixes k :: "'h"   and m :: "'h \<Rightarrow> 'g option" shows "op_map_lookup k m \<equiv> m k"
has_derivative_sqnorm_at: fixes a :: "'a" shows "((\<lambda>x. (norm x)\<^sup>2) has_derivative (\<lambda>x. 2 *\<^sub>R (a \<bullet> x))) (at a)"
leg1_uniquely_isomorphic_1: shows "tab\<^sub>1 r \<star> p\<^sub>1 \<cong> tab\<^sub>1 (r \<star> s) \<star> cmp.chine"
lang_deriv: fixes c :: "'b"   and r :: "'b rexp" shows "lang (deriv c r) = Deriv c (lang r)"
S_alt_def: fixes p :: "proj2" shows "(p \<in> S) = (proj2_rep p \<bullet> (M *v proj2_rep p) = 0)"
1: shows "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y (Fo y) (\<eta>o y)"
\<tau>exec_methI: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'a instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'b"   and h :: "'c"   and stk :: "'a val list"   and loc :: "'a val list"   and pc :: "nat"   and xcp :: "'a option"   and s' :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option" assumes "JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P ins xt t h (stk, loc, pc, xcp) \<lbrace>\<rbrace> h s'"   and "xcp = None \<Longrightarrow> JVM_heap_base.\<tau>instr typeof_addr P h stk (ins ! pc)" shows "JVM_heap_base.\<tau>exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P ins xt t h (stk, loc, pc, xcp) s'"
efficient_code_aux: fixes n :: "nat"   and ps :: "(nat \<times> nat) list" assumes "0 < n"   and "set ps = (\<lambda>p. (p, multiplicity p n - 1)) ` prime_factors n"   and "distinct ps" shows "f n = (\<Prod>(p, d)\<leftarrow>ps. f_prime_power p (Suc d))"
top_lifted_distrib_lattice_top_def: shows "top \<equiv> Abs_lifted_distrib_lattice_top (\<lambda>f. top)"
assms_4: shows "nodes G1 \<subseteq> nodes G2"
simps_6: fixes Extend :: "'a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'b"   and Length :: "'b \<Rightarrow> nat"   and satisfies0 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and r :: "'a \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> bool"   and \<AA> :: "'b"   and k :: "'a"   and \<phi> :: "('d, 'a) aformula" shows "Formula_Operations.satisfies_gen Extend Length satisfies0 r \<AA> (FEx k \<phi>) = (\<exists>P. r k P (Length \<AA>) \<and> Formula_Operations.satisfies_gen Extend Length satisfies0 r (Extend k 0 \<AA> P) \<phi>)"
pgwt_deducible: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set" assumes "public_ground_wf_term t" shows "M \<turnstile>\<^sub>c t"
weak_partial_order_top_axioms: shows "weak_partial_order_top L"
vdomain_atD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "\<exists>b\<in>elts (\<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>)). \<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = b"
assms: shows "\<forall>fm'. ListMem fm' fmList \<longrightarrow> agree fm fm'"
simps_2: fixes uv :: "mreal" shows "mreal_plus MInfty uv = MInfty"
abs_fset_eq_iff: fixes xs :: "'b list"   and ys :: "'b list" shows "(abs_fset xs = abs_fset ys) = list_eq xs ys"
is_binqueue_trans: fixes l :: "nat"   and x :: "('c, 'd) bintree option"   and xs :: "('c, 'd) bintree option list" assumes "is_binqueue l (x # xs)" shows "is_binqueue (Suc l) xs"
dom_is_obj: fixes Obj :: "'a set"   and Arr :: "'b set"   and Dom :: "'b \<Rightarrow> 'a"   and Cod :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "graph Obj Arr Dom Cod"   and "x \<in> Arr" shows "Dom x \<in> Obj"
lex_pm_keys_leE: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd"   and x :: "'c" assumes "lex_pm s t"   and "x \<in> keys s" obtains y :: "'c" where "y \<in> keys t" and "y \<le> x"
InitConf_ChiRel_Ancestor: fixes T :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and S :: "'a" assumes "T \<in> InitConf A"   and "(S, T) \<in> ChiRel A" shows "S \<in> InitConf A"
measure_return: fixes X :: "'h set"   and M :: "'h measure"   and x :: "'h" assumes "X \<in> sets M" shows "Sigma_Algebra.measure (return M x) X = indicat_real X x"
fconf_hext: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and fs :: "'a \<Rightarrow> val option"   and E :: "'a \<Rightarrow> ty option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "P,h \<turnstile> fs (:\<le>) E"   and "h \<unlhd> h'" shows "P,h' \<turnstile> fs (:\<le>) E"