hcju/isabelleps huggingface dataset:
vectorize_monomial: fixes v :: "'a"   and c :: "'c" shows "local.punit.vectorize_poly (monomial c v) = monomial (monomial c (local.punit.pp_of_term v)) (local.punit.component_of_term v)"
netlimit_at_vector: fixes a :: "'a" shows "netlimit (at a) = a"
S_V: fixes l :: "'s"   and u :: "'c \<Rightarrow> real" assumes "(l, u) \<in> S" shows "u \<in> V"
set_uprod_simps: fixes x :: "'a"   and y :: "'a" shows "set_uprod (Upair x y) = {x, y}"
CIComplete: fixes T :: "(ZF, ZF) Axioms"   and \<phi> :: "(ZF, ZF) Language" assumes "ZFAxioms T"   and "L\<lbrakk>\<phi>\<rbrakk>\<^bsub>CI T\<^esub> \<rightarrow> IBool True" shows "\<phi> \<in> Axioms.Theory T"
inject_1: fixes x1 :: "'f"   and y1 :: "'f" shows "(Link x1 = Link y1) = (x1 = y1)"
disj: shows "h\<^sub>0 \<bottom> h\<^sub>1"
connected_component_eq_empty: fixes S :: "'a set"   and x :: "'a" shows "(connected_component_set S x = {}) = (x \<notin> S)"
sub_list_comp: fixes f :: "'g \<Rightarrow> 'f"   and g :: "'h \<Rightarrow> 'g"   and ps :: "('e, 'h) fm list" shows "sub_list f (sub_list g ps) = sub_list (f \<circ> g) ps"
recalc_tableau_lvars: fixes t :: "(nat \<times> linear_poly) list" assumes "\<triangle> t" shows "\<forall>v. \<exists>v'. (\<forall>x\<in>rvars t. v x = v' x) \<and> v' \<Turnstile>\<^sub>t t"
Gets_imp_knows_Spy: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> bankerb_gets" shows "X \<in> knows Spy evs"
\<sigma>_ineqs_neq: fixes X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S" shows "ineq_model \<sigma> X F"
ide_char: fixes f :: "'s" assumes "arr f" shows "ide f = (Dom f = Cod f \<and> Fun f = (\<lambda>x\<in>Dom f. x))"
exec_step_ind_Init_None_Called: fixes sh :: "sheap"   and C :: "char list"   and Cs :: "char list list"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list" assumes "sh C = None" shows "exec_step_ind (StepC C Cs) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, Calling C Cs) # frs, sh(C \<mapsto> (sblank P C, Prepared)))"
eq2: shows "int (card X) - int j gchoose (card X - j) = 1"
NSLIM_diff: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and l :: "'d"   and g :: "'c \<Rightarrow> 'd"   and m :: "'d" assumes "f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l"   and "g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m" shows "(\<lambda>x. f x - g x) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l - m"
atMost_def: fixes u :: "'c list" shows "suffix_order.atMost u \<equiv> {x. suffix x u}"
LTLUntil_1: fixes w :: "nat \<Rightarrow> 'a set" assumes "w \<Turnstile> \<phi>" shows "\<exists>j. \<G>\<^sub>F\<^sub>G \<phi> w \<Turnstile>\<^sub>P af\<^sub>G \<phi> (prefix j w)"
set_mult_card_single_el_eq: fixes J :: "'a set"   and x :: "'a" assumes "J \<subseteq> carrier G"   and "x \<in> carrier G" shows "card (x <# J) = card J"
transpose_mult: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and B :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "(A * B)\<^sup>T = B\<^sup>T * A\<^sup>T"
presCons_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresCons h hA (fromMOD MOD)" shows "presCons (fromIMor h) (fromIMorAbs hA) MOD"
of_real_euler_mascheroni: shows "of_real euler_mascheroni = euler_mascheroni"
matrix_vector_mult_def: fixes m :: "(('b, 'd) vec, 'c) vec"   and x :: "('b, 'd) vec" shows "m *v x \<equiv> \<chi>i. \<Sum>j\<in>UNIV. m $ i $ j * x $ j"
intro: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs"   and "l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs"   and "l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs" shows "l_set_disconnected_nodes_get_parent\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr known_ptrs get_parent get_parent_locs"
possExImpliesNecEx_v1: shows "\<lfloor>\<^bold>\<diamond>(\<lambda>v. \<exists>x. \<forall>xa. \<P> xa v \<longrightarrow> xa x) \<^bold>\<rightarrow> \<^bold>\<box>(\<lambda>v. \<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa v \<longrightarrow> xa x))\<rfloor>"
ring_hom_mult: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
frontier_of_convex_hull: fixes S :: "'b set" assumes "card S = Suc DIM('b)" shows "frontier (convex hull S) = \<Union> {convex hull (S - {a}) |a. a \<in> S}"
left_idem: fixes a :: "'a word"   and b :: "'a word" shows "signed.max a (signed.max a b) = signed.max a b"
neg_1_not_0: shows "- (1::'a) \<noteq> (0::'a)"
Int_def: fixes A :: "'a set"   and B :: "'a set" shows "A \<inter> B = {x \<in> A. x \<in> B}"
sets_restrict_space_count_space: fixes A :: "'b set"   and B :: "'b set" shows "sets (restrict_space (count_space A) B) = sets (count_space (A \<inter> B))"
inf_lower: fixes A :: "'a set"   and x :: "'a" assumes "A \<subseteq> carrier L"   and "x \<in> A" shows "\<Sqinter>A \<sqsubseteq> x"
Tinv_measure_preserving: assumes "I.invertible_qmpt" shows "I.Tinv \<in> measure_preserving M M"
assms_3: shows "c \<longlonglongrightarrow> c0"
assms: fixes x1 :: "'a llist"   and x2 :: "'a llist" assumes "X x1 x2" shows "(\<exists>xs. x1 = LNil \<and> x2 = xs \<and> lfinite xs) \<or> (\<exists>xs ys x zs. x1 = LCons x xs \<and> x2 = prepend zs (LCons x ys) \<and> (prepend_cong1 (X xs) ys \<or> emb xs ys))"
Spec: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and r :: "('e \<times> 'e) set"   and s :: "'e" shows "\<Gamma>\<turnstile>Spec r \<down> Normal s"
ih: fixes ta :: "('s, 'v) tm"   and sa :: "('s, 'v) tm" assumes "{#size ta, size sa#} < {#size t, size s#}"   and "ta >\<^sub>t sa" shows "subst \<rho> ta >\<^sub>t subst \<rho> sa"
cdie: assumes "d\<inverse> \<in> {e, m, b, ov, s, f, d, m\<inverse>, b\<inverse>, ov\<inverse>, s\<inverse>, f\<inverse>, d\<inverse>}" shows "d\<inverse> O e = d\<inverse>"
n_star_sim_right: fixes y :: "'a"   and x :: "'a" assumes "!y \<cdot> x = x \<cdot> !y" shows "!y \<cdot> x\<^sup>\<star> = (!y \<cdot> x)\<^sup>\<star> \<cdot> !y"
sup_Sec_def: fixes d1 :: "Sec"   and d2 :: "Sec" shows "d1 \<squnion> d2 = (if d1 = High \<or> d2 = High then High else Low)"
angelic_def: fixes Q :: "'e \<Rightarrow> 'f"   and p :: "'f" shows "{: Q :} p = {s. Q s \<sqinter> p \<noteq> \<bottom>}"
chine_composite_2: shows "Chn (SPN f \<circ> SPN g \<circ> SPN h) = Maps.pbdom SPN_fgh.\<mu>.leg0 (SPN_fgh.\<nu>.leg1 \<odot> SPN_fgh.\<nu>\<pi>.prj\<^sub>1)"
IfE: fixes b :: "bexp"   and c1 :: "com"   and c2 :: "com"   and s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int" assumes "(IF b THEN c1 ELSE c2, s) \<Rightarrow> t" obtains "bval b s" and "(c1, s) \<Rightarrow> t"   | "\<not> bval b s" and "(c2, s) \<Rightarrow> t"
bnds_cos_def: fixes prec :: "nat"   and lx :: "float"   and ux :: "float" shows "bnds_cos prec lx ux = (let lpi = float_round_down prec (lb_pi prec); upi = float_round_up prec (ub_pi prec); k = floor_fl (float_divr prec (lx + lpi) (2 * lpi)); lx = float_plus_down prec lx (- k * 2 * (if k < 0 then lpi else upi)); ux = float_plus_up prec ux (- k * 2 * (if k < 0 then upi else lpi)) in if - lpi \<le> lx \<and> ux \<le> 0 then (lb_cos prec (- lx), ub_cos prec (- ux)) else if 0 \<le> lx \<and> ux \<le> lpi then (lb_cos prec ux, ub_cos prec lx) else if - lpi \<le> lx \<and> ux \<le> lpi then (min (lb_cos prec (- lx)) (lb_cos prec ux), Float 1 0) else if 0 \<le> lx \<and> ux \<le> 2 * lpi then (Float (- 1) 0, max (ub_cos prec lx) (ub_cos prec (- (ux - 2 * lpi)))) else if - 2 * lpi \<le> lx \<and> ux \<le> 0 then (Float (- 1) 0, max (ub_cos prec (lx + 2 * lpi)) (ub_cos prec (- ux))) else (Float (- 1) 0, Float 1 0))"
height_bound: shows "\<forall>ua\<in>V. relabel_effect f l u ua \<le> 2 * card V - 1"
W_3: shows "f -` U \<inter> (topspace T \<inter> f -` X) = W \<inter> f -` X"
simps_2: fixes \<Gamma> :: "char list set"   and v :: "char list" shows "wfs \<Gamma> (Random v) = (v \<in> \<Gamma>)"
finite_imageI: fixes F :: "'a set"   and h :: "'a \<Rightarrow> 'b" assumes "finite F" shows "finite (h ` F)"
PreEnvironmentJView_axioms: shows "PreEnvironmentJView jkbp spr_jview"
Goal: fixes \<Psi>' :: "'b"   and \<Psi>\<^sub>Q' :: "'b"   and Pa :: "('a, 'b, 'c) psi"   and \<alpha>' :: "'a action"   and P'a :: "('a, 'b, 'c) psi"   and Qa :: "('a, 'b, 'c) psi"   and A\<^sub>Q' :: "name list" assumes "\<Psi>' \<otimes> \<Psi>\<^sub>Q' \<rhd> Pa \<longmapsto> \<alpha>' \<prec> P'a"   and "extractFrame Qa = \<langle>A\<^sub>Q', \<Psi>\<^sub>Q'\<rangle>"   and "bn \<alpha>' \<sharp>* Qa"   and "bn \<alpha>' \<sharp>* subject \<alpha>'"   and "A\<^sub>Q' \<sharp>* \<Psi>'"   and "A\<^sub>Q' \<sharp>* Pa"   and "A\<^sub>Q' \<sharp>* \<alpha>'"   and "distinct A\<^sub>Q'" shows "\<Psi>' \<rhd> Pa \<parallel> Qa \<longmapsto> \<alpha>' \<prec> P'a \<parallel> Qa"
ord_le_eq_subst: fixes a :: "'e"   and b :: "'e"   and f :: "'e \<Rightarrow> 'f"   and c :: "'f" assumes "a \<le> b"   and "f b = c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a \<le> c"
assms: shows "[], 0 \<turnstile> \<Psi> \<triangleright> [] \<hookrightarrow>\<^bsup>k\<^esup> \<S>"
DataValue_def: shows "DataValue = fst \<circ> Rep_data"
Merge_def: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option"   and A :: "'a set" assumes "merkle_interface h bo m" shows "merkle_interface.Merge m A = (if A = {} \<or> infinite A then None else Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) (Some (SOME x. x \<in> A)) A)"
angle_construction_3: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "C \<noteq> B"   and "A' \<noteq> B'" shows "\<exists>C'. Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"
dom_split2_def: fixes P :: "('d \<Rightarrow> 'e) \<times> ('d \<Rightarrow> 'e)"   and p :: "'d \<Rightarrow> 'f decision option" shows "P \<Delta> p = (\<lambda>x. case p x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow y\<rfloor> \<Rightarrow> \<lfloor>allow (fst P x)\<rfloor> | \<lfloor>deny y\<rfloor> \<Rightarrow> \<lfloor>deny (snd P x)\<rfloor>)"
parCasesB': fixes P :: "pi"   and Q :: "pi"   and b :: "subject"   and y :: "name"   and P' :: "pi" assumes "P \<parallel> Q \<longmapsto> b\<guillemotleft>y\<guillemotright> \<prec> P'" obtains P :: "pi" and a :: "subject" and x :: "name" and P'a :: "pi" and Q :: "pi" where "Pa__ \<parallel> Qa__ = Pb__ \<parallel> Qb__" and "b\<guillemotleft>y\<guillemotright> \<prec> P' = a__\<guillemotleft>x__\<guillemotright> \<prec> P'a \<parallel> Qb__" and "Pb__ \<longmapsto> a__\<guillemotleft>x__\<guillemotright> \<prec> P'a" and "x__ \<sharp> Pb__" and "x__ \<sharp> Qb__" and "x__ \<sharp> a__"   | Q :: "pi" and a :: "subject" and x :: "name" and Q' :: "pi" and P :: "pi" where "Pa__ \<parallel> Qa__ = P \<parallel> Q" and "b\<guillemotleft>y\<guillemotright> \<prec> P' = a\<guillemotleft>x\<guillemotright> \<prec> P \<parallel> Q'" and "Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> Q'" and "x \<sharp> P" and "x \<sharp> Q" and "x \<sharp> a"
exch_alw_InvCapsNonneg: fixes s :: "('b, 't, 'loc) Combined.configuration stream" assumes "FullSpec s" shows "alw (\<lambda>s. cri.InvCapsNonneg (exchange_config (shd s))) s"
inj_map: fixes f :: "'e \<Rightarrow> 'f" assumes "inj f" shows "inj (map_bset f)"
valid_sop: fixes D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and \<theta> :: "nat \<Rightarrow> nat option" assumes "(D\<^sub>1, g\<^sub>1) = (D, f)"   and "D \<subseteq> dom \<theta>" shows "f \<theta> = f (\<theta> |` D)"
homeomorphism_image1: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "homeomorphism S T f g" shows "f ` S = T"
steps_0: fixes l :: "cell list"   and r :: "cell list"   and p :: "(action \<times> nat) list \<times> nat"   and n :: "nat" shows "steps (0, l, r) p n = (0, l, r)"
add_diff_cancel_left: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "c + a - (c + b) = a - b"
is_volatile_Write\<^sub>s\<^sub>b_def: fixes r :: "'b memref" shows "is_volatile_Write\<^sub>s\<^sub>b r = (case r of Write\<^sub>s\<^sub>b volatile a x v xa xb xc xd \<Rightarrow> volatile | _ \<Rightarrow> False)"
simps_1: fixes left :: "'c \<Rightarrow> 'd \<Rightarrow> ('a option \<times> 'c) spmf"   and right :: "'c \<Rightarrow> 'e \<Rightarrow> ('b option \<times> 'c) spmf"   and s :: "'c"   and a :: "'d" shows "(left \<oplus>\<^sub>O\<^sup>S right) s (Inl a) = map_spmf (apfst (map_option Inl)) (left s a)"
scount_infinite: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" assumes "alw (ev P) \<omega>" shows "scount P \<omega> = \<infinity>"
assms_2: shows "x \<in> set (local.filter_min_aux xs ys)"
Quotient3_ocircline: shows "Quotient3 ocircline_eq_clmat abs_ocircline rep_ocircline"
divide_numeral_1: fixes a :: "'c" shows "a / Numeral1 = a"
intros: fixes n :: "nat"   and w :: "'b list" assumes "0 < n"   and "n < \<^bold>|w\<^bold>|" shows "mismatch_pair w (rotate n w) \<in> rotate_rel w"
Abs_fps_inverse: fixes y :: "nat \<Rightarrow> 'b" assumes "y \<in> {f. True}" shows "($) (Abs_fps y) = y"
isPath_distD: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "Graph.isPath c u p v" shows "Graph.dist c u (length p) v"
imp_intro: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" assumes "\<phi> \<pi>l \<Longrightarrow> \<psi> \<pi>l" shows "imp \<phi> \<psi> \<pi>l"
language_entails: fixes w :: "'interpretation stream" shows "(w \<in> F.language) = (!!) w \<Turnstile>\<^sub>p \<phi>"
vsubst_preserves_fresh: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "good X"   and "fresh zs z X"   and "zs \<noteq> ys \<or> z \<noteq> y1" shows "fresh zs z (X #[y1 // y]_ys)"
qfree_lift_nnf_qe: fixes qe :: "'a fm \<Rightarrow> 'a fm"   and \<phi> :: "'a fm" assumes "\<And>\<phi>. nqfree \<phi> \<Longrightarrow> qfree (qe \<phi>)" shows "qfree (lift_nnf_qe qe \<phi>)"
beukers_nn_integral1_different: assumes "s < r" shows "beukers_nn_integral1 r s = ennreal ((\<Sum>k\<in>{s<..r}. 1 / real (k\<^sup>2)) / real (r - s))"
untime_hd_eq_filter_hd: fixes s :: "'b message_af list" assumes "untime s \<noteq> []" shows "Msg (hd (untime s)) = hd (filter (\<lambda>x. x \<noteq> NoMsg) s)"
Lock1Synchronized: fixes V :: "nat"   and xs :: "'addr val list"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and e :: "'addr expr1"   and h :: "'heap" assumes "V < length xs" shows "uf,P,t \<turnstile>1 \<langle>sync\<^bsub>V\<^esub> (addr a) e,(h, xs)\<rangle> -\<lbrace>(Lock, a), SyncLock a\<rbrace>\<rightarrow> \<langle>insync\<^bsub>V\<^esub> (a) e,(h, xs[V := Addr a])\<rangle>"
Cons: 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 e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es :: "expr list"   and es' :: "expr list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>e # es,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>Val v # es',s\<^sub>2\<rangle>"
frechet_derivative_plus: fixes x :: "'c \<Rightarrow> 'd"   and a :: "'c"   and y :: "'c \<Rightarrow> 'd" assumes "x differentiable at a"   and "y differentiable at a" shows "frechet_derivative (\<lambda>xa. x xa + y xa) (at a) = (\<lambda>xa. frechet_derivative x (at a) xa + frechet_derivative y (at a) xa)"
sub: shows "flow0 x ` (existence_ivl0 x \<inter> {0..}) \<subseteq> D"
wt_elim_cases_13: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and va :: "Term.var"   and v :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (va:=v)\<Colon>T" obtains Ta :: "ty" and T' :: "ty" where "T = Inl T'" and "E,dt\<Turnstile>va\<Colon>=Ta" and "va \<noteq> LVar This" and "E,dt\<Turnstile>v\<Colon>-T'" and "prg E\<turnstile>T'\<preceq>Ta"
disc_eq_case_2: fixes option :: "'a option" shows "(option \<noteq> None) = (case option of None \<Rightarrow> False | \<lfloor>uu_\<rfloor> \<Rightarrow> True)"
h: shows "abc_steps_l (0, args) (ap [+] dummy_abc (length args)) stp = (length (ap [+] dummy_abc (length args)), args @ rec_exec recf args # 0 \<up> m)"
vsum_comm: fixes W1 :: "'c set"   and W2 :: "'c set" assumes "subspace K W1 V"   and "subspace K W2 V" shows "subspace_sum W1 W2 = subspace_sum W2 W1"
Cons_2: fixes ys :: "'a list"   and zs :: "'a list" assumes "a # xs = ys @ x # zs" shows "thesis"
add_closed: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p\<^sub>1 :: "'c point"   and p\<^sub>2 :: "'c point" assumes "ell_field R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "ell_field.on_curve R a b p\<^sub>1"   and "ell_field.on_curve R a b p\<^sub>2" shows "ell_field.on_curve R a b (ell_field.add R a p\<^sub>1 p\<^sub>2)"
bind_mono_1: fixes M :: "'b nres"   and M' :: "'b nres"   and f :: "'b \<Rightarrow> 'c nres"   and f' :: "'b \<Rightarrow> 'c nres" assumes "M \<le> M'"   and "\<And>x. RETURN x \<le> M \<Longrightarrow> f x \<le> f' x" shows "M \<bind> f \<le> M' \<bind> f'"
pgen_lossless_gpv_PauseI: fixes out :: "'h"   and \<I> :: "('h, 'i) \<I>"   and fail :: "ennreal"   and c :: "('j, 'h, 'i) rpv" assumes "out \<in> outs_\<I> \<I>"   and "\<And>r. r \<in> responses_\<I> \<I> out \<Longrightarrow> pgen_lossless_gpv fail \<I> (c r)" shows "pgen_lossless_gpv fail \<I> (Pause out c)"
relpow_commute: fixes R :: "('b \<times> 'b) set"   and n :: "nat" shows "R O R ^^ n = R ^^ n O R"
simps_4: fixes x2 :: "t list"   and x1 :: "ct list" shows "Type x2 \<noteq> TopType x1"
fps_shift_zero: fixes n :: "nat" shows "fps_shift n 0 = 0"
subdigraphD_2: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>" shows "digraph \<alpha> \<CC>"
inv_from_def: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and m :: "nat" shows "local.inv_from p A m \<equiv> \<forall>i j. i < n \<longrightarrow> j < n \<longrightarrow> m < j \<longrightarrow> p A i j"
genv_cons: fixes v :: "bval"   and v' :: "val"   and \<rho> :: "(nat \<times> bval) list"   and \<rho>' :: "(nat \<times> val) list"   and x :: "nat" assumes "v \<in> good v'"   and "good_env \<rho> \<rho>'" shows "good_env ((x, v) # \<rho>) ((x, v') # \<rho>')"
cobounded2: fixes a :: "'c"   and b :: "'c" shows "inf a b \<le> b"
A_proj_wf_trms: fixes l :: "'lbl" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (N2 (proj_unl l A))"
\<delta>_prod_sng1_def: fixes r :: "('a, 'c) ta_rule"   and \<delta>2 :: "('b, 'c) ta_rule set" shows "\<delta>_prod_sng1 r \<delta>2 \<equiv> case r of q1 \<rightarrow> l qs1 \<Rightarrow> {r_prod r q2 \<rightarrow> l qs2 |q2 qs2. length qs1 = length qs2 \<and> q2 \<rightarrow> l qs2 \<in> \<delta>2}"
adjuster_lincomb: fixes w :: "'b vec"   and n :: "nat"   and us :: "'b vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us" shows "adjuster n w us = module.lincomb (module_vec TYPE('b) n) (\<lambda>u. - (w \<bullet>c u) / (u \<bullet>c u)) (set us)"
borel_measurable_inverse: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
real_degree_2_factorization_exists: fixes p :: "real poly" shows "\<exists>qs. p = prod_list qs \<and> (\<forall>q\<in>set qs. degree q \<le> 2)"
L_transform_Tree\<^sub>\<alpha>_preserves_hereditarily_fs: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) FL_Formula.Tree\<^sub>\<alpha>" assumes "FL_Formula.hereditarily_fs t\<^sub>\<alpha>" shows "Formula.hereditarily_fs (L_transform_Tree\<^sub>\<alpha> t\<^sub>\<alpha>)"
simps: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and V :: "nat set"   and s :: "nat set"   and t :: "'b option list set" assumes "getIJ getIJ" shows "New_max.projectTable V (s, t) = (s \<inter> V, restrict V ` t)"
fun_arity_gt_zero: fixes f :: "'fun" assumes "f \<in> \<Union> (range \<Sigma>n) - \<C>" shows "0 < arity f"
pc_bound_def: fixes bp :: "instr array" shows "pc_bound bp \<equiv> max (Max (max_next_pcs (list_of_array bp)) + 1) (array_length bp + 1)"
rot_same_dir_1: fixes x1 :: "'a"   and a :: "'a"   and b :: "'a"   and x2 :: "'a"   and y :: "'a" assumes "x1 \<in> {a<--<b}"   and "x2 \<in> {x1<--<b}" shows "(0 < y \<bullet> rot (a - b)) = (0 < y \<bullet> rot (x1 - x2))"
in_subterms_Union: fixes t :: "('a, 'b) Term.term"   and M :: "('a, 'b) Term.term set" assumes "t \<in> M" shows "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M"
aff_dim_negative_iff: fixes S :: "'a set" shows "(aff_dim S < 0) = (S = {})"