hcju/isabelleps huggingface dataset:
lens_scene_override: fixes X :: "'a \<Longrightarrow> 'b"   and s\<^sub>1 :: "'b"   and s\<^sub>2 :: "'b" assumes "mwb_lens X" shows "s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 on \<lbrakk>X\<rbrakk>\<^sub>\<sim> = s\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2"
inf_sup_aci_3: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.lattice inf less_eq less sup" shows "inf x (inf y z) = inf y (inf x z)"
span_explicit: fixes b :: "'a set" assumes "b \<subseteq> S" shows "subspace.span b = {x. (\<exists>t\<subseteq>S. (\<exists>r. x = (\<Sum>a\<in>t. r a *\<^sub>R a)) \<and> finite t \<and> t \<subseteq> b) \<and> x \<in> S}"
one_fun_def: shows "1 = (\<lambda>x. 1::'b)"
set_valued_functor_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" shows "set_valued_functor C S F \<equiv> category C \<and> replete_set_category S \<and> functor C S F"
h2: assumes "Av + Bv * sqrt Cv = 0" shows "Av\<^sup>2 = Bv\<^sup>2 * Cv"
strand_sem_Send_split_1: fixes M :: "('fun, 'var) Term.term set"   and T :: "('fun, 'var) Term.term list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "\<lbrakk>M; map Send T\<rbrakk>\<^sub>c \<theta>"   and "t \<in> set T" shows "\<lbrakk>M; [send\<langle>t\<rangle>\<^sub>s\<^sub>t]\<rbrakk>\<^sub>c \<theta>"
sig_red_zero_monom_mult: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and r :: "'t \<Rightarrow>\<^sub>0 'b"   and c :: "'b"   and t :: "'a" assumes "sig_red_zero sing_reg F r" shows "sig_red_zero sing_reg F (monom_mult c t r)"
Cons_7: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"
upper_closure: shows "\<pi>\<^sub>*\<^bsub>G\<^esub> \<in> carrier \<Y>\<^bsub>G\<^esub> \<rightarrow> carrier \<X>\<^bsub>G\<^esub>"
assoc'_in_hom_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "partial_magma.ide V h"   and "src f = trg g"   and "src g = trg h" shows "partial_magma.in_hom V (bicategory.\<a>' V H \<a> src trg f g h) (H (partial_magma.dom V f) (H (partial_magma.dom V g) (partial_magma.dom V h))) (H (H (partial_magma.cod V f) (partial_magma.cod V g)) (partial_magma.cod V h))"
assms_5: shows "vars_of_lit (subst_lit L \<sigma>) = {}"
rep_eq: fixes x :: "'c \<Rightarrow> 'b"   and xa :: "('c, 'a) uexpr" shows "\<lbrakk>uop x xa\<rbrakk>\<^sub>e = (\<lambda>b. x (\<lbrakk>xa\<rbrakk>\<^sub>e b))"
intros_2: fixes A :: "'d set"   and a :: "'d" assumes "finite A" shows "finite (insert a A)"
length_add2: fixes x :: "real"   and y :: "real" shows "\<forall>ts v. 0 \<le> x \<and> 0 \<le> y \<longrightarrow> \<parallel>ext v\<parallel> = x + y \<longrightarrow> (\<exists>va u. v=va\<parallel>u \<and> \<parallel>ext va\<parallel> = x \<and> \<parallel>ext u\<parallel> = y)"
cramer_lemma_transpose: fixes k :: "'b"   and x :: "('a, 'b) vec"   and A :: "(('a, 'b) vec, 'b) vec" shows "det (\<chi>i. if i = k then \<Sum>i\<in>UNIV. x $ i *s row i A else row i A) = x $ k * det A"
RedCallExternal: fixes s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and a :: "'addr"   and hU :: "htype"   and P :: "'addr J_prog"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and D :: "String.literal"   and t :: "'thread_id"   and vs :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and va :: "'addr extCallRet"   and h' :: "'heap"   and ta' :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and e' :: "'addr expr"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" assumes "typeof_addr (hp s) a = \<lfloor>hU\<rfloor>"   and "P \<turnstile> class_type_of hU sees M: Ts\<rightarrow>T = Native in D"   and "P,t \<turnstile> \<langle>a\<bullet>M(vs),hp s\<rangle> -ta\<rightarrow>ext \<langle>va,h'\<rangle>"   and "ta' = extTA ta"   and "e' = extRet2J (addr a\<bullet>M(map Val vs)) va"   and "s' = (h', lcl s)" shows "extTA,P,t \<turnstile> \<langle>addr a\<bullet>M(map Val vs),s\<rangle> -ta'\<rightarrow> \<langle>e',s'\<rangle>"
BIT_inv2: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and s :: "('a list \<times> bool list \<times> 'a list) pmf" assumes "x \<noteq> y"   and "z \<in> {x, y}" shows "BIT_inv s z [x, y] = (s = type0 [x, y] z (other z x y))"
prj_chine_assoc_1: shows "\<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>1 \<cdot> \<mu>_\<nu>\<pi>_\<rho>.chine_assoc = \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>1\<^sub>1"
R'_prop: fixes x :: "'a"   and y :: "'a" assumes "x \<in> S"   and "y \<in> S"   and "R x y" shows "R' (g x) (g y)"
l2_inv6'D: fixes Rb :: "fid_t"   and B :: "agent"   and A :: "agent"   and x :: "l2_state"   and gnx :: "msg" assumes "guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>"   and "in_progress (progress x Rb) xsk"   and "sklvl1.guessed_frame Rb xgnx = Some gnx"   and "x \<in> l2_inv6'" shows "sklvl1.guessed_frame Rb xsk = Some (Exp gnx (NonceF (Rb $ ny)))"
of_int_diff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and w :: "int"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (w - z) = minus (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z)"
ord_resolve_rename_unique: fixes CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset"   and \<sigma>' :: "'s"   and E' :: "'a literal multiset" assumes "ord_resolve_rename CAs DA AAs As \<sigma> E"   and "ord_resolve_rename CAs DA AAs As \<sigma>' E'" shows "\<sigma> = \<sigma>' \<and> E = E'"
starfun_n_def: fixes F :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" shows "*fn* F = Ifun (star_n F)"
orbit_trans: fixes s :: "'e"   and f :: "'e \<Rightarrow> 'e"   and t :: "'e"   and u :: "'e" assumes "s \<in> orbit f t"   and "t \<in> orbit f u" shows "s \<in> orbit f u"
\<dd>_mono: fixes r :: "'a language"   and s :: "'a language"   and a :: "'a" assumes "r \<le> s" shows "\<dd> r a \<le> \<dd> s a"
pred_entailment_mono: fixes P :: "'BExp set"   and P' :: "'BExp set"   and P'' :: "'BExp set" assumes "P \<turnstile> P'"   and "P \<subseteq> P''" shows "P'' \<turnstile> P'"
jpTraces_step_inv: fixes t :: "'s Trace"   and s :: "'s"   and jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol" assumes "t \<leadsto> s \<in> jpTraces jp" shows "t \<in> jpTraces jp \<and> (\<exists>eact\<in>set (envAction (tLast t)). \<exists>aact. (\<forall>a. aact a \<in> set (actJP jp t a)) \<and> s = envTrans eact aact (tLast t))"
siso_cont: fixes c :: "('test, 'atom, 'choice) cmd"   and i :: "nat"   and s :: "'state" assumes "siso c"   and "i < brn c" shows "siso (local.cont c s i)"
Restrp_strict_order: shows "strict_ordered_set UNIV ((\<sqsubset>) \<restriction> A)"
A_trusts_NS2: fixes A :: "agent"   and NA :: "msg"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>NA, Agent B, Key K, X\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> ns_shared" shows "Says Server A (Crypt (shrK A) \<lbrace>NA, Agent B, Key K, X\<rbrace>) \<in> set evs"
Healthy_func: fixes F :: "'d upred \<Rightarrow> 'e upred"   and \<H>\<^sub>1 :: "'d upred \<Rightarrow> 'd upred"   and \<H>\<^sub>2 :: "'e upred \<Rightarrow> 'e upred"   and P :: "'d upred" assumes "F \<in> \<lbrakk>\<H>\<^sub>1\<rbrakk>\<^sub>H \<rightarrow> \<lbrakk>\<H>\<^sub>2\<rbrakk>\<^sub>H"   and "P is \<H>\<^sub>1" shows "\<H>\<^sub>2 (F P) = F P"
vwalk_joinI_vwalk: fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph"   and q :: "'a list" assumes "vwalk p G"   and "vwalk q G"   and "joinable p q" shows "vwalk (p \<oplus> q) G"
wprepare_loop_start_on_rightmost_Oc: fixes m :: "nat"   and lm :: "nat list"   and b :: "cell list"   and list :: "cell list" assumes "wprepare_loop_start_on_rightmost m lm (b, Oc # list)" shows "wprepare_loop_start_on_rightmost m lm (Oc # b, list)"
rk_ab\<gamma>: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points"   and Q :: "Points"   and a :: "Points"   and b :: "Points"   and c :: "Points"   and R :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}"   and "rk {Q, A', a} = 2"   and "rk {Q, B', b} = 2"   and "rk {Q, C', c} = 2"   and "rk {P, Q, R} = 2"   and "rk {P, R} = 2"   and "rk {Q, R} = 2"   and "rk {R, A, a} = 2"   and "rk {R, B, b} = 2"   and "rk {R, C, c} = 2"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "rk {C', P} = 2" shows "rk {a, b, \<gamma>} = 2"
rel_fun_def: fixes A :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
run_one_step_basic_br_if_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x9 :: "nat"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Br_if x9) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
assms: shows "\<forall>n\<ge>limit. A (Suc n) < A n"
approx_le_bound: fixes z :: "real star"   and f :: "real star"   and g :: "real star" assumes "z \<le> f"   and "f \<approx> g"   and "g \<le> z" shows "f \<approx> z"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "'a mpoly" shows "mapping_of (swap_poly x xa xb) = swapPoly\<^sub>0 x xa (mapping_of xb)"
A: shows "(a, b) \<in> (unlabel ars)\<^sup>*"
HC_intros_2: fixes F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula" shows "AX0 \<turnstile>\<^sub>H (F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H"
assms_2: shows "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>3"
dims1_def: fixes dims :: "nat list"   and vars :: "nat set" shows "partial_state.dims1 dims vars = nths dims vars"
widen_Boolean2: fixes G :: "prog"   and S :: "ty" assumes "G\<turnstile>S\<preceq>PrimT Boolean" shows "S = PrimT Boolean"
Cast: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and n :: "nat"   and v :: "val"   and s1 :: "state"   and s2 :: "state"   and T :: "ty" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1"   and "s2 = abupd (raise_if (\<not> G,snd s1\<turnstile>v fits T) ClassCast) s1" shows "G\<turnstile>Norm s0 \<midarrow>Cast T e-\<succ>v\<midarrow>n\<rightarrow> s2"
t: shows "isLub \<real> {s \<in> \<real>. s < x} t"
succ_rvar_valuation: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and s' :: "('a, 'b) state"   and x :: "nat" assumes "PivotUpdateMinVars eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update"   and "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> PivotUpdateMinVars.gt_state eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update s s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"   and "x \<in> rvars (\<T> s')" shows "\<langle>\<V> s'\<rangle> x = \<langle>\<V> s\<rangle> x \<or> \<langle>\<V> s'\<rangle> x = the (\<B>\<^sub>l s x) \<or> \<langle>\<V> s'\<rangle> x = the (\<B>\<^sub>u s x)"
prefix_fininf_set_range: fixes u :: "'a list"   and v :: "'a stream" assumes "u \<le>\<^sub>F\<^sub>I v" shows "set u \<subseteq> sset v"
merge_restrict: fixes A :: "nat set"   and J :: "nat set"   and I :: "nat set"   and xx :: "'a option list"   and n :: "nat"   and t :: "'a option list" assumes "A \<inter> J = {}"   and "A \<subseteq> I"   and "length xx = n"   and "length t = n"   and "restrict J xx = xx" shows "restrict A (merge xx t) = restrict A t"
Union_iff: fixes A :: "'b"   and C :: "'b set set" shows "(A \<in> \<Union> C) = (\<exists>X\<in>C. A \<in> X)"
None_eq_map_option_iff: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e option" shows "(\<bottom> = map_option f x) = (x = \<bottom>)"
rel_stateT_def: fixes S :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and M :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "('c, 'e) stateT"   and m' :: "('d, 'f) stateT" shows "rel_stateT S M m m' = (S ===> M) (run_state m) (run_state m')"
Diff_triv_mset: fixes M :: "'c multiset"   and N :: "'c multiset" assumes "M \<inter># N = {#}" shows "M - N = M"
ex_fin_conv: fixes A :: "'a fset" shows "(\<exists>x. x |\<in>| A) = (A \<noteq> {||})"
h0: assumes "subspace S" shows "affine S \<and> (0::'a) \<in> S"
n_less_m_mult_n: fixes n :: "nat"   and m :: "nat" assumes "0 < n"   and "Suc 0 < m" shows "n < m * n"
eq_val_eq_idealTr: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier (Vr K v) - {\<zero>}"   and "y \<in> carrier (Vr K v)"   and "v x \<le> v y" shows "Vr K v \<diamondsuit>\<^sub>p y \<subseteq> Vr K v \<diamondsuit>\<^sub>p x"
converse_rtranclE: fixes x :: "'b"   and z :: "'b"   and r :: "('b \<times> 'b) set" assumes "(x, z) \<in> r\<^sup>*" obtains "x = z"   | y :: "'b" where "(x, y) \<in> r" and "(y, z) \<in> r\<^sup>*"
det_bound: fixes A :: "'b mat"   and n :: "nat"   and x :: "'b" assumes "A \<in> carrier_mat n n"   and "A \<in> Bounded_mat x" shows "\<bar>det A\<bar> \<le> det_bound n x"
prems_3: shows "lesvars upds \<inter> varacom ({A} WHILE b DO C) = {}"
mult_zero_left: fixes times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and a :: "'e" assumes "class.mult_zero times zero" shows "times zero a = zero"
Nil_less_Cons: fixes a :: "'b"   and x :: "'b list" shows "list_less [] (a # x)"
dvd_ideal_generated_singleton': fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "dvd.dvd times b a" shows "ring.ideal_generated plus zero uminus times {a} \<subseteq> ring.ideal_generated plus zero uminus times {b}"
lambda_zero: shows "(\<lambda>h. 0::'c) = (*) (0::'c)"
starrel_in_star: fixes x :: "nat \<Rightarrow> 'b" shows "starrel `` {x} \<in> star"
msubseteq_replicate_msetE: fixes A :: "'b multiset"   and n :: "nat"   and a :: "'b" assumes "A \<subseteq># replicate_mset n a" obtains m :: "nat" where "m \<le> n" and "A = replicate_mset m a"
card_ge_0_finite: fixes A :: "'b set" assumes "0 < card A" shows "finite A"
simps: fixes M :: "('a, 'b) insert_message set"   and x :: "('a, 'b) insert_message"   and y :: "('a, 'b) insert_message" shows "depends_on M x y = (x \<in> M \<and> y \<in> M \<and> I x \<in> deps (Insert y))"
matrix_Tensor_elements: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and M1 :: "'b list list"   and M2 :: "'b list list" assumes "Matrix_Tensor.mult id f" shows "\<forall>i j. (i < mult.row_length M1 * mult.row_length M2 \<and> j < length M1 * length M2) \<and> mat (mult.row_length M1) (length M1) M1 \<and> mat (mult.row_length M2) (length M2) M2 \<longrightarrow> mult.Tensor f M1 M2 ! j ! i = f (M1 ! (j div length M2) ! (i div mult.row_length M2)) (M2 ! (j mod length M2) ! (i mod mult.row_length M2))"
induct: fixes P :: "CL \<Rightarrow> CL \<Rightarrow> bool"   and a0 :: "CL"   and a1 :: "CL" assumes "\<And>x y z. P (` ` S x y) z"   and "\<And>x y. P (` K x) y"   and "\<And>x. P I x"   and "\<And>u. P S u"   and "\<And>u. P K u"   and "\<And>va u. P (` S va) u"   and "\<And>va u. P (` I va) u"   and "\<And>vc va u. P (` ` K vc va) u"   and "\<And>vc va u. P (` ` I vc va) u"   and "\<And>vd ve vc va u. P (` ` ` vd ve vc va) u" shows "P a0 a1"
l_new_document_get_tag_name_def: fixes get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_document_get_tag_name get_tag_name_locs \<equiv> \<forall>h new_document_ptr h' r ptr'. h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr \<longrightarrow> h \<turnstile> new_document \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_tag_name_locs ptr' \<longrightarrow> r h h'"
elims_2: fixes x :: "('b, 'c) Combinators list" assumes "NetsCollected2 x" obtains xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs" and "xs = [] \<or> first_bothNet xa \<noteq> firstList xs \<and> NetsCollected2 xs"   | "x = []"
axiom_and_def: fixes p :: "fol fm"   and q :: "fol fm" shows "axiom_and p q \<equiv> Thm (Iff (And p q) (Imp (Imp p (Imp q Falsity)) Falsity))"
term_ok_var: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and idn :: "variable"   and \<tau> :: "typ" shows "term_ok \<Theta> (Fv idn \<tau>) = typ_ok \<Theta> \<tau>"
hyps: shows "\<Prod>\<^sub># (image_mset Mp b) =m \<Prod>\<^sub># b"
Loss_msg1: fixes a :: "nat \<Rightarrow> aType list"   and lose :: "nat \<Rightarrow> bool list"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and x :: "nat \<Rightarrow> aType list"   and i :: "nat \<Rightarrow> ECall_Info list" assumes "msg (Suc 0) a"   and "Loss lose a i2 x i" shows "msg (Suc 0) x"
mult_left_cancel: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.semiring_no_zero_divisors_cancel plus zero times"   and "c \<noteq> zero" shows "(times c a = times c b) = (a = b)"
well_def_wp_sublinear: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "well_def prog" shows "sublinear (wp prog)"
assms: shows "max_Var_floatariths ea \<le> i"
subgraphI: fixes H :: "('b, 'c) pre_digraph"   and G :: "('b, 'c) pre_digraph" assumes "verts H \<subseteq> verts G"   and "arcs H \<subseteq> arcs G"   and "compatible G H"   and "wf_digraph H"   and "wf_digraph G" shows "subgraph H G"
Formula_axioms: shows "Formula SUC LESS assigns nvars Extend CONS SNOC Length extend size zero alphabet eval downshift upshift add cut len restrict Restrict lformula0 FV0 find0 wf0 decr0 (\<Turnstile>\<^sub>0) nullable0 lderiv0 rderiv0"
vwalk_induct: fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph"   and P :: "'a list \<Rightarrow> bool" assumes "vwalk p G"   and "\<And>u. u \<in> verts G \<Longrightarrow> P [u]"   and "\<And>u v es. \<lbrakk>u \<rightarrow>\<^bsub>G\<^esub> v; P (v # es)\<rbrakk> \<Longrightarrow> P (u # v # es)" shows "P p"
exhaust: fixes y :: "('f, 'g, 'h) Resumption.resumption" obtains x1 :: "'f option" where "y = Resumption.resumption.Done x1"   | x21 :: "'g" and x22 :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" where "y = Resumption.resumption.Pause x21 x22"
trss_stack_comp: fixes s :: "'f list"   and c :: "'f list multiset"   and w :: "('g, 'h) label list"   and s' :: "'f list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and r :: "'f list" assumes "((s, c), w, s', c') \<in> trcl (trss fg)" shows "((s @ r, c), w, s' @ r, c') \<in> trcl (trss fg)"
eq_general_inverses: fixes B :: "'c set"   and k :: "'c \<Rightarrow> 'd"   and A :: "'d set"   and h :: "'d \<Rightarrow> 'c"   and \<gamma> :: "'c \<Rightarrow> 'e"   and \<phi> :: "'d \<Rightarrow> 'e" assumes "\<And>y. y \<in> B \<Longrightarrow> k y \<in> A \<and> h (k y) = y"   and "\<And>x. x \<in> A \<Longrightarrow> h x \<in> B \<and> k (h x) = x \<and> \<gamma> (h x) = \<phi> x" shows "sum \<phi> A = sum \<gamma> B"
elims: fixes x :: "('c, 'd) fmap"   and xa :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and y :: "('c, 'd) fmap" assumes "exec_plan x xa = y" obtains s :: "('c, 'd) fmap" where "x = s__" and "xa = []" and "y = s__"   | s :: "('c, 'd) fmap" and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap" and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" where "x = s" and "xa = a # as" and "y = exec_plan (state_succ s a) as"
prems_2: shows "\<not> list_all (\<lambda>x. x = dir.Left) (a # p)"
arg_bounded: fixes z :: "complex" shows "- pi < arg z \<and> arg z \<le> pi"
mono_epigraph: fixes S :: "('b \<times> real) set" assumes "\<forall>x y z. (x, y) \<in> S \<and> y \<le> z \<longrightarrow> (x, z) \<in> S"   and "closed S" shows "\<exists>g. Epigraph UNIV g = S"
simps_1: fixes n :: "nat"   and \<rho> :: "(nat \<times> val) list" shows "E (ENat n) \<rho> = {v. v = VNat n}"
Un_insert_left: fixes a :: "'g"   and B :: "'g set"   and C :: "'g set" shows "insert a B \<union> C = insert a (B \<union> C)"
nn_integral_Markov_inequality: fixes u :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and A :: "'b set"   and c :: "ennreal" assumes "u \<in> borel_measurable M"   and "A \<in> sets M" shows "emeasure M ({x \<in> space M. 1 \<le> c * u x} \<inter> A) \<le> c * set_nn_integral M A u"