hcju/isabelleps huggingface dataset:
in_second_append_list: fixes a :: "'a list"   and i :: "nat"   and b :: "'a list" assumes "length a \<le> i"   and "i < length (a @ b)" shows "(a @ b) ! i \<in> set b"
vconsI: fixes a :: "V"   and xs :: "V"   and x :: "V" assumes "a \<in>\<^sub>\<circ> vinsert \<langle>vcard xs, x\<rangle> xs" shows "a \<in>\<^sub>\<circ> xs #\<^sub>\<circ> x"
Kc: fixes S :: "'a set" assumes "S \<in> K" shows "closedin U S"
diff_delta_ln_upper_13_def: shows "diff_delta_ln_upper_13 \<equiv> \<lambda>x. (x - 1) ^ 13 / ((1 + 42 * x + 525 * x ^ 4 + 700 * x ^ 3 + 315 * x\<^sup>2 + 126 * x ^ 5 + 7 * x ^ 6)\<^sup>2 * x)"
set_berlekamp_basis_eq: fixes u :: "'b mod_ring poly" shows "set (berlekamp_basis u) = (Poly \<circ> list_of_vec) ` set (find_base_vectors (berlekamp_resulting_mat u))"
ptensor_mat_mult_vec: fixes m1 :: "'a mat"   and v1 :: "'a vec"   and m2 :: "'a mat"   and v2 :: "'a vec" assumes "m1 \<in> carrier_mat st.d1 st.d1"   and "v1 \<in> carrier_vec st.d1"   and "m2 \<in> carrier_mat st.d2 st.d2"   and "v2 \<in> carrier_vec st.d2" shows "st.ptensor_vec (m1 *\<^sub>v v1) (m2 *\<^sub>v v2) = st.ptensor_mat m1 m2 *\<^sub>v st.ptensor_vec v1 v2"
antipar_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 \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "adjunction_data_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "src g = trg f"
Inr_eqvt: fixes p :: "perm"   and y :: "'c" shows "p \<bullet> Inr y = Inr (p \<bullet> y)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   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"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors_di :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_di_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_parent type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs"   and "l_get_host get_host get_host_locs"   and "l_get_disconnected_document get_disconnected_document get_disconnected_document_locs"   and "l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_parent_locs get_host get_host_locs get_disconnected_document get_disconnected_document_locs get_ancestors_di get_ancestors_di_locs" shows "l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_host get_host_locs get_disconnected_document get_disconnected_document_locs get_ancestors_di get_ancestors_di_locs"
vcomp_vempty_right: fixes r :: "V" shows "r \<circ>\<^sub>\<circ> 0 = 0"
plus_com_lemma: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A" shows "a \<squnion>\<^bsub>f\<^esub> b \<sqsubseteq>\<^bsub>r\<^esub> b \<squnion>\<^bsub>f\<^esub> a"
Fadd_subseteq_Eadd: fixes Fadd :: "('a \<times> 'a) set" shows "(Fadd \<inter> E' = {} \<and> Fadd \<subseteq> E) = (Fadd \<subseteq> Eadd)"
mult_left_mono: fixes plus :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and zero :: "'i"   and less_eq :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and less :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and times :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and a :: "'i"   and b :: "'i"   and c :: "'i" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (times c a) (times c b)"
Dy_def: shows "Dy_pair = {(x, y). x \<in> cbox a b \<and> y \<in> cbox (g2 x) (g1 x)}"
sumlist_def: fixes xs :: "'a list" shows "R.sumlist xs \<equiv> foldr (+) xs (0::'a)"
Bignorabimus_init: fixes x :: "variable"   and \<alpha> :: "game" assumes "x \<notin> BVG \<alpha>" shows "{x} \<in> Bignorabimus \<alpha>"
eucl_down_vec_def: fixes p :: "int"   and x :: "('a, 'b) vec" shows "eucl_down p x = (\<chi>i. eucl_down p (x $ i))"
prv_all_imp: fixes x :: "'var"   and \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<notin> Fvars \<phi>"   and "bprv (all x (imp \<phi> \<chi>))" shows "bprv (imp \<phi> (all x \<chi>))"
simps_1: fixes t :: "nat"   and k :: "nat" shows "var_to_dimacs (Operator t k) = 1 + t + k * Suc h"
sets_embed_measure: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" assumes "inj f" shows "sets (embed_measure M f) = {f ` A |A. A \<in> sets M}"
mult2_alt_sE: fixes X :: "'a multiset"   and Y :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "(X, Y) \<in> mult2_alt_s ns s" obtains X1 :: "'a multiset" and X2 :: "'a multiset" and Y1 :: "'a multiset" and Y2 :: "'a multiset" where "X = X1 + X2" and "Y = Y1 + Y2" and "(X1, Y1) \<in> multpw ns" and "Y2 \<noteq> {#}" and "\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s)"
OCom_iff_weak_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" shows "(operational_complete (TRel\<^sup>*) \<and> weak_reduction_simulation (TRel\<^sup>+) Target) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel) \<and> (\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+) \<and> (\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel\<^sup>*) \<and> weak_reduction_simulation Rel (STCal Source Target))"
smallomegaD_filterlim_at_infinity: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> \<omega>[F](g)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'c)" shows "LIM x F. f x / g x :> at_infinity"
eval_wlp_Abort: fixes P :: "'b \<Rightarrow> real" shows "wlp Abort P = (\<lambda>s. 1)"
abs_eq: fixes x :: "((real, 'a) vec, 'a) vec" shows "to_mtx x\<^sup>-\<^sup>1 = to_mtx (matrix_inv x)"
induct: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x1 :: "'a list"   and x2 :: "'a list"   and Pa :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "list_emb P x1 x2"   and "\<And>ys. Pa [] ys"   and "\<And>xs ys y. \<lbrakk>list_emb P xs ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa xs (y # ys)"   and "\<And>x y xs ys. \<lbrakk>P x y; list_emb P xs ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa (x # xs) (y # ys)" shows "Pa x1 x2"
poly_isCont: fixes x :: "'b"   and p :: "'b poly" shows "isCont (poly p) x"
Es: shows "\<forall>s1 s2. s1 = s2 on Es \<longrightarrow> E s1 = E s2"
length_aforms_errD: fixes vs :: "real list"   and e :: "nat \<Rightarrow> real"   and VS :: "((real \<times> real pdevs) \<times> real) list" assumes "vs \<in> aforms_err e VS" shows "length vs = length VS"
assms: shows "nop_onto A (2\<^sub>\<nat>) f"
older_seniors_monotonic_Suc: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "older_seniors x (Suc n) \<subseteq> older_seniors x n"
m_closed: fixes x :: "'a mod_ring poly"   and y :: "'a mod_ring poly" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<in> carrier R"
resPres: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and x :: "name" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "\<And>R S y. (R, S) \<in> Rel \<Longrightarrow> (\<lparr>\<nu>y\<rparr>R, \<lparr>\<nu>y\<rparr>S) \<in> Rel'" shows "\<lparr>\<nu>x\<rparr>P \<leadsto>\<^sup>^<Rel'> \<lparr>\<nu>x\<rparr>Q"
case_3: fixes f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> instr list \<Rightarrow> 'b"   and f4 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and x31 :: "'c"   and x32 :: "'c"   and x33 :: "instr list" shows "(case Prog\<^sub>s\<^sub>b x31 x32 x33 of Write\<^sub>s\<^sub>b x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Read\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f2 x xa xb xc | Prog\<^sub>s\<^sub>b x xa xb \<Rightarrow> f3 x xa xb | Ghost\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f4 x xa xb xc) = f3 x31 x32 x33"
sel: fixes x :: "char list" shows "as_string (Name x) = x"
dom_runit: fixes a :: "'c local.term set" assumes "ide a" shows "local.dom \<r>[a] = a \<otimes> \<I>"
mset_diff_cancel1elem: fixes a :: "'c"   and B :: "'c multiset" assumes "a \<notin># B" shows "{#a#} - B = {#a#}"
one_dim_iso_of_one: shows "one_dim_iso (1::'c) = (1::'b)"
iallE: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and t :: "nat" assumes "\<box> t I. P t" obtains "P t"   | "t \<notin> I"
cong__nlt: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Cong A B C D" shows "\<not> A B Lt C D"
new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t h = (new_shadow_root_ptr, h')"   and "ptr \<noteq> cast new_shadow_root_ptr" shows "get ptr h = get ptr h'"
INF1_I: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b" assumes "\<And>x. x \<in> A \<Longrightarrow> B x b" shows "(\<Sqinter> (B ` A)) b"
clamp_cancel_cbox: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> cbox a b" shows "clamp a b x = x"
e1_not_empty: shows "nat_to_ce_set e_1 \<noteq> {}"
poincare_line_2: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> w" shows "on_circline (poincare_line z w) w"
cases: fixes a1 :: "bexp"   and a2 :: "com"   and a3 :: "char list \<Rightarrow> int"   and a4 :: "nat" assumes "Its a1 a2 a3 a4" obtains b :: "bexp" and s :: "char list \<Rightarrow> int" and c :: "com" where "a1 = b__" and "a2 = ca__" and "a3 = s__" and "a4 = 0" and "\<not> bval b__ s__"   | b :: "bexp" and s :: "char list \<Rightarrow> int" and c :: "com" and s' :: "char list \<Rightarrow> int" and n :: "nat" where "a1 = b" and "a2 = c" and "a3 = s" and "a4 = Suc n" and "bval b s" and "(c, s) \<Rightarrow> s'" and "Its b c s' n"
face_of_translation_eq: fixes a :: "'b"   and T :: "'b set"   and S :: "'b set" shows "((+) a ` T face_of (+) a ` S) = (T face_of S)"
inf_apply: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "(f \<sqinter> g) x = f x \<sqinter> g x"
empty_sets: fixes M :: "'a measure" shows "{} \<in> fmeasurable M"
L_infinity_space: fixes M :: "'b measure" shows "space\<^sub>N (\<LL> \<infinity> M) = {f \<in> borel_measurable M. \<exists>C. AE x in M. \<bar>f x\<bar> \<le> C}"
uniform_limit_EM_remainder: fixes a :: "int"   and G :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and A :: "'a set"   and n :: "nat"   and f :: "'a \<Rightarrow> real \<Rightarrow> 'b" assumes "\<And>y. real_of_int a \<le> y \<Longrightarrow> (G has_real_derivative g y) (at y within {real_of_int a..})"   and "\<And>a' b y. \<lbrakk>y \<in> A; real_of_int a \<le> a'; a' \<le> b\<rbrakk> \<Longrightarrow> (\<lambda>t. pbernpoly n t *\<^sub>R f y t) integrable_on {a'..b}"   and "convergent (\<lambda>y. G (real y))"   and "\<forall>\<^sub>F x in at_top. \<forall>y\<in>A. norm (f y x) \<le> g x" shows "uniform_limit A (\<lambda>b s. EM_remainder' n (f s) (real_of_int a) (real b)) (\<lambda>s. EM_remainder n (f s) a) sequentially"
le_add_diff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_cancel_comm_monoid_diff plus zero less_eq less minus"   and "less_eq a b" shows "less_eq c (minus (plus b c) a)"
assms_3: shows "\<forall>\<^sub>F z in nhds 0. eval_fps f z = eval_fps g z"
connected_component_of_eq_empty: fixes X :: "'a topology"   and x :: "'a" shows "(connected_component_of_set X x = {}) = (x \<notin> topspace X)"
intro: fixes x :: "'e \<Longrightarrow> 'f" assumes "weak_lens x"   and "wb_lens_axioms x" shows "wb_lens x"
sylvester_mat_sub_dim_2: fixes m :: "nat"   and n :: "nat"   and p :: "'c poly"   and q :: "'c poly" shows "dim_col (sylvester_mat_sub m n p q) = m + n"
coset_join2: fixes x :: "'a"   and H :: "'a set" assumes "x \<in> carrier G"   and "subgroup H (add_monoid G)"   and "x \<in> H" shows "H #>\<^bsub>add_monoid G\<^esub> x = H"
invertible_row_add_iterate: fixes n :: "nat"   and A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'e" assumes "n < nrows A" shows "\<exists>P. invertible P \<and> row_add_iterate A n i j = P ** A"
\<omega>_limit_point_def: fixes x :: "'a"   and p :: "'a" shows "\<omega>_limit_point x p = ({0..} \<subseteq> existence_ivl0 x \<and> (\<exists>s. s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity> \<and> (flow0 x \<circ> s) \<longlonglongrightarrow> p))"
sel_1: fixes x1 :: "'e"   and x2 :: "'f" shows "fst (x1, x2) = x1"
chooset: fixes p :: "fm"   and q :: "fm"   and B :: "(MIR.num \<times> int) list"   and d :: "int"   and bs :: "real list"   and i :: "int" assumes "qfree p"   and "chooset p = (q, B, d)" shows "(\<exists>x. Ifm (real_of_int x # bs) p) = (\<exists>x. Ifm (real_of_int x # bs) q) \<and> (\<lambda>(t, k). (Inum (real_of_int i # bs) t, k)) ` set B = (\<lambda>(t, k). (Inum (real_of_int i # bs) t, k)) ` set (\<rho> q) \<and> \<delta> q = d \<and> 0 < d \<and> iszlfm q (real_of_int i # bs) \<and> (\<forall>(e, c)\<in>set B. numbound0 e \<and> 0 < c)"
s_as_upd: shows "s = s(r \<mapsto> ls)"
simps_2: fixes t :: "'d" shows "dbm_entry_bound (Lt t) = t"
subset_lemma: fixes A :: "'d set"   and B :: "'d set" assumes "A \<inter> B = {}"   and "A \<subseteq> B" shows "A = {}"
union_commute: fixes M :: "'k multiset"   and N :: "'k multiset" shows "M + N = N + M"
ured: shows "s' \<degree> t' \<rightarrow>\<^sub>\<beta>\<^sup>* u"
dollar0_left: fixes Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "($ 0 \<and>* Q) = Q"
diagonal_to_Smith_PQ_JNF_with_types: fixes A :: "'b mat"   and S :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" assumes "A \<in> carrier_mat CARD('c) CARD('d)"   and "S \<in> carrier_mat CARD('c) CARD('d)"   and "P \<in> carrier_mat CARD('c) CARD('c)"   and "Q \<in> carrier_mat CARD('d) CARD('d)"   and "diagonal_to_Smith_PQ_JNF A bezout = (P, S, Q)"   and "isDiagonal_mat A"   and "is_bezout_ext bezout" shows "S = P * A * Q \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat S"
rep_eq: fixes x :: "('c, 'b) commute"   and xa :: "'c uprod" shows "case_uprod x xa = (case rep_uprod xa of (xa, xb) \<Rightarrow> apply_commute x xa xb)"
rel_nabla_bin_unfold: fixes X :: "('a \<times> 'a) set"   and Q :: "('a \<times> 'a) set" shows "rel_nabla_bin X Q \<subseteq> relfdia X (rel_nabla_bin X Q) \<union> rdom Q"
MSF_eq: fixes E' :: "(nat \<times> int \<times> nat) set" shows "s.MSF E' = minimum_spanning_forest \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E'\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>"
reduction_doesnt_break_subsets_R: fixes p :: "real poly"   and subsets :: "(nat list \<times> nat list) list"   and qs :: "real poly list"   and signs :: "rat list list" assumes "p \<noteq> 0"   and "all_list_constr_R subsets (length qs)"   and "satisfy_equation_R p qs subsets signs"   and "invertible_mat (M_mat_R signs subsets)" shows "all_list_constr_R (reduction_subsets_R p qs signs subsets (M_mat_R signs subsets)) (length qs)"
power_0: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" shows "power.power one times a 0 = one"
setdist_eq_0_sing_1: fixes x :: "'b"   and S :: "'b set" shows "(setdist {x} S = 0) = (S = {} \<or> x \<in> closure S)"
nprv_eql_sucI: fixes F :: "'fmla set"   and t :: "'trm"   and t' :: "'trm" assumes "nprv F (eql (suc t) (suc t'))"   and "t \<in> atrm"   and "t' \<in> atrm"   and "F \<subseteq> fmla"   and "finite F" shows "nprv F (eql t t')"
ih_15: shows "InvariantNoDecisionsWhenUnit (getF state') (getM state') (currentLevel (getM state'))"
Sup_finite_5_def: fixes A :: "finite_5 set" shows "\<Squnion> A = (if a\<^sub>5 \<in> A \<or> a\<^sub>4 \<in> A \<and> (a\<^sub>2 \<in> A \<or> a\<^sub>3 \<in> A) then a\<^sub>5 else if a\<^sub>3 \<in> A then a\<^sub>3 else if a\<^sub>2 \<in> A then a\<^sub>2 else if a\<^sub>4 \<in> A then a\<^sub>4 else a\<^sub>1)"
filter_mset_eq_conv: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b multiset"   and N :: "'b multiset" shows "(filter_mset P M = N) = (N \<subseteq># M \<and> (\<forall>b\<in>#N. P b) \<and> (\<forall>a\<in>#M - N. \<not> P a))"
wfLexLessState: fixes decisionVars :: "nat set"   and F0 :: "Literal list list" assumes "finite decisionVars" shows "wf (lexLessState F0 decisionVars)"
ker_f_def: shows "ker_f = {x. f x = (0::'c)}"
shift1_head: fixes p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "head (shift1 p) = head p"
Inr_in_sum_iff: fixes b :: "V"   and A :: "V"   and B :: "V" shows "(ZFC_Cardinals.Inr b \<in> elts (A \<Uplus> B)) = (b \<in> elts B)"
mult_zero_left: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d" assumes "class.mult_zero times zero" shows "times zero a = zero"
group_eq_subgrp_rcoset_un: fixes H :: "'g set"   and gs :: "'g list" assumes "Subgroup H"   and "is_rcoset_replist H gs" shows "G = (\<Union>g\<in>set gs. H + {g})"
H_1: shows "rep_Bonk_Schramm_extension x \<in> extended_distance_set"
winker_object_def: shows "\<rho> = \<gamma> \<squnion> \<gamma> \<squnion> \<gamma>"
degree_def: shows "MPoly_Type.degree \<equiv> map_fun mapping_of id (\<lambda>p v. Max (insert 0 ((\<lambda>m. lookup m v) ` keys p)))"
derivation_ge_empty: fixes i :: "nat" shows "derivation_ge [] i"
submonoid_axioms: shows "submonoid (R\<setminus>\<pp>) R (\<cdot>) \<one>"
sup_idem: fixes x :: "'b" shows "x \<sqinter> x = x"
sorted_append1: fixes xs :: "('b, 'c) Combinators list"   and l :: "'b set list"   and ys :: "('b, 'c) Combinators list" assumes "all_in_list xs l"   and "singleCombinators xs"   and "all_in_list ys l"   and "singleCombinators ys"   and "FWNormalisationCore.sorted (xs @ ys) l" shows "FWNormalisationCore.sorted xs l \<and> FWNormalisationCore.sorted ys l \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. smaller x y l)"
simps_3: fixes p1 :: "8 word"   and p2 :: "8 word" shows "simple_proto_conjunct (Proto p1) (Proto p2) = (if p1 = p2 then Some (Proto p1) else None)"
fimage_vinsert: fixes r :: "V"   and a :: "V"   and A :: "V" shows "r `\<^sub>\<bullet> vinsert a A = r `\<^sub>\<bullet> set {a} \<union>\<^sub>\<circ> r `\<^sub>\<bullet> A"
exec_move_CAS1: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and D :: "String.literal"   and F :: "String.literal"   and e' :: "'addr expr1"   and e'' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (e\<bullet>compareAndSwap(D\<bullet>F, e', e'')) h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
someI2_bex: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<exists>a\<in>A. P a"   and "\<And>x. x \<in> A \<and> P x \<Longrightarrow> Q x" shows "Q (SOME x. x \<in> A \<and> P x)"
natsetf: shows "{w \<in> space M. \<not> w !! 0} \<in> sets (nat_filtration (Suc 0))"
p_coprime_left: fixes a :: "nat" shows "coprime p a = (\<not> p dvd a)"
single_is_union: fixes a :: "'c"   and M :: "'c multiset"   and N :: "'c multiset" shows "(\<LM> a  \<RM> = M + N) = (\<LM> a  \<RM> = M \<and> N = \<Empt> \<or> M = \<Empt> \<and> \<LM> a  \<RM> = N)"
epred_numeral_def: shows "epred_numeral = enat \<circ> pred_numeral"
CauchyI': fixes dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'e" assumes "class.metric_space dist uniformity open"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (X m) (X n) < e" shows "uniform_space.Cauchy uniformity X"
l2_inv2_def: shows "l2_inv2 \<equiv> {s. \<forall>A B KE. Auth A B \<langle>Number 0, KE\<rangle> \<in> chan s \<longrightarrow> A \<notin> bad s \<and> B \<notin> bad s \<longrightarrow> (\<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progress (progress s Ra) xpkE \<and> KE = epubKF (Ra $ kE))}"
ennsqrt_eq_iff_square: fixes x :: "ennreal"   and y :: "ennreal" shows "(ennsqrt x = y) = (x = y * y)"
le_cases3: fixes x :: "'b"   and y :: "'b"   and z :: "'b" obtains "x \<le> y" and "y \<le> z"   | "y \<le> x" and "x \<le> z"   | "x \<le> z" and "z \<le> y"   | "z \<le> y" and "y \<le> x"   | "y \<le> z" and "z \<le> x"   | "z \<le> x" and "x \<le> y"
chine_simps_1: shows "C.arr HfHgh.chine"
sg3: shows "stop t = [] \<or> stop t = [stop_vc]"
hbx: shows "\<forall>l\<in>C. \<bar>f l - g l\<bar> \<le> x"
simps_103: fixes x91 :: "expr"   and x92 :: "char list"   and x93 :: "char list list"   and x94 :: "expr"   and x4 :: "val" shows "x91\<bullet>x92{x93} := x94 \<noteq> Val x4"
chart_basis_inv_def: shows "local.chart_basis_inv \<equiv> (id ---> Proj) local.chart_basis_inv_nonzero"
vfsequence_vcpower_two_vpair: fixes a :: "V"   and b :: "V"   and A :: "V" shows "([a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> A ^\<^sub>\<times> 2\<^sub>\<nat>) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> A \<times>\<^sub>\<circ> A)"
binary_product_diagram_axioms: shows "binary_product_diagram (\<cdot>) a0 a1"
npMul: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "(x \<cdot>\<^sub>r y)^\<^bsup>R n\<^esup> = x^\<^bsup>R n\<^esup> \<cdot>\<^sub>r y^\<^bsup>R n\<^esup>"
that: fixes u :: "'a list"   and v :: "'a stream" assumes "w = u @- v"   and "fold (count cs) u k = l" shows "thesis"
trans: assumes "ge_game (x, y)"   and "ge_game (y, z)" shows "ge_game (x, z)"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F H \<Phi>\<^sub>H \<sigma>\<tau>.map\<^sub>0 \<sigma>\<tau>.map\<^sub>1"
hurwitz_zeta_conv_suminf: assumes "0 < 1"   and "1 < Re s" shows "hurwitz_zeta 1 s = (\<Sum>n. (of_nat n + complex_of_real 1) powr - s)"
check_InitialData: shows "([V0 0, V1 0], DSpace) \<in> data"