hcju/isabelleps huggingface dataset:
NCisSD2: fixes a :: "('b set set, 'c) Combinators"   and p :: "('b set set, 'c) Combinators list"   and s :: "('b set set, 'c) Combinators" assumes "\<not> member DenyAll a"   and "OnlyTwoNets (a # p)"   and "NetsCollected2 (a # p)"   and "NetsCollected (a # p)"   and "noDenyAll p"   and "allNetsDistinct (a # p)"   and "s \<in> set p" shows "disjSD_2 a s"
Key_synth_eq: fixes K :: "nat"   and H :: "msg set" shows "(Key K \<in> synth H) = (Key K \<in> H)"
simps_1: shows "avl \<langle>\<rangle> = True"
allowed_redex_coincide: fixes t :: "'a trm"   and C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and \<sigma>' :: "('a \<times> 'a trm) list" assumes "allowed_redex t C \<sigma>"   and "t \<in> subterms_of_cl (cl_ecl C)"   and "coincide_on \<sigma> \<sigma>' (vars_of_cl (cl_ecl C))"   and "well_constrained C" shows "allowed_redex t C \<sigma>'"
real_roots_of_int_poly_def: fixes p :: "int poly" shows "real_roots_of_int_poly p = map real_of (roots_of_real_alg p)"
simps: fixes k :: "complex"   and a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex" shows "k *\<^sub>s\<^sub>m (a, b, c, d) = (k * a, k * b, k * c, k * d)"
hcomp_simps_3: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.hseq \<nu> \<mu>" shows "B.dom (\<nu> \<star>\<^sub>B \<mu>) = B.dom \<nu> \<star>\<^sub>B B.dom \<mu>"
step_load: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and x :: "'var"   and H :: "'henv"   and y :: "'dyn"   and d :: "'dyn"   and \<Sigma> :: "'dyn list"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = Inca.instr.ILoad x"   and "heap_get H (x, y) = Some d" shows "Sinca.step (State F H (Frame f pc (y # \<Sigma>) # st)) (State F H (Frame f (Suc pc) (d # \<Sigma>) # st))"
assms: shows "poly q2 x \<noteq> (0::'a)"
NN_unique2: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and N :: "'c"   and \<phi> :: "'c" assumes "Repr_Neg var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv bprv enc N"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (all (Generic_Syntax.Variable var 11) (imp (Repr_Neg.NN var Var FvarsT Fvars subst N (enc \<phi>) (Var (Generic_Syntax.Variable var 11))) (eql (enc (Syntax_with_Connectives_False.neg imp fls \<phi>)) (Var (Generic_Syntax.Variable var 11)))))"
nospoofing: shows "\<forall>p. \<exists>ips. map_of ipassmt (Iface (p_iiface p)) = Some ips \<and> p_src p \<in> ipcidr_union_set (set ips)"
prefixes_prod_encode: fixes s :: "nat \<Rightarrow> nat option"   and t :: "nat"   and j :: "nat" assumes "total_cons s"   and "eval (total_cons.r_prefixes s) [t, j] \<down>" obtains b :: "nat" where "eval (total_cons.r_prefixes s) [t, j] \<down>= b"
stacks_eq_iff: shows "(stack s = []) = (tj_stack s = [])"
fs_carrier: shows "set (map of_int_hom.vec_hom fs) \<subseteq> carrier_vec n"
cast_element_ptr_not_character_data_ptr_2: fixes character_data_ptr :: "(_) character_data_ptr"   and element_ptr :: "(_) element_ptr" shows "cast character_data_ptr \<noteq> cast element_ptr"
l7: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - y = - y \<sqinter> - x"
assms_5: fixes x :: "'a"   and Z :: "'a set" assumes "A (f x)"   and "B (sum f Z)" shows "A (f x + sum f Z)"
size_annos_same: fixes C1 :: "'a acom"   and C2 :: "'b acom" assumes "strip C1 = strip C2" shows "length (annos C1) = length (annos C2)"
undefined: fixes a :: "(('a\<^sub>m, 'a\<^sub>h) T\<^sub>m, 'a\<^sub>h T\<^sub>h, 'b\<^sub>m, 'b\<^sub>h) F\<^sub>m"   and b :: "(('a\<^sub>m, 'a\<^sub>h) T\<^sub>m, 'a\<^sub>h T\<^sub>h, 'b\<^sub>m, 'b\<^sub>h) F\<^sub>m" assumes "root_hash_F (root_hash_T rha) rhb a \<noteq> root_hash_F (root_hash_T rha) rhb b"   and "a \<in> {x. set1_F\<^sub>m x \<subseteq> {x. set1_T\<^sub>m x \<subseteq> A} \<and> set3_F\<^sub>m x \<subseteq> B}" shows "merge_F (root_hash_T rha) (merge_T rha ma) rhb mb a b = None"
flowD_finite_IN: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "flow_attainability \<Delta>"   and "flow \<Delta> f"   and "x \<noteq> sink \<Delta>" shows "d_IN f x \<noteq> \<top>"
map_pmf_const: fixes c :: "'e"   and M :: "'f pmf" shows "map_pmf (\<lambda>_. c) M = return_pmf c"
inject_2: fixes \<Delta> :: "('a, 'b) network_scheme"   and x21 :: "'c"   and x22 :: "'c"   and y21 :: "'c"   and y22 :: "'c" assumes "antiparallel_edges \<Delta>" shows "(antiparallel_edges.Edge x21 x22 = antiparallel_edges.Edge y21 y22) = (x21 = y21 \<and> x22 = y22)"
hyps_1: shows "a \<rightarrow>\<^sub>\<beta>\<^sup>* b"
induct: fixes x1 :: "ty \<times> ty \<times> Syntax.tvar"   and x2 :: "ty"   and P :: "ty \<times> ty \<times> Syntax.tvar \<Rightarrow> ty \<Rightarrow> bool" assumes "subst_type_graph x1 x2"   and "\<And>subst_type_sum t' x. P (One, t', x) One"   and "\<And>subst_type_sum t1 t2 t' x. \<lbrakk>subst_type_graph (t1, t', x) (subst_type_sum (t1, t', x)); P (t1, t', x) (subst_type_sum (t1, t', x)); subst_type_graph (t2, t', x) (subst_type_sum (t2, t', x)); P (t2, t', x) (subst_type_sum (t2, t', x))\<rbrakk> \<Longrightarrow> P (Fun t1 t2, t', x) (Fun (subst_type_sum (t1, t', x)) (subst_type_sum (t2, t', x)))"   and "\<And>subst_type_sum t1 t2 t' x. \<lbrakk>subst_type_graph (t1, t', x) (subst_type_sum (t1, t', x)); P (t1, t', x) (subst_type_sum (t1, t', x)); subst_type_graph (t2, t', x) (subst_type_sum (t2, t', x)); P (t2, t', x) (subst_type_sum (t2, t', x))\<rbrakk> \<Longrightarrow> P (Syntax.Sum t1 t2, t', x) (Syntax.Sum (subst_type_sum (t1, t', x)) (subst_type_sum (t2, t', x)))"   and "\<And>subst_type_sum t1 t2 t' x. \<lbrakk>subst_type_graph (t1, t', x) (subst_type_sum (t1, t', x)); P (t1, t', x) (subst_type_sum (t1, t', x)); subst_type_graph (t2, t', x) (subst_type_sum (t2, t', x)); P (t2, t', x) (subst_type_sum (t2, t', x))\<rbrakk> \<Longrightarrow> P (Syntax.Prod t1 t2, t', x) (Syntax.Prod (subst_type_sum (t1, t', x)) (subst_type_sum (t2, t', x)))"   and "\<And>subst_type_sum y t' x t. \<lbrakk>atom y \<sharp> (t', x); subst_type_graph (t, t', x) (subst_type_sum (t, t', x)); P (t, t', x) (subst_type_sum (t, t', x))\<rbrakk> \<Longrightarrow> P (Mu y t, t', x) (Mu y (subst_type_sum (t, t', x)))"   and "\<And>subst_type_sum y t' x. P (Alpha y, t', x) (if y = x then t' else Alpha y)"   and "\<And>subst_type_sum t t' x. \<lbrakk>subst_type_graph (t, t', x) (subst_type_sum (t, t', x)); P (t, t', x) (subst_type_sum (t, t', x))\<rbrakk> \<Longrightarrow> P (AuthT t, t', x) (AuthT (subst_type_sum (t, t', x)))" shows "P x1 x2"
submap_sat_precond_submap: fixes s1 :: "('c, bool) fmap"   and s2 :: "('c, bool) fmap"   and as :: "(('c, bool) fmap \<times> ('c, bool) fmap) list" assumes "s1 \<subseteq>\<^sub>f s2"   and "sat_precond_as s1 as" shows "sat_precond_as s2 as"
fgi_2: fixes j :: "'b" assumes "j \<in> J" shows "f (g j) = j"
NSTD_lemma2'': fixes c :: "nat" assumes "NSTD c" shows "rec_exec rec_NSTD [c] = Suc 0"
mset_eq_mset_set_imp_distinct: fixes A :: "'a set"   and xs :: "'a list" assumes "finite A"   and "mset_set A = mset xs" shows "distinct xs"
assms_1: shows "Fn\<cdot>f \<triangleleft>\<triangleright>\<^bsub>Suc n\<^esub> CFn\<cdot>g"
simps_6: fixes vb :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and vd :: "'b"   and ve :: "'b list" shows "[] \<simeq> {[], vd # ve, vb} = False"
r_match_alt: fixes L :: "'a set list"   and l :: "'a list" shows "r_match L l = (length L = length l \<and> (\<forall>i<length l. l ! i \<in> L ! i))"
path_transs2_1: fixes g :: "'b ugraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b"   and w :: "'b" assumes "path g u p v"   and "(w, v) \<in> edges g" shows "path g u (p @ [(v, w)]) w"
get_owner_document_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "known_ptrs h"   and "type_wf h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_owner_document ptr"
diff: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'d" assumes "module_hom s1 s2 f" shows "f (x - y) = f x - f y"
rem_implicit_pres_ops_init: fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.I (rem_implicit_pres_ops prob) = ast_problem.I prob"
glob_agree_def: fixes g :: "'a global_scheme"   and tg :: "'b tg_scheme" shows "glob_agree g tg = (tg_mut tg = g_mut g \<and> tg_t tg = typeof (g_val g))"
Limit_vifunion_in_Vset_if_VLambda_in_VsetI: fixes \<alpha> :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V" assumes "Limit \<alpha>"   and "VLambda I A \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<Union>\<^sub>\<circ> (VLambda I A `\<^sub>\<circ> I) \<in>\<^sub>\<circ> Vset \<alpha>"
cut_less_mono: fixes t :: "'a"   and t' :: "'a"   and I :: "'a set" assumes "t \<le> t'" shows "I \<down>< t \<subseteq> I \<down>< t'"
Sem_no_zero_height_derivsAux: fixes c :: "IMP" shows "\<forall>s t.  s , c \<rightarrow>\<^sub>0  t  \<longrightarrow> False"
CAS1Red2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and v :: "'addr val"   and D :: "String.literal"   and F :: "String.literal"   and e3 :: "'addr expr1" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>Val v\<bullet>compareAndSwap(D\<bullet>F, e, e3),s\<rangle> -ta\<rightarrow> \<langle>Val v\<bullet>compareAndSwap(D\<bullet>F, e', e3),s'\<rangle>"
Lyndon_mono_fac: fixes ws :: "'a list list" assumes "Lyndon_mono ws" shows "ws = LynFac (concat ws)"
setops\<^sub>s\<^sub>s\<^sub>t_unlabel_dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_eq: fixes A :: "('c strand_label \<times> ('a, 'b) stateful_strand_step) list" shows "setops\<^sub>s\<^sub>s\<^sub>t (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A)) = setops\<^sub>s\<^sub>s\<^sub>t (unlabel A)"
K_block_design_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and sizes :: "int set" shows "K_block_design point_set block_collection sizes \<equiv> proper_design point_set block_collection \<and> K_block_design_axioms block_collection sizes"
coprime_minus_left_iff: fixes a :: "'a"   and b :: "'a" shows "coprime (- a) b = coprime a b"
Cons_1: fixes t :: "'a event list"   and u :: "'a event list" assumes "A \<inter> C = {}"   and "r setinterleaves ((t, u), C)" shows "trace_hide r A setinterleaves ((trace_hide t A, trace_hide u A), C)"
C\<rho>x: fixes x :: "'a" shows "C\<rho> *\<^sub>V ket x = (case \<rho> x of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i)"
err_semilat_eslI_aux: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "Semilat A r f" shows "err_semilat (esl (A, r, f))"
simps_1: fixes q :: "hf" shows "MN.nextl q [] = q"
liveVal'D: fixes g :: "'g"   and vs :: "'val list"   and v :: "'val"   and vs' :: "'val list" assumes "step.liveVal' g vs"   and "vs = v # vs'" shows "step.liveVal g v"
in_out_arcs_reflexive_uniq: fixes v :: "'a"   and e :: "'b" assumes "v \<in> verts G"   and "e \<in> in_arcs G v" shows "\<exists>!e'. e' \<in> out_arcs G v \<and> head G e' = tail G e"
no_better_subset_pref: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "rational_preference carrier relation"   and "x \<succeq>[relation] y" shows "no_better_than y carrier relation \<subseteq> no_better_than x carrier relation"
independent_empty: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "module scale" shows "\<not> module.dependent scale {}"
fun_merge_part: fixes as :: "'c set list"   and i :: "nat"   and a :: "'c"   and fs :: "('c \<Rightarrow> 'd) list" assumes "\<forall>i<length as. \<forall>j<length as. i \<noteq> j \<longrightarrow> as ! i \<inter> as ! j = {}"   and "i < length as"   and "a \<in> as ! i" shows "fun_merge fs as a = (fs ! i) a"
combine_options_def: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c option"   and y :: "'c option" shows "combine_options f x y = (case x of None \<Rightarrow> y | Some x \<Rightarrow> case y of None \<Rightarrow> Some x | Some y \<Rightarrow> Some (f x y))"
sum_telescope: fixes f :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "(\<Sum>i\<le>i. f i - f (Suc i)) = f 0 - f (Suc i)"
le_log_of_power: fixes b :: "real"   and n :: "nat"   and m :: "real" assumes "b ^ n \<le> m"   and "1 < b" shows "real n \<le> Transcendental.log b m"
lower_uminus: fixes A :: "'a interval" shows "lower (- A) = - upper A"
S1RPCUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [RPCNext crCh rmCh rst p]_(r p) \<and> $S1 rmhist p \<longrightarrow> unchanged r p"
gcsort_sort_inv: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and n :: "nat"   and t :: "nat \<times> nat list \<times> 'c list"   and p :: "nat"   and t' :: "nat \<times> nat list \<times> 'c list" assumes "index_less index key"   and "index_mono index key"   and "add_inv n t"   and "n \<le> p"   and "t' \<in> gcsort_set index key p t"   and "sort_inv key t" shows "sort_inv key t'"
differentiable_inverse: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set" assumes "f differentiable at x within s"   and "f x \<noteq> (0::'d)" shows "(\<lambda>x. inverse (f x)) differentiable at x within s"
graph_homomorphism_def: fixes G\<^sub>1 :: "('d, 'e) labeled_graph"   and G\<^sub>2 :: "('d, 'f) labeled_graph"   and f :: "('e \<times> 'f) set" shows "graph_homomorphism G\<^sub>1 G\<^sub>2 f = (vertices G\<^sub>1 = Domain f \<and> graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> f `` vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2 \<and> univalent f \<and> edge_preserving f (edges G\<^sub>1) (edges G\<^sub>2))"
max_pos_imp_Interp: fixes C :: "'a literal multiset"   and N :: "'a literal multiset set"   and A :: "'a" assumes "C \<in> N"   and "C \<noteq> {#}"   and "Max_mset C = Pos A"   and "S C = {#}" shows "Interp N C \<Turnstile> C"
read_only_share_all_shared: fixes a :: "nat"   and sb :: "'b memref list"   and \<S> :: "nat \<Rightarrow> bool option" assumes "a \<in> read_only (share sb \<S>)" shows "a \<in> read_only \<S> \<union> all_shared sb"
progress_simps_10: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and \<phi> :: "Formula.formula"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula.Exists \<phi>) j = Monitor.progress \<sigma> P \<phi> j"
IH: assumes "concurrent_ops_commute (prefix @ xs @ [x])"   and "concurrent_set x xs"   and "distinct (prefix @ x # xs)" shows "apply_operations (prefix @ xs @ [x]) = apply_operations (prefix @ x # xs)"
select_convs_4: fixes verts :: "'d set"   and arcs :: "'c set"   and tail :: "'c \<Rightarrow> 'd"   and head :: "'c \<Rightarrow> 'd"   and more :: "'e" shows "head \<lparr>verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = head"
set_status_nth: fixes xs :: "'a list"   and n :: "nat"   and i :: "nat"   and \<sigma> :: "'b status"   and f :: "'b" assumes "length xs = n"   and "i \<in> set (status \<sigma> (f, n))" shows "i < length xs \<and> xs ! i \<in> set xs"
bchoice_iff: fixes S :: "'b set"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(\<forall>x\<in>S. \<exists>y. Q x y) = (\<exists>f. \<forall>x\<in>S. Q x (f x))"
SN: shows "SN_both_mono_ordered_semiring_1 1 (\<lambda>x y. y < x) pos_arctic"
chf_lm_2: fixes x :: "nat"   and A :: "nat set" shows "(x \<in> A) = (chf A x = 0)"
path_from_to_first': fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and xs :: "'c list"   and x :: "'c"   and xs' :: "'c list"   and w :: "'c" assumes "Digraph G"   and "v \<leadsto>(xs @ x # xs')\<leadsto>\<^bsub>G\<^esub> w" shows "v \<notin> set xs'"
exp\<omega>\<^sub>O: fixes n :: "Ord" shows "\<langle>exp\<omega> n\<rangle>\<^sub>O = (\<lambda>z s l. \<langle>n\<rangle>\<^sub>O s (\<lambda>x z. l (\<lambda>n. \<langle>n\<rangle>\<^sub>N x z)) (\<lambda>f z. l (\<lambda>n. f n z)) z)"
res_eq: shows "[:resultant f g:] = s * f + t * g"
LeftDerivationFix_cut_appendix': fixes \<alpha> :: "symbol list"   and \<delta> :: "symbol list"   and i :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<beta> :: "symbol list"   and \<delta>' :: "symbol list" assumes "LeftDerivationFix (\<alpha> @ \<delta>) i D j (\<beta> @ \<delta>')"   and "LeftDerivation \<alpha> D \<beta>"   and "i < length \<alpha>"   and "j < length \<beta>" shows "LeftDerivationFix \<alpha> i D j \<beta>"
gseqE: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "(f, g) \<in> gseq" obtains i :: "nat" where "\<forall>i. f i \<in> A" and "\<forall>i. g i \<in> A" and "size (g i) < size (f i)" and "\<forall>j<i. f j = g j"
f_aggregate_Nil: fixes k :: "nat"   and ag :: "'b list \<Rightarrow> 'b" shows "f_aggregate [] k ag = []"
finite_bij_enumerate: fixes S :: "'a set" assumes "finite S" shows "bij_betw (enum S) {..<card S} S"
simps_1: fixes w :: "'a list" shows "Lyndon_rec w 0 = True"
not_square_less_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "\<not> less (times a a) zero"
base: fixes stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option" assumes "\<tau>exec_move ci P t e h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "Q stk' loc' pc' xcp'"
hyps_5: shows "cbelow_on (dgrad_p_set d m) (\<prec>\<^sub>p) (z + r) (\<lambda>a b. red F a b \<or> red F b a) (p + r) (q' + r)"
LocalSecretsComposition_ine2_s: fixes s :: "Secrets"   and PQ :: "specID"   and P :: "specID"   and Q :: "specID" assumes "sKS s \<in> LocalSecrets PQ"   and "subcomponents PQ = {P, Q}"   and "correctCompositionLoc PQ"   and "\<not> ine P (sE s)"   and "sKS s \<notin> LocalSecrets P"   and "sKS s \<notin> LocalSecrets Q" shows "ine Q (sE s)"
simps_3: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "local.less_eq_list (x # xs) (y # ys) = (x < y \<or> x = y \<and> local.less_eq_list xs ys)"
induct: fixes P :: "rat vec \<Rightarrow> (rat vec \<times> rat) list \<Rightarrow> rat list \<Rightarrow> rat vec \<Rightarrow> bool"   and a0 :: "rat vec"   and a1 :: "(rat vec \<times> rat) list"   and a2 :: "rat list"   and a3 :: "rat vec" assumes "\<And>f mus g'. P f [] mus g'"   and "\<And>f g ng n_gs mus g'. (\<And>x. x = f \<bullet> g / ng \<Longrightarrow> P f n_gs (x # mus) (- x \<cdot>\<^sub>v g + g')) \<Longrightarrow> P f ((g, ng) # n_gs) mus g'" shows "P a0 a1 a2 a3"
Bleast_code: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" shows "Bleast (set xs) P = (case filter P (sort xs) of [] \<Rightarrow> abort_Bleast (set xs) P | x # xs \<Rightarrow> x)"
sgn_abs: fixes abs :: "'k \<Rightarrow> 'k"   and sgn :: "'k \<Rightarrow> 'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and a :: "'k" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (sgn a) = zero_neq_one.of_bool one zero (a \<noteq> zero)"
map_filter_alt: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "map f (filter P xs) = concat (map (\<lambda>x. if P x then [f x] else []) xs)"
flush_all_until_volatile_write_Read\<^sub>s\<^sub>b_commute: fixes i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat"   and sb :: "'b memref list"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and \<D>' :: "'c"   and \<R>' :: "'e"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, Read\<^sub>s\<^sub>b volatile a t v # sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p, is, \<theta>, sb, \<D>', \<O>, \<R>')]) m = flush_all_until_volatile_write ts m"
lookup_mult_scalar_monomial_right_plus: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and p :: "'d \<Rightarrow>\<^sub>0 'f"   and v :: "'c"   and c :: "'f"   and t :: "'d" assumes "term_powerprod pair_of_term term_of_pair" shows "lookup (term_powerprod.mult_scalar pair_of_term term_of_pair p (monomial c v)) (term_powerprod.splus pair_of_term term_of_pair t v) = lookup p t * c"
bt_augment_Node: fixes a :: "'d"   and v :: "'c"   and ts :: "('d, 'c) bintree list"   and q :: "('c, 'd) pq" shows "bt_augment (Node a v ts) q = PQ.push v a (fold bt_augment ts q)"
Hwit21_def: fixes b :: "'n"   and c :: "'q" shows "Hwit21 b c = wit2_G (wit1_F2 b) (wit_F3 b c)"
intv2: fixes i :: "'b" assumes "\<I> i" shows "i \<in> END i"
acyclic_insert: fixes y :: "'a"   and x :: "'a"   and r :: "('a \<times> 'a) set" shows "acyclic (insert (y, x) r) = (acyclic r \<and> (x, y) \<notin> r\<^sup>*)"
id: shows "rat_of_int (d fs (Suc j)) = gs.d (Suc j)"
rank_ge_imp_is_solution: fixes A :: "(('c, 'd) vec, 'b) vec"   and b :: "('c, 'b) vec" assumes "(if \<exists>a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'c) then mod_type_class.to_nat (GREATEST a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'c)) + 1 else 0) \<le> rank A" shows "is_solution (solve_consistent_rref (Gauss_Jordan A) (P_Gauss_Jordan A *v b)) A b"
incX_def: fixes r :: "'a point_scheme" shows "incX r = \<lparr>xpos = xpos r + 1, ypos = ypos r, \<dots> = point.more r\<rparr>"
nnegD2: fixes P :: "'a \<Rightarrow> 'b" assumes "nneg P" shows "(\<lambda>s. 0::'b) \<le> P"
c: shows "change_respecting (cms\<^sub>1 ! k, mem\<^sub>1) (cms\<^sub>1' ! k, mem\<^sub>1') (differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k)"
cfinite_def: fixes r :: "('a \<times> 'a) set" shows "BNF_Cardinal_Arithmetic.cfinite r = finite (Field r)"
unique_connecting_path_tl: fixes v :: "'a"   and u :: "'a"   and w :: "'a" assumes "v \<in> V"   and "u \<in> set (w \<leadsto> v)"   and "u \<rightarrow> w" shows "tl (w \<leadsto> v) = u \<leadsto> v"
select_convs_1: fixes mag :: "'a"   and dim :: "(int, 'b) dimvec"   and more :: "'c" shows "mag \<lparr>mag = mag, dim = dim, \<dots> = more\<rparr> = mag"
prems_3: fixes a :: "'c"   and a' :: "'c" assumes "a \<in> insert x F"   and "a' \<in> insert x F"   and "a \<noteq> a'" shows "B a \<inter> B a' = {}"