hcju/isabelleps huggingface dataset:
sum_SDG_summary_edge_matched: fixes n :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "n s-p\<rightarrow>\<^bsub>sum\<^esub> n'" obtains ns :: "'node SDG_node list" where "matched n ns n'" and "n \<in> set ns" and "get_proc (parent_node (last ns)) = p"
split_face_edges_f21: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list"   and f12 :: "face"   and f21 :: "face"   and vs2 :: "nat list" assumes "pre_split_face f ram1 ram2 vs"   and "(f12, f21) = split_face f ram1 ram2 vs"   and "vs \<noteq> []"   and "vs2 = between (vertices f) ram2 ram1"   and "vs2 \<noteq> []" shows "\<E> f21 = {(last vs2, ram1), (ram1, hd vs), (last vs, ram2), (ram2, hd vs2)} \<union> Edges vs \<union> Edges vs2"
face_cycle_succ_permutes: shows "face_cycle_succ permutes arcs G"
simps: fixes l :: "'b tree"   and m :: "'b"   and r :: "'b tree" shows "get_min \<langle>l, m, r\<rangle> = (if l = \<langle>\<rangle> then m else get_min l)"
star_plus: fixes y :: "'a" shows "y\<^sup>\<star> * y = y\<^sup>+"
ArrayInitE: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: (CLEAR x[], s) \<Rightarrow> t" obtains "t = s(x := \<lambda>_. 0)"
dist_les_5: shows "ValTT \<notsqsubseteq> ValFF"
bheight_RBT_Impl_diag: fixes t :: "('c, 'd) rbt" shows "bheight (RBT_Impl_diag t) = bheight t"
af_letter_subst: fixes \<phi> :: "'b ltln"   and \<nu> :: "'b set" shows "af_letter \<phi> \<nu> = subst \<phi> (\<lambda>\<psi>. Some (af_letter \<psi> \<nu>))"
plus_poly_mapping_def: shows "(+) \<equiv> map_fun poly_mapping.lookup (map_fun poly_mapping.lookup Abs_poly_mapping) (\<lambda>f1 f2 k. f1 k + f2 k)"
rComm2: fixes M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and K :: "'a"   and P'' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "\<Psi> \<rhd> !P \<longmapsto> K\<lparr>N\<rparr> \<prec> P''"   and "\<And>C. Prop C \<Psi> (!P) (K\<lparr>N\<rparr>) P''"   and "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* C"   and "distinct xvec" shows "Prop C \<Psi> (P \<parallel> !P) (\<tau>) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> P'')"
list_app_eq_cases: fixes xs\<^sub>1 :: "'c list"   and xs\<^sub>2 :: "'c list"   and ys\<^sub>1 :: "'c list"   and ys\<^sub>2 :: "'c list" assumes "xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>1 @ ys\<^sub>2" obtains "xs\<^sub>1 = take (length xs\<^sub>1) ys\<^sub>1" and "xs\<^sub>2 = drop (length xs\<^sub>1) ys\<^sub>1 @ ys\<^sub>2"   | "ys\<^sub>1 = take (length ys\<^sub>1) xs\<^sub>1" and "ys\<^sub>2 = drop (length ys\<^sub>1) xs\<^sub>1 @ xs\<^sub>2"
sum_map_pmf_def: fixes fa :: "'e \<Rightarrow> 'g"   and fb :: "'f \<Rightarrow> 'h" shows "sum_map_pmf fa fb = (\<lambda>a. case a of Inl e \<Rightarrow> Inl (fa e) | Inr e \<Rightarrow> Inr (fb e))"
get_ancestors_si_reads: fixes h :: "(_) heap"   and node_ptr :: "(_) object_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h" shows "reads get_ancestors_si_locs (get_ancestors_si node_ptr) h h'"
set_lfp_def: fixes s :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "set_lfp s f = lfp (\<lambda>s'. f s' \<union> s)"
hyperpow_two_gt_one: fixes r :: "'b star" assumes "1 < r" shows "1 < r pow 2"
build_index_loc_axioms: shows "build_index_loc hm_ops ls_ops ls_ops"
no_empty_in_non_overlapping: fixes p :: "'b set set" assumes "is_non_overlapping p" shows "{} \<notin> p"
SD_efficientD: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'b pmf" assumes "pref_profile_wf agents alts R"   and "SD_efficient R p"   and "q \<in> lotteries_on alts"   and "\<And>i. i \<in> agents \<Longrightarrow> SD (R i) p q"   and "\<exists>i\<in>agents. \<not> SD (R i) q p" shows "False"
disc_19: fixes x4 :: "'a" shows "isSnapshot (Snapshot x4)"
DBM_up_sound': fixes v :: "'e \<Rightarrow> nat"   and n :: "nat"   and u :: "'e \<Rightarrow> 'f"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry" assumes "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "u \<in> [up M]\<^bsub>v,n\<^esub>" shows "u \<in> [M]\<^bsub>v,n\<^esub>\<^sup>\<up>"
IS_BELOW_ID_def: fixes R :: "('b \<times> 'b) set" shows "IS_BELOW_ID R \<equiv> R \<subseteq> Id"
conforms_init_lvars: fixes G :: "prog"   and declC :: "qtname"   and sig :: "sig"   and dm :: "qtname \<times> methd"   and s :: "st"   and pvs :: "val list"   and pTsa :: "ty list"   and x :: "abopt"   and L :: "lenv"   and statT :: "ref_ty"   and invC :: "qtname"   and a' :: "val"   and sm :: "ref_ty \<times> mhead"   and e :: "expr" assumes "wf_mhead G (pid declC) sig (mhead (mthd dm))"   and "wf_prog G"   and "list_all2 (conf G s) pvs pTsa"   and "G\<turnstile>pTsa[\<preceq>]parTs sig"   and "(x, s)\<Colon>\<preceq>(G, L)"   and "methd G declC sig = Some dm"   and "isrtype G statT"   and "G\<turnstile>invC\<preceq>\<^sub>C declC"   and "G,s\<turnstile>a'\<Colon>\<preceq>RefT statT"   and "invmode (mhd sm) e = IntVir \<longrightarrow> a' \<noteq> Null"   and "invmode (mhd sm) e \<noteq> IntVir \<longrightarrow> (\<exists>statC. statT = ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC) \<or> (\<forall>statC. statT \<noteq> ClassT statC \<and> declC = Object)"   and "invC = invocation_class (invmode (mhd sm) e) s a' statT"   and "declC = invocation_declclass G (invmode (mhd sm) e) s a' statT sig"   and "x \<noteq> Some (Jump Ret)" shows "init_lvars G declC sig (invmode (mhd sm) e) a' pvs (x, s)\<Colon>\<preceq>(G, \<lambda>k. case k of EName (VNam v) \<Rightarrow> (table_of (lcls (mbody (mthd dm)))(pars (mthd dm) [\<mapsto>] parTs sig)) v | EName Res \<Rightarrow> Some (resTy (mthd dm)) | This \<Rightarrow> if is_static (mthd sm) then None else Some (Class declC))"
Bij_inv_into_mem: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c" assumes "f \<in> Bij S"   and "x \<in> S" shows "inv_into S f x \<in> S"
last_snoc: fixes xs :: "'b list"   and x :: "'b" shows "last (xs @ [x]) = x"
comb_sat_props: shows "1 < length qs \<longrightarrow> p \<noteq> 0 \<and> 0 < length qs \<longrightarrow> satisfies_properties_R p qs (get_subsets_R (snd (combine_systems_R p (take (length qs div 2) qs, calculate_data_R p (take (length qs div 2) qs)) (drop (length qs div 2) qs, calculate_data_R p (drop (length qs div 2) qs))))) (get_signs_R (snd (combine_systems_R p (take (length qs div 2) qs, calculate_data_R p (take (length qs div 2) qs)) (drop (length qs div 2) qs, calculate_data_R p (drop (length qs div 2) qs))))) (get_matrix_R (snd (combine_systems_R p (take (length qs div 2) qs, calculate_data_R p (take (length qs div 2) qs)) (drop (length qs div 2) qs, calculate_data_R p (drop (length qs div 2) qs)))))"
VPi_I: fixes A :: "V"   and b :: "V \<Rightarrow> V"   and B :: "V \<Rightarrow> V" assumes "\<And>x. x \<in> elts A \<Longrightarrow> b x \<in> elts (B x)" shows "VLambda A b \<in> elts (VPi A B)"
bind_advantage_def: fixes \<eta> :: "nat"   and \<A> :: "nat \<times> nat \<Rightarrow> (nat \<times> nat \<times> (nat \<times> nat) \<times> nat \<times> nat \<times> nat) spmf" shows "rivest_commit.bind_advantage \<eta> \<A> \<equiv> spmf (rivest_commit.bind_game \<eta> \<A>) True"
rep_eq: fixes x :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "('a, 'b) mapping" shows "mapping.rep (Mapping.filter x xa) = (\<lambda>k. case mapping.rep xa k of None \<Rightarrow> None | Some v \<Rightarrow> if x k v then Some v else None)"
has_products_def: fixes I :: "'d set" shows "has_products I \<equiv> I \<noteq> UNIV \<and> (\<forall>J D. discrete_diagram J (\<cdot>) D \<and> Collect (partial_magma.arr J) = I \<longrightarrow> (\<exists>a. has_as_product J D a))"
list_nth_split: fixes m :: "nat"   and cs :: "'c list"   and n :: "nat" assumes "m < length cs"   and "n < m"   and "1 < length cs" shows "\<exists>xs ys zs. cs = xs @ cs ! n # ys @ cs ! m # zs"
CFG_call_edge_params: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and i :: "nat"   and ins :: "'c list"   and outs :: "'c list"   and s :: "(('c \<Rightarrow> 'd option) \<times> 'e) list"   and s' :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "i < length ins"   and "(p, ins, outs) \<in> set procs"   and "local.CFG.pred (kind a) s"   and "local.CFG.pred (kind a) s'"   and "\<forall>V\<in>ParamUses (sourcenode a) ! i. local.CFG.state_val s V = local.CFG.state_val s' V" shows "local.CFG.params fs (local.CFG.state_val s) ! i = local.CFG.params fs (local.CFG.state_val s') ! i"
v: shows "\<not> SN_on (E\<^sup>* O R O E\<^sup>*) {v}"
simps: fixes f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and aa :: "'b" shows "segmentp f a b aa = (aa = f a \<and> f a \<noteq> b \<or> (\<exists>x. aa = f x \<and> segmentp f a b x \<and> f x \<noteq> b))"
r_2: shows "(w \<circ> r) \<longlonglongrightarrow> liminf w"
prems_4: shows "msgs (S t i) cid \<noteq> []"
simps_2: fixes x :: "'a"   and AB :: "'a tree"   and CD :: "'a tree" shows "splay x \<langle>AB, x, CD\<rangle> = \<langle>AB, x, CD\<rangle>"
document_ptr_kinds_eq2_h2: shows "document_ptr_kinds h' |\<subseteq>| document_ptr_kinds h2"
assms_2: shows "f\<lparr>ArrDom\<rparr> = A"
gauss_jordan_single_def: fixes A :: "'a mat" shows "gauss_jordan_single A = fst (gauss_jordan A (0\<^sub>m (dim_row A) 0))"
assms_1: shows "congruence M (1, 0, 0, - 1) = cor k' *\<^sub>s\<^sub>m (1, 0, 0, - 1)"
simps_7: fixes C :: "ctx"   and e2 :: "exp"   and e3 :: "exp"   and e :: "exp" shows "plug (CIf1 C e2 e3) e = EIf (plug C e) e2 e3"
fstate_poss_def: fixes p :: "lambda" shows "fstate_poss p = proc_poss p \<times> NList (Pow (proc_poss p)) (call_list_lengths p) \<times> smaps_over (vars p \<times> UNIV) (proc_poss p) \<times> UNIV"
fpr_first_in_path': fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl"   and path' :: "(ctx \<times> cld) list"   and path :: "(ctx \<times> cld) list"   and fqn :: "fqn"   and ctxcld :: "ctx \<times> cld" assumes "find_path_rec_f P ctx cl path' = Some path"   and "cl = cl_fqn fqn \<and> find_cld_f P ctx fqn = Some ctxcld" shows "ctxcld \<in> set path"
assms_1: shows "\<forall>E F G. Ball X \<lbrakk>E, F, G\<rbrakk> \<longrightarrow> \<lbrakk>E, F, G\<rbrakk> p"
mat_plus: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'c list list"   and m2 :: "'c list list"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "mat nr nc m1"   and "mat nr nc m2" shows "mat nr nc (mat_plusI pl m1 m2)"
"case": fixes f :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> 'a"   and x1 :: "bool"   and x2 :: "bool"   and x3 :: "bool"   and x4 :: "bool"   and x5 :: "bool"   and x6 :: "bool"   and x7 :: "bool"   and x8 :: "bool" shows "(case Char x1 x2 x3 x4 x5 x6 x7 x8 of Char x xa xb xc xd xe xf xg \<Rightarrow> f x xa xb xc xd xe xf xg) = f x1 x2 x3 x4 x5 x6 x7 x8"
val_of_divide': fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "y \<noteq> \<zero>"   and "val_Zp y \<le> val_Zp x" shows "val_Zp (local.divide x y) = val_Zp x - val_Zp y"
edges_ins_edge_ss: fixes g :: "'b ugraph"   and e :: "'b \<times> 'b" shows "edges g \<subseteq> edges (ins_edge e g)"
get_elements_by_class_name_pure: fixes ptr :: "(_) object_ptr"   and name :: "char list"   and h :: "(_) heap" shows "pure (get_elements_by_class_name ptr name) h"
simps_1: fixes l :: "('c \<times> nat) option"   and t :: "'c" shows "lock_actions_ok l t [] = True"
smult_add_left: fixes a :: "'b"   and b :: "'b"   and p :: "'b poly" shows "Polynomial.smult (a + b) p = Polynomial.smult a p + Polynomial.smult b p"
permutes_compose: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and q :: "'c \<Rightarrow> 'c" assumes "p permutes S"   and "q permutes S" shows "q \<circ> p permutes S"
j_3: fixes i :: "nat" assumes "i < length YS" shows "j i < length XS"
"1_2": shows "col \<in> {a..b} \<rightarrow> {..<k}"
base_in_hom_2: shows "\<guillemotleft>r : r \<Rightarrow> r\<guillemotright>"
vals_insert: fixes a :: "'d"   and v :: "'c"   and xs :: "('d, 'c) bintree option list" shows "set (vals (Binomial_Queue.insert a v xs)) = set (vals xs) \<union> {v}"
L\<^sub>1\<^sub>2: shows "instance_of\<^sub>l\<^sub>s L\<^sub>1 L\<^sub>2"
eqcelem_imp_iff: fixes x :: "'f"   and y :: "'f"   and A :: "'f cset" assumes "x = y" shows "cin x A = cin y A"
substitute_mono1': fixes f :: "'a \<Rightarrow> 'a ttree"   and f' :: "'a \<Rightarrow> 'a ttree"   and T :: "'a set"   and t :: "'a ttree" assumes "f \<sqsubseteq> f'" shows "substitute f T t \<sqsubseteq> substitute f' T t"
least_power_of_permutation_2: fixes p :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "permutation p" shows "0 < least_power p a"
coretract_property: fixes y :: "'b" assumes "y \<in> carrier \<Y>" shows "y \<sqsubseteq>\<^bsub>\<Y>\<^esub> \<pi>\<^sup>* (\<pi>\<^sub>* y)"
invC: shows "invC = invocation_class (invmode sm e) s a' statT"
nres_order_simps_5: fixes X :: "'b set"   and Y :: "'b set" shows "(RES X \<le> RES Y) = (X \<subseteq> Y)"
simps_2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "foldseq f s (Suc n) = f (s 0) (foldseq f (\<lambda>k. s (Suc k)) n)"
MPLS_def: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" shows "MPLS PROB \<equiv> (\<lambda>(s, as). Inf (PLS s as)) ` {(s, as). s \<in> valid_states PROB \<and> as \<in> valid_plans PROB}"
get_owner_document_owner_document_in_heap_node: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr"   and owner_document :: "(_) document_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> local.CD.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr () \<rightarrow>\<^sub>r owner_document" shows "owner_document |\<in>| document_ptr_kinds h"
iMODb_subset_imp_iMOD_subset: fixes r' :: "nat"   and m' :: "nat"   and c' :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "[ r', mod m', c' ] \<subseteq> [ r, mod m, c ]"   and "0 < c'" shows "[ r', mod m' ] \<subseteq> [ r, mod m ]"
add_uminus_conv_diff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "plus a (uminus b) = minus a b"
mult_cancel_right1: 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 c :: "'b"   and b :: "'b" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(c = times b c) = (c = zero \<or> b = one)"
qax: shows "\<exists>b. \<forall>a. q b a \<and> q 0 0 \<and> q 1 a \<and> q a 1 \<or> q 0 1 \<and> q 1 0 \<and> q a b \<and> q 1 1"
map_add_def: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option" shows "m1 ++ m2 = (\<lambda>x. case m2 x of None \<Rightarrow> m1 x | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>y\<rfloor>)"
scf_ok: fixes weights :: "'a weights"   and i :: "nat"   and n :: "nat"   and f :: "'a" assumes "class.weighted weights"   and "i < n" shows "scf weights (f, n) i \<le> 1"
mod_diff_left_eq: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a mod c - b) mod c = (a - b) mod c"
mojmir_to_rabin: fixes \<G> :: "'a ltl set"   and \<psi> :: "'a ltl" assumes "Only_G \<G>" shows "mojmir_to_rabin \<Sigma> \<up>af\<^sub>G (Abs \<psi>) w {q. \<G> \<up>\<Turnstile>\<^sub>P q}"
uint_up_ucast: fixes w :: "'b word" assumes "is_up UCAST('b \<rightarrow> 'c)" shows "uint (UCAST('b \<rightarrow> 'c) w) = uint w"
power_down_pos: fixes x :: "real"   and p :: "nat"   and n :: "nat" assumes "0 < x" shows "0 < power_down p x n"
value_eq_false: fixes a :: "value option"   and b :: "value option" shows "(value_eq a b = false) = (\<exists>x y. a = Some x \<and> b = Some y \<and> x \<noteq> y)"
star_fps_rep_Cons: fixes f :: "'d list \<Rightarrow> 'c"   and v :: "'d"   and va :: "'d list" shows "star_fps_rep f (v # va) = f []\<^sup>\<star> \<cdot> \<Sum> {f y \<cdot> star_fps_rep f z |y z. v # va = y @ z \<and> y \<noteq> []}"
Cons_1: fixes state :: "State"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "InvariantUniqQ (getQ state)" shows "let state' = notifyWatches_loop literal Wl' newWl state in InvariantUniqQ (getQ state')"
one_plus_i: assumes "x = 1 + \<i>\<^sub>\<int>" shows "thesis"
adopt_node: shows "h \<turnstile> adopt_node owner_document node_ptr \<rightarrow>\<^sub>h h2"
H: shows "(\<Delta> @ TVarB X Q # \<Gamma>)\<turnstile>S<:T"
caratheodory_aff_dim: fixes p :: "'c set" shows "convex hull p = {x. \<exists>S. finite S \<and> S \<subseteq> p \<and> int (card S) \<le> aff_dim p + 1 \<and> x \<in> convex hull S}"
intro: fixes \<eta> :: "'c \<Rightarrow> 'd"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and M' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d" assumes "monoid_homomorphism \<eta> M composition unit M' composition' unit'"   and "injective_map \<eta> M M'" shows "monoid_monomorphism \<eta> M composition unit M' composition' unit'"
drop_at_shift: fixes n :: "nat"   and index :: "nat"   and D :: "'a list" assumes "n \<le> index"   and "index < length D" shows "drop n D ! (index - n) = D ! index"
to_interval_def: fixes a :: "'a" shows "to_interval a \<equiv> Interval a a"
isomorphic_sym: shows "\<exists>T. FGModuleIso G (\<star>) W (\<cdot>) T V"
id_oracle_def: fixes s :: "'i"   and x :: "'j" shows "1\<^sub>I s x = Pause x (\<lambda>x. Done (x, s))"
CR4_def: fixes \<tau> :: "ty" shows "CR4 \<tau> \<equiv> \<forall>t. NEUT t \<and> NORMAL t \<longrightarrow> t \<in> RED \<tau>"
IH_1: fixes d :: "nat"   and t' :: "'a tree"   and n' :: "nat"   and h' :: "nat" assumes "ins3 n d x l = Unbal2 t' n' h'"   and "bal_i n (height l + d)" shows "\<Phi> t' - \<Phi> l \<le> 6 * e * real (height l)"
lsorted_code_2: fixes x :: "'a" shows "lsorted (LCons x LNil) = True"
get_alloc: fixes xs :: "'b list"   and h :: "heap"   and ys :: "'b list" shows "Array.get (snd (Array.alloc xs h)) (fst (Array.alloc ys h)) = xs"
primepow_gt_Suc_0: fixes n :: "nat" assumes "primepow n" shows "Suc 0 < n"
to_hma\<^sub>m_def: fixes a :: "'b mat" shows "Mod_Type_Connect.to_hma\<^sub>m a = (\<chi>i j. a $$ (mod_type_class.to_nat i, mod_type_class.to_nat j))"
set_pmf_eq: fixes M :: "'b pmf" shows "set_pmf M = {x. pmf M x \<noteq> 0}"
not_null_not_col: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "QCongAACute a"   and "\<not> QCongANullAcute a"   and "a A B C" shows "\<not> Col A B C"
select_convs_6: fixes Obj :: "'h set"   and Mor :: "'g set"   and Dom :: "'g \<Rightarrow> 'h"   and Cod :: "'g \<Rightarrow> 'h"   and Id :: "'h \<Rightarrow> 'g"   and Comp :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and more :: "'i" shows "(;;\<^bsub>\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr>\<^esub>) = Comp"
netgmap_netmask_subnets': fixes sr :: "'e \<Rightarrow> 'c \<times> 'd"   and s1 :: "'e net_state"   and n1 :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'c"   and s2 :: "'e net_state"   and n2 :: "net_tree"   and s :: "'e net_state" assumes "netgmap sr s1 = netmask (net_tree_ips n1) (\<sigma>, snd (netgmap sr s1))"   and "netgmap sr s2 = netmask (net_tree_ips n2) (\<sigma>, snd (netgmap sr s2))"   and "s = SubnetS s1 s2" shows "netgmap sr s = netmask (net_tree_ips (n1 \<parallel> n2)) (\<sigma>, snd (netgmap sr s))"
a_in_A: shows "\<forall>\<tau>\<in>set a. \<tau> \<in> A"
Gromov_converging_at_boundary_def: fixes u :: "nat \<Rightarrow> 'c" shows "Gromov_converging_at_boundary u = (\<forall>a M. \<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. M \<le> Gromov_product_at a (u m) (u n))"
rsp: shows "eq_onp (\<lambda>f. \<forall>\<^sub>\<infinity>n. f (- int n) = (0::'a)) (\<lambda>_. 0::'a) (\<lambda>_. 0::'a)"
fplus_maybe_simps_3: fixes x :: "'c"   and ys :: "'c\<cdot>maybe" shows "fplus\<cdot>(Just\<cdot>x)\<cdot>ys = Just\<cdot>x"
hn_merge3_1: fixes XX :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and R1 :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and x :: "'c"   and x' :: "'d"   and Rm :: "'c \<Rightarrow> 'd \<Rightarrow> assn" assumes "NO_MATCH (hn_invalid XX) R2"   and "hn_ctxt R1 x x' \<or>\<^sub>A hn_ctxt R2 x x' \<Longrightarrow>\<^sub>t hn_ctxt Rm x x'" shows "hn_invalid R1 x x' \<or>\<^sub>A hn_ctxt R2 x x' \<Longrightarrow>\<^sub>t hn_invalid Rm x x'"
canon_ang_1: fixes \<alpha> :: "real" shows "- pi < \<downharpoonright>\<alpha>\<downharpoonleft>"
Max_wrt_among_map_relation_bij: fixes f :: "'alt \<Rightarrow> 'a"   and A :: "'alt set" assumes "bij f" shows "f ` Max_wrt_among (R i) A = Max_wrt_among (map_relation (inv f) (R i)) (f ` A)"
that: fixes i :: "nat" assumes "\<And>j. i < j \<Longrightarrow> component_of_term (seq j) < component_of_term (seq i)" shows "thesis"
\<mu>_sq_non_zero: shows "0 < \<mu>\<^sup>2"
mtf_id: fixes x :: "'b"   and xs :: "'b list" assumes "x \<notin> set xs" shows "mtf x xs = xs"
lnull_inf_llist: fixes f :: "nat \<Rightarrow> 'b" shows "\<not> lnull (inf_llist f)"
fam_morphisms_are_maps: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and \<FF>' :: "'c set \<Rightarrow> 'e set"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and b' :: "'e"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e"   and U :: "'c set" assumes "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"   and "is_open U" shows "Set_Theory.map (fam_morphisms U) (\<FF> U) (\<FF>' U)"
assms_4: shows "InvariantNoDecisionsWhenUnit (getF state) (getM state) (currentLevel (getM state))"
A71_NSources_L1: shows "\<forall>C\<in>AbstrLevel level1. C \<noteq> sA92 \<and> C \<noteq> sA91 \<and> C \<noteq> sA81 \<longrightarrow> sA71 \<notin> Sources level1 C"
diagram_determines_ide_1: fixes D :: "'j \<Rightarrow> 'c" assumes "diagram (\<cdot>\<^sub>J) (\<cdot>) D" shows "J_C.ide (J_C.MkIde D)"
map_int_pair_def: fixes f :: "int \<Rightarrow> int \<Rightarrow> 'a"   and g :: "val \<Rightarrow> 'a"   and x :: "val" shows "map_int_pair f g x = (case x of <|IntVal a, IntVal b|> \<Rightarrow> f a b | <|IntVal a, _|> \<Rightarrow> g x | <|_, val2|> \<Rightarrow> g x | _ \<Rightarrow> g x)"
simps_1: fixes cs :: "'edge list" shows "valid_path_aux cs [] = True"
2: shows "Fvars (subst \<chi> \<langle>neg \<phi>\<rangle> yy) = {}"
prime_factor_exists: fixes n :: "nat" assumes "1 < n" shows "\<exists>p. prime p \<and> p dvd n"