hcju/isabelleps huggingface dataset:
Stone_Weierstrass_special: fixes f :: "'a \<Rightarrow> real"   and e :: "real" assumes "continuous_on S f"   and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x"   and "0 < e"   and "e < 1 / 3" shows "\<exists>g\<in>R. \<forall>x\<in>S. \<bar>f x - g x\<bar> < 2 * e"
consec_re: shows "\<forall>c. |re c| = 2 \<longrightarrow> (\<exists>n. Rep_nat_int (re c) = {n, n + 1})"
bounded_equiintegral_over_thin_tagged_partial_division: fixes F :: "('a \<Rightarrow> 'b) set"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and \<epsilon> :: "real" assumes "F equiintegrable_on cbox a b"   and "f \<in> F"   and "0 < \<epsilon>"   and "\<And>h x. \<lbrakk>h \<in> F; x \<in> cbox a b\<rbrakk> \<Longrightarrow> norm (h x) \<le> norm (f x)" obtains \<gamma> :: "'a \<Rightarrow> 'a set" where "gauge \<gamma>" and "\<And>c i S h. \<lbrakk>c \<in> cbox a b; i \<in> Basis; S tagged_partial_division_of cbox a b; \<gamma> fine S; h \<in> F; \<And>x K. (x, K) \<in> S \<Longrightarrow> K \<inter> {x. x \<bullet> i = c \<bullet> i} \<noteq> {}\<rbrakk> \<Longrightarrow> (\<Sum>(x, K)\<in>S. norm (integral K h)) < \<epsilon>"
conforms_xconf: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lname \<Rightarrow> ty option"   and x' :: "abopt" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "\<forall>a. x' = Some (Xcpt (Loc a)) \<longrightarrow> G,s\<turnstile>Addr a\<Colon>\<preceq>Class (SXcpt Throwable)"   and "x' = Some (Jump Ret) \<longrightarrow> locals s Result \<noteq> None" shows "(x', s)\<Colon>\<preceq>(G, L)"
cbp: shows "0 < content (cbox (a, c) (b, d))"
eval_fps_sin: fixes c :: "'a"   and z :: "'a" shows "eval_fps (fps_sin c) z = sin (c * z)"
Edges_rev_disj: fixes xs :: "nat list" assumes "distinct xs" shows "Edges (rev xs) \<inter> Edges xs = {}"
assms_5: fixes X :: "'a set" assumes "X \<in> \<F>" shows "polytope X"
wf_aux: fixes V :: "'v set"   and ws :: "'v \<Rightarrow> 'v list option"   and u :: "'v"   and p :: "'v list" assumes "inv (None, V, ws)"   and "ws u = Some p" shows "(V \<union> E `` {u}, V) \<in> finite_psupset (E\<^sup>* `` U0) \<or> V \<union> E `` {u} = V \<and> card (E `` {u} - V \<union> (dom ws - {u})) < card (dom ws)"
merge_on_unit: shows "merge_on UNIV id (=) merge_unit"
numeral_num_of_nat: fixes n :: "nat" assumes "0 < n" shows "numeral (num_of_nat n) = n"
round_robin_NoneD: fixes final :: "'c \<Rightarrow> bool"   and r :: "('b, 'a, 'c, 'd, 'e, 'f) semantics"   and convert_RA :: "'b \<Rightarrow>f nat \<Rightarrow> 'f list"   and n0 :: "nat"   and \<sigma> :: "'a list \<times> nat"   and s :: "('b, 'a) locks \<times> (('a \<Rightarrow> ('c \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('a \<Rightarrow> 'e wait_set_status option) \<times> 'a set" assumes "multithreaded_base.round_robin final r convert_RA n0 \<sigma> s = None"   and "Round_Robin.round_robin_invar \<sigma> (dom (thr s))" shows "multithreaded_base.active_threads final r s = {}"
circ_mult_increasing: fixes x :: "'a" shows "x \<le> x \<sqinter> top"
P2: shows "F' E' J CongA A B C \<and> \<not> E' F' OS D' J \<and> Coplanar D' E' F' J \<and> D' E' J CongA G' H' I'"
state_prop_def: fixes P :: "'g \<Rightarrow> bool" shows "\<lceil>P\<rceil> = (\<lambda>\<sigma>. P (\<sigma> 0))"
prems_7: shows "\<forall>l1 l2. PProbl (l1 @ l2) = (PProbl l1 \<and> PProbl l2)"
case3: fixes y :: "game" assumes "zin y (left_options x)" shows "\<not> ge_game (plus_game y (neg_game x), zero_game)"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "A.arr f"   and "A.cod f = b" shows "b \<cdot>\<^sub>A f = f"
init_item_dot: fixes r :: "symbol \<times> symbol list"   and k :: "nat" shows "item_dot (init_item r k) = 0"
is_orthogonal_sym: fixes \<psi> :: "'c"   and \<phi> :: "'c" shows "is_orthogonal \<psi> \<phi> = is_orthogonal \<phi> \<psi>"
ruler: fixes xs\<^sub>1 :: "'b list"   and ys :: "'b list"   and xs\<^sub>2 :: "'b list" assumes "xs\<^sub>1 \<le>p ys"   and "xs\<^sub>2 \<le>p ys" shows "xs\<^sub>1 \<le>p xs\<^sub>2 \<or> xs\<^sub>2 \<le>p xs\<^sub>1"
Ints_cases: fixes q :: "'b" assumes "q \<in> \<int>" obtains z :: "int" where "q = of_int z"
get_disconnected_nodes_reads: fixes document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "reads (get_disconnected_nodes_locs document_ptr) (get_disconnected_nodes document_ptr) h h'"
triv_path: fixes u :: "'v"   and v :: "'v" shows "path u [w (u, v)] v"
type_wf_impl: shows "type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = DocumentClass.type_wf"
lookup_subtensor1: fixes i :: "nat"   and "is" :: "nat list"   and A :: "'b tensor" assumes "i # is \<lhd> dims A" shows "lookup (subtensor A i) is = lookup A (i # is)"
rel_def: shows "rel \<equiv> rel_sector \<union> rel_token \<union> rel_pubk_less \<union> rel_pubk_more \<union> rel_hash \<union> rel_dec \<union> rel_enc \<union> rel_sep \<union> rel_con \<union> rel_id_pubk_less \<union> rel_id_pubk_more \<union> rel_id_pubk_prik \<union> rel_id_hash"
triv_init_disc_filtr_prob_space_axioms: shows "triv_init_disc_filtr_prob_space M F"
Derives1_drop: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "Derives1 a i r b" shows "drop (Suc i) a = drop (i + length (snd r)) b"
Eq3: shows "\<bar>VC p t - VC q t\<bar> = \<bar>IC p i t - IC q (i + 1) t\<bar>"
t_wcode_def: shows "t_wcode = t_wcode_prepare |+| t_wcode_main |+| t_wcode_adjust"
scale_right_imp_eq: fixes x :: "('c, 'd) vec"   and a :: "'c"   and b :: "'c" assumes "x \<noteq> 0"   and "a *s x = b *s x" shows "a = b"
less_prefix_def: fixes x :: "'a prefix"   and y :: "'a prefix" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
assms_7: shows "kind a = (\<lambda>s. True)\<^sub>\<surd>"
IH: assumes "A - Max_wrt_among le A \<subseteq> carrier" shows "is_weak_ranking (weak_ranking_aux (A - Max_wrt_among le A))"
i_in2: shows "\<iota> \<in> Inf_from (active_subset (Liminf_llist Ns))"
Rep_Assertion: fixes x :: "'b Assertion" shows "Rep_Assertion x \<in> assertion"
lookup_update': fixes ks :: "'c list"   and v :: "'d"   and t :: "('c, 'd) trie" shows "lookup_trie (update_trie ks v t) = lookup_trie t(ks \<mapsto> v)"
get_host_ok: fixes get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) 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_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and shadow_root_ptr :: "(_) shadow_root_ptr" assumes "l_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_document get_disconnected_document_locs known_ptr type_wf get_host get_host_locs get_shadow_root get_shadow_root_locs get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_tag_name heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h" shows "h \<turnstile> ok get_host shadow_root_ptr"
latestAct_less: fixes n' :: "nat"   and n :: "nat"   and nid :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "latestAct_cond nid t n n'" shows "n' \<le> \<langle>nid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>"
simps_1: fixes \<alpha> :: "'a" shows "bn (L_action.Act \<alpha>) = bn \<alpha>"
nprt_neg: 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 inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "lattice_ab_group_add.nprt zero inf (uminus x) = uminus (lattice_ab_group_add.pprt zero sup x)"
distinct_1: shows "0\<^sub>r \<noteq> 1\<^sub>r"
ibind_strict_2: fixes bind :: "'a \<Rightarrow> (unit \<Rightarrow> 'b) \<Rightarrow> 'b"   and return :: "unit \<Rightarrow> 'a"   and ASSERT :: "bool \<Rightarrow> 'a"   and ASSUME :: "bool \<Rightarrow> 'a"   and f :: "unit \<Rightarrow> 'b" assumes "generic_Assert bind return ASSERT ASSUME" shows "bind top f = top"
abs_eq: fixes xa :: "('a, 'b) pfun"   and x :: "'b set" assumes "eq_onp (\<lambda>f. finite (pdom f)) xa xa" shows "ffun.Abs_pfun xa \<rhd>\<^sub>f x = ffun.Abs_pfun (xa \<rhd>\<^sub>p x)"
trg_vcomp: fixes \<nu> :: "'d"   and \<mu> :: "'d" assumes "D.seq \<nu> \<mu>" shows "trg\<^sub>D (\<nu> \<cdot>\<^sub>D \<mu>) = trg\<^sub>D \<nu>"
plus_monomial_less_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and c :: "'b"   and u :: "'a" shows "plus_monomial_less p c u = p + monomial c u"
eval_to: fixes t :: "db"   and t' :: "db" assumes "t \<Down> t'" shows "val t'"
bounded_def: fixes S :: "'c set" shows "bounded S = (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
same_proc_call_unique_target: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and a' :: "'a"   and Q\<^sub>1 :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r\<^sub>1 :: "'e"   and p :: "'f"   and fs\<^sub>1 :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and Q\<^sub>2 :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r\<^sub>2 :: "'e"   and fs\<^sub>2 :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "valid_edge a'"   and "kind a = Q\<^sub>1:r\<^sub>1\<hookrightarrow>\<^bsub>p\<^esub>fs\<^sub>1"   and "kind a' = Q\<^sub>2:r\<^sub>2\<hookrightarrow>\<^bsub>p\<^esub>fs\<^sub>2" shows "targetnode a = targetnode a'"
assms: assumes "\<Phi>" shows "S \<le> \<Down> R S'"
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> xs" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> xs). xs\<lparr>a\<rparr> = b"
ChamberComplexIsomorphism_def: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" shows "ChamberComplexIsomorphism X Y f \<equiv> ChamberComplexMorphism X Y f \<and> ChamberComplexIsomorphism_axioms X Y f"
ODord_le_trans: fixes X :: "'c Order set"   and Y :: "'c Order set"   and Z :: "'c Order set" assumes "X \<in> ODnums"   and "Y \<in> ODnums"   and "Z \<in> ODnums"   and "X \<sqsubseteq> Y"   and "Y \<sqsubseteq> Z" shows "X \<sqsubseteq> Z"
lang\<^sub>M\<^sub>2\<^sub>L_rexp_of''_norm: fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n (norm \<phi>)" shows "\<Phi>.lang\<^sub>M\<^sub>2\<^sub>L n \<phi> = lang enum_class.enum n (rexp_of'' n (norm \<phi>)) - {[]}"
P: shows "\<forall>P. eventually P net \<longrightarrow> Inf (f ` Collect P) \<le> y"
VPG_MSTn: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and k :: "nat" assumes "VP_G G" shows "MST_validn k"
hf_induct: fixes P :: "hf \<Rightarrow> bool"   and z :: "hf" assumes "P 0"   and "\<And>x y. \<lbrakk>P x; P y; x \<^bold>\<notin> y\<rbrakk> \<Longrightarrow> P (y \<triangleleft> x)" shows "P z"
delete_conv': fixes k :: "'a"   and al :: "('a \<times> 'b) list" shows "map_of (delete k al) = (map_of al)(k := None)"
liminf_subseq_mono: fixes r :: "nat \<Rightarrow> nat"   and X :: "nat \<Rightarrow> 'a" assumes "strict_mono r" shows "liminf X \<le> liminf (X \<circ> r)"
in_unit_table: fixes v :: "'b option list"   and n :: "nat" shows "(v \<in> unit_table n) = wf_tuple n {} v"
insert: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
ineM_ine: fixes P :: "specID"   and M :: "chanID set"   and E :: "Expression" assumes "ineM P M E" shows "ine P E"
upair_inv_def: shows "upair_inv \<equiv> map_fun rep_uprod id (\<lambda>(x, y). if x = y then x else undefined)"
int_simps_13: fixes P :: "'a \<Rightarrow> bool" shows "\<turnstile> (P \<longrightarrow> P) = #True"
put_M_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) node_ptr"   and setter :: "('ab \<Rightarrow> 'ab) \<Rightarrow> 'z RNode_scheme \<Rightarrow> 'z RNode_scheme"   and v :: "'ab" assumes "h \<turnstile> ok put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr setter v" shows "ptr |\<in>| node_ptr_kinds h"
cinner_commute: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<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 dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b"   and y :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x y = cnj (cinner y x)"
top_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b" assumes "class.linordered_comm_monoid_add_top plus less_eq less top" shows "plus top x = x"
Can_Inv: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Can V src trg t" shows "bicategorical_language.Can V src trg (bicategorical_language.Inv V t)"
mult_le_mono1_hmset: fixes i :: "hmultiset"   and j :: "hmultiset"   and k :: "hmultiset" assumes "i \<le> j" shows "i * k \<le> j * k"
bv_to_nat_type: fixes w :: "bit list" shows "bv_to_nat (norm_unsigned w) = bv_to_nat w"
dep_def: fixes PROB :: "(('b, 'c) fmap \<times> ('b, 'd) fmap) set"   and v1 :: "'b"   and v2 :: "'b" shows "dep PROB v1 v2 \<equiv> (\<exists>a. a \<in> PROB \<and> (v1 \<in> fmdom' (fst a) \<and> v2 \<in> fmdom' (snd a) \<or> v1 \<in> fmdom' (snd a) \<and> v2 \<in> fmdom' (snd a))) \<or> v1 = v2"
simps: fixes a :: "'a"   and r :: "('b \<times> 'a) set" shows "(a \<in> Range r) = (\<exists>a b. a = b \<and> (a, b) \<in> r)"
gen_shleg_poly_def: fixes n :: "nat"   and k :: "nat" shows "gen_shleg_poly n k = (pderiv ^^ k) ([:0, 1, - 1:] ^ n)"
l6_4_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A P B"   and "\<not> Bet A P B" shows "Tarski_neutral_dimensionless.Out Bet P A B"
opn: fixes T :: "'b" assumes "T \<in> C" shows "open (f T)"
5: shows "integral Dy_pair (partial_vector_derivative (\<lambda>a. F a \<bullet> i) j) = integral (cbox a b) (\<lambda>x. F (x, g1 x) \<bullet> i - F (x, g2 x) \<bullet> i)"
keys_plus_eqI: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and q :: "'c \<Rightarrow>\<^sub>0 'd" assumes "keys p \<inter> keys q = {}" shows "keys (p + q) = keys p \<union> keys q"
sopen_sclose_eq_t: fixes t :: "sterm"   and n :: "nat"   and s :: "char list"   and p :: "char list" assumes "lc t" shows "{n \<rightarrow> [Fvar s,Fvar p]} {n \<leftarrow> [s,p]} t = t"
vint_fsingleton_in_fproduct_vint: fixes a :: "int" shows "[a\<^sub>\<int>]\<^sub>\<circ> \<in>\<^sub>\<circ> \<int>\<^sub>\<circ> ^\<^sub>\<times> 1\<^sub>\<nat>"
K: fixes x :: "nat" shows "norm (f x) \<le> K"
ChiRel_HAStates_NoCycles: fixes s :: "'a"   and t :: "'a"   and a :: "('a, 'b, 'c) hierauto" assumes "(s, t) \<in> ChiRel a" shows "(t, s) \<notin> ChiRel a"
path_conforms_with_strategy_irrelevant_updates: fixes p :: "Player"   and P :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "G'.path_conforms_with_strategy p P \<sigma>"   and "\<And>v. v \<in> lset P \<Longrightarrow> \<sigma> v = \<sigma>' v" shows "G'.path_conforms_with_strategy p P \<sigma>'"
df_system_def: fixes s\<^sub>0 :: "'a"   and lstep :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "df_system s\<^sub>0 lstep \<equiv> \<forall>s. system.reachable s\<^sub>0 lstep s \<longrightarrow> (\<exists>s' l. lstep l s s')"
ratfps_nth_altdef: fixes x :: "'b ratfps"   and n :: "nat" shows "ratfps_nth x n = fps_of_ratfps x $ n"
fls_base_factor_to_fps_base: fixes f :: "'b fls" assumes "f \<noteq> 0" shows "fls_base_factor_to_fps f $ 0 \<noteq> (0::'b)"
integral_negligible: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "negligible S" shows "integral S f = (0::'b)"
to_tree_order_ptrs_in_heap: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "l_to_tree_order_wf heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_child_nodes"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"   and "ptr' \<in> set to" shows "ptr' |\<in>| object_ptr_kinds h"
mon_n_same_proc: fixes fg :: "('b, 'a, 'c, 'd, 'e) flowgraph_rec_scheme"   and n :: "'b"   and n' :: "'b" assumes "proc_of fg n = proc_of fg n'" shows "mon_n fg n = mon_n fg n'"
horizontal_homs_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c" shows "horizontal_homs V src trg \<equiv> (category V \<and> endofunctor V src) \<and> endofunctor V trg \<and> horizontal_homs_axioms V src trg"
at_least_one_in_maximal: fixes C :: "'b fm set set"   and V :: "'b fm set"   and p :: "'b fm" assumes "consistency C"   and "V \<in> C"   and "maximal V C" shows "p \<in> V \<or> (\<^bold>\<not> p) \<in> V"
simps_4: fixes l :: "dag"   and v :: "ref"   and va :: "dag"   and vb :: "ref"   and vc :: "dag" shows "reduced (Node l v (Node va vb vc)) = (l \<noteq> Node va vb vc \<and> reduced l \<and> reduced (Node va vb vc))"
C2: shows "num_args t - Suc (length p) < num_args t"
prod_mset_diff: fixes B :: "'a multiset"   and A :: "'a multiset" assumes "B \<subseteq># A"   and "(0::'a) \<notin># B" shows "\<Prod>\<^sub># (A - B) = \<Prod>\<^sub># A div \<Prod>\<^sub># B"
distinct_71: fixes x81 :: "'b ltlr"   and x82 :: "'b ltlr"   and x91 :: "'b ltlr"   and x92 :: "'b ltlr" shows "x81 U\<^sub>r x82 \<noteq> x91 R\<^sub>r x92"
root_main'_simps: fixes x :: "int"   and n :: "int" shows "root_main' x n = (let xpm = x ^ pm; xp = xpm * x in if xp \<le> n then (x, xp = n) else root_main' ((n div xpm + x * int pm) div int p) n)"
indep_var_compose: fixes M1 :: "'b measure"   and X1 :: "'a \<Rightarrow> 'b"   and M2 :: "'b measure"   and X2 :: "'a \<Rightarrow> 'b"   and Y1 :: "'b \<Rightarrow> 'c"   and N1 :: "'c measure"   and Y2 :: "'b \<Rightarrow> 'c"   and N2 :: "'c measure" assumes "indep_var M1 X1 M2 X2"   and "Y1 \<in> M1 \<rightarrow>\<^sub>M N1"   and "Y2 \<in> M2 \<rightarrow>\<^sub>M N2" shows "indep_var N1 (Y1 \<circ> X1) N2 (Y2 \<circ> X2)"
intro: fixes V :: "'b set"   and norm :: "'b \<Rightarrow> real" assumes "\<And>x. x \<in> V \<Longrightarrow> 0 \<le> norm x"   and "\<And>x a. x \<in> V \<Longrightarrow> norm (a \<cdot> x) = \<bar>a\<bar> * norm x"   and "\<And>x y. \<lbrakk>x \<in> V; y \<in> V\<rbrakk> \<Longrightarrow> norm (x + y) \<le> norm x + norm y" shows "seminorm V norm"
GmultD: fixes g :: "'g"   and v :: "'v" shows "g *\<cdot> v = (1::'f) \<delta>\<delta> g \<cdot> v"
Abs_freelist_Cons: fixes x :: "'b"   and xs :: "'b list" shows "Abs_freelist (x # xs) = Abs_freeletter x + Abs_freelist xs"
finfun_Diag_update2: fixes f :: "('a, 'b) finfun"   and g :: "('a, 'c) finfun"   and a :: "'a"   and c :: "'c" shows "finfun_Diag f (finfun_update g a c) = finfun_update (finfun_Diag f g) a (finfun_apply f a, c)"
Rep: fixes x :: "'b word" shows "uint x \<in> uints LENGTH('b)"
connected_if_composable: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "regular_weak_composition V H"   and "H \<nu> \<mu> \<noteq> partial_magma.null V" shows "weak_composition.sources V H \<nu> = weak_composition.targets V H \<mu>"
card_b: fixes j :: "nat"   and k :: "nat"   and i :: "nat" assumes "j \<le> k" shows "card (b k (j, i)) = enum (d j) (Suc i) - enum (d j) i"
path_trans_2: fixes s :: "'c"   and l :: "'a"   and sh :: "'c"   and p :: "'a list"   and s' :: "'c" assumes "astep s l sh"   and "lts.path sh p s'" shows "lts.path s (l # p) s'"
accept_def: fixes w :: "nat \<Rightarrow> 'L" shows "accept w \<equiv> \<exists>r. is_acc_run r \<and> (\<forall>i. L (r i) (w i))"