hcju/isabelleps huggingface dataset:
model_to_dimacs_model_append: fixes M :: "nat \<Rightarrow> bool"   and vs :: "nat list"   and vs' :: "nat list" shows "set (model_to_dimacs_model M (vs @ vs')) = set (model_to_dimacs_model M vs) \<union> set (model_to_dimacs_model M vs')"
iFROM_infinite: fixes n :: "nat" shows "infinite [n\<dots>]"
distinct_17: fixes x2 :: "'c"   and x61 :: "'c"   and x62 :: "('b, 'c) fm" shows "Nom x2 \<noteq> (\<^bold>@ x61 x62)"
affG: fixes X :: "'a set" assumes "X \<in> \<G>" shows "aff_dim X \<le> aff_dim T"
processedD1: fixes a :: "'e"   and b :: "'e"   and xs :: "'e list"   and ps :: "('e \<times> 'e) list" assumes "processed (a, b) xs ps" shows "a \<in> set xs"
iFROM_conv: fixes n :: "nat" shows "[n\<dots>] = UNIV \<oplus> n"
strategyproofness_2: shows "pmf (sds R10) d + (pmf (sds R10) b + pmf (sds R10) a) \<le> pmf (sds R12) d + (pmf (sds R12) b + pmf (sds R12) a)"
farey_fractions_bij: fixes N :: "nat" shows "bij_betw (\<lambda>(a, b). Rat.Fract (int a) (int b)) {(a, b) |a b. 0 < a \<and> a \<le> b \<and> b \<le> N \<and> coprime a b} (farey_fractions N)"
strict_ex_refine_lemma_v: fixes X :: "'b program set"   and F :: "'b program"   and G :: "'b program" assumes "strict_ex_prop X" shows "(\<forall>H. F ok H \<and> G ok H \<and> F \<squnion> H \<in> welldef \<and> F \<squnion> H \<in> X \<longrightarrow> G \<squnion> H \<in> X) = (F \<in> welldef \<inter> X \<longrightarrow> G \<in> X)"
deg: fixes f :: "'a mod_ring poly" assumes "irreducible f"   and "f dvd v" shows "d < degree f"
ev_HLD_sset: fixes A :: "'b set"   and w :: "'b stream" shows "ev (HLD A) w = (sset w \<inter> A \<noteq> {})"
nolhs: shows "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s'"
rev_append: fixes xs :: "'d list"   and ys :: "'d list" shows "rev (xs @ ys) = rev ys @ rev xs"
in_empty_table: fixes x :: "'a" shows "x \<notin> empty_table"
ps: shows "ps = xs @ (ic, q_in_bs, p, q) # ys"
SigmaD1: fixes a :: "'c"   and b :: "'d"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" assumes "(a, b) \<in> Sigma A B" shows "a \<in> A"
less_append_cases: fixes xs :: "nat list"   and ys :: "nat list"   and vs :: "nat list"   and ws :: "nat list" assumes "xs @ ys <\<^sub>v vs @ ws"   and "length xs = length vs" obtains "xs <\<^sub>v vs" and "ys \<le>\<^sub>v ws"   | "xs \<le>\<^sub>v vs" and "ys <\<^sub>v ws"
compilePrim_Prop1: fixes l :: "int"   and code :: "(int \<times> Instr) list"   and p :: "Prim"   and OUT :: "(int \<times> Instr) list \<times> int"   and code1 :: "(int \<times> Instr) list"   and l1 :: "int" assumes "(l, code, p, OUT) \<in> compilePrim"   and "OUT = (code1, l1)" shows "l < l1 \<and> (\<forall>ll<l. code1\<down>ll = code\<down>ll) \<and> (\<forall>ll\<ge>l. ll < l1 \<longrightarrow> (\<exists>ins. code1\<down>ll = Some ins))"
\<phi>_mapsto: fixes b :: "'d"   and a :: "'d" assumes "D.ide b"   and "D.ide a" shows "inD \<in> D.hom b a \<rightarrow> HomD.set (b, a)"
ni: shows "n = pickFT cl \<longrightarrow> i \<noteq> 0"
iso_is_arr: fixes f :: "'a" assumes "hom'.iso f" shows "hom'.arr f"
direct_decompD_1: fixes A :: "'a set"   and ss :: "'a set list"   and qs :: "'a list" assumes "direct_decomp A ss"   and "qs \<in> listset ss" shows "sum_list qs \<in> A"
vv_transitive: fixes v :: "'a" assumes "vector v" shows "transitive (v * v\<^sup>T)"
abs_eq: fixes xa :: "hmultiset zmultiset"   and x :: "hmultiset zmultiset" shows "ZHMSet xa + ZHMSet x = ZHMSet (xa + x)"
prog_unclocked: fixes env :: "v sem_env"   and s :: "'b state"   and prog :: "top0 list"   and s' :: "'b state"   and r :: "(v sem_env, v) result"   and cnt1 :: "nat"   and cnt2 :: "nat" shows "(evaluate_prog False env s prog (s', r) \<longrightarrow> r \<noteq> Rerr (Rabort Rtimeout_error) \<and> clock s = clock s') \<and> evaluate_prog False env (update_clock (\<lambda>_. cnt1) s) prog (update_clock (\<lambda>_. cnt1) s', r) = evaluate_prog False env (update_clock (\<lambda>_. cnt2) s) prog (update_clock (\<lambda>_. cnt2) s', r)"
Busemann_function_triangle: fixes xi :: "'b Gromov_completion"   and x :: "'b"   and z :: "'b"   and y :: "'b" shows "Busemann_function_at xi x z \<le> Busemann_function_at xi x y + Busemann_function_at xi y z"
assms: shows "completely_specified_rel_over rel (nodes (from_rel rel q0))"
boundedE: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> inf b c" obtains "a \<le> b" and "a \<le> c"
assms: fixes x :: "'x" assumes "x \<in> indets p" shows "poly_deg (f x) \<le> 1"
smult_l_minus: fixes x :: "'a Matrix.vec"   and a :: "'a" assumes "True"   and "x \<in> carrier_vec n" shows "- a \<cdot>\<^sub>v x = \<ominus>\<^bsub>NC.V\<^esub> (a \<cdot>\<^sub>v x)"
atoms_PLUS: fixes xs :: "'b rexp list" shows "atoms (PLUS xs) = \<Union> (atoms ` set xs)"
seq_char: fixes g :: "'a"   and f :: "'a" shows "hhom\<^sub>C.seq g f = (hhom\<^sub>C.arr f \<and> hhom\<^sub>C.arr g \<and> C.seq g f)"
PROJ_case: fixes i :: "nat"   and l :: "nat list" shows "PROJ i l < ack 0 (sum_list l)"
type_universe_nonempty: fixes t :: "pdf_type" shows "type_universe t \<noteq> {}"
pmf_le_0_iff: fixes M :: "'a pmf"   and p :: "'a" shows "(pmf M p \<le> 0) = (pmf M p = 0)"
chain_boundary_0: fixes p :: "nat" shows "chain_boundary p 0 = 0"
hb_completionD: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) list"   and ttas :: "('thread_id \<times> ('l, 'thread_id, 'x, 'm, 'w, ('addr, 'thread_id) obs_event action) thread_action) list"   and s' :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and t :: "'thread_id"   and x :: "'x"   and ta :: "('l, 'thread_id, 'x, 'm, 'w, ('addr, 'thread_id) obs_event action) thread_action"   and x' :: "'x"   and m' :: "'m"   and i :: "nat" assumes "hb_completion s E"   and "s -\<triangleright>ttas\<rightarrow>* s'"   and "non_speculative P (w_values P (\<lambda>_. {}) (map snd E)) (llist_of (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas)))"   and "thr s' t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "t \<turnstile> \<langle>x, shr s'\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "actions_ok s' t ta"   and "non_speculative P (w_values P (w_values P (\<lambda>_. {}) (map snd E)) (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))) (llist_of (take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>))" shows "\<exists>ta' x'' m''. t \<turnstile> \<langle>x, shr s'\<rangle> -ta'\<rightarrow> \<langle>x'', m''\<rangle> \<and> actions_ok s' t ta' \<and> take i \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> = take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> \<and> ta_hb_consistent P (E @ concat (map (\<lambda>(t, ta). map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas) @ map (Pair t) (take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) (llist_of (map (Pair t) (drop i \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub>))) \<and> (i < length \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> \<longrightarrow> i < length \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub>) \<and> (if \<exists>ad al v. \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> ! i = NormalAction (ReadMem ad al v) then sim_action else (=)) (\<lbrace>ta\<rbrace>\<^bsub>o\<^esub> ! i) (\<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> ! i)"
inj_map_strong: fixes x :: "'c multiset"   and xa :: "'c multiset"   and f :: "'c \<Rightarrow> 'd"   and fa :: "'c \<Rightarrow> 'd" assumes "\<And>z za. \<lbrakk>z \<in># x; za \<in># xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "image_mset f x = image_mset fa xa" shows "x = xa"
pow_cong: fixes m :: "int"   and x :: "int"   and n :: "nat" assumes "residues m" shows "(x mod m) [^]\<^bsub>residue_ring m\<^esub> n = x ^ n mod m"
Catch: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and R :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c\<^sub>1 Q,R"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Catch c\<^sub>1 c\<^sub>2 Q,A"
simps: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "('d, 'e, 'f) configuration"   and a2 :: "('d, 'e, 'f) event list"   and a3 :: "('d, 'e, 'f) configuration" assumes "distributed_system channel" shows "distributed_system.trace channel trans send recv a1 a2 a3 = ((\<exists>c. a1 = c \<and> a2 = [] \<and> a3 = c) \<or> (\<exists>c ev c' t c''. a1 = c \<and> a2 = ev # t \<and> a3 = c'' \<and> distributed_system.next channel trans send recv c ev c' \<and> distributed_system.trace channel trans send recv c' t c''))"
all_inclG_4: shows "K1 \<subseteq> carrier G"
map_eq_conv': fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and g :: "'c \<Rightarrow> 'a"   and ys :: "'c list" shows "(map f xs = map g ys) = (length xs = length ys \<and> (\<forall>i<length xs. f (xs ! i) = g (ys ! i)))"
next_nrec_pos: fixes s :: "('c, 'd) configuration stream" assumes "holds (\<lambda>c. \<forall>t. 0 \<le> zcount (c_records c) t) s"   and "next s" shows "nxt (holds (\<lambda>c. \<forall>t. 0 \<le> zcount (c_records c) t)) s"
bal_tab_def: shows "bal_tab = [0, 1, 1, 2, 4, 6, 10, 16, 25, 40, 64, 101, 161, 256, 406, 645, 1024, 1625, 2580, 4096, 6501, 10321, 16384, 26007, 41285, 65536, 104031, 165140, 262144, 416127, 660561, 1048576, 1664510, 2642245, 4194304, 6658042, 10568983, 16777216, 26632170, 42275935, 67108864, 106528681, 169103740, 268435456, 426114725, 676414963, 1073741824, 1704458900, 2705659852, 4294967296]"
Union_subsetI: fixes A :: "'b set set"   and B :: "'b set set" assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> B \<and> x \<subseteq> y" shows "\<Union> A \<subseteq> \<Union> B"
cf_similarI: fixes x :: "'a itrm"   and y :: "'a itrm" assumes "x \<in> CF"   and "y \<in> CF"   and "opaque x = opaque y"   and "CF_pure x \<leftrightarrow> CF_pure y" shows "x \<cong> y"
F1rel_cong: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and S1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and S2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and S3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "R1 = S1"   and "R2 = S2"   and "R3 = S3" shows "F1rel R1 R2 R3 = F1rel S1 S2 S3"
wf_subst_compose: fixes \<theta>1 :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<theta>2 :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>1"   and "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>2"   and "subst_domain \<theta>1 \<inter> subst_domain \<theta>2 = {}"   and "subst_domain \<theta>1 \<inter> range_vars \<theta>2 = {}" shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t (\<theta>1 \<circ>\<^sub>s \<theta>2)"
factorgroup_is_group: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "H \<lhd> G" shows "Group.group (G Mod H)"
LeadsTo_UN: fixes I :: "'c set"   and F :: "'d program"   and A :: "'c \<Rightarrow> 'd set"   and B :: "'d set" assumes "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i \<longmapsto>w B" shows "F \<in> \<Union> (A ` I) \<longmapsto>w B"
set_mkIde: fixes A :: "'t set" assumes "A \<subseteq> S'.Univ"   and "|A| <o \<AA>" shows "S'.set (S'.mkIde A) = A"
assms: fixes c :: "('a, 'b, 'c) cmd" assumes "c \<in> set cl" shows "finished c"
alw_sdrop: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream"   and n :: "nat" assumes "alw \<phi> xs" shows "alw \<phi> (sdrop n xs)"
unequal: fixes r :: "'a ref"   and r' :: "'a ref" shows "(r \<noteq> r') = r =!= r'"
lconjby_eq_conv_rconjby_eq: fixes w :: "'a"   and x :: "'a"   and y :: "'a" assumes "w = lconjby x y" shows "y = rconjby x w"
prodag_sameTr1: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and B :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)"   and "\<forall>k\<in>I. A k = B k" shows "carr_prodag I A = carr_prodag I B"
induct: fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'a"   and a2 :: "'a list" assumes "\<And>i a. P i a []"   and "\<And>a v va. P 0 a (v # va)"   and "\<And>i a b bs. P i a bs \<Longrightarrow> P (Suc i) a (b # bs)" shows "P a0 a1 a2"
is_empty_Mapping: fixes xs :: "('c \<times> 'd) list" shows "Mapping.is_empty (AList_Mapping.Mapping xs) = List.null xs"
cop_F_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and XH :: "'d set" assumes "Contracts Xd Xh Pd Ch" shows "Contracts.cop_F Pd Ch ds XH = XH \<union> Contracts.CD_on Pd ds (- Rf (Contracts.CH Ch) XH)"
rep_single: fixes m :: "nat" shows "Rep_nat_int (Abs_nat_int {m..m}) = {m}"
atom_map_def_var: fixes x :: "'a" shows "atom_map x = \<down> x \<inter> Collect atom"
substAbs_compose2: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Z :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var" assumes "goodAbs A"   and "good Y"   and "good Z"   and "ys \<noteq> zs \<or> y \<noteq> z"   and "fresh ys y Z" shows "A $[Y / y]_ys $[Z / z]_zs = A $[Z / z]_zs $[Y #[Z / z]_zs / y]_ys"
compose_inv_into_id: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "compose A (restrict (inv_into A f) B) f = identity A"
fun_type_id_eq: fixes f :: "'fun"   and X :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and g :: "'fun"   and Y :: "('fun, 'atom) Term.term list" assumes "\<Gamma> (Fun f X) = Fun g Y" shows "f = g"
not_open_singleton: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a" assumes "class.perfect_space open" shows "\<not> open {x}"
restrict_view: fixes v :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "restrict v f c \<le> lan v"
throw_of_spec: fixes e :: "'a exp"   and e' :: "'a exp" assumes "throw_of e = \<lfloor>e'\<rfloor>" shows "e = throw e'"
ACI_norm_PLUS: fixes xs :: "'c rexp list"   and ys :: "'c rexp list" assumes "list_all2 (\<lambda>r s. \<guillemotleft>r\<guillemotright> = \<guillemotleft>s\<guillemotright>) xs ys" shows "\<guillemotleft>PLUS xs\<guillemotright> = \<guillemotleft>PLUS ys\<guillemotright>"
old_set_rel_sv_eq: fixes R :: "('a \<times> 'b) set" assumes "single_valued R" shows "\<langle>R\<rangle>old_set_rel = \<langle>R\<rangle>set_rel"
opp_fold_A_morph_A_A: shows "ChamberComplexMorphism A A opp_fold_A"
IH: fixes x :: "nat" shows "(\<Sum>i<n. x div base ^ i mod base * base ^ i) = x mod base ^ n"
nn_integral_cong: fixes M :: "'h measure"   and u :: "'h \<Rightarrow> ennreal"   and v :: "'h \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> u x = v x" shows "integral\<^sup>N M u = integral\<^sup>N M v"
EX_State_SA: fixes SA :: "('d, 'e, 'f) seqauto" shows "\<exists>S. S \<in> States SA"
transpos_inj: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "inj_on (\<tau>\<^bsub>i j\<^esub>) {i. i \<le> n}"
add_arc_simps_3: fixes G :: "('d, 'c) pre_digraph"   and a :: "'c" shows "tail (pre_digraph.add_arc G a) = tail G"
0: assumes "Arr t" shows "Nml \<^bold>\<lfloor>t\<^bold>\<rfloor> \<and> Src \<^bold>\<lfloor>t\<^bold>\<rfloor> = Src t \<and> Trg \<^bold>\<lfloor>t\<^bold>\<rfloor> = Trg t \<and> Dom \<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>Dom t\<^bold>\<rfloor> \<and> Cod \<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>Cod t\<^bold>\<rfloor>"
surjective: fixes r :: "'a State_scheme" shows "r = \<lparr>getF = getF r, getM = getM r, getConflictFlag = getConflictFlag r, getC = getC r, \<dots> = more r\<rparr>"
Pred_eqvt: fixes p :: "perm"   and f :: "'d"   and \<phi> :: "'b" shows "p \<bullet> Pred f \<phi> = Pred (p \<bullet> f) (p \<bullet> \<phi>)"
list2FWpolicys_eq_el: fixes Filter :: "((int \<times> int) set set, int) Combinators list"   and p :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "Filter \<noteq> []" shows "C (list2policyR Filter) p = C (list2FWpolicy (rev Filter)) p"
mult_left_sub_dist_sup_right: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "x * z \<le> x * (y \<squnion> z)"
is_basis_imp_full_rank: fixes A :: "(('b, 'd) vec, 'c) vec" assumes "is_basis (columns A)"   and "card (columns A) = ncols A" shows "rank A = ncols A"
subpath0_linepath: fixes u :: "real"   and t :: "real"   and t' :: "real" shows "subpath 0 u (linepath t t') = linepath t (t + u * (t' - t))"
stabilizer_supergrp_P: fixes P :: "'a set"   and p :: "nat" assumes "finite (carrier (G\<lparr>carrier := P__\<rparr>))"   and "P \<in> groupP.subgroups_of_size p" shows "subgroup P (G\<lparr>carrier := P__, carrier := group_action.stabilizer (G\<lparr>carrier := P__\<rparr>) (groupP.conjugation_action p) P\<rparr>)"
keys_rep_subset: fixes u :: "'a"   and r :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" assumes "u \<in> keys (local.punit.pmdl.rep r)" obtains t :: "'a" and v :: "'a" where "t \<in> Keys (Poly_Mapping.range r)" and "v \<in> Keys (keys r)" and "u = local.punit.splus t v"
card_info_def: fixes x :: "'d set" shows "card_info x = SPEC top"
fps_compose_sum_distrib: fixes f :: "'b \<Rightarrow> 'a fps"   and S :: "'b set"   and a :: "'a fps" shows "sum f S oo a = (\<Sum>i\<in>S. f i oo a)"
equals_dom_if_value_is_ide: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b" assumes "constant_transformation A B g"   and "partial_magma.ide B g" shows "constant_transformation.map A B g = constant_functor.map A B (partial_magma.dom B g)"
RG_el_decomp_aezdeltafun: fixes G :: "'a set"   and c :: "('b, 'a) aezfun" assumes "Group G"   and "c \<in> aezfun_setspan G" shows "\<exists>ras. set (map snd ras) \<subseteq> G \<and> c = (\<Sum>(r, a)\<leftarrow>ras. r \<delta>\<delta> a)"
ad_ae_q: shows "{a - complex_of_real d, a + complex_of_real e} \<subseteq> path_image q"
decrtm: fixes bs :: "'b list"   and t :: "tm"   and m :: "nat"   and vs :: "'b list" assumes "tmboundslt (length bs) t"   and "tmbound m t"   and "m \<le> length bs" shows "Itm vs (removen m bs) (decrtm m t) = Itm vs bs t"
soundness: fixes p :: "'a fm"   and M :: "('a, 'b) kripke"   and s :: "'b" assumes "\<turnstile> p" shows "M, s \<Turnstile> p"
lm063: fixes g :: "'a \<Rightarrow> 'b"   and X :: "'a set"   and f :: "'b \<Rightarrow> 'c" assumes "inj_on g X" shows "sum f (g ` X) = sum (f \<circ> g) X"
new_update: fixes s :: "Store"   and l :: "Location"   and x :: "Value"   and t :: "New" shows "new (s\<langle>l := x\<rangle>) t = new s t"
lookup_map_default_neq: fixes x :: "'b"   and y :: "'b"   and d :: "'c"   and f :: "'c \<Rightarrow> 'c"   and m :: "('b, 'c) mapping" assumes "x \<noteq> y" shows "lookup (map_default x d f m) y = lookup m y"
full_SetCompr_eq: fixes f :: "'e \<Rightarrow> 'd" shows "{u. \<exists>x. u = f x} = range f"
assms_4: shows "A B C D' E' F' SumA G H I"
Gromov_converging_at_boundary_bounded_perturbation: fixes u :: "nat \<Rightarrow> 'c"   and x :: "'c Gromov_completion"   and v :: "nat \<Rightarrow> 'c"   and C :: "real" assumes "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x"   and "x \<in> Gromov_boundary"   and "\<And>n. dist (u n) (v n) \<le> C" shows "(\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> x"
wt_sym_ge\<^sub>h: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and f :: "'b" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "of_nat \<epsilon> - of_nat \<delta> * hmset_of_enat (arity_sym f) \<le> wt_sym f"
lagrange_finite: fixes H :: "'a set" assumes "finite (carrier H')"   and "subgroup H H'" shows "card (rcosets\<^bsub>H'\<^esub> H) * card H = order H'"
invh_baliL: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree"   and a :: "'c" assumes "invh l"   and "invh r"   and "bheight l = bheight r" shows "invh (baliL l a r)"
fds_const_one: shows "fds_const (1::'a) = 1"
minus_unique: fixes y :: "'c"   and x :: "'c"   and y' :: "'c" assumes "y \<oplus>\<^bsub>H\<^esub> x = \<zero>\<^bsub>H\<^esub>"   and "x \<oplus>\<^bsub>H\<^esub> y' = \<zero>\<^bsub>H\<^esub>"   and "x \<in> carrier H"   and "y \<in> carrier H"   and "y' \<in> carrier H" shows "y = y'"
diff_add_cancel: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.group_add minus plus zero uminus" shows "plus (minus a b) b = a"
current_leI: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "\<And>e. g e \<le> f e"   and "\<And>x. x \<notin> A \<Gamma> \<Longrightarrow> d_OUT g x \<le> d_IN g x" shows "current \<Gamma> g"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "cycle (x # xs) = (x \<notin> set xs \<and> cycle xs)"
dist_triangle3: fixes x :: "'b"   and y :: "'b"   and a :: "'b" shows "dist x y \<le> dist a x + dist a y"
neutr_eq_iff: fixes a :: "'a"   and b :: "'a" shows "(1 = a \<cdot> b) = (a = 1 \<and> b = 1)"
RT: fixes x :: "'M"   and y :: "'M" assumes "x \<in> S"   and "norm (y - x) < R x" shows "y \<in> T"
PO_m1_ifresh: shows "reach m1 \<subseteq> m1_inv1_ifresh"
bot_FD: fixes Q :: "'a process" shows "\<bottom> \<sqsubseteq>\<^sub>F\<^sub>D Q"
dehomogenize_zero: fixes x :: "'c" shows "dehomogenize x 0 = 0"
override_on_apply_in: fixes a :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "a \<in> A" shows "(f ++\<^bsub>A\<^esub> g) a = g a"
sel_2: fixes x1 :: "'a"   and x2 :: "'a rose_tree list" shows "children (RNode x1 x2) = x2"
foldD_Un_Int: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B"   and "A \<subseteq> D"   and "B \<subseteq> D" shows "foldD D (\<cdot>) e A \<cdot> foldD D (\<cdot>) e B = foldD D (\<cdot>) e (A \<union> B) \<cdot> foldD D (\<cdot>) e (A \<inter> B)"
length_Tensor: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and M1 :: "'a list list"   and M2 :: "'a list list" assumes "Matrix_Tensor.mult id f" shows "length (mult.Tensor f M1 M2) = length M1 * length M2"
greaterThan_borel: fixes a :: "'c" shows "{a<..} \<in> sets borel"
that: fixes X1 :: "'a multiset"   and X2 :: "'a multiset" assumes "X = X1 + X2"   and "(X1, Y1) \<in> multpw ns"   and "(X2, Y2) \<in> multpw ns" shows "thesis"
finite_Image_subset: fixes A :: "('b \<times> 'a) set"   and B :: "'b set"   and C :: "('b \<times> 'a) set" assumes "finite (A `` B)"   and "C \<subseteq> A" shows "finite (C `` B)"
all_security_requirements_fulfilled_def: fixes M :: "'a SecurityInvariant_configured list"   and G :: "'a graph" shows "TopoS_Composition_Theory.all_security_requirements_fulfilled M G \<equiv> \<forall>m\<in>set M. c_sinvar m G"