hcju/isabelleps huggingface dataset:
pprefix_closed_def: fixes A :: "'b llist set" shows "pprefix_closed A \<equiv> \<forall>t\<in>A. \<forall>s. s \<le> t \<and> s \<noteq> LNil \<longrightarrow> s \<in> A"
S2: shows "log2 1 = (LEAST z. 1 < 2 ^ (z + 1))"
prems: shows "range_vars \<theta> \<subseteq> fv (Var v) \<union> V"
Cons_3: shows "set (a # u) \<inter> set v = {}"
\<tau>: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) G H \<tau>"
supp_PVar: fixes x :: "name"   and T :: "ty" shows "supp (PVar x T) = supp x"
integ_exis: shows "\<forall>(k, \<gamma>)\<in>two_chain_boundary two_chain. line_integral_exists F {j} \<gamma>"
eval_simps'_3: fixes f :: "'a local.term" assumes "Arr f" shows "trg \<lbrace>f\<rbrace> = \<lbrace>Trg f\<rbrace>"
xor_left_comm: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<oplus> y \<oplus> z = y \<oplus> x \<oplus> z"
SKIP_def: shows "\<bottom> = mk_program (UNIV, {}, UNIV)"
propSepViews: shows "properSeparationOfViews ES1 ES2 \<V> \<V>1 \<V>2"
monoid_add_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "monoid_add_hom hom \<equiv> zero_hom hom \<and> semigroup_add_hom hom"
assms_2: shows "v\<^sup>\<star> * e\<^sup>T = e\<^sup>T"
f13: shows "(s'' \<otimes> s' \<otimes> (r'' \<otimes> r) \<oplus> s'' \<otimes> s \<otimes> (r'' \<otimes> r'), s'' \<otimes> s \<otimes> (s'' \<otimes> s')) \<in> carrier rel"
case_prodE': fixes c :: "'b \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'b \<times> 'd"   and z :: "'e" assumes "(case p of (a, b) \<Rightarrow> c a b) z" obtains x :: "'b" and y :: "'d" where "p = (x, y)" and "c x y z"
neighborhoods_def: fixes x :: "'c" shows "target.neighborhoods x \<equiv> {U. is_open\<^sub>Y U \<and> x \<in> U}"
wf_rderivs: fixes n :: "nat"   and r :: "'b rexp"   and ws :: "'a list" assumes "local.wf n r" shows "local.wf n (rderivs ws r)"
eucl_le_atMost: fixes a :: "'c" shows "{x. \<forall>i\<in>Basis. x \<bullet> i \<le> a \<bullet> i} = {..a}"
normalize_length_lt: fixes p :: "'a list" assumes "lead_coeff p = \<zero>"   and "0 < length p" shows "length (local.normalize p) < length p"
mtx_vec_mult_add_rdistl: fixes A :: "'b sq_mtx"   and x :: "(real, 'b) vec"   and y :: "(real, 'b) vec" shows "A *\<^sub>V (x + y) = A *\<^sub>V x + A *\<^sub>V y"
span_extend_basis: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'b set" assumes "vector_space scale"   and "\<not> module.dependent scale B" shows "module.span scale (vector_space.extend_basis scale B) = UNIV"
mem_rel_interior: fixes x :: "'a"   and S :: "'a set" shows "(x \<in> rel_interior S) = (\<exists>T. open T \<and> x \<in> T \<inter> S \<and> T \<inter> affine hull S \<subseteq> S)"
facetrel_card: fixes y :: "'c set"   and x :: "'c set" assumes "y \<lhd> x" shows "card (x - y) = 1"
s: shows "s \<sqsubseteq> t \<cdot> \<delta>"
le_alg_mult: shows "n \<le> alg_mult p x"
order_trans_rules_25: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b"   and "b = c" shows "a \<le> c"
Rep_scene_inverse: fixes x :: "'a scene" shows "Abs_scene (Rep_scene x) = x"
has_integral_neg_iff: fixes f :: "'c \<Rightarrow> 'd"   and k :: "'d"   and S :: "'c set" shows "((\<lambda>x. - f x) has_integral k) S = (f has_integral - k) S"
generalizes_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and C :: "'b literal multiset"   and D :: "'b literal multiset" shows "substitution_ops.generalizes subst_atm C D = (\<exists>\<sigma>. substitution_ops.subst_cls subst_atm C \<sigma> = D)"
parts_msg_mono: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> H" shows "parts_msg X \<subseteq> parts H"
induced_automorphism_unique_automorphism: fixes s :: "'a \<Rightarrow> 'a" assumes "ChamberComplexAutomorphism X s"   and "s \<noteq> id"   and "fixespointwise s (C0 \<inter> D0)" shows "s = \
"
card1_eE: fixes S :: "'b set" assumes "finite S"   and "\<exists>y. y \<in> S" shows "1 \<le> card S"
assms_1: shows "a<x>.P \<longmapsto> b\<guillemotleft>y\<guillemotright> \<prec> P'"
intros_10: fixes \<tau> :: "'a OCL_Types.type" shows "Set \<tau> \<sqsubset> Collection \<tau>"
assms: shows "((\<sigma>, NodeS i s R), ii\<not>ni:arrive(m), \<sigma>', NodeS i' s' R') \<in> onode_sos S"
red_diff: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and r :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.red F (p - q) r" obtains p' :: "'a \<Rightarrow>\<^sub>0 'c" and q' :: "'a \<Rightarrow>\<^sub>0 'c" where "(punit.red F)\<^sup>*\<^sup>* p p'" and "(punit.red F)\<^sup>*\<^sup>* q q'" and "r = p' - q'"
ars2: fixes y :: "'a"   and x :: "'a" shows "ad (y \<cdot> x) + ad (y \<cdot> ad (ad x)) = ad (y \<cdot> ad (ad x))"
WCFG_WhileTrue: fixes b :: "expr"   and c' :: "cmd" shows "while (b) c' \<turnstile> (_ 0 _) -(\<lambda>s. interpret b s = Some true)\<^sub>\<surd>\<rightarrow> (_ 0 _) \<oplus> 2"
aux_scc: fixes x :: "'v" assumes "x \<noteq> u"   and "x \<in> set (takeWhile ((\<noteq>) u) (tj_stack s))"   and "u \<in> set (tj_stack s)" shows "\<exists>y. finished s x = Some y"
that: fixes S :: "'node set" assumes "S \<in> slices p"   and "S \<subseteq> Q"   and "\<And>p'. p' \<in> S \<inter> W \<Longrightarrow> quorum_of p' Q" shows "thesis"
stut_angle_action: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" assumes "NSTUTINV P" shows "STUTINV \<diamond>\<langle>P\<rangle>_v"
sums_fds_perzeta: fixes s :: "complex"   and q :: "real" assumes "1 < Re s" shows "(\<lambda>m. exp (complex_of_real (2 * pi) * \<i> * of_nat (Suc m) * complex_of_real q) / of_nat (Suc m) powr s) sums eval_fds (fds_perzeta q) s"
rep_eq: fixes x :: "'a \<Rightarrow> 'b" shows "pp.mapping_of (pp_of_fun x) = Abs_poly_mapping x"
subst_fml_hp: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<alpha> :: "('sf, 'sc, 'sz) hp"   and \<phi> :: "('sf, 'sc, 'sz) formula" assumes "is_interp I" shows "(Padmit \<sigma> \<alpha> \<longrightarrow> hpsafe \<alpha> \<longrightarrow> ssafe \<sigma> \<longrightarrow> (\<forall>\<nu> \<omega>. ((\<nu>, \<omega>) \<in> prog_sem I (Psubst \<alpha> \<sigma>)) = ((\<nu>, \<omega>) \<in> prog_sem (local.adjoint I \<sigma> \<nu>) \<alpha>))) \<and> (Fadmit \<sigma> \<phi> \<longrightarrow> fsafe \<phi> \<longrightarrow> ssafe \<sigma> \<longrightarrow> (\<forall>\<nu>. (\<nu> \<in> fml_sem I (Fsubst \<phi> \<sigma>)) = (\<nu> \<in> fml_sem (local.adjoint I \<sigma> \<nu>) \<phi>)))"
cast_chunk_scast_assemble_id: fixes n :: "nat"   and m :: "nat"   and x :: "'b word" assumes "n = LENGTH('a)"   and "m = LENGTH('b)"   and "n * 2 = m" shows "UCAST('a \<rightarrow> 'b) (SCAST('b \<rightarrow> 'a) x) || (UCAST('a \<rightarrow> 'b) (SCAST('b \<rightarrow> 'a) (x >> n)) << n) = x"
matrix_to_iarray_matrix_matrix_mult: fixes A :: "(('b, 'e) vec, 'd) vec"   and B :: "(('b, 'c) vec, 'e) vec" shows "matrix_to_iarray (A ** B) = matrix_to_iarray A **i matrix_to_iarray B"
Int_Diff: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and B :: "'a set" assumes "finite A" shows "sum g A = sum g (A \<inter> B) + sum g (A - B)"
assms_2: shows "y1 = y2 \<or> fresh ys y1 X"
map_indets_one: fixes f :: "'e \<Rightarrow> 'c" shows "map_indets f 1 = 1"
stern_brocot_recurse_iterate: shows "stern_brocot_recurse = stern_brocot_iterate"
geral_U_is_expected_value_of_ocU: fixes p :: "'a pmf" assumes "set_pmf p \<subseteq> outcomes" shows "local.general_U p = measure_pmf.expectation p local.ocU"
T_cost: shows "T\<^sub>p_on_rand' (Partial_Cost_Model.embed (rTS [])) Db qs = ts"
matrix_rank_le_cp_rank: fixes I :: "nat set"   and A :: "'a tensor" shows "mrank (matricize I A) \<le> cprank A"
sets_fx: fixes x :: "'a" assumes "x \<in> space M" shows "sets (f x) = sets N"
hyps: shows "local.eval (local.monom b n) a = b \<otimes> a [^] n"
switch_vchop1: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and c :: "cars"   and v' :: "view" shows "v=v1--v2 \<and> v = c > v' \<longrightarrow> (\<exists>v1' v2'. v1 = c > v1' \<and> v2 = c > v2' \<and> v'=v1'--v2')"
simps_9: shows "get_axiom Adassign = diff_assign_axiom"
idinfo_hash_hash: fixes n :: "nat"   and X :: "msg"   and s :: "(agent \<times> msg) set" assumes "(Spy, \<langle>n, Hash (Hash X)\<rangle>) \<in> s"   and "s\<^sub>0 \<Turnstile> s" shows "\<langle>n, Hash X\<rangle> \<in> spied s"
vpa_valid_call_list_valid_return_list_vpra: 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 cs :: "'a list"   and cs' :: "'a list"   and n :: "'b"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.valid_path_aux kind get_return_edges cs cs'"   and "CFG.valid_call_list sourcenode kind valid_edge get_proc cs n"   and "CFG.valid_return_list targetnode kind valid_edge get_proc cs' n'" shows "CFG.valid_path_rev_aux kind get_return_edges cs' (rev cs)"
mk_db_def: fixes t :: "(String.literal \<times> event_data list set) list" shows "Monitor_Impl.mk_db t = Monitor.mk_db (\<Union>n\<in>set (map fst t). Pair n ` the (map_of t n))"
hyps: fixes y :: "hmultiset"   and e :: "nat"   and N :: "hmultiset" assumes "y < M"   and "local.well_base\<^sub>h y"   and "local.aligned\<^sub>d e y"   and "local.aligned\<^sub>d e N"   and "y < N" shows "local.decode e y < local.decode e N"
redT_updTs'_eq_None_conv: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and ts' :: "'f \<Rightarrow> ('i \<times> 'j \<Rightarrow>f nat) option"   and tas :: "('f, 'i, 'k) new_thread_action list"   and t :: "'f" assumes "\<And>t. (ts t = None) = (ts' t = None)" shows "(redT_updTs' ts tas t = None) = (redT_updTs ts' tas t = None)"
length_update_matchF_step: fixes I :: "\<I>"   and mr :: "mregex"   and mrs :: "mregex list"   and nt :: "nat"   and entry :: "nat \<times> 'a option list set list \<times> 'a option list set"   and st :: "(nat \<times> 'a option list set list \<times> 'a option list set) list \<times> (mregex, 'a option list set) mapping" shows "length (fst (update_matchF_step I mr mrs nt entry st)) = Suc (length (fst st))"
length_conv_length_it: fixes xs :: "'a list" shows "length xs = length_it xs"
maskshift_eq_not_mask_generic: shows "mask len << LENGTH('a) - len = ~~ mask (LENGTH('a) - len)"
type_definition_IF1: shows "type_definition Rep_IF1 Abs_IF1 (car_init1 undefined)"
preserves_reflects_arr: fixes f :: "'a \<times> 'a" shows "B.hseq (fst f) (snd f) = B.VV.arr f"
elliptic_isometry_def: fixes f :: "'b \<Rightarrow> 'b" shows "elliptic_isometry f = (isometry f \<and> (\<forall>x. bounded {(f ^^ n) x |n. True}))"
finite_braun_indices: fixes t :: "'a tree" shows "finite (braun_indices t)"
spans_image: fixes V :: "'b set"   and B :: "'b set" assumes "V \<subseteq> vs1.span B" shows "f ` V \<subseteq> vs2.span (f ` B)"
that: fixes xs' :: "'b list"   and xs'' :: "'b list" assumes "map f xs' = ys"   and "map f xs'' = zs"   and "xs = xs' @ xs''" shows "thesis"
Units_minus_one_closed: shows "\<ominus>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub> \<in> Units R"
A11_DAcc_level1: shows "DAcc level1 sA11 = {sA21, sA22, sA23}"
msetext_dersh_map_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt (f y) (f x)"   and "msetext_dersh gt ys xs" shows "msetext_dersh gt (map f ys) (map f xs)"
Cs'1: shows "Cs' = Cs @ [D, D'] @ rev (C'' # Cs'')"
rem_initial_equal: fixes b :: "bit"   and w :: "bit list" assumes "length (rem_initial b w) = length w" shows "rem_initial b w = w"
zhmset_of_sum_list: fixes Ms :: "hmultiset list" shows "zhmset_of (sum_list Ms) = sum_list (map zhmset_of Ms)"
fg: shows "f = g \<or> (\<exists>a b d. (a, d) \<in> c \<and> (b, d) \<in> c \<and> f = Abs a \<and> g = Abs b)"
card_length: fixes xs :: "'b list" shows "card (set xs) \<le> length xs"
split_relation_def: fixes xs :: "'b list" shows "split_relation xs \<equiv> \<lambda>(as, bs) i. i \<le> length xs \<and> as = take i xs \<and> bs = drop i xs"
Min_snd_hd: fixes q :: "('c \<times> 'd) list" assumes "q \<noteq> []"   and "sorted (map snd q)" shows "Min (snd ` set q) = snd (hd q)"
recognises_must_be_step: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and h :: "String.literal \<times> value list"   and ts :: "(String.literal \<times> value list) list" assumes "recognises_execution e s r (h # ts)" shows "\<exists>t s' p d'. step e s r (fst h) (snd h) = Some (t, s', p, d')"
invariant_starE: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and I :: "'a \<Rightarrow> bool" assumes "rel\<^sup>*\<^sup>* x y"   and "invariant rel I"   and "I x" shows "I y"
Rep_filter_iff_eventually: fixes F :: "'b filter"   and P :: "'b \<Rightarrow> bool" shows "Rep_filter F P = eventually P F"
inv_rule: fixes T :: "('a, 'b) TS_scheme"   and I :: "'a set"   and P :: "'a set" assumes "init T \<subseteq> I"   and "I \<subseteq> P"   and "{I} TS.trans T {> I}" shows "reach T \<subseteq> P"
strategy_attracts_via_lset: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and A :: "'c set"   and W :: "'c set" assumes "vmc_path G P v0 p \<sigma>"   and "ParityGame.strategy_attracts_via G p \<sigma> v0 A W" shows "lset P \<inter> W \<noteq> {}"
eventually_False: fixes F :: "'a filter" shows "(\<forall>\<^sub>F x in F. False) = (F = bot)"
ereal_ineq_diff_add: fixes b :: "ereal"   and a :: "ereal" assumes "b \<noteq> - \<infinity>"   and "b \<le> a" shows "a = b + (a - b)"
l: shows "\<forall>e>0. \<forall>\<^sub>F n in sequentially. \<forall>i\<in>Basis. dist (blinfun_apply (f (r n)) i) (blinfun_apply l i) < e"
Max_ge: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "x \<in> A" shows "less_eq x (linorder.Max less_eq A)"
div_times_less_eq_dividend: fixes m :: "nat"   and n :: "nat" shows "m div n * n \<le> m"
\<omega>_of_Cantor_Cons: fixes \<alpha> :: "'a"   and \<alpha>s :: "'a list"   and m :: "nat"   and ms :: "nat list" shows "\<omega>_of_Cantor (\<alpha> # \<alpha>s) (m # ms) = replicate m \<alpha> @ \<omega>_of_Cantor \<alpha>s ms"
iMOD_not_empty: fixes r :: "nat"   and m :: "nat" shows "[ r, mod m ] \<noteq> {}"
not_malformed: fixes x :: "char list"   and e :: "'b environment" assumes "x \<in> env_dom e" shows "\<exists>fun. e = Env fun"
\<psi>_def: fixes ba :: "'c \<times> 'c" shows "\<psi> ba = inv_into (C.hom (fst ba) (snd ba)) (\<phi> ba)"
EndPhase0_HInv3_q: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and p :: "Proc"   and d :: "Disk" assumes "EndPhase0 s s' q"   and "HInv3_L s' p q d" shows "HInv3_R s' p q d"
\<C>_is_code: fixes \<C> :: "'b list set"   and xs :: "'b list list"   and ys :: "'b list list" assumes "code \<C>"   and "xs \<in> lists \<C>"   and "ys \<in> lists \<C>"   and "concat xs = concat ys" shows "xs = ys"
monotonic: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and A :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and B :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> P \<leadsto>[A] Q"   and "A \<subseteq> B" shows "\<Psi> \<rhd> P \<leadsto>[B] Q"
component_of_term_of_pair: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and t :: "'d"   and k :: "'e" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.component_of_term pair_of_term (term_of_pair (t, k)) = k"
elims: fixes x :: "opinl"   and y :: "dynamic list \<Rightarrow> dynamic" assumes "eval_inl x = y" obtains "x = OpAddNum" and "y = eval_AddNum"   | "x = OpMulNum" and "y = eval_MulNum"
arr_def: fixes f :: "'s" shows "arr f \<equiv> domains f \<noteq> {} \<or> codomains f \<noteq> {}"
assms: shows "run_one_step d i (s, vs, ves, Local x51 x52 x53 x54) = (s', vs', res)"
updates_1: fixes ii :: "nat"   and c :: "int" assumes "ii \<le> m" shows "d (fs[i := fs ! i - c \<cdot>\<^sub>v fs ! j]) ii = d fs ii"
Quotient_part_equivp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "Quotient R Abs Rep T" shows "part_equivp R"
ik_dy_def: fixes m :: "msg" shows "ik_dy m \<equiv> {(s, s'). m \<in> synth (analz (ik s)) \<and> s' = s\<lparr>ik := ik s \<union> {m}\<rparr>}"
closed_Un: fixes S :: "'a set"   and T :: "'a set" assumes "closed S"   and "closed T" shows "closed (S \<union> T)"
project_set_clw_def: fixes X :: "'c set"   and b :: "'c"   and y :: "real" shows "project_set_clw X b y = sets_of_coll (X ::: clw_rel appr_rel) \<bind> (\<lambda>XS. FORWEAK XS (RETURN op_empty_coll) (\<lambda>X. op_project_set X b y \<bind> (\<lambda>P. RETURN (mk_coll P))) (\<lambda>X Y. RETURN (Y \<union> X)))"
pst_getmin_ismin: fixes t :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree" assumes "invpst t"   and "t \<noteq> \<langle>\<rangle>" shows "is_min2 (pst_getmin t) (Tree2.set_tree t)"
left_spec_def: fixes xs :: "int list"   and j :: "nat" shows "left_spec xs j = (if \<exists>i<j. xs ! i < xs ! j then Some (GREATEST i. i < j \<and> xs ! i < xs ! j) else None)"
2: shows "list_of w \<le>\<^sub>F\<^sub>I stream_of_llist (llist_of_stream u)"
stable_on_CD_on: fixes ds :: "'b set"   and X :: "'a set" assumes "TruePref_tax.stable_on ds X" shows "TruePref_tax.CD_on ds X = X"
f_inv: fixes x :: "'a" assumes "x \<in> src.carrier" shows "f' (f x) = x"
ThinChamberComplex_axioms: shows "ThinChamberComplex X"
simps_2: fixes T :: "typ"   and body :: "Core.term"   and lev :: "nat"   and u :: "Core.term" shows "subst_bv1 (Abs T body) lev u = Abs T (subst_bv1 body (lev + 1) u)"
mat_of_row_dim_1: fixes y :: "'b Matrix.vec" shows "dim_row (mat_of_row y) = 1"
intros_1: fixes ys :: "'a list" shows "interleavings [] ys ys"
partition_cong: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> bool"   and g :: "'a \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x = g x" shows "partition f xs = partition g ys"
Abs_Msg: shows "msg = Abs_Msg (msgrel `` {U})"
rs_2: fixes j :: "nat" assumes "j < n" shows "q j \<le> s j \<and> s j \<le> q j + 1"
aff: fixes C :: "'a set" assumes "C \<in> \<M>" shows "aff_dim C \<le> d"
completion_def: fixes Cn :: "'b set \<Rightarrow> 'b set"   and \<gamma> :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set set"   and A :: "'b set"   and \<phi> :: "'b" assumes "Tarskian_logic Cn" shows "Tarskian_logic.completion Cn \<gamma> A \<phi> \<equiv> if Tarskian_logic.remainder Cn A \<phi> = {} then {A} else {B \<in> Tarskian_logic.remainder Cn A \<phi>. \<Inter> (\<gamma> A \<phi>) \<subseteq> B}"
facts_6: fixes n :: "nat" assumes "2 * card (uverts H) \<le> n" shows "0 < 1 / real (card (uverts H) ^ card (uverts H)) * (real n ^ card (uverts H) * p n ^ card (uedges H))"
split: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'g option \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> ('i \<Rightarrow> ('g, 'h, 'i) Resumption.resumption) \<Rightarrow> 'f"   and resumption :: "('g, 'h, 'i) Resumption.resumption" shows "P (case resumption of Resumption.resumption.Done x \<Rightarrow> f1 x | Resumption.resumption.Pause x xa \<Rightarrow> f2 x xa) = ((\<forall>x1. resumption = Resumption.resumption.Done x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. resumption = Resumption.resumption.Pause x21 x22 \<longrightarrow> P (f2 x21 x22)))"
set_borel_measurable_def: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "set_borel_measurable M A f \<equiv> (\<lambda>x. indicat_real A x *\<^sub>R f x) \<in> borel_measurable M"
not_prime_elem_one: shows "\<not> prime_elem (1::'a)"
sup_distl: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "times_prod' x (sup_prod y z) = sup_prod (times_prod' x y) (times_prod' x z)"
extended_ord_def: fixes ord :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and s :: "'c option \<Rightarrow>\<^sub>0 nat"   and t :: "'c option \<Rightarrow>\<^sub>0 nat" assumes "extended_ord_pm_powerprod ord ord_strict" shows "extended_ord_pm_powerprod.extended_ord ord_strict s t = (ord_strict (restrict_indets_pp s) (restrict_indets_pp t) \<or> restrict_indets_pp s = restrict_indets_pp t \<and> lookup s None \<le> lookup t None)"
cases: fixes a :: "'b llist" assumes "lfinite a" obtains "a = LNil"   | xs :: "'b llist" and x :: "'b" where "a = LCons x xs" and "lfinite xs"
in_image_mset: fixes y :: "'b"   and f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset" shows "(y \<in># image_mset f M) = (y \<in> f ` set_mset M)"
mirror_elem_eq_conv: fixes a :: "nat"   and I :: "nat set"   and b :: "nat" assumes "a \<le> iMin I + Max I"   and "b \<le> iMin I + Max I" shows "(mirror_elem a I = mirror_elem b I) = (a = b)"
gdpr_scenario_def: shows "gdpr_scenario \<equiv> Infrastructure ex_graph local_policies"
amult_commute: fixes x :: "ant"   and y :: "ant" shows "x * y = y * x"
param_opt_def: fixes i :: "'a ifex"   and t :: "'a ifex"   and e :: "'a ifex" shows "param_opt i t e = (if i = Trueif then Some t else if i = Falseif then Some e else if t = Trueif \<and> e = Falseif then Some i else if t = e then Some t else if e = Trueif \<and> i = t then Some Trueif else if t = Falseif \<and> i = e then Some Falseif else None)"
simps_2: fixes t2 :: "'c Tree" shows "merge E t2 = t2"
FactRing_def: fixes R :: "('d, 'e) ring_scheme"   and I :: "'d set" shows "R Quot I = \<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, monoid.mult = rcoset_mult R I, one = I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>, zero = I, add = (<+>\<^bsub>R\<^esub>)\<rparr>"
red_single_nonzero2: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and f :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "punit.red_single p q f t" shows "f \<noteq> 0"
add_implies_diff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and c :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.cancel_comm_monoid_add plus minus zero"   and "plus c b = a" shows "c = minus a b"
field_less_half_sum: fixes x :: "'b"   and y :: "'b" assumes "x < y" shows "x < (x + y) / (2::'b)"
canonical_retraction_simplex_im: fixes A :: "'a set set"   and C :: "'a set" assumes "A \<in> \<A>"   and "chamber C"   and "C \<in> A" shows "canonical_retraction A C \<turnstile> X = A"
edom_mono: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "x \<sqsubseteq> y" shows "edom x \<subseteq> edom y"
assms_4: shows "tabulation (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (F r) (D.inv (\<Phi> (r, f)) \<cdot>\<^sub>D F \<rho>) (F f) (F g)"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_subst_cons: fixes a :: "'d strand_label \<times> ('e, 'f) stateful_strand_step"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<sigma> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (a # A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<sigma>) = (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<sigma>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<sigma>)"
0: fixes xs :: "'a stream" assumes "\<psi>1 xs" shows "\<psi>2 xs"
wf_implies_dom: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "wf (set_of R)" shows "All (Wellfounded.accp R)"