hcju/isabelleps huggingface dataset:
ext_inject: fixes c_work :: "'a \<Rightarrow> 'b zmultiset"   and c_pts :: "'a \<Rightarrow> 'b zmultiset"   and c_imp :: "'a \<Rightarrow> 'b zmultiset"   and more :: "'c"   and c_work' :: "'a \<Rightarrow> 'b zmultiset"   and c_pts' :: "'a \<Rightarrow> 'b zmultiset"   and c_imp' :: "'a \<Rightarrow> 'b zmultiset"   and more' :: "'c" shows "(\<lparr>c_work = c_work, c_pts = c_pts, c_imp = c_imp, \<dots> = more\<rparr> = \<lparr>c_work = c_work', c_pts = c_pts', c_imp = c_imp', \<dots> = more'\<rparr>) = (c_work = c_work' \<and> c_pts = c_pts' \<and> c_imp = c_imp' \<and> more = more')"
elims_2: fixes x :: "'b list"   and xa :: "'b list \<times> 'b list \<times> 'b list" assumes "is_awalk_cyc_decomp x xa" obtains p :: "'b list" and q :: "'b list" and r :: "'b list" and s :: "'b list" where "x = p" and "xa = (q, r, s)" and "p = q @ r @ s \<and> (\<exists>u v w. awalk u q v \<and> awalk v r v \<and> awalk v s w) \<and> 0 < length r \<and> (\<exists>u. distinct (awalk_verts u q))"
e2ennreal_Liminf: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> ereal" assumes "F \<noteq> \<bottom>" shows "e2ennreal (Liminf F f) = Liminf F (\<lambda>n. e2ennreal (f n))"
undual_leq: fixes x' :: "'b dual"   and y' :: "'b dual" shows "(undual x' \<sqsubseteq> undual y') = (y' \<sqsubseteq> x')"
transpos_mem: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and l :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j"   and "l \<le> n" shows "transpos i j l \<le> n"
assms_3: shows "subst_domain \<theta>1 \<inter> subst_domain \<theta>2 = {}"
subdprodl_impl_def: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and y :: "nat list" shows "subdprodl_impl a b x y = (\<forall>k\<le>length a. take k a \<bullet> take k x \<le> b \<bullet> y)"
transpos_eq: fixes k :: "nat" shows "(\<tau>\<^bsub>0 0\<^esub>) k = k"
real_card_permutations_of_multiset_remove: fixes x :: "'a"   and A :: "'a multiset" assumes "x \<in># A" shows "real (card (permutations_of_multiset (A - {#x#}))) = real (card (permutations_of_multiset A) * count A x) / real (size A)"
simps_24: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "'a" shows "(case a\<^sub>4 of a\<^sub>1 \<Rightarrow> f1 | a\<^sub>2 \<Rightarrow> f2 | a\<^sub>3 \<Rightarrow> f3 | a\<^sub>4 \<Rightarrow> f4 | a\<^sub>5 \<Rightarrow> f5) = f4"
diff_left: fixes a :: "'e \<Rightarrow>\<^sub>L 'd"   and a' :: "'e \<Rightarrow>\<^sub>L 'd"   and b :: "'e" shows "blinfun_apply (a - a') b = blinfun_apply a b - blinfun_apply a' b"
move_keeps_owner: fixes v :: "view"   and ts :: "traffic"   and ts' :: "traffic" shows "own v = own (move ts ts' v)"
diff_card_le_card_Diff: fixes B :: "'a set"   and A :: "'a set" assumes "finite B" shows "card A - card B \<le> card (A - B)"
tendsto_cnj: fixes g :: "'a \<Rightarrow> complex"   and a :: "complex"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. cnj (g x)) \<longlongrightarrow> cnj a) F"
minimality_offending_flows_mono_edges_graph_extend: fixes V :: "'a set"   and E :: "('a \<times> 'a) set"   and E' :: "('a \<times> 'a) set"   and Fadd :: "('a \<times> 'a) set"   and F' :: "('a \<times> 'a) set"   and nP :: "'a \<Rightarrow> 'b" assumes "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "E' \<subseteq> E"   and "Fadd \<inter> E' = {}"   and "F' \<in> set_offending_flows \<lparr>nodes = V, edges = E'\<rparr> nP" shows "\<forall>(e1, e2)\<in>F'. \<not> sinvar (add_edge e1 e2 (delete_edges \<lparr>nodes = V, edges = E\<rparr> (F' \<union> Fadd))) nP"
GF_advice_a2_helper: fixes X :: "'b ltln set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" assumes "\<forall>\<psi>\<in>X. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)"   and "w \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<nu>" shows "w \<Turnstile>\<^sub>n \<phi>"
E: shows "rev xs = us @ y # vs \<and> (\<forall>u\<in>set us. key u \<noteq> k) \<and> key y = k \<and> ys = filter (\<lambda>x. key x = k) vs"
c_def: shows "c = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 0)"
semAssignSC_aux: fixes E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and DA :: "'b \<Rightarrow> 'e"   and x :: "'b"   and lH :: "('e \<times> 'a \<times> nat) set"   and \<iota> :: "nat"   and e :: "'a"   and BMap :: "'c \<Rightarrow> bool"   and d :: "'e" assumes "WHATWHERE_Secure_Programs.dH_indistinguishable E DA (DA x) (WHATWHERE.htchLoc lH \<iota>) e e" shows "WHATWHERE.NDC (MWLs_semantics.MWLsSteps_det E BMap) DA d (x:=\<^bsub>\<iota>\<^esub> e) \<or> WHATWHERE.IDC (MWLs_semantics.MWLsSteps_det E BMap) E DA d (x:=\<^bsub>\<iota>\<^esub> e) (WHATWHERE.htchLoc lH (pp (x:=\<^bsub>\<iota>\<^esub> e)))"
assms_3: fixes x :: "real" assumes "x \<in> {a..<b}" shows "\<exists>y\<in>{x<..b}. dist (f y) (f x) \<le> M * (y - x)"
simps_3: fixes a :: "'a"   and b :: "'a" shows "less_eq_option (Some a) (Some b) = less_eq0 a b"
fresh_2: fixes a :: "atom"   and var :: "Syntax.var" shows "a \<sharp> Var var = a \<sharp> var"
proots_count_of_real: fixes p :: "real poly"   and s :: "real set" assumes "p \<noteq> 0" shows "proots_count (map_poly of_real p) (of_real ` s) = proots_count p s"
ennreal_indicator_less: fixes A :: "'b set"   and x :: "'b"   and B :: "'b set" shows "(indicator A x \<le> indicator B x) = (x \<in> A \<longrightarrow> x \<in> B)"
modular_1_aux': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ; (y \<cdot> - (x\<^sup>T ; z)) \<cdot> z = 0"
invar_restore: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option"   and C :: "nat set"   and N :: "nat set"   and d :: "nat" assumes "nf_invar' c src dst PRED C N d"   and "C \<noteq> {}" shows "nf_invar c src dst PRED C N d"
has_derivative_inverse_basic_x: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and g' :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and T :: "'b set" assumes "(f has_derivative f') (at x)"   and "bounded_linear g'"   and "g' \<circ> f' = id"   and "isCont g (f x)"   and "g (f x) = x"   and "open T"   and "f x \<in> T"   and "\<And>y. y \<in> T \<Longrightarrow> f (g y) = y" shows "(g has_derivative g') (at (f x))"
face_cycle_succ_not_uv: fixes a :: "'b" assumes "a \<in> arcs G"   and "a \<notin> {uv, rev_G uv}" shows "GM.face_cycle_succ a = proj_arcs_H (H.face_cycle_succ a)"
assms_1: shows "linear_split xs p = (ls, (sub, sep) # rs)"
execn_strip: shows "\<Gamma>\<turnstile> \<langle>strip_guards F c,s\<rangle> =n\<Rightarrow> t"
ex_unrest: fixes x :: "'a \<Longrightarrow> 'b"   and P :: "'b upred" assumes "x \<sharp> P" shows "(\<exists> x \<bullet> P) = P"
isomorphic_implies_hpar_2: fixes f :: "'a"   and f' :: "'a" assumes "B.isomorphic f f'" shows "B.ide f'"
map_delete: fixes m :: "'b"   and a :: "'v" assumes "Q_invar m" shows "Q_lookup (Q_delete a m) = (Q_lookup m)(a := None)"
fast_bv_to_nat_def: fixes bs :: "bit list" shows "bv_to_nat (\<one> # bs) \<equiv> numeral (fast_bv_to_nat_helper bs num.One)"
unit_r_inv_ex: fixes x :: "'a" assumes "unit x" shows "\<exists>y. x ** y = one"
mem_val_w32_mod_cache: fixes asi :: "word8"   and a :: "word36"   and s :: "'b sparc_state"   and new_cache :: "cpu_cache" shows "mem_val_w32 asi a s = mem_val_w32 asi a (s\<lparr>cache := new_cache\<rparr>)"
Inr_not_Inl: fixes b :: "'f"   and a :: "'e" shows "Inr b \<noteq> Inl a"
zero_not_one: shows "\<zero>\<^bsub>M\<^esub> \<noteq> \<one>\<^bsub>M\<^esub>"
simps_5: fixes uu :: "int \<Rightarrow> int \<Rightarrow> bool"   and uv :: "value option"   and va :: "String.literal" shows "MaybeBoolInt uu uv (Some (value.Str va)) = invalid"
Suc_ile_eq: fixes m :: "int"   and n :: "eint" shows "(eint (m + 1) \<le> n) = (eint m < n)"
id_take_nth_drop: fixes i :: "nat"   and xs :: "'a list" assumes "i < length xs" shows "xs = take i xs @ xs ! i # drop (Suc i) xs"
assms_3: shows "X \<in> sets (Pi\<^sub>M J M)"
neumann_3B_b: fixes u :: "'a pmf"   and v :: "'a pmf"   and \<alpha> :: "real" assumes "u \<succeq>[\<R>] v \<and> (v, u) \<notin> \<R>"   and "\<alpha> \<in> {0<..<1}" shows "u \<succeq>[\<R>] mix_pmf \<alpha> u v \<and> (mix_pmf \<alpha> u v, u) \<notin> \<R>"
axioms_1: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "isomorphic_categories A B" shows "category A"
weakCongEE_2: fixes P :: "pi"   and Q :: "pi" assumes "P \<simeq> Q" shows "Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P"
Inf_from_def: fixes Inf :: "'b inference set"   and N :: "'b set" shows "inference_system.Inf_from Inf N = {\<iota> \<in> Inf. set (prems_of \<iota>) \<subseteq> N}"
min_repU1: fixes w1 :: "32 Word.word"   and x :: "real"   and w2 :: "32 Word.word"   and y :: "real" assumes "w1 \<equiv>\<^sub>L x"   and "w2 \<equiv>\<^sub>L y" shows "wmin w1 w2 \<equiv>\<^sub>L x"
vsfwm: shows "list_all simple_match_valid (map (fst \<circ> snd) ard)"
finite_norm_ACI_juncts_2: fixes B :: "('a, 'k) aformula set"   and f :: "('a, 'k) aformula \<Rightarrow> ('a, 'k) aformula" assumes "finite B" shows "finite {f \<phi> |\<phi>. nf_ACI \<phi> \<and> conjuncts \<phi> \<subseteq> B}"
gen_def: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and cs :: "'a list"   and w :: "'b" shows "gen P cs w \<equiv> \<forall>c\<in>set cs. P c w"
linmap_im_linspan: fixes A :: "'b set"   and N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "R module N"   and "f \<in> mHom R M N"   and "H \<subseteq> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "g \<in> {j. j \<le> n} \<rightarrow> H" shows "f (l_comb R M n s g) \<in> linear_span R N A (f ` H)"
Or_propos: fixes xs :: "'c ltl list" shows "propos (Or xs) = \<Union> {propos x |x. x \<in> set xs}"
P1: fixes F :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool" shows "|~ \<box>F \<longrightarrow> \<circle>F"
\<pi>_def: fixes x :: "'a" shows "\<pi> x = (if x \<in> agents then (unidx \<circ> f \<circ> idx) x else x)"
inv_11: shows "upw \<or> \<mu>_small fs i"
MF: shows "Ifm (x # bs) (minusinf (rlfm (And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))) = False"
simps_3: fixes x :: "'c \<times> 'd \<times> bool"   and xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool"   and i :: "'c" shows "insert (x # xs) e (Some i) = (if fst x = i then Some (x # insert_body xs e) else insert xs e (Some i) \<bind> (\<lambda>t. Some (x # t)))"
l3_step3_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and K :: "msg" shows "l3_step3 Ra A B K \<equiv> {(s, s'). guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress s Ra = Some {xpkE, xskE} \<and> guessed_frame Ra xsk = Some K \<and> implAuth B A (Aenc K (epubKF (Ra $ kE))) \<in> ik s \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xpkE, xskE, xsk}), l1_state.signals := if can_signal s A B then addSignal (l1_state.signals s) (Commit A B \<langle>epubKF (Ra $ kE), K\<rangle>) else l1_state.signals s, secret := {x. x = K \<and> Ra = test} \<union> secret s\<rparr>}"
z_closed: shows "\<^bold>1\<^sub>o\<^sub>w \<in> U"
assms_2: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) F G \<tau>"
ex_basis_span: fixes V :: "'a vec set" assumes "V \<subseteq> carrier_vec n" shows "\<exists>W\<subseteq>carrier_vec n. lin_indpt W \<and> local.span V = local.span W \<and> dim_span V = card W"
cons_pre_rule: fixes P :: "assn"   and P' :: "assn"   and c :: "'d Heap"   and Q :: "'d \<Rightarrow> assn" assumes "P \<Longrightarrow>\<^sub>A P'"   and "<P'> c <Q>" shows "<P> c <Q>"
S1: shows "c_cons (c_pair x y) z \<noteq> 0"
a1: shows "nextnat s = (LEAST i. s i \<noteq> s 0)"
assms: fixes f :: "'t \<Rightarrow>\<^sub>0 'b" assumes "f \<in> F" shows "dgrad_p_set_le d {f} G"
oth_class_taut_8_a: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" shows "[\<phi> \<^bold>& \<psi> \<^bold>\<rightarrow> \<chi> \<^bold>\<rightarrow> (\<phi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<chi>)) in v]"
face_cycle_succ_permutes: shows "face_cycle_succ permutes arcs (with_proj (list_digraph G_list))"
has_field_derivative_stirling_sum'_real: fixes j :: "nat"   and x :: "real"   and m :: "nat" assumes "0 < j"   and "0 < x" shows "(stirling_sum' j m has_real_derivative stirling_sum' (Suc j) m x) (at x)"
relative_to_relative_to: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" shows "P relative_to S relative_to T = P relative_to S \<inter> T"
const_map_is_Some: fixes y :: "'c"   and A :: "'d set"   and x :: "'d"   and z :: "'c" shows "(const_map y A x = Some z) = (z = y \<and> x \<in> A)"
alphaAbs_ex_qAFresh_imp_qFresh: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" assumes "alphaAbs_ex_qAFresh xs x X xs' x' X'" shows "alphaAbs_ex_qFresh xs x X xs' x' X'"
M_to_int_mod_ring: fixes x :: "'a mod_ring" shows "M (to_int_mod_ring x) = to_int_mod_ring x"
h1: shows "{x - a |x. x \<in> s} = (+) (- a) ` s"
simps_1: fixes k :: "nat"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "exact_aux k 0 qs = qs"
viewR_correct_2: fixes t :: "('a, 'b) FingerTree" assumes "t \<noteq> FingerTree.empty" shows "\<exists>a s. FingerTree.viewR t = Some (a, s) \<and> FingerTree.toList t = FingerTree.toList s @ [a]"
uminus_closed: fixes U :: "'a set"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "uminus_ow U uminus"   and "a \<in> U" shows "uminus a \<in> U"
arr_cod: fixes f :: "'s" assumes "arr f" shows "arr (cod f)"
hmining_def: fixes t :: "nat \<Rightarrow> cnf" shows "hmining t \<equiv> \<lambda>n. \<exists>nid\<in>actHn (t n). mining (\<sigma>\<^bsub>nid\<^esub>t n)"
in_borel_measurable: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" shows "(f \<in> borel_measurable M) = (\<forall>S\<in>sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"
inner_simps_5: fixes r :: "real"   and x :: "'a"   and y :: "'a" shows "r *\<^sub>R x \<bullet> y = r * (x \<bullet> y)"
compare_Integer_bottom_iff: fixes x :: "Integer"   and y :: "Integer" shows "(compare\<cdot>x\<cdot>y = \<bottom>) = (x = \<bottom> \<or> y = \<bottom>)"
hd_prog_idem: fixes p :: "'b"   and xs :: "'b memref list" shows "hd_prog (hd_prog p xs) xs = hd_prog p xs"
widen_subcls: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" assumes "P \<turnstile> C \<preceq>\<^sup>* D" shows "P \<turnstile> Class C \<le> Class D"
is_float_ivl_err_2: fixes x :: "real interval"   and a :: "real"   and b :: "real pdevs"   and c :: "real" assumes "ivl_err x = ((a, b), c)"   and "is_float (lower x)"   and "is_float (upper x)" shows "is_float c"
StepQ: fixes cfg1 :: "('p, 'v, 's) configuration"   and Q :: "'p set"   and cfg2 :: "('p, 'v, 's) configuration"   and msg :: "('p, 'v) message"   and cfg3 :: "('p, 'v, 's) configuration" assumes "qReachable cfg1 Q cfg2"   and "cfg2 \<turnstile> msg \<mapsto> cfg3"   and "\<exists>p\<in>Q. isReceiverOf p msg" shows "qReachable cfg1 Q cfg3"
inangle_distincts: fixes P :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "P InAngle A B C" shows "A \<noteq> B \<and> C \<noteq> B \<and> P \<noteq> B"
swapBinp_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and binp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs set option" shows "binp %%[x \<and> y]_xs \<equiv> lift (swapAbs xs x y) binp"
e: shows "enat (lfind P lst) < enat (Suc (lfind P lst))"
cyclic_on_orbit': fixes f :: "'e \<Rightarrow> 'e"   and x :: "'e" assumes "permutation f" shows "cyclic_on f (orbit f x)"
assms: shows "snd (rep_nat_term u) < snd (rep_nat_term v)"
resCasesB': fixes x' :: "name"   and A :: "pi"   and a :: "name"   and y' :: "name"   and A' :: "pi" assumes "<\<nu>x'>A \<longmapsto> a<\<nu>y'> \<prec> A'" obtains P :: "pi" and aa :: "name" and b :: "name" and P' :: "pi" where "<\<nu>x'>A = <\<nu>b>Pa__" and "a<\<nu>y'> \<prec> A' = aa__<\<nu>b> \<prec> P'__" and "Pa__ \<longmapsto> aa__[b] \<prec> P'__" and "aa__ \<noteq> b"   | P :: "pi" and aa :: "name" and x :: "name" and P' :: "pi" and y :: "name" where "<\<nu>x'>A = <\<nu>y>P" and "a<\<nu>y'> \<prec> A' = aa<\<nu>x> \<prec> (<\<nu>y>P')" and "P \<longmapsto> aa<\<nu>x> \<prec> P'" and "y \<noteq> aa" and "y \<noteq> x" and "x \<sharp> P" and "x \<noteq> aa"
grd_def: fixes S :: "'b \<Rightarrow> 'a" shows "grd S = - S \<bottom>"
assms_2: shows "\<delta> \<equiv> max 0 (1 - \<alpha>)"
Card_order_iff_ordIso_card_of: fixes r :: "('a \<times> 'a) set" shows "Card_order r = (r =o |Field r|)"
suffix_Cons: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "xs \<le>s y # ys = (xs = y # ys \<or> xs \<le>s ys)"
simps_1: fixes S :: "Runs.var set"   and S' :: "Runs.var set" shows "in_progressS (Some S) S' = (S' \<subseteq> S)"
assms: shows "ineq_model \<I> X [(t, t')]"
openin_topology_generated_by: fixes S :: "'c set set"   and s :: "'c set" assumes "openin (topology_generated_by S) s" shows "generate_topology_on S s"
blocks1_max_vars: fixes n :: "nat"   and Ts :: "ty list"   and e :: "(nat, 'a, 'b) exp" shows "max_vars (blocks1 n Ts e) = max_vars e + length Ts"
RootEx_def: fixes F :: "('d, 'e, 'f) seqauto set"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option" shows "RootEx F G = (\<exists>!A. A \<in> F \<and> A \<notin> \<Union> (ran G))"
padic_set_res_coherent: fixes f :: "nat \<Rightarrow> int"   and p :: "int"   and m :: "nat"   and n :: "nat" assumes "f \<in> padic_set p"   and "m \<le> n"   and "prime p" shows "residue (p ^ m) (f n) = f m"
inf_aforms_def: fixes prec :: "nat"   and xs :: "('a \<times> 'a pdevs) list" shows "inf_aforms prec xs = map (Inf_aform' prec) xs"
rep_eq: fixes x :: "'b fset" shows "fset (fcompress x) = compress (fset x)"
map_graph_fn_id_1: fixes X :: "('a, 'b) labeled_graph" shows "map_graph_fn X id = restrict X"
var_closed_2: fixes K :: "'a set" assumes "subring K R" shows "polynomial K X"
diff_add_self_ennreal: fixes b :: "ennreal"   and a :: "ennreal" shows "b - a + a = (if a \<le> b then b else a)"
mkVarChannelProc_cl_inv: fixes g :: "'a gState_scheme"   and p :: "pState"   and v :: "procVarDecl" assumes "cl_inv (g, p)" shows "cl_inv (mkVarChannelProc v g p)"
mat_addrow_def: fixes ad :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mul :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "mat_addrow_gen ad mul a k l A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = i then ad (mul a (A $$ (l, j))) (A $$ (i, j)) else A $$ (i, j))"
oreturnOk_def: fixes x :: "'d" shows "oreturnOk x = K (Some (Inr x))"
act_assoc: fixes x :: "'a"   and y :: "'a"   and p :: "'b" assumes "D x y \<and> Dla (x \<oplus> y) p" shows "\<alpha> (x \<oplus> y) p = \<alpha> x (\<alpha> y p)"
ct: shows "ct \<in> {ct \<in> Pi\<^sub>E S K. proper ct}"
intros_5: fixes dims :: "nat list"   and P :: "complex mat"   and Q :: "complex mat"   and S :: "com"   and M :: "nat \<Rightarrow> complex mat" assumes "state_sig.is_quantum_predicate dims P"   and "state_sig.is_quantum_predicate dims Q"   and "state_sig.hoare_partial dims Q S (adjoint (M 0) * P * M 0 + adjoint (M 1) * Q * M 1)" shows "state_sig.hoare_partial dims (adjoint (M 0) * P * M 0 + adjoint (M 1) * Q * M 1) (While M S) P"
FGx: shows "((F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) \<circ> (G::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)) (x::'a::refinement_lattice) = (d::'a::refinement_lattice) \<sqinter> (c::'a::refinement_lattice) ; c\<^sup>\<star> ; d \<sqinter> c ; x"
lbisimulation_this: shows "lbisimulation (br pidgc_\<alpha> (\<lambda>_. True)) pid_system_automaton system_automaton"
relcomp_unfold: fixes r :: "('a \<times> 'c) set"   and s :: "('c \<times> 'b) set" shows "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
graph_in_degree_r_imp_reg: fixes G :: "('a, 'b) pre_digraph"   and \<r> :: "nat" assumes "graph G"   and "\<And>v. v \<in> verts G \<Longrightarrow> in_degree G v = \<r>" shows "regular_graph G (int \<r>)"
intros: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'c"   and d :: "'d" assumes "R1 a b"   and "R2 c d" shows "rel_prod R1 R2 (a, c) (b, d)"
OrdNotEqP_I: fixes x :: "tm"   and y :: "tm"   and H :: "fm set" assumes "insert (x EQ y) H \<turnstile> Fls"   and "H \<turnstile> OrdP x"   and "H \<turnstile> OrdP y" shows "H \<turnstile> x NEQ y"
ts_inv_ok_upd_invs: fixes ts :: "'g \<Rightarrow> ('h \<times> 'i \<Rightarrow>f nat) option"   and I :: "'g \<Rightarrow> 'j option"   and tas :: "('g, 'h, 'k) new_thread_action list"   and P :: "'j \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'k \<Rightarrow> bool" assumes "ts_inv_ok ts I" shows "ts_inv_ok (redT_updTs ts tas) (upd_invs I P tas)"
M: shows "CU.Model GE_wtFsym GE_wtPsym (length \<circ> GE_arOf) (length \<circ> GE_parOf) GE_\<Phi> D eintF eintP"
share_all_empty: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and \<S> :: "nat \<Rightarrow> bool option" assumes "\<forall>i p is xs sb \<D> \<O> \<R>. i < length ts \<longrightarrow> ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>) \<longrightarrow> sb = []" shows "share_all_until_volatile_write ts \<S> = \<S>"
Ax: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" assumes "F \<in> \<Gamma>" shows "\<Gamma> \<turnstile>\<^sub>H F"
nj_set_def: shows "nj_set = {nj. \<exists>j\<in>{0..<m}. enat (Suc nj) < llength Ns \<and> prems_of \<iota> ! j \<notin> active_subset (lnth Ns nj) \<and> (\<forall>k>nj. enat k < llength Ns \<longrightarrow> prems_of \<iota> ! j \<in> active_subset (lnth Ns k))}"
inverse2: fixes f :: "'b \<Rightarrow> 'a Heap" shows "U (C f) = f"
complete_iff_height: fixes kdt :: "'a kdt" shows "KD_Tree.complete kdt = (min_height kdt = height kdt)"
that: fixes u :: "dbtm"   and t :: "dbtm" assumes "y = \<lbrakk>quot_dbtm u\<rbrakk>e"   and "x = \<lbrakk>quot_dbtm t\<rbrakk>e"   and "x' = \<lbrakk>quot_dbtm (subst_dbtm u (decode_Var v) t)\<rbrakk>e" shows "thesis"
homotopic_with_prod_topology: fixes p :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X1 :: "'b topology"   and Y1 :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and q :: "('d \<Rightarrow> 'e) \<Rightarrow> bool"   and X2 :: "'d topology"   and Y2 :: "'e topology"   and g :: "'d \<Rightarrow> 'e"   and g' :: "'d \<Rightarrow> 'e"   and r :: "('b \<times> 'd \<Rightarrow> 'c \<times> 'e) \<Rightarrow> bool" assumes "homotopic_with p X1 Y1 f f'"   and "homotopic_with q X2 Y2 g g'"   and "\<And>i j. \<lbrakk>p i; q j\<rbrakk> \<Longrightarrow> r (\<lambda>(x, y). (i x, j y))" shows "homotopic_with r (prod_topology X1 X2) (prod_topology Y1 Y2) (\<lambda>z. (f (fst z), g (snd z))) (\<lambda>z. (f' (fst z), g' (snd z)))"
S_list_pgallery_decomp1: fixes ss :: "'a permutation list"   and Cs :: "'a set list" assumes "set ss = (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "Cs \<noteq> []"   and "pgallery (C0 # Cs)" shows "\<exists>s\<in>set ss. \<exists>C\<in>set Cs. \<forall>(f, g)\<in>fundfoldpairs. s = Abs_induced_automorph f g \<longrightarrow> C \<in> g \<turnstile> \<C>"
initial: fixes p :: "'state" assumes "init p" shows "p \<in> R.nodes"
subfield_polynomial_hom: fixes K :: "'a set"   and p :: "'a list" assumes "subfield K R"   and "\<one>\<^bsub>S\<^esub> \<noteq> \<zero>\<^bsub>S\<^esub>"   and "p \<in> carrier (K [X]\<^bsub>R\<^esub>)" shows "map h p \<in> carrier (h ` K [X]\<^bsub>S\<^esub>)"
simps_15: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case T_f32 of T_i32 \<Rightarrow> f1 | T_i64 \<Rightarrow> f2 | T_f32 \<Rightarrow> f3 | T_f64 \<Rightarrow> f4) = f3"