hcju/isabelleps huggingface dataset:
uminus_sheaf_spec_in_sheaf_spec: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and s :: "'c set \<Rightarrow> ('c \<times> 'c) set"   and U :: "'c set set" assumes "comm_ring R addition multiplication zero unit"   and "s \<in> comm_ring.sheaf_spec R addition multiplication zero unit U"   and "U \<subseteq> comm_ring.spectrum R addition multiplication zero unit" shows "comm_ring.uminus_sheaf_spec R addition multiplication zero U s \<in> comm_ring.sheaf_spec R addition multiplication zero unit U"
asAbs_equal_iff_alphaAbs: fixes qA :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and qB :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodAbs qA \<or> qGoodAbs qB" shows "(asAbs qA = asAbs qB) = (qA $= qB)"
prefixes_def: fixes s :: "'a list set" shows "prefixes s \<equiv> {x. \<exists>n y. 0 < n \<and> y \<in> s \<and> take n y = x}"
im_set_un1: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set"   and A1 :: "'c set"   and A2 :: "'c set" assumes "\<forall>l\<in>A. f l \<in> B"   and "A = A1 \<union> A2" shows "f ` (A1 \<union> A2) = f ` A1 \<union> f ` A2"
in_results_gpv_restrict_gpvD: fixes x :: "'a"   and \<I> :: "('b, 'c) \<I>"   and \<I>' :: "('b, 'c) \<I>"   and gpv :: "('a, 'b, 'c) gpv" assumes "x \<in> results_gpv \<I> (restrict_gpv \<I>' gpv)" shows "x \<in> results_gpv \<I> gpv"
step: shows "direct_computation.concurrent_step (ts, m, \<S>) (ts', m', \<S>')"
simps_77: fixes f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c"   and f4 :: "'c"   and f5 :: "char list \<Rightarrow> 'c"   and f6 :: "'c"   and f7 :: "char list \<Rightarrow> 'c"   and f8 :: "'c"   and f9 :: "'c" shows "(case Log of Accept \<Rightarrow> f1 | Drop \<Rightarrow> f2 | Log \<Rightarrow> f3 | Reject \<Rightarrow> f4 | Call x \<Rightarrow> f5 x | Return \<Rightarrow> f6 | Goto x \<Rightarrow> f7 x | Empty \<Rightarrow> f8 | Unknown \<Rightarrow> f9) = f3"
hd_weak_ranking: fixes x :: "'a"   and y :: "'a" assumes "x \<in> hd (weak_ranking le)"   and "y \<in> carrier" shows "le y x"
lub_set: fixes S :: "'b set set" shows "lub S = \<Union> S"
simps_4: fixes M :: "('fun, 'var) Term.term set"   and uu :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; \<langle>uu: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>c' = (\<lambda>\<I>. Unifier \<I> t t' \<and> \<lbrakk>M; S\<rbrakk>\<^sub>c' \<I>)"
list_exhaust_NSC: fixes xs :: "'b list" obtains "xs = []"   | x :: "'b" where "xs = [x__]"   | x :: "'b" and y :: "'b" and ys :: "'b list" where "xs = x # y # ys"
nsets_mono: fixes A :: "'a set"   and B :: "'a set"   and n :: "nat" assumes "A \<subseteq> B" shows "[A]\<^bsup>n\<^esup> \<subseteq> [B]\<^bsup>n\<^esup>"
nonsat_push_decr_nonsat_potential: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and e :: "nat \<times> nat" assumes "Height_Bounded_Labeling c s t f l"   and "Network.nonsat_push_precond c f l e" shows "Network.nonsat_potential c (Network.push_effect c f e) l < Network.nonsat_potential c f l"
connected_componentI: fixes T :: "'c set"   and S :: "'c set"   and x :: "'c"   and y :: "'c" assumes "connected T"   and "T \<subseteq> S"   and "x \<in> T"   and "y \<in> T" shows "connected_component S x y"
funion_fsingleton_iff: fixes A :: "'a fset"   and B :: "'a fset"   and x :: "'a" shows "(A |\<union>| B = {|x|}) = (A = {||} \<and> B = {|x|} \<or> A = {|x|} \<and> B = {||} \<or> A = {|x|} \<and> B = {|x|})"
nextl_app: fixes Q :: "hf set"   and xs :: "'a list"   and ys :: "'a list" shows "nextl Q (xs @ ys) = nextl (nextl Q xs) ys"
lossless_S2: fixes m2 :: "bool \<times> bool"   and out2 :: "bool" shows "lossless_spmf (S2_14_OT m2 out2)"
strong_exhaust_1: fixes y :: "exp"   and c :: "'b" obtains x1 :: "Vars.var" where "y = Var x1"   | x21 :: "exp" and x22 :: "Vars.var" where "y = App x21 x22"   | x31 :: "assn" and x32 :: "exp" where "set (bn x31) \<sharp>* c" and "y = LetA x31 x32"   | x41 :: "Vars.var" and x42 :: "exp" where "{atom x41} \<sharp>* c" and "y = Lam [x41]. x42"   | x5 :: "bool" where "y = Bool x5"   | x61 :: "exp" and x62 :: "exp" and x63 :: "exp" where "y = (x61 ? x62 : x63)"
ub_add_nat: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'b"   and p :: "nat" assumes "m \<le> n + 1" shows "sum g {m..n + p} = sum g {m..n} + sum g {n + 1..n + p}"
fst2: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm" assumes "A >> A'" shows "Fst (SimplyTyped.Pair A B) >> A'"
cupcake_list_correct0: fixes env :: "v sem_env"   and es :: "exp0 list"   and rs :: "(v, v) result list"   and s :: "'b state" assumes "list_all2_shortcircuit (\<lambda>e r. cupcake_evaluate_single env e r \<and> (is_cupcake_all_env env \<longrightarrow> is_cupcake_exp e \<longrightarrow> (\<forall>s. \<exists>s'. Big_Step_Unclocked_Single.evaluate env s e (s', r)))) es rs"   and "is_cupcake_all_env env"   and "list_all is_cupcake_exp es" shows "\<exists>s'. Big_Step_Unclocked_Single.evaluate_list (Big_Step_Unclocked_Single.evaluate env) s es (s', sequence_result rs)"
locked_def: fixes Quorum :: "process set set"   and s :: "v_state" assumes "quorum_process Quorum" shows "quorum_process.locked Quorum s = {v. \<exists>r. quorum_process.locked_in Quorum s r v}"
union_eq_subsetD: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<union> B = C" shows "A \<union> B \<subseteq> C \<and> C \<subseteq> A \<union> B"
AbstrLevels_S6_A22: fixes i :: "AbstrLevelsID" assumes "sS6 \<in> AbstrLevel i" shows "sA22 \<notin> AbstrLevel i"
frt: fixes t :: "'trm" assumes "set (map snd txs) \<subseteq> var"   and "set (r # map fst txs) \<subseteq> trm"   and "set [] \<subseteq> fmla"   and "t \<in> set (r # map fst txs)" shows "set (getFrN (map snd txs) (r # map fst txs) [] (length txs)) \<inter> FvarsT t = {}"
h_def: shows "h = Cn 2 (r_smn 1 2) [r_dummy 1 (r_const (encode r_univuniv2)), recf.Id 2 0, recf.Id 2 1]"
fst_uminus_aform: fixes Y :: "'b \<times> 'c pdevs" shows "fst (uminus_aform Y) = - fst Y"
the_elem_surj: shows "(\<lambda>X. the_elem (h ` X)) ` carrier (R Quot a_kernel R S h) = h ` carrier R"
hyps_4: shows "mds\<^sub>2'''__ \<le> mds\<^sub>1'"
intro: fixes G :: "('c, 'd) ParityGame_scheme"   and v0 :: "'c" assumes "\<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v0 \<rightarrow>\<^bsub>G\<^esub> w)" shows "vmc_path_no_deadend_axioms G v0"
assms: shows "StableNoDecomp.stable_pair_on UNIV (XD, XH)"
exec_elim_cases_10: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>While b c,s\<rangle> \<Rightarrow> t" obtains f :: "'f" where "sb__ = Fault f" and "t = Fault f"   | s :: "'e" and s' :: "('e, 'f) xstate" where "sb__ = Normal sc__" and "sc__ \<in> b" and "\<Gamma>\<turnstile> \<langle>c,Normal sc__\<rangle> \<Rightarrow> s'" and "\<Gamma>\<turnstile> \<langle>While b c,s'\<rangle> \<Rightarrow> t"   | s :: "'e" where "sb__ = Normal sd__" and "t = Normal sd__" and "sd__ \<notin> b"   | "sb__ = Stuck" and "t = Stuck"   | s :: "'e" where "sb__ = Abrupt s" and "t = Abrupt s"
reads_writes_preserved: fixes SR :: "('x \<Rightarrow> 'x \<Rightarrow> bool) set"   and getter :: "('x, 'y, 'z) prog"   and h :: "'x"   and h' :: "'x"   and SW :: "('x, 'aa, 'ab) prog set"   and setter :: "('x, 'aa, 'ac) prog"   and x :: "'z" assumes "reads SR getter h h'"   and "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "\<And>h h'. \<forall>w\<in>SW. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>SR. r h h')" shows "h \<turnstile> getter \<rightarrow>\<^sub>r x = h' \<turnstile> getter \<rightarrow>\<^sub>r x"
foldli_cong: fixes l :: "'e list"   and l' :: "'e list"   and \<sigma> :: "'f"   and \<sigma>' :: "'f"   and c :: "'f \<Rightarrow> bool"   and c' :: "'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and f' :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "l = l'"   and "\<sigma> = \<sigma>'"   and "c = c'"   and "\<And>\<sigma> x. \<lbrakk>x \<in> set l'; c' \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> = f' x \<sigma>" shows "foldli l c f \<sigma> = foldli l' c' f' \<sigma>'"
simps: fixes st :: "nat"   and ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_double_case_inv st ires rs (l, r) = (if st = Suc 0 then wcode_on_left_moving_1 ires rs (l, r) else if st = Suc (Suc 0) then wcode_on_checking_1 ires rs (l, r) else if st = 3 then wcode_erase1 ires rs (l, r) else if st = 4 then wcode_on_right_moving_1 ires rs (l, r) else if st = 5 then wcode_goon_right_moving_1 ires rs (l, r) else if st = 6 then wcode_backto_standard_pos ires rs (l, r) else if st = 13 then wcode_before_double ires rs (l, r) else False)"
that: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "m1 = EnvT f"   and "m2 = StateT g"   and "(eq_onp ((=) s) ===> M) f g" shows "thesis"
bounded_linear_inner_left_comp: fixes g :: "'d \<Rightarrow> 'e"   and b :: "'e" assumes "bounded_linear g" shows "bounded_linear (\<lambda>x. g x \<bullet> b)"
isoE: fixes f :: "'a \<times> 'a" assumes "VV.iso f" obtains g :: "'a \<times> 'a" where "VV.inverse_arrows f g"
PowI: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "A \<in> Pow B"
notin': shows "\<nexists>k'. cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'"
rep_eq: fixes x :: "('a, 'c, 'b) gen_cfg_wf" shows "gen_wf_inEdges x = gen_inEdges (Rep_gen_cfg_wf x)"
fcrf_def: fixes \<Gamma> :: "'e Gamma"   and ur :: "('s \<times> 's) set" shows "fcrf \<Gamma> ur \<equiv> \<forall>c\<in>C\<^bsub>\<V>\<^esub> \<inter> \<Upsilon>\<^bsub>\<Gamma>\<^esub>. \<forall>v\<in>V\<^bsub>\<V>\<^esub> \<inter> \<nabla>\<^bsub>\<Gamma>\<^esub>. \<forall>s\<in>S\<^bsub>SES\<^esub>. \<forall>s'\<in>S\<^bsub>SES\<^esub>. reachable SES s \<and> s ([c] @ [v])\<Longrightarrow>\<^bsub>SES\<^esub> s' \<longrightarrow> (\<exists>s''\<in>S\<^bsub>SES\<^esub>. \<exists>\<delta>. (\<forall>d\<in>set \<delta>. d \<in> N\<^bsub>\<V>\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>\<^esub>) \<and> s (\<delta> @ [v])\<Longrightarrow>\<^bsub>SES\<^esub> s'' \<and> (s', s'') \<in> ur)"
enat_of_option_inj: fixes x :: "nat option"   and y :: "nat option" shows "(Common.enat_of_option x = Common.enat_of_option y) = (x = y)"
\<ll>'_ide_simp: fixes a :: "'d" assumes "D.ide a" shows "D.\<ll>' a = D.lunit' a"
some_GB_pmdl_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "pmdl (some_GB F) = pmdl F"
tab_simps_4: shows "local.dom h.tab = tab\<^sub>1 h"
frontier_properties: assumes "simply_connected S" shows "if bounded S then connected (frontier S) else \<forall>C\<in>components (frontier S). \<not> bounded C"
npepow_correct: fixes xs :: "'a list"   and e :: "pexpr"   and n :: "nat" assumes "in_carrier xs" shows "peval xs (npepow e n) = peval xs (PExpr2 (PPow e n))"
f1_is_pr: shows "PrimRecOp g (\<lambda>x y z. h z y x) \<in> PrimRec2"
eq_size_0: fixes t :: "'c tree" shows "(size t = 0) = (t = \<langle>\<rangle>)"
almost_full_onD: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and f :: "nat \<Rightarrow> 'c" assumes "almost_full_on P A"   and "\<And>i. f i \<in> A" obtains i :: "nat" and j :: "nat" where "i < j" and "P (f i) (f j)"
bst_union: fixes t1 :: "('a \<times> 'b) tree"   and t2 :: "('a \<times> 'b) tree" assumes "Tree2.bst t1"   and "Tree2.bst t2" shows "Tree2.bst (local.union t1 t2)"
lossless_random_oracle: fixes p :: "'b spmf"   and \<sigma> :: "'c \<Rightarrow> 'b option"   and x :: "'c" assumes "lossless_spmf p" shows "lossless_spmf (random_function.random_oracle p \<sigma> x)"
let_to_bind_conv: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'c nres" shows "Let x f = RETURN x \<bind> f"
measurable_lfp_coinduct: fixes P :: "'b measure \<Rightarrow> bool"   and M :: "'b measure"   and F :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c" assumes "P M"   and "sup_continuous F"   and "\<And>M A. \<lbrakk>P M; \<And>N. P N \<Longrightarrow> A \<in> N \<rightarrow>\<^sub>M count_space UNIV\<rbrakk> \<Longrightarrow> F A \<in> M \<rightarrow>\<^sub>M count_space UNIV" shows "lfp F \<in> M \<rightarrow>\<^sub>M count_space UNIV"
is_monomial_monomial: fixes p :: "'c \<Rightarrow>\<^sub>0 'd" assumes "is_monomial p" obtains c :: "'d" and t :: "'c" where "c \<noteq> (0::'d)" and "p = monomial c t"
isomorphic_transitive: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "category C"   and "category.isomorphic C f g"   and "category.isomorphic C g h" shows "category.isomorphic C f h"
Rep_List: fixes x :: "'b SList.list" shows "Rep_List x \<in> List"
cltn2_id_is_K2_isometry: shows "is_K2_isometry cltn2_id"
that: fixes c :: "real" assumes "0 < c"   and "\<forall>\<^sub>F x in F. R (f x) (c * \<bar>g x\<bar>)" shows "thesis"
get_proc_get_return_edge: fixes a :: "'edge"   and a' :: "'edge" assumes "valid_edge a"   and "a' \<in> get_return_edges a" shows "get_proc (sourcenode a) = get_proc (targetnode a')"
insert_subsetI: fixes x :: "'b"   and A :: "'b set"   and X :: "'b set" assumes "x \<in> A"   and "order.greater_eq A X" shows "order.greater_eq A (insert x X)"
assms_2: shows "real_of_float x \<le> pi / 2"
cond2_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and bs :: "'b list"   and y :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "cond2 f bs y xs ys \<equiv> cond1 f bs y xs ys \<or> (\<exists>b. b \<in> set bs \<and> snd (f b y) \<and> snd (mul_ext f (remove1 b xs) ys))"
dvd_all_coeffs_iff: fixes p :: "'b poly"   and x :: "'b" shows "(\<forall>pi\<in>set (coeffs p). x dvd pi) = (\<forall>i. x dvd coeff p i)"
rt_eq: shows "return_time p2 (flow0 x0 h) + h = return_time p2 x0"
dg_Par_Cod_vrange: fixes \<alpha> :: "V" shows "\<R>\<^sub>\<circ> (dg_Par \<alpha>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> dg_Par \<alpha>\<lparr>Obj\<rparr>"
has_integral_offset: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and i :: "'a"   and c :: "real" assumes "has_bochner_integral (lebesgue_on {a..b}) f i" shows "has_bochner_integral (lebesgue_on {a - c..b - c}) (\<lambda>x. f (x + c)) i"
get_scdom_component_owner_document_same: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "DocumentClass.type_wf h"   and "DocumentClass.known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" obtains owner_document :: "(_) document_ptr" where "h \<turnstile> get_owner_document ptr' \<rightarrow>\<^sub>r owner_document" and "cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r owner_document \<in> set sc"
conc_epsilon_2: fixes A :: "'b list set" shows "A @@ {[]} = A"
app_insf_Seq: fixes s :: "hf"   and k :: "hf"   and y :: "hf" assumes "Seq s k" shows "app (insf s k y) k = y"
singleton: fixes x :: "'a" shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>n \<eta> x = x"
E23: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<circle>F \<longrightarrow> \<diamond>F"
associative: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> M"   and "b \<in> M"   and "c \<in> M" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
image_Collect_subsetI: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "\<And>x. P x \<Longrightarrow> f x \<in> B" shows "f ` {x. P x} \<subseteq> B"
Compl_step3: fixes I :: "'a set"   and s :: "'a set" assumes "I \<noteq> {}"   and "finite I"   and "\<forall>x\<in>I. x \<in> s \<or> (\<exists>sl. sl \<noteq> [] \<and> tl sl \<noteq> [] \<and> (sl ! 0, sl ! (length sl - 1)) = ({x}, s) \<and> (\<forall>i<length sl - 1. \<turnstile>\<N>\<^bsub>(sl ! i, sl ! (i + 1))\<^esub>))" shows "\<exists>lI. set lI = {x \<in> I. x \<notin> s} \<and> (\<exists>Sj. length Sj = length lI \<and> nodup_all lI \<and> (\<forall>j<length Sj. Sj ! j \<noteq> [] \<and> tl (Sj ! j) \<noteq> [] \<and> (Sj ! j ! 0, Sj ! j ! (length (Sj ! j) - 1)) = ({lI ! j}, s) \<and> (\<forall>i<length (Sj ! j) - 1. \<turnstile>\<N>\<^bsub>(Sj ! j ! i, Sj ! j ! (i + 1))\<^esub>)))"
mat_gt_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sd :: "nat"   and A :: "'a mat"   and B :: "'a mat" shows "ord.mat_gt less_eq gt sd A B = (ord.mat_ge less_eq A B \<and> (\<exists>i<sd. \<exists>j<sd. gt (A $$ (i, j)) (B $$ (i, j))))"
coprime_r_zero2: fixes qs :: "real poly list" shows "poly (coprime_r qs) (real_of_int (- crb (prod_list qs))) = 0"
unowned_shared: shows "- (\<Union>(uu_, uu_, uu_, uu_, uu_, \<O>, uu_)\<in>set ts\<^sub>s\<^sub>b. \<O>) \<subseteq> dom \<S>\<^sub>s\<^sub>b"
basis_reduction_add_rows_def: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" shows "local.basis_reduction_add_rows upw i fs = (if upw then local.basis_reduction_add_rows_loop i fs i else fs)"
nat_mask_eq: fixes n :: "nat" shows "nat (mask n) = mask n"
results_gpv_bind_gpv: fixes \<Gamma> :: "('e, 'f) \<I>"   and gpv :: "('g, 'e, 'f) gpv"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "results_gpv \<Gamma> (gpv \<bind> f) = (\<Union>x\<in>results_gpv \<Gamma> gpv. results_gpv \<Gamma> (f x))"
pdevs_inner2_def: fixes X :: "'b pdevs"   and n :: "'b"   and l :: "'b" shows "pdevs_inner2 X n l = unop_pdevs (\<lambda>x. inner2 x n l) X"
elements_four_block_mat: fixes A :: "'b mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'b mat"   and nc2 :: "nat"   and C :: "'b mat"   and nr2 :: "nat"   and D :: "'b mat" assumes "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2" shows "elements_mat (four_block_mat A B C D) \<subseteq> elements_mat A \<union> elements_mat B \<union> elements_mat C \<union> elements_mat D"
strict_monoI: fixes f :: "'a word \<Rightarrow> 'c" assumes "\<And>x y. x <s y \<Longrightarrow> f x < f y" shows "signed.strict_mono f"
d0_def: shows "st2.d0 = prod_list st2.dims0"
Chain3: fixes At :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> bool"   and x0 :: "'a"   and y0 :: "'b"   and z0 :: "'c"   and Follows :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "At x0 y0 z0 0"   and "\<And>x y z n. At x y z n \<Longrightarrow> \<exists>x' y' z'. At x' y' z' (Suc n) \<and> Follows x' y' z' x y z" obtains f :: "nat \<Rightarrow> 'a" and g :: "nat \<Rightarrow> 'b" and h :: "nat \<Rightarrow> 'c" where "f 0 = x0" and "g 0 = y0" and "h 0 = z0" and "\<And>n. At (f n) (g n) (h n) n" and "\<And>n. Follows (f (Suc n)) (g (Suc n)) (h (Suc n)) (f n) (g n) (h n)"
absolutely_integrable_mult_Fejer_kernel_reflected_part5: fixes f :: "real \<Rightarrow> real"   and d :: "real"   and n :: "nat"   and t :: "real"   and c :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x"   and "d \<le> pi" shows "(\<lambda>x. Fejer_kernel n x * (f (t + x) + f (t - x) - c)) absolutely_integrable_on {0..d}"
z_def: shows "z \<equiv> Cn 4 r_result1 [Cn 4 r_length [recf.Id 4 3], recf.Id 4 2, Cn 4 r_take [Cn 4 S [recf.Id 4 0], recf.Id 4 3]]"
fminus_finsert_absorb: fixes x :: "'b"   and A :: "'b fset" assumes "x |\<notin>| A" shows "finsert x A |-| {|x|} = A"
fold_eq_nfoldli: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and l :: "'d list"   and s :: "'c" shows "RETURN (fold f l s) = nfoldli l (\<lambda>_. True) (\<lambda>x s. RETURN (f x s)) s"
add_closed_converse_right: fixes g :: "'g"   and x :: "'g" assumes "g + x \<in> G"   and "g \<in> G" shows "x \<in> G"
impr_impl_times: fixes a :: "'b N"   and b :: "'b N" shows "(a l\<rightarrow> b) * a = a * (a r\<rightarrow> b)"
valid_def: fixes ds :: "'b set"   and P :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool"   and Q :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool" shows "valid ds P Q = (Q ds {} \<and> (\<forall>fp. P ds fp \<and> Q ds fp \<longrightarrow> Q ds (cop_F ds fp)))"
heun_remainder2_def: fixes p :: "real"   and x :: "real \<Rightarrow> 'c"   and f :: "real \<times> 'c \<Rightarrow> 'c"   and f'' :: "real \<times> 'c \<Rightarrow> (real \<times> 'c) \<Rightarrow>\<^sub>L (real \<times> 'c) \<Rightarrow>\<^sub>L 'b"   and t :: "real"   and h :: "real"   and s :: "real" shows "heun_remainder2 p x f f'' t h s = (h ^ 3 * p / 4) *\<^sub>R f'' (t + s * h * p, x t + (s * h * p) *\<^sub>R f (t, x t)) $ (1, f (t, x t)) $ (1, f (t, x t))"
simps_2: fixes l :: "'c tree"   and x :: "'c"   and r :: "'c tree" shows "complete \<langle>l, x, r\<rangle> = (height l = height r \<and> complete l \<and> complete r)"
deg_pm_zero: shows "deg_pm 0 = (0::'a)"
Cn_union: fixes A :: "'a set"   and B :: "'a set" shows "Cn (Cn A \<union> Cn B) = Cn (A \<union> B)"
equ_1: shows "subtopology (nsphere n) (equator n) = nsphere (n - Suc 0)"
exN: fixes u :: "'M"   and v :: "'M" assumes "cbox u v \<in> \<D>" shows "\<exists>n. \<forall>i\<in>Basis. v \<bullet> i - u \<bullet> i = 2 * c / 2 ^ n"
hcomp_obj_arr: fixes b :: "'a"   and f :: "'a" assumes "obj b"   and "arr f"   and "b = trg f" shows "b \<star> f = f"
float_gt1_scale: fixes m :: "int"   and e :: "int" assumes "1 \<le> Float m e" shows "0 \<le> e + (bitlen m - 1)"
mHom_mId_bijec: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'i"   and g :: "'i \<Rightarrow> 'e" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N"   and "g \<in> mHom R N M"   and "compose (carrier M) g f = mId\<^bsub>M\<^esub> "   and "compose (carrier N) f g = mId\<^bsub>N\<^esub> " shows "bijec\<^bsub>M,N\<^esub> f"
reachable_imp_dfs: fixes y :: "'a"   and xs :: "'a list" assumes "y \<in> reach (set \<Sigma>) \<delta> q\<^sub>0"   and "list_all (\<lambda>q. q \<in> reach (set \<Sigma>) \<delta> q\<^sub>0) xs"   and "y \<in> reachable (set xs)" shows "\<exists>y'. id y' = id y \<and> y' \<in> dfs {} xs"
spectral_radius_ev: fixes A :: "((complex, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" shows "\<exists>ev v. eigen_vector A v ev \<and> cmod ev = HMA_Connect.spectral_radius A"
consis_sb: shows "sharing_consistent \<S> \<O> (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
c_def: shows "(poly p has_sgnx c) (at_right a)"
elims: fixes x :: "nat"   and xa :: "real"   and xb :: "nat"   and y :: "real" assumes "power_up x xa xb = y" obtains p :: "nat" and xc :: "real" where "x = p__" and "xa = xc__" and "xb = 0" and "y = 1"   | p :: "nat" and xc :: "real" and n :: "nat" where "x = p" and "xa = xc" and "xb = Suc n" and "y = (if odd n then truncate_up p ((power_up p xc (Suc n div 2))\<^sup>2) else truncate_up p (xc * power_up p xc n))"
not_le_imp_less: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.linorder less_eq less"   and "\<not> less_eq y x" shows "less x y"
SN_rel_imp_SN_relto: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" assumes "SN_rel R S" shows "SN (S\<^sup>* O R O S\<^sup>*)"
A_mk: shows "A $ m $ mod_type_class.from_nat k \<noteq> (0::'a)"