hcju/isabelleps huggingface dataset:
intro: fixes adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "adopt_node = l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node get_owner_document get_parent remove_child get_disconnected_nodes set_disconnected_nodes"   and "adopt_node_locs = l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node_locs remove_child_locs set_disconnected_nodes_locs" shows "l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_owner_document get_parent remove_child remove_child_locs get_disconnected_nodes set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs"
tok_prikey_secret: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_tok_prik" shows "Tok_PriKey n \<notin> spied s"
intro: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set" assumes "nominal_ts satisfies transition"   and "effect_nominal_ts_axioms effect_apply L" shows "effect_nominal_ts satisfies transition effect_apply L"
compact_space_product_topology: fixes X :: "'a \<Rightarrow> 'b topology"   and I :: "'a set" shows "compact_space (product_topology X I) = (topspace (product_topology X I) = {} \<or> (\<forall>i\<in>I. compact_space (X i)))"
axioms_1: fixes carrier :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "finite_total_preorder_on carrier le" shows "total_preorder_on carrier le"
is_functor: shows "functor (\<cdot>) CC.comp \<Delta>.map"
connectedD: fixes A :: "'b set"   and U :: "'b set"   and V :: "'b set" assumes "connected A"   and "open U"   and "open V"   and "U \<inter> V \<inter> A = {}"   and "A \<subseteq> U \<union> V" shows "U \<inter> A = {} \<or> V \<inter> A = {}"
size_Diff_singleton: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "size (remove1_mset x M) = size M - 1"
abs_eq: fixes x :: "('a, 'b) RBT.rbt" shows "RBT_Mapping.Mapping x = mapping.Mapping (RBT.lookup x)"
Lim_at_imp_Lim_at_within: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and x :: "'a"   and S :: "'a set" assumes "f \<midarrow>x\<rightarrow> l" shows "(f \<longlongrightarrow> l) (at x within S)"
dord_pm_zero_min: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and s :: "'a \<Rightarrow>\<^sub>0 'b" assumes "class.linorder less_eq less"   and "\<And>t. ord t t" shows "linorder.dord_pm ord 0 s"
eucl_le_Basis_list_iff: fixes x :: "'b"   and y :: "'b" shows "(x \<le> y) = (\<forall>i<DIM('b). x \<bullet> Basis_list ! i \<le> y \<bullet> Basis_list ! i)"
sep_conj_comI: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "(P ** Q) s" shows "(Q ** P) s"
trace_rest_return_pmf_None: fixes rest :: "('f, 'c, 'a, 'd, 'b, 'e, 'g) rest_scheme"   and tr :: "('a \<times> 'b \<times> 'c list + 'd \<times> 'e \<times> 'c list) list" shows "trace_rest' rest (return_pmf None) tr = (\<lambda>_. return_pmf None, \<lambda>_. return_pmf None)"
simps: fixes as :: "('a \<times> 'b atom) set"   and bi :: "((nat \<Rightarrow> 'b option) \<times> (nat \<Rightarrow> 'b option)) \<times> (nat \<Rightarrow> 'a) \<times> (nat \<Rightarrow> 'a)" shows "as \<Turnstile>\<^sub>i bi = (\<forall>I v. (I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as \<longrightarrow> (I, v) \<Turnstile>\<^sub>i\<^sub>b bi)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "idom_hom hom" shows "field_hom hom"
simps_2: fixes e :: "aexp"   and sa :: "'av Abs_State.st" shows "aval'' e (Some sa) = aval' e sa"
vars_def: shows "vars \<equiv> (inp, mid, out, q1, q2)"
foreach_refine_gen: fixes adjs :: "('v \<times> nat) list"   and u :: "'v"   and Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option" assumes "set adjs \<subseteq> {(v, d). (u, v) \<in> edges g \<and> w {u, v} = d}" shows "foreach u adjs (Q, \<pi>) = (Qigen Q \<pi> u adjs, \<pi>'gen Q \<pi> u adjs)"
stream_of_llist_def: shows "stream_of_llist = unfold_stream lhd ltl"
sympartp_sympartp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "sympartp (sympartp r) = sympartp r"
msub_addition: fixes M :: "('e, 'a, 'f) Module_scheme"   and M1 :: "('e, 'a, 'f) Module_scheme"   and x :: "'e"   and y :: "'e" assumes "R module M"   and "R module M1"   and "msubmodule R M M1"   and "x \<in> carrier M1"   and "y \<in> carrier M1" shows "x \<plusminus>\<^bsub>M1\<^esub> y = x \<plusminus>\<^bsub>M\<^esub> y"
unstream_simps_1: fixes g :: "('e, 'f) generator"   and s :: "'f" assumes "generator g s = Done" shows "unstream g s = []"
filtercomap: fixes f' :: "'b \<Rightarrow> 'c"   and F'' :: "'b filter"   and g' :: "'b \<Rightarrow> 'c"   and h' :: "'d \<Rightarrow> 'b" assumes "f' \<in> O[F''](g')" shows "(\<lambda>x. f' (h' x)) \<in> O[filtercomap h' F''](\<lambda>x. g' (h' x))"
prime_ge_1_nat: fixes p :: "nat" assumes "prime p" shows "1 \<le> p"
xobj: shows "X \<in> obj\<^bsub>C\<^esub>"
cases: fixes x :: "'b list" obtains x :: "'b" and xs :: "'b list" where "xa__ = x # xs"   | "xa__ = []"
ltl_ldrop: fixes n :: "enat"   and xs :: "'b llist" shows "ltl (ldrop n xs) = ldrop n (ltl xs)"
w_in_hom_1: shows "\<guillemotleft>\<mu>.chine : src r\<^sub>0 \<rightarrow> src s\<^sub>0\<guillemotright>"
unique_half_chamber_system_g: fixes f' :: "'a \<Rightarrow> 'a"   and g' :: "'a \<Rightarrow> 'a" assumes "OpposedThinChamberComplexFoldings X f' g' C0"   and "g' ` C0 = D0" shows "g' \<turnstile> folding_g.\<C> = g \<turnstile> folding_g.\<C>"
subs: shows "P \<turnstile> Ts' [\<le>] (T # Ts)"
infplays_augment_one: fixes \<sigma> :: "'b list \<Rightarrow> 'b"   and p :: "'b list" shows "strategy_progression \<sigma> (augment_list \<sigma> p) = strategy_progression \<sigma> p"
simps_1: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and x :: "'b"   and xs :: "'b list" shows "not_MT C (x # xs) = (if dom (C x) = {} then not_MT C xs else True)"
Cons_1: fixes i :: "nat"   and j :: "nat"   and thesis :: "bool" assumes "\<And>i' j'. \<lbrakk>i' < j'; i' < length xs; j' < length xs; filter P xs ! i = xs ! i'; filter P xs ! j = xs ! j'\<rbrakk> \<Longrightarrow> thesis"   and "i < j"   and "i < length (filter P xs)"   and "j < length (filter P xs)" shows "thesis"
neg_distr_cond_bset_left_unique: fixes D :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and tytok :: "'f itself" assumes "left_unique D" shows "neg_distr_cond_bset C D tytok"
part_equiv_trans: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "part_equiv R"   and "(x, y) \<in> R"   and "(y, z) \<in> R" shows "(x, z) \<in> R"
Inr: fixes sh' :: "'b" shows "lunstream (append_raw g h sh) (Inr sh') = lunstream h sh'"
mem_box_cart_2: fixes x :: "(real, 'a) vec"   and a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" shows "(x \<in> cbox a b) = (\<forall>i. a $ i \<le> x $ i \<and> x $ i \<le> b $ i)"
lconcat_simps_1: shows "lconcat LNil = LNil"
member_def: fixes dxs :: "'d dlist" shows "member dxs = List.member (list_of_dlist dxs)"
append: fixes xs :: "'a list"   and ys :: "'a list"   and r :: "('a \<times> 'a) set"   and zs :: "'a list" assumes "(xs, ys) \<in> genPrefix r" shows "(xs, ys @ zs) \<in> genPrefix r"
False: shows "\<not> (\<forall>C. T \<noteq> Class C)"
flow_cong: fixes Y :: "('a \<Rightarrow>\<^sub>L 'a) set"   and x0a :: "'a"   and S :: "real set"   and g :: "real \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a"   and t :: "real"   and t0 :: "real"   and x0 :: "'a \<Rightarrow>\<^sub>L 'a" assumes "UNIV = Y"   and "Y.existence_ivl0 x0a = S"   and "\<And>x t. \<lbrakk>x \<in> Y; t \<in> S\<rbrakk> \<Longrightarrow> Y.vareq x0a t o\<^sub>L x = g t x"   and "t \<in> Y.mvar.existence_ivl x0a t0 x0" shows "Y.mvar.flow x0a t0 x0 t = ll_on_open.flow S g Y t0 x0 t"
init_rev_wrap_4: fixes P :: "'b list \<Rightarrow> bool" shows "\<exists>y. P y \<equiv> Ex_rev_wrap P"
ide_in_hom: fixes a :: "'s" shows "ide a = \<guillemotleft>a : a \<rightarrow> a\<guillemotright>"
theNFTBoth_sym: fixes cl :: "('a, 'b, 'c) cmd list"   and dl :: "('d, 'e, 'f) cmd list" shows "theNFTBoth cl dl = theNFTBoth dl cl"
disjoint_interval_4: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "(box a b \<inter> box c d = {}) = (\<exists>i\<in>Basis. b \<bullet> i \<le> a \<bullet> i \<or> d \<bullet> i \<le> c \<bullet> i \<or> b \<bullet> i \<le> c \<bullet> i \<or> d \<bullet> i \<le> a \<bullet> i)"
image_ident: fixes Y :: "'c set" shows "{x |. x \<in> Y} = Y"
some_trg_cod: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "some_trg (cod \<mu>) = some_trg \<mu>"
asIMOD_igSubstIPresIGWls: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSubstIPresIGWls (asIMOD SEM)"
minus_uexpr_def: fixes u :: "('e, 'f) uexpr"   and v :: "('e, 'f) uexpr" shows "u - v = bop (-) u v"
inject: fixes x :: "'b \<Rightarrow> 'c \<times> 'b"   and ya :: "'b \<Rightarrow> 'c \<times> 'b" shows "(State x = State ya) = (x = ya)"
right_refines_DC: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "(a \<Sqinter> b) \<sqsubseteq> b"
goodInp_def: fixes inp :: "'i \<Rightarrow> ('j, 'k, 'varSort, 'var, 'l) qTerm set option" shows "goodInp inp \<equiv> liftAll good inp \<and> |{i. inp i \<noteq> None}| <o |UNIV|"
HPhase0Read_HInv5_q2: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase0Read s s' p d"   and "p \<noteq> q"   and "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s q) < mbal (disk s d qq) \<and> \<not> hasRead s q d qq" shows "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s' q) < mbal (disk s' d qq) \<and> \<not> hasRead s' q d qq"
S: shows "S = (\<forall><:S\<^sub>1. S\<^sub>2)"
PUpdateApply_def: fixes U :: "'b pupdate"   and D :: "'b data" shows "(U !! D) = Rep_pupdate U D"
const_mono: fixes ord :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'f option" shows "monotone ord option_ord (\<lambda>f. c)"
healthy_monoD2: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real" assumes "healthy t"   and "sound P"   and "sound Q"   and "P \<tturnstile> Q" shows "t P \<tturnstile> t Q"
bisim1Seq2: fixes P :: "'a prog"   and h :: "'heap"   and e2 :: "'addr expr1"   and exs :: "'addr expr1 \<times> 'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e1 :: "'addr expr1" assumes "P,e2,h \<turnstile> exs \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e1;; e2,h \<turnstile> exs \<leftrightarrow> (stk, loc, Suc (length (compE2 e1) + pc), xcp)"
neq_Nil_conv: fixes xs :: "'f list" shows "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
MClkRetry_def: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and p :: "PrIds" shows "MClkRetry send rcv cst p = ($(cst!p) = #clkB \<and> res<$(rcv!p)> = #RPCFailure \<and> Call rcv p MClkRelayArg<arg<send!p>> \<and> unchanged (cst!p, rtrner send!p))"
of_nat_eq_1_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero n = one) = (n = 1)"
induct: fixes P4 :: "(cell \<times> interface) list \<Rightarrow> bool"   and P1 :: "sdiagram \<Rightarrow> bool"   and P2 :: "cell \<Rightarrow> bool"   and P3 :: "cell \<times> interface \<Rightarrow> bool"   and cell_interface_prod_list :: "(cell \<times> interface) list" assumes "\<And>x. P4 x \<Longrightarrow> P1 (SDiagram x)"   and "\<And>x. P2 (Filler x)"   and "\<And>x1a x2 x3. P2 (Basic x1a x2 x3)"   and "\<And>x1a x2. P1 x2 \<Longrightarrow> P2 (Exists_sdia x1a x2)"   and "\<And>x1a x2 x3 x4. \<lbrakk>P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P2 (Choose_sdia x1a x2 x3 x4)"   and "\<And>x1a x2 x3. P1 x2 \<Longrightarrow> P2 (Loop_sdia x1a x2 x3)"   and "\<And>x1a x2. P2 x1a \<Longrightarrow> P3 (x1a, x2)"   and "P4 []"   and "\<And>x1a x2. \<lbrakk>P3 x1a; P4 x2\<rbrakk> \<Longrightarrow> P4 (x1a # x2)" shows "P4 cell_interface_prod_list"
R: shows "equivalence_of_categories (\<cdot>) (\<cdot>) F (\<lambda>f. f \<otimes> \<I>) \<eta> \<epsilon>"
set_borel_integral_eq_integral_1: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "set_integrable lborel S f" shows "f integrable_on S"
seq_extract_out_of_range: fixes A :: "nat set"   and xs :: "'b list" assumes "A \<inter> dom\<^sub>l xs = {}" shows "A \<upharpoonleft>\<^sub>l xs = []"
lit_atoms_cases: fixes xa :: "'b literal" shows "atoms_of_lit xa = (case xa of xda\<^sup>+ \<Rightarrow> xda | xea\<inverse> \<Rightarrow> xea)"
wp_whileI': fixes R :: "((char list \<Rightarrow> int \<Rightarrow> int) \<times> (char list \<Rightarrow> int \<Rightarrow> int)) set"   and I :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s\<^sub>0 :: "char list \<Rightarrow> int \<Rightarrow> int"   and b :: "bexp"   and \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "wf R"   and "I s\<^sub>0"   and "\<And>s. I s \<Longrightarrow> if bval b s then wp \<pi> c (\<lambda>s'. I s' \<and> (s', s) \<in> R) s else Q s" shows "wp \<pi> (WHILE b DO c) Q s\<^sub>0"
assms_3: shows "op\<^sub>2 \<in> set (\<Pi>\<^sub>\<O>)"
topological_space_axioms: shows "topological_space Spec is_zariski_open"
properfactor_prod_l: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group.comm_monoid G"   and "properfactor G a b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "properfactor G a (c \<otimes>\<^bsub>G\<^esub> b)"
sub_non_positive: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (neg_numeral.sub minus plus one n m) zero = (n \<le> m)"
span_induct': fixes p :: "'b"   and B :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "p \<in> span B"   and "P (0::'b)"   and "\<And>a q p. \<lbrakk>a \<in> span B; P a; p \<in> B; q \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> P (a + q *s p)" shows "P p"
get_return_edge_call: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "get_return_edges a \<noteq> {}"
bisimilar_def: fixes a :: "('h, 'i) Stream"   and b :: "('h, 'j) Stream" shows "(a \<approx> b) = (unstream\<cdot>a = unstream\<cdot>b \<and> a \<noteq> \<bottom> \<and> b \<noteq> \<bottom>)"
cases: fixes Enc :: "'a \<Rightarrow> 'b"   and a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc" assumes "encoding.indRelLp Enc a1 a2" obtains S :: "'a" where "a1 = TargetTerm (Enc S)" and "a2 = SourceTerm S"
span_insert_subset: fixes A :: "'a set"   and B :: "'a set"   and r :: "'a" assumes "More_Modules.ideal A \<subseteq> More_Modules.ideal B"   and "r \<in> More_Modules.ideal B" shows "More_Modules.ideal (insert r A) \<subseteq> More_Modules.ideal B"
interchange: fixes h :: "'d"   and g :: "'d"   and h' :: "'d"   and g' :: "'d" assumes "D.seq h g"   and "D.seq h' g'" shows "(h \<otimes>\<^sub>D h') \<cdot>\<^sub>D (g \<otimes>\<^sub>D g') = h \<cdot>\<^sub>D g \<otimes>\<^sub>D h' \<cdot>\<^sub>D g'"
multrow_mat_def: fixes n :: "nat"   and k :: "nat"   and a :: "'b" shows "multrow_mat n k a = mat n n (\<lambda>(i, j). if k = i \<and> k = j then a else if i = j then 1::'b else (0::'b))"
dims_A: fixes ws :: "nat \<Rightarrow> real" shows "dims (A ws) = replicate (2 * N_half) (last rs)"
fMax_def: shows "fMax \<equiv> semilattice_fset.F max"
intro: fixes R :: "('a, 'b) ring_scheme" assumes "domain R" shows "UP_domain R"
assms_2: shows "\<not> Carmichael_number n \<or> n < 3"
subst_closed_except_id: fixes t :: "'a"   and S :: "name fset"   and env :: "(name, 'a) fmap" assumes "closed_except t S"   and "fdisjnt (fmdom env) S" shows "subst t env = t"
pred_sexp_t1: fixes a :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set" assumes "a \<in> sexp"   and "N \<in> sexp" shows "(a, Scons a N) \<in> pred_sexp\<^sup>+"
LN_4: fixes k :: "nat"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "k' + 1 + ka__ = k' + k"   and "\<S>\<bullet>\<C> \<turnstile> lfilledk : ts _> ts'" shows "k < length (label \<C>)"
normal_rconjby_closed: fixes H :: "'g set"   and g :: "'g"   and h :: "'g" assumes "Group H \<and> order.greater_eq G H"   and "normal H"   and "g \<in> G"   and "h \<in> H" shows "rconjby g h \<in> H"
numeral_4_eq_4: shows "4 = Suc (Suc (Suc (Suc 0)))"
eq_monom_sum_var: fixes m :: "'b monom"   and n :: "'b monom" shows "(m = n) = (\<forall>x. sum_var m x = sum_var n x)"
assms: fixes p :: "nat"   and k :: "nat" assumes "prime p"   and "0 < k" shows "f (p ^ k) = f p ^ k"
prefixE: fixes xs :: "'b list"   and ys :: "'b list" assumes "prefix xs ys" obtains zs :: "'b list" where "ys = xs @ zs"
IH: assumes "x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F" shows "fv (\<theta> x) \<subseteq> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>)"
assms_4: fixes x :: "'c" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x" shows "h \<turnstile> g x \<rightarrow>\<^sub>r y = h' \<turnstile> g x \<rightarrow>\<^sub>r y'"
coboundedI1: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<le> a" shows "c \<le> max a b"
M_4: fixes vs :: "nat \<Rightarrow> nat" shows "\<exists>vs'. ((Suc 0, vs), 0, vs') \<in> (step M)\<^sup>* \<and> vs' 0 = vs 0 \<and> vs' 1 = vs 1 \<and> vs' 2 = g (vs 0) (vs 1)"
ProcRec1: fixes \<Gamma> :: "'f \<Rightarrow> ('g, 'f, 'h) com option"   and \<Theta> :: "('g set \<times> 'f \<times> 'g set \<times> 'g set) set"   and P :: "'e \<Rightarrow> 'g set"   and p :: "'f"   and Q :: "'e \<Rightarrow> 'g set"   and A :: "'e \<Rightarrow> 'g set"   and F :: "'h set" assumes "\<forall>Z. \<Gamma>,\<Theta> \<union> (\<Union>Z. {(P Z, p, Q Z, A Z)})\<turnstile>\<^bsub>/F \<^esub>(P Z) the (\<Gamma> p) (Q Z),(A Z)"   and "p \<in> dom \<Gamma>" shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P Z) Call p (Q Z),(A Z)"
conn: fixes y :: "'b" assumes "y \<in> T" shows "connected {x \<in> S. f x = y}"
compact_imp_compactin_subtopology: fixes X :: "'a topology"   and K :: "'a set"   and S :: "'a set" assumes "compactin X K"   and "K \<subseteq> S" shows "compactin (subtopology X S) K"
assms_1: fixes a :: "'a"   and b :: "'a" assumes "P a b" shows "P b a"
assms_4: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and zs :: "'a list" assumes "xs = x # y # z # zs" shows "P"
c1_manifold_atlas_eucl: fixes k :: "enat" shows "c_manifold charts_eucl k"
matching_srcs_filterW: fixes filterW :: "'i simple_rule list"   and s :: "'i word"   and d :: "'i word" assumes "filterW = filter (\<lambda>r. simple_conn_matches (match_sel r) c) rs" shows "(s \<in> wordinterval_to_set (matching_srcs d filterW Empty_WordInterval)) = (runFw s d c filterW = Decision FinalAllow)"
term_variants_pred_inv_1: fixes P :: "'c \<Rightarrow> 'c list"   and f :: "'c"   and T :: "('c, 'd) Term.term list"   and h :: "'c"   and S :: "('c, 'd) Term.term list" assumes "term_variants_pred P (Fun f T) (Fun h S)" shows "length T = length S"
swap_at_base_simps_1: fixes x :: "'a"   and y :: "'a" assumes "sort_of (atom x) = sort_of (atom y)" shows "(atom x \<rightleftharpoons> atom y) \<bullet> x = y"
Map_dom: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "A_B.arr f" shows "A_B.Map (A_B.dom f) = A_B.Dom f"
prems: shows "\<forall>P\<in>insert P B. card P \<le> k"
right_adjoint_is_ide: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.is_right_adjoint f" shows "Span.ide f"
to_rat_real_def: fixes x :: "real" shows "to_rat x = (if x \<in> \<rat> then THE y. x = real_of_rat y else 0)"
compactin_subset_topspace: fixes X :: "'a topology"   and S :: "'a set" assumes "compactin X S" shows "S \<subseteq> topspace X"
assoc_simps_1: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.arr \<a>\<^sub>D[f, g, h]"
\<alpha>_props_2: shows "map_to_set m = set (map (\<lambda>(k, v). (k, \<alpha> v)) lc)"
advantage2_def: fixes \<A> :: "'a \<Rightarrow> (nat \<times> nat) spmf" shows "local.dis_log_alt.advantage2 \<A> \<equiv> spmf (local.dis_log_alt.dis_log2 \<A>) True"
dflt_None_set_def: fixes S :: "'a set" shows "dflt_None_set S \<equiv> if S = {} then None else Some S"
OAlist_lookup_ntm_single: fixes k :: "'c"   and v :: "'b"   and ko :: "'c nat_term_order"   and k' :: "'c" shows "OAlist_lookup_ntm (oalist_of_list_ntm ([(k, v)], ko)) k' = (if k = k' then v else (0::'b))"
mult_locally_cancelL: fixes s :: "('a \<times> 'a) set"   and X :: "'a multiset"   and Z :: "'a multiset"   and Y :: "'a multiset" assumes "trans s"   and "locally_irrefl s (X + Z)"   and "locally_irrefl s (Y + Z)" shows "((Z + X, Z + Y) \<in> mult s) = ((X, Y) \<in> mult s)"
simps_1: fixes nodes :: "'b set"   and edges :: "('b \<times> 'b) set"   and more :: "'c" shows "nodes \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = nodes"
smooth_on_id: fixes k :: "enat"   and S :: "'c set" shows "k-smooth_on S (\<lambda>x. x)"
ucast_nat_def: fixes x :: "'c word" shows "word_of_nat (unat x) = ucast x"
\<alpha>_def: fixes \<mu>\<nu>\<tau> :: "'a \<times> 'a \<times> 'a" shows "B.\<alpha> \<mu>\<nu>\<tau> \<equiv> \<a>\<^sub>B[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)]"
map\<^sub>0_def: fixes a :: "'a" shows "F.map\<^sub>0 a \<equiv> src\<^sub>D (HoF.map a)"
rep_eq: shows "Rep_blinop 1 = 1\<^sub>L"
add_right_imp_eq: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and a :: "'d"   and c :: "'d" assumes "class.cancel_semigroup_add plus"   and "plus b a = plus c a" shows "b = c"
sorted_wrt_true: fixes xs :: "'a list" shows "sorted_wrt (\<lambda>_ _. True) xs"