hcju/isabelleps huggingface dataset:
bit_drop_bit_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "bit (drop_bit n a) = bit a \<circ> (+) n"
AddSA_CompFun_PseudoHA_ran: fixes S :: "'d"   and RootSA :: "('d, 'e, 'f) seqauto"   and SA :: "('d, 'e, 'f) seqauto"   and D :: "'f data" assumes "S \<in> States RootSA"   and "States RootSA \<inter> States SA = {}" shows "ran (CompFun (PseudoHA RootSA D [++] (S, SA))) = {{}, {SA}}"
perm_eqI: fixes f :: "'a perm"   and g :: "'a perm" assumes "\<And>a. f \<langle>$\<rangle> a = g \<langle>$\<rangle> a" shows "f = g"
insert_eq_iff: fixes a :: "'c"   and A :: "'c set"   and b :: "'c"   and B :: "'c set" assumes "a \<notin> A"   and "b \<notin> B" shows "({a} \<union> A = {b} \<union> B) = (if a = b then A = B else \<exists>C. A = {b} \<union> C \<and> b \<notin> C \<and> B = {a} \<union> C \<and> a \<notin> C)"
rbt_comp_lookup: shows "local.rbt_comp_lookup = ord.rbt_lookup (lt_of_comp c)"
update_assignment_alt: fixes u :: "('c \<times> 'd) list"   and as :: "'c \<Rightarrow> 'd" shows "update_assignment u as = update_assignment_alt u as"
V_alt_def: shows "{u. \<forall>c. 0 < v c \<and> v c \<le> n \<longrightarrow> (0::'d) \<le> u c} = interp.V"
segment_inc: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<prec> b = (a \<in> segment D b)"
set_eq_iff: fixes A :: "'c set"   and B :: "'c set" shows "(A = B) = (\<forall>x. (x \<in> A) = (x \<in> B))"
cod_simps_3: shows "f.cod.leg1 = fgh.\<mu>.leg1"
IH2: shows "interp_ins (before @ (oid, ref) # op_list) = xs @ ys @ zs"
bal_exec: fixes t :: "'a tree23"   and as :: "'a act list" assumes "bal t" shows "bal (exec as t)"
T_meas: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "qmpt M T" shows "T \<in> M \<rightarrow>\<^sub>M M"
conj_zero_right: fixes x :: "'a" shows "x \<^bold>\<squnion> \<one> = \<one>"
assms_11: shows "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"
gen_span_closed: fixes gens :: "'a set" assumes "gens \<subseteq> carrier G" shows "\<langle>gens\<rangle> \<subseteq> carrier G"
alw_safe_combined: fixes s :: "('b, 't, 'loc) Combined.configuration stream" assumes "FullSpec s" shows "alw (holds safe_combined) s"
nstd_case3: fixes c :: "cell list"   and a :: "nat"   and b :: "cell list" assumes "\<forall>rs n. c \<noteq> Oc \<up> Suc rs @ Bk \<up> n" shows "NSTD (trpl_code (a, b, c))"
assms: shows "\<Psi> \<rhd> P \<sim>\<^sub>s Q"
sel_3: fixes x1 :: "((('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) list"   and x2 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x3 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x4 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x5 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x6 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list" shows "transaction_selects (Transaction x1 x2 x3 x4 x5 x6) = x3"
\<psi>_naturality: fixes f :: "'c"   and x :: "'c"   and x' :: "'c"   and g :: "'d"   and y' :: "'d"   and y :: "'d"   and h :: "'d" assumes "\<guillemotleft>f : x \<rightarrow>\<^sub>C x'\<guillemotright>"   and "\<guillemotleft>g : y' \<rightarrow>\<^sub>D y\<guillemotright>"   and "\<guillemotleft>h : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "f \<cdot>\<^sub>C \<psi> x h \<cdot>\<^sub>C F g = \<psi> x' (G f \<cdot>\<^sub>D h \<cdot>\<^sub>D g)"
ring_simprules_1: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
a: shows "n x \<cdot> n z = n y \<cdot> n z"
"1_1": assumes "p \<noteq> 0"   and "find_adds fs (lt p) = None" shows "(red (set fs))\<^sup>*\<^sup>* (tail p) (trd_aux fs (tail p) (r + monomial (lc p) (lt p)) - (r + monomial (lc p) (lt p)))"
i_2: shows "remdups_adj (map ((\<inter>) K) \<GG>) ! j = map ((\<inter>) K) \<GG> ! i"
ls2: shows "xs = take i xs @ xs ! i # drop (Suc i) xs"
inputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "Early_Semantics.residual" assumes "a<b>.P \<longmapsto>\<^sub>e Rs" obtains x :: "name" and aa :: "name" and u :: "name" and Pa :: "pi" where "a = aa \<and> [b].P = [x].Pa" and "Rs = aa<u> \<prec>\<^sub>e Pa[x::=u]" and "x \<noteq> aa" and "x \<noteq> u"
simps_1: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('b, 'a, 'c) Stmt"   and uu :: "'b ModeUpd" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B" shows "sifum_types.has_annotated_stop (c@[uu]) = (if c = Stop then True else sifum_types.has_annotated_stop c)"
rplusinf_\<Upsilon>: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and x :: "real" assumes "isrlfm p"   and "\<not> Ifm (a # bs) (plusinf p)"   and "Ifm (x # bs) p" shows "\<exists>(s, m)\<in>set (\<Upsilon> p). x \<le> Inum (a # bs) s / real_of_int m"
z_def: shows "z = [] \<odot> 0\<^sup>\<infinity>"
get_scdom_component_different_owner_documents: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and owner_document :: "(_) document_ptr"   and ptr' :: "(_) object_ptr"   and owner_document' :: "(_) document_ptr" assumes "heap_is_wellformed h"   and "DocumentClass.type_wf h"   and "DocumentClass.known_ptrs h"   and "h \<turnstile> get_owner_document ptr \<rightarrow>\<^sub>r owner_document"   and "h \<turnstile> get_owner_document ptr' \<rightarrow>\<^sub>r owner_document'"   and "owner_document \<noteq> owner_document'" shows "set |h \<turnstile> get_scdom_component ptr|\<^sub>r \<inter> set |h \<turnstile> get_scdom_component ptr'|\<^sub>r = {}"
assms_2: fixes x :: "'a" assumes "x \<in> X" shows "supp x \<subseteq> S"
algebraic_nth_root: fixes n :: "nat"   and x :: "'a"   and y :: "'a" assumes "n \<noteq> 0"   and "algebraic x"   and "y ^ n = x" shows "algebraic y"
vertices_cases: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and v :: "'b \<times> nat list" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "v |\<in>| Solved_Task.vertices freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" obtains c :: "'b" where "c__ \<in> set conclusions" and "v = (c__, [])"   | c :: "'b" and "is" :: "nat list" where "c \<in> set conclusions" and "is \<in> it_paths (Abstract_Task.globalize freshenLC subst_renameLCs anyP antecedent [fidx (Abstract_Task.conc_forms conclusions) c, 0] (freshenLC v_away) (Abstract_Task.to_it freshenLC renameLCs lconsts subst subst_lconsts anyP antecedent consequent rules assumptions conclusions (Solved_Task.ts freshenLC renameLCs lconsts subst subst_lconsts antecedent consequent rules assumptions c)))" and "v = (c, 0 # is)"
orthogonal_clauses_2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and a :: "'b"   and x :: "'b"   and c :: "real" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner"   and "real_inner.orthogonal inner a x" shows "real_inner.orthogonal inner a (scaleR c x)"
remove1_insort: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and xs :: "'k list" assumes "class.linorder less_eq less" shows "remove1 x (linorder.insort_key less_eq (\<lambda>x. x) x xs) = xs"
is_aligned_nth: fixes p :: "'b word"   and m :: "nat" shows "is_aligned p m = (\<forall>n<m. \<not> p !! n)"
hd_zip: fixes xs :: "'b list"   and ys :: "'c list" assumes "xs \<noteq> []"   and "ys \<noteq> []" shows "hd (zip xs ys) = (hd xs, hd ys)"
all: fixes us :: "'a list list" assumes "us \<in> lists G\<^sub>+ \<and> concat us = \<epsilon>" shows "us = \<epsilon>"
simps_1: fixes uu :: "'b \<Rightarrow> 'b" shows "update_rightmost uu Leaf = Leaf"
\<tau>Move2_compP2: fixes P :: "'addr J1_prog"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and body :: "'addr expr1"   and D :: "String.literal"   and xcp :: "'addr option"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and frs :: "'addr frame list" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = \<lfloor>body\<rfloor> in D" shows "\<tau>Move2 (compP2 P) (xcp, h, (stk, loc, C, M, pc) # frs) = (case xcp of None \<Rightarrow> \<tau>move2 P h stk body pc xcp \<or> pc = length (compE2 body) | \<lfloor>a\<rfloor> \<Rightarrow> pc < Suc (length (compE2 body)))"
set_plus_rearrange: fixes a :: "'c"   and C :: "'c set"   and b :: "'c"   and D :: "'c set" shows "a +o C + b +o D = (a + b) +o (C + D)"
assms: shows "\<forall>A\<in>carrier_mat 1 2. admits_diagonal_reduction A"
WhileCondThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and c :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
exhaust: fixes y :: "Javatype" obtains "y = BoolT"   | "y = IntgT"   | "y = ShortT"   | "y = ByteT"   | "y = NullT"   | x6 :: "Arraytype" where "y = ArrT x6"   | x7 :: "CTypeId" where "y = CClassT x7"   | x8 :: "ATypeId" where "y = AClassT x8"   | x9 :: "ITypeId" where "y = InterfaceT x9"
equalityI: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
prim_pyth_triple_commute: fixes x :: "nat"   and y :: "nat"   and z :: "nat" shows "prim_pyth_triple x y z = prim_pyth_triple y x z"
borel_measurable_uminus_eq: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" shows "((\<lambda>x. - f x) \<in> borel_measurable M) = (f \<in> borel_measurable M)"
ipurge_r_def: fixes execs :: "'dom_t \<Rightarrow> 'action_t list list"   and u :: "'dom_t" shows "ipurge_r execs u \<equiv> \<lambda>d. if intermediary d u then [] else if ind_source d u then SOME alpha. realistic_execution alpha else if d = u then remove_gateway_communications u (execs u) else execs d"
ChiPlus_ChiRel: fixes S :: "'a"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "(S, T) \<in> ChiRel A" shows "(S, T) \<in> ChiPlus A"
AssumeH_4: fixes A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and H :: "fm set" shows "insert A1 (insert A2 (insert A3 (insert A4 H))) \<turnstile> A4"
prefix_singleton_FF: fixes x :: "'r"   and ys :: "[:'r:]" shows "(prefix\<cdot>[:x:]\<cdot>ys = FF) = (x \<noteq> \<bottom> \<and> (ys = [::] \<or> (\<exists>z zs. z \<noteq> \<bottom> \<and> zs \<noteq> \<bottom> \<and> ys = z :# zs \<and> x \<noteq> z)))"
refl: fixes ts :: "traffic" shows "ts \<^bold>\<Rightarrow> ts"
rel_star_contl: fixes X :: "('a \<times> 'b) set"   and Y :: "('b \<times> 'b) set" shows "X ; Y\<^sup>* = (\<Union>i. X ; rel_d.power Y i)"
amap_fm_list_disj: fixes h :: "'b \<Rightarrow> 'a fm"   and fs :: "'b fm list" shows "amap\<^sub>f\<^sub>m h (list_disj fs) = Disj fs (amap\<^sub>f\<^sub>m h)"
simps: fixes s :: "('c, 'd option) insert_message list"   and n :: "nat"   and \<sigma>' :: "'d"   and i :: "'c" shows "create_insert s n \<sigma>' i = nth_visible s n \<bind> (\<lambda>p. nth_visible s (n + 1) \<bind> (\<lambda>q. Inr (Insert (InsertMessage p i q \<sigma>'))))"
freshSets_5: fixes x :: "name"   and xvec :: "name list"   and C :: "'d" shows "(x # xvec) \<sharp>* C = (x \<sharp> C \<and> xvec \<sharp>* C)"
enum_alt_word_def: shows "enum_alt \<equiv> alt_from_ord enum"
Greatest_equality: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "class.order less_eq less"   and "P x"   and "\<And>y. P y \<Longrightarrow> less_eq y x" shows "order.Greatest less_eq P = x"
anga_not_flat: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" assumes "QCongAACute a" shows "QCongAnFlat a"
usesOf_def: fixes m :: "('b, 'a set) mapping"   and n :: "'b" shows "usesOf m n \<equiv> case Mapping.lookup m n of None \<Rightarrow> {} | Some x \<Rightarrow> id x"
rev_is_Nil_conv: fixes xs :: "'f list" shows "(rev xs = []) = (xs = [])"
intros_2: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>*\<^sup>* a c"
convergent_LIMSEQ_iff: fixes X :: "nat \<Rightarrow> 'b" shows "convergent X = X \<longlonglongrightarrow> lim X"
HighSec_def: shows "HighSec = (\<lambda>s t. noLowDPs s \<longrightarrow> s \<equiv>\<^sub>mkId (snd s) t)"
min_inf: fixes f :: "atom fm"   and xs :: "'a list" assumes "nqfree f" shows "\<exists>x. \<forall>y\<le>x. DLO.I (inf\<^sub>- f) xs = DLO.I f (y # xs)"
simps_3: fixes head :: "'b"   and x :: "'b"   and xs :: "'b list" shows "succ_rel (head # x # xs) = {(head, Some x)} \<union> succ_rel (x # xs)"
LLL_with_assms_axioms: shows "LLL_with_assms n m fs_init \<alpha>"
unl_Rex: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and w :: "'c" shows "(\<exists>x. A x) w \<equiv> \<exists>x. A x w"
cases: fixes a1 :: "trm"   and a2 :: "trm" assumes " a1 \<guillemotright> a2" obtains t :: "trm" and s :: "trm" where "a1 = App t s" and "a2 = t"
bdd_above_mono: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c set"   and A :: "'c set" assumes "class.preorder less_eq less"   and "preorder.bdd_above less_eq B"   and "A \<subseteq> B" shows "preorder.bdd_above less_eq A"
self_domain_iff_ide: fixes a :: "'a" shows "(a \<in> domains a) = ide a"
index_less: fixes x :: "'b"   and xs :: "'b list"   and n :: "nat" assumes "x \<in> set xs"   and "length xs \<le> n" shows "index xs x < n"
mutual_information_distr: fixes S :: "'d measure"   and T :: "'e measure"   and X :: "'a \<Rightarrow> 'd"   and Px :: "'d \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'e"   and Py :: "'e \<Rightarrow> real"   and Pxy :: "'d \<times> 'e \<Rightarrow> real" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "distributed M S X (\<lambda>x. ennreal (Px x))"   and "\<And>x. x \<in> space S \<Longrightarrow> 0 \<le> Px x"   and "distributed M T Y (\<lambda>x. ennreal (Py x))"   and "\<And>y. y \<in> space T \<Longrightarrow> 0 \<le> Py y"   and "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>x. ennreal (Pxy x))"   and "\<And>x y. \<lbrakk>x \<in> space S; y \<in> space T\<rbrakk> \<Longrightarrow> 0 \<le> Pxy (x, y)" shows "mutual_information b S T X Y = LINT x|(S \<Otimes>\<^sub>M T). Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
assms: shows "ct_suffix (xs @ ys) (to_ct_list zs)"
that_1: fixes c :: "('a, 'b) chart" assumes "c \<in> charts" shows "k-smooth_compat c1 c"
less_irrefl: fixes x :: "'t" shows "\<not> x \<prec>\<^sub>t x"
simps: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and C :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.mset_ecl (C, \<sigma>) = {#basic_superposition.mset_lit (subst_lit x \<sigma>). x \<in># mset_set (cl_ecl C)#}"
sup_bot_left: fixes sup :: "'l \<Rightarrow> 'l \<Rightarrow> 'l"   and less_eq :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and less :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and bot :: "'l"   and x :: "'l" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup bot x = x"
snd_matrix_to_iarray_echelon_form_of_column_k: fixes k :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" assumes "k < ncols A"   and "i \<le> nrows A" shows "snd (echelon_form_of_column_k bezout (A, i) k) = fst (snd (echelon_form_of_column_k_iarrays (matrix_to_iarray A, i, bezout) k))"
deMorgan1: fixes X :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Y :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "not (X and Y) = (not X or not Y)"
HMA_I_def: shows "HMA_Connect.HMA_I = (\<lambda>i a. i = Bij_Nat.to_nat a)"
push_forward_in_tangent_space: fixes p :: "'a" assumes "p \<in> sub.carrier" shows "inclusion.push_forward ` sub.tangent_space p \<subseteq> tangent_space p"
srtc_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "relation.srtc r a b \<equiv> (\<lambda>a b. r a b \<or> r b a)\<^sup>*\<^sup>* a b"
extensional0_restrict0: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "extensional0 A (restrict0 A f)"
nt_match_list_simp: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and a :: "action"   and p :: "'d"   and ms :: "'c negation_type list" shows "nt_match_list \<gamma> a p ms = ((\<forall>m\<in>set (getPos ms). matches \<gamma> (Match m) a p) \<and> (\<forall>m\<in>set (getNeg ms). matches \<gamma> (MatchNot (Match m)) a p))"
fix_eq: fixes F :: "'r \<rightarrow> 'r" shows "fix\<cdot>F = F\<cdot>(fix\<cdot>F)"
\<alpha>_def: shows "\<alpha> = \<Squnion> (value_flow \<Psi> ` {f. flow \<Psi> f})"
gen_mono: fixes I :: "'a set"   and J :: "'a set"   and P :: "'a set set" assumes "I \<subseteq> J" shows "gen P I \<subseteq> gen P J"
apartment_chamber: fixes Y :: "'a set set"   and C :: "'a set" assumes "Y \<in> \<A>"   and "SimplicialComplex.maxsimp Y C" shows "chamber C"
fstx: shows "fst x = lowest_vertex (fst X, nlex_pdevs (snd X))"
bind_commute_pmf: fixes A :: "'b pmf"   and B :: "'c pmf"   and C :: "'b \<Rightarrow> 'c \<Rightarrow> 'a pmf" shows "A \<bind> (\<lambda>x. B \<bind> C x) = B \<bind> (\<lambda>y. A \<bind> (\<lambda>x. C x y))"
homeomorphic_space_sym: fixes X :: "'d topology"   and Y :: "'e topology" shows "(X homeomorphic_space Y) = (Y homeomorphic_space X)"
IQ2: fixes l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "I l s"   and "bval b s" shows "pre C I l s \<and> 1 + preT C E s + time C s \<le> E s \<and> S s = S (postQ C s) on \<Union> (funStar SS ` (LVE \<union> LVQ))"
is_standard_operator'_def: shows "is_standard_operator' \<equiv> \<lambda>(uu_, uu_, effects, uu_). list_all is_standard_effect' effects"
axioms_1: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<Z> \<alpha>"
lookup_default': fixes x :: "'c"   and d :: "'b"   and m :: "('c, 'b) mapping"   and y :: "'c" shows "lookup (Mapping.default x d m) y = (if x = y then Some (lookup_default d m x) else lookup m y)"
cd_not_pd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and l :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_cdi prob l \<pi> k"   and "\<pi> l \<noteq> \<pi> k" shows "\<not> IFC_def.is_pd prob (\<pi> l) (\<pi> k)"
simple_path_image_uncountable: fixes g :: "real \<Rightarrow> 'a" assumes "simple_path g" shows "uncountable (path_image g)"
finite_set_fold_singleton: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and x :: "'c" shows "Finite_Set.fold f z {x} = f x z"
id_1: shows "(\<forall>j\<in>set (\<sigma> (f, Suc (length bef + length aft))). Fun f (bef @ s # aft) \<succ> (bef @ t # aft) ! j) = True"
path_distinct: fixes l1 :: "'vtx"   and l2 :: "'vtx"   and xs :: "('vtx \<times> 'lbl \<times> 'vtx) list" assumes "path l1 l2 xs" shows "\<exists>xs'. distinct xs' \<and> path l1 l2 xs' \<and> map (\<lambda>(s, l, t). l) xs' \<preceq> map (\<lambda>(s, l, t). l) xs"
elementary_monoidal_category_axioms: shows "elementary_monoidal_category (\<cdot>) (\<otimes>) \<I> lunit runit assoc"
non_span_det_zero: fixes G :: "'a vec list" assumes "length G = n"   and "\<not> carrier_vec n \<subseteq> local.span (set G)"   and "set G \<subseteq> carrier_vec n" shows "det (mat_of_rows n G) = (0::'a)"
path_valid_node_1: 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 n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n as n'" shows "CFG.valid_node sourcenode targetnode valid_edge n"
unit_simps_1: shows "arr (local.inv \<epsilon>\<^sub>0)"
finestpart_def: fixes X :: "'c set" shows "finestpart X = (\<lambda>x. {x}) ` X"
that_1: shows "bitlen \<bar>x\<bar> \<le> m"
set_disconnected_nodes_types_preserved: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and w :: "((_) heap, exception, unit) prog"   and document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "l_set_disconnected_nodes type_wf set_disconnected_nodes set_disconnected_nodes_locs"   and "w \<in> set_disconnected_nodes_locs document_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"