hcju/isabelleps huggingface dataset:
map_simp: fixes f :: "(bool, bool list) concrete_category.arr" assumes "J.arr f" shows "E.map f = C.dom e"
ID1: fixes x :: "'e" shows "ID\<cdot>x = x"
case_swap: fixes f :: "'f \<Rightarrow> 'e \<Rightarrow> 'd"   and p :: "'f \<times> 'e" shows "(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> f x y)"
index_mat_swaprows_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and k :: "nat"   and l :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "swaprows k l A $$ (i, j) = (if k = i then A $$ (l, j) else if l = i then A $$ (k, j) else A $$ (i, j))"
lookup_except_eq_zeroI: fixes t :: "'a"   and S :: "'a set"   and p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "t \<in> S" shows "lookup (except p S) t = (0::'b)"
Leaf_def: shows "Leaf \<equiv> \<Lambda> a. btree_abs\<cdot>(sinl\<cdot>(up\<cdot>a))"
borel_measurable_simple_function_limit_increasing: fixes f :: "'a \<Rightarrow> real" shows "(f \<in> borel_measurable lebesgue \<and> (\<forall>x. 0 \<le> f x)) = (\<exists>g. (\<forall>n x. 0 \<le> g n x \<and> g n x \<le> f x) \<and> (\<forall>n x. g n x \<le> g (Suc n) x) \<and> (\<forall>n. g n \<in> borel_measurable lebesgue) \<and> (\<forall>n. finite (range (g n))) \<and> (\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x))"
cartesian_category_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "cartesian_category C \<equiv> category_with_binary_products C \<and> category_with_terminal_object C"
LEQ_0: fixes k :: "'k"   and idx :: "'n" shows "LEQ k 0 idx"
last_message_hold_init_append: fixes xs :: "'b message_af list"   and ys :: "'b message_af list"   and m :: "'b message_af" shows "last_message_hold_init (xs @ ys) m = last_message_hold_init xs m @ last_message_hold_init ys (last_message (m # xs))"
vquot_dbtm_empty: fixes t :: "dbtm" shows "vquot_dbtm {} t = quot_dbtm t"
boolEx: shows "E\<turnstile>e\<Colon>-PrimT Boolean"
distinct_disc_4: fixes event :: "('a, 'b, 'c) event" assumes "isRecv event" shows "\<not> isTrans event"
signed_take_bit_Suc: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not (Suc n) a = plus (modulo a (numeral.numeral one plus (num.Bit0 num.One))) (times (numeral.numeral one plus (num.Bit0 num.One)) (ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n (divide a (numeral.numeral one plus (num.Bit0 num.One)))))"
assms: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
not_nil_if_in_set: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "xs \<noteq> []"
induct: fixes P :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list \<Rightarrow> ('c \<times> ('d, 'c) Term.term) list \<Rightarrow> bool"   and a0 :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and a1 :: "('c \<times> ('d, 'c) Term.term) list" assumes "\<And>bs. P [] bs"   and "\<And>f ss g ts E bs. (\<And>x2. decompose (Fun f ss) (Fun g ts) = Some x2 \<Longrightarrow> P (x2 @ E) bs) \<Longrightarrow> P ((Fun f ss, Fun g ts) # E) bs"   and "\<And>x t E bs. \<lbrakk>t = Var x \<Longrightarrow> P E bs; \<lbrakk>t \<noteq> Var x; x \<notin> vars_term t\<rbrakk> \<Longrightarrow> P (subst_list (subst x t) E) ((x, t) # bs)\<rbrakk> \<Longrightarrow> P ((Var x, t) # E) bs"   and "\<And>v va x E bs. (x \<notin> vars_term (Fun v va) \<Longrightarrow> P (subst_list (subst x (Fun v va)) E) ((x, Fun v va) # bs)) \<Longrightarrow> P ((Fun v va, Var x) # E) bs" shows "P a0 a1"
cwiseext_compat_snoc: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and x :: "'b" assumes "cwiseext gt ys xs" shows "cwiseext gt (ys @ [x]) (xs @ [x])"
UN_empty: fixes B :: "'d \<Rightarrow> 'c set" shows "\<Union> (B ` {}) = {}"
rangeI: fixes f :: "'f \<Rightarrow> 'e"   and x :: "'f" shows "f x \<in> range f"
blub: fixes x :: "nat" assumes "Q = P"   and "n = m"   and "x = x" shows "thesis"
finite_ordLess_infinite2: fixes A :: "'f set"   and B :: "'g set" assumes "finite A"   and "infinite B" shows "|A| <o |B|"
essential_\<E>: fixes \<Gamma> :: "('a, 'b) web_scheme"   and S :: "'a set" shows "essential \<Gamma> (B \<Gamma>) (\<E>\<^bsub>\<Gamma>\<^esub> S) = essential \<Gamma> (B \<Gamma>) S"
map2_val_compat'I: fixes f :: "('a \<times> 'c) list \<times> 'o \<Rightarrow> ('a \<times> 'd) list \<times> 'o" assumes "\<And>zs. oalist_inv zs \<Longrightarrow> oalist_inv (f zs)"   and "\<And>zs. oalist_inv zs \<Longrightarrow> snd (f zs) = snd zs"   and "\<And>zs. oalist_inv zs \<Longrightarrow> fst ` set (fst (f zs)) \<subseteq> fst ` set (fst zs)" shows "map2_val_compat' f"
6: shows "coreflexive (((d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)[[top]]) * (d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top))"
set_intros_2: fixes y :: "'k"   and x22 :: "'k list"   and x21 :: "'k" assumes "y \<in> set x22" shows "y \<in> set (x21 # x22)"
term_ok'_bind_fv2: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term"   and v :: "variable"   and T :: "typ"   and lev :: "nat" assumes "term_ok' \<Sigma> t" shows "term_ok' \<Sigma> (bind_fv2 (v, T) lev t)"
assms_1: shows "heterogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec"
qtable_mmulti_join: fixes n :: "nat"   and P :: "'a option list \<Rightarrow> bool"   and A_pos :: "nat set list"   and Q_pos :: "('a option list \<Rightarrow> bool) list"   and L_pos :: "'a option list set list"   and A_neg :: "nat set list"   and Q_neg :: "('a option list \<Rightarrow> bool) list"   and L_neg :: "'a option list set list"   and C :: "nat set"   and L :: "'a option list set list"   and Q :: "'a option list \<Rightarrow> bool" assumes "list_all3 (\<lambda>A Qi X. qtable n A P Qi X \<and> wf_set n A) A_pos Q_pos L_pos"   and "list_all3 (\<lambda>A Qi X. qtable n A P Qi X \<and> wf_set n A) A_neg Q_neg L_neg"   and "C = \<Union> (set A_pos)"   and "L = L_pos @ L_neg"   and "A_pos \<noteq> []"   and "\<Union> (set A_neg) \<subseteq> \<Union> (set A_pos)"   and "\<And>x. \<lbrakk>wf_tuple n C x; P x\<rbrakk> \<Longrightarrow> list_all (\<lambda>A. P (restrict A x)) A_pos"   and "\<And>x. \<lbrakk>wf_tuple n C x; P x\<rbrakk> \<Longrightarrow> list_all (\<lambda>A. P (restrict A x)) A_neg"   and "\<And>x. \<lbrakk>wf_tuple n C x; P x\<rbrakk> \<Longrightarrow> Q x = (list_all2 (\<lambda>A Qi. Qi (restrict A x)) A_pos Q_pos \<and> list_all2 (\<lambda>A Qi. \<not> Qi (restrict A x)) A_neg Q_neg)" shows "qtable n C P Q (mmulti_join n A_pos A_neg L)"
counit_simps_6: fixes a :: "'a" assumes "C.obj a" shows "D.cod (counit a) = G.map\<^sub>0 a"
y_1: fixes i :: "nat" assumes "i \<in> {1..k2}" shows "0 \<le> u2 i \<and> x2 i \<in> S"
lm095: fixes R :: "('c \<times> 'd) set" shows "R \<subseteq> Domain R \<times> Range R"
Bseq_Suc_iff: fixes f :: "nat \<Rightarrow> 'b" shows "Bseq (\<lambda>n. f (Suc n)) = Bseq f"
rquot_def: fixes y :: "'a"   and x :: "'a" shows "rquot y x = (THE z. D x z \<and> x \<oplus> z = y)"
vfsequence_vrange_vcpower: fixes xs :: "V"   and A :: "V" assumes "vfsequence xs"   and "\<R>\<^sub>\<circ> xs \<subseteq>\<^sub>\<circ> A" shows "xs \<in>\<^sub>\<circ> A ^\<^sub>\<times> vcard xs"
base_spset_2: shows "V = R_scalar_mult.RSpan UNIV (\<sharp>\<cdot>) vs"
fin_dim_def: shows "fin_dim = (\<exists>A. finite A \<and> A \<subseteq> carrier V \<and> gen_set A)"
Aii: shows "A $$ (i, i) = (0::'a)"
polyadd_head: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "Reflected_Multivariate_Polynomial.degree p \<noteq> Reflected_Multivariate_Polynomial.degree q" shows "head (p +\<^sub>p q) = (if Reflected_Multivariate_Polynomial.degree p < Reflected_Multivariate_Polynomial.degree q then head q else head p)"
readable: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure"   and Mkt :: "('c, 'd) discrete_market"   and pos_stock :: "'d" assumes "disc_market_pos_stock M F Mkt pos_stock" shows "\<forall>asset\<in>stocks Mkt. borel_adapt_stoch_proc F (prices Mkt asset)"
B_trusts_OR2: fixes B :: "agent"   and Nb :: "nat"   and M :: "nat"   and A :: "agent"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Nonce Nb, Nonce M, Nonce M, Agent A, Agent B\<rbrace> \<in> parts (knows Spy evs)"   and "B \<notin> bad"   and "evs \<in> orb" shows "\<exists>X. Says B Server \<lbrace>Nonce M, Agent A, Agent B, X, Crypt (shrK B) \<lbrace>Nonce Nb, Nonce M, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs"
big_refl: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "f \<in> L F f"
min_0L: fixes n :: "nat" shows "min 0 n = 0"
reachable_def: fixes m :: "'a"   and y :: "'d" shows "mut_m.reachable m y = (\<lambda>s. \<exists>x. x \<in> roots (s (mutator m)) \<union> ghost_honorary_root (s (mutator m)) \<union> mut_m.tso_store_refs m s \<and> (x reaches y) s)"
transp_on_irreflp_on_imp_antisymp_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "transp_on P A"   and "irreflp_on P A" shows "antisymp_on P\<^sup>=\<^sup>= A"
ZassenhausTr0: fixes H :: "'a set"   and H1 :: "'a set"   and K :: "'a set"   and K1 :: "'a set" assumes "G \<guillemotright> H "   and "G \<guillemotright> H1 "   and "G \<guillemotright> K "   and "G \<guillemotright> K1 "   and "\<natural>H \<triangleright> H1"   and "\<natural>K \<triangleright> K1" shows "\<natural>H \<inter> K \<triangleright> H \<inter> K1"
FSwImorph_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" shows "FSwImorph h hA termMOD MOD = termFSwImorph h hA MOD"
assms: shows "\<forall>x1\<in>X. x1 \<noteq> {} \<and> (\<forall>x2\<in>X - {x1}. x1 \<inter> x2 = {})"
sees: shows "P \<turnstile> C sees F,b:T in D"
adjoint_def: fixes A :: "'c mat" shows "adjoint A = mat (dim_col A) (dim_row A) (\<lambda>(i, j). conjugate (A $$ (j, i)))"
cond_B_def: fixes x :: "nat list" shows "cond_B x = (\<forall>k\<le>m. take k a \<bullet> take k x \<le> b \<bullet> map (max_y (take k x)) [0..<n])"
simps_6: fixes a :: "'a poly" shows "num_params (Neg a) = num_params a"
less_def: fixes xs :: "nat list"   and ys :: "nat list" shows "(xs <\<^sub>v ys) = (xs \<le>\<^sub>v ys \<and> \<not> ys \<le>\<^sub>v xs)"
put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: fixes document_ptr :: "(_) document_ptr"   and document :: "('n, 'l, 'm, 'u option) RDocument_scheme"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "put document_ptr document h = h'" shows "document_ptr |\<in>| document_ptr_kinds h'"
cycle_root_end_empty: fixes r :: "'a"   and x :: "'a"   and e :: "'a" assumes "terminating_path_root_end r x e"   and "many_strongly_connected x" shows "x = 0"
le_sup_iff: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "(x \<union># y \<subseteq># z) = (x \<subseteq># z \<and> y \<subseteq># z)"
ordinal_minus_le_eq: fixes x :: "ordinal"   and y :: "ordinal"   and z :: "ordinal" shows "(x - y \<le> z) = (x \<le> y + z)"
lQuot_def: fixes x :: "'b"   and A :: "'b list set" shows "lQuot x A = {xs. x # xs \<in> A}"
rtranclp_mono: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "r \<le> s" shows "r\<^sup>*\<^sup>* \<le> s\<^sup>*\<^sup>*"
s4: fixes t :: "'a"   and c :: "complex" assumes "t \<in> {x. f x = (0::'b)}" shows "c *\<^sub>C t \<in> {x. f x = (0::'b)}"
comp_apply: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "(f \<circ> g) x = f (g x)"
WODNonInterferenceGraph: fixes prog :: "cmd" shows "NonInterferenceIntraGraph src trg knd (lift_valid_edge (valid_edge prog) sourcenode targetnode kind (_Entry_) (_Exit_)) NewEntry (lift_Def (Defs prog) (_Entry_) (_Exit_) H L) (lift_Use (Uses prog) (_Entry_) (_Exit_) H L) id (CFG_wf.wod_backward_slice src trg (lift_valid_edge (valid_edge prog) sourcenode targetnode kind (_Entry_) (_Exit_)) (lift_Def (Defs prog) (_Entry_) (_Exit_) H L) (lift_Use (Uses prog) (_Entry_) (_Exit_) H L)) NewExit H L (LDCFG_node.Node (_Entry_)) (LDCFG_node.Node (_Exit_))"
finsum_empty: fixes f :: "'f \<Rightarrow> 'c" shows "finsum M f {} = \<zero>\<^bsub>M\<^esub>"
typing_weaken: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and M :: "'a trm"   and \<tau> :: "PreSimplyTyped.type"   and y :: "'a"   and \<sigma> :: "PreSimplyTyped.type" assumes "\<Gamma> \<turnstile> M : \<tau>"   and "y \<notin> fvs M" shows "\<Gamma>(y \<mapsto> \<sigma>) \<turnstile> M : \<tau>"
a: shows "\<forall>x\<in>F. g x = f x"
enn2ereal_strict_mono: fixes x :: "ennreal"   and y :: "ennreal" assumes "x < y" shows "enn2ereal x < enn2ereal y"
digitToList_def: shows "digitToList \<equiv> rec_Digit nodeToList (\<lambda>a b. nodeToList a @ nodeToList b) (\<lambda>a b c. nodeToList a @ nodeToList b @ nodeToList c) (\<lambda>a b c d. nodeToList a @ nodeToList b @ nodeToList c @ nodeToList d)"
add_imply: fixes q :: "('a, 'b) fm"   and ps :: "('a, 'b) fm list" assumes "\<turnstile> q" shows "ps \<turnstile> q"
holds_eq1: fixes x :: "'a" shows "holds ((=) x) = HLD {x}"
last_changes_implies_send_when_msgs_nonempty_trace: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and j :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "i < j"   and "j \<le> length t"   and "Marker \<in> set (msgs (S t i) cid)"   and "Marker \<in> set (msgs (S t j) cid)"   and "last (msgs (S t i) cid) = Marker"   and "last (msgs (S t j) cid) \<noteq> Marker"   and "channel cid = Some (p, q)" shows "\<exists>k u u' m. i \<le> k \<and> k < j \<and> t ! k = Send cid p q u u' m"
fmlookup_drop_set: fixes A :: "'d set"   and m :: "('d, 'c) fmap"   and x :: "'d" shows "fmlookup (fmdrop_set A m) x = (if x \<notin> A then fmlookup m x else None)"
\<eta>o_def: fixes y :: "'b" shows "G.\<eta>o y = (SOME u. initial_arrow_to_functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) G y (G.Fo y) u)"
pos_cost_pos_awalk_cost: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c"   and c :: "'d \<Rightarrow> real" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v"   and "\<And>e. e \<in> arcs G \<Longrightarrow> 0 \<le> c e" shows "0 \<le> wf_digraph.awalk_cost c p"
fmdom'_add: fixes m :: "('a, 'b) fmap"   and n :: "('a, 'b) fmap" shows "fmdom' (m ++\<^sub>f n) = fmdom' m \<union> fmdom' n"
FAssReds1: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D} := e\<^sub>2,s,b\<rangle> \<rightarrow>* \<langle>e'\<bullet>F{D} := e\<^sub>2,s',b'\<rangle>"
xcpt_correct: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "bytecode"   and xt :: "ex_table"   and pc :: "nat"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and xcp :: "nat"   and \<sigma>' :: "jvm_state" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = (mxs, mxl\<^sub>0, ins, xt) in C"   and "P,T,mxs,length ins,xt \<turnstile> ins ! pc,pc :: \<Phi> C M"   and "fst (exec_step P h stk loc C M pc ics frs sh) = \<lfloor>xcp\<rfloor>"   and "\<lfloor>\<sigma>'\<rfloor> = exec (P, None, h, (stk, loc, C, M, pc, ics) # frs, sh)"   and "P,\<Phi> |- (None, h, (stk, loc, C, M, pc, ics) # frs, sh) [ok]" shows "P,\<Phi> |- \<sigma>' [ok]"
induct: fixes x1 :: "('c, 'd) tm"   and x2 :: "('c, 'd) tm"   and P :: "('c, 'd) tm \<Rightarrow> ('c, 'd) tm \<Rightarrow> bool" assumes "x1 \<rightarrow>\<^sub>e\<^sub>m\<^sub>b x2"   and "\<And>t1 t2. P (App t1 t2) t1"   and "\<And>t1 t2. P (App t1 t2) t2"   and "\<And>t s u. \<lbrakk>t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s; P t s\<rbrakk> \<Longrightarrow> P (App t u) (App s u)"   and "\<And>t s u. \<lbrakk>t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s; P t s\<rbrakk> \<Longrightarrow> P (App u t) (App u s)" shows "P x1 x2"
show_law_float: fixes r :: "float" shows "show_law showsp_float r"
simps: fixes a1 :: "'b DBMEntry"   and a2 :: "'b DBMEntry" shows "(a1 \<prec> a2) = ((\<exists>uu_. a1 = Lt uu_ \<and> a2 = \<infinity>) \<or> (\<exists>uv_. a1 = Le uv_ \<and> a2 = \<infinity>) \<or> (\<exists>a b. a1 = Le a \<and> a2 = Le b \<and> a < b) \<or> (\<exists>a b. a1 = Le a \<and> a2 = Lt b \<and> a < b) \<or> (\<exists>a b. a1 = Lt a \<and> a2 = Le b \<and> a \<le> b) \<or> (\<exists>a b. a1 = Lt a \<and> a2 = Lt b \<and> a < b))"
s_def: shows "s = (\<lambda>i k. sum t {i..<i + k})"
verticesFrom_length: fixes f :: "face"   and v :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f" shows "|verticesFrom f v| = |vertices f|"
convex_lsc_hull: fixes f :: "'b \<Rightarrow> ereal" assumes "Lower_Semicontinuous.convex_on UNIV f" shows "Lower_Semicontinuous.convex_on UNIV (lsc_hull f)"
elims: fixes x :: "'a trm"   and y :: "Preliminaries.var set" assumes "nv2T x = y" obtains x :: "Preliminaries.var" where "xa__ = Var x" and "y = {x}"   | f :: "'a" and Tl :: "'a trm list" where "xa__ = Fn f Tl" and "y = {}"
objI_src: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "horizontal_homs V src trg"   and "partial_magma.arr V a"   and "src a = a" shows "horizontal_homs.obj V src a"
Units_closed: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.monoid G"   and "x \<in> Units G" shows "x \<in> carrier G"
weak_transition_not_tau_iff: fixes \<alpha> :: "'act"   and P :: "'state"   and P' :: "'state" assumes "\<alpha> \<noteq> \<tau>" shows "P \<Rightarrow>\<langle>\<alpha>\<rangle> P' = P \<Rightarrow>{\<alpha>} P'"
luckyFind_eval_augment: fixes amount :: "nat"   and var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" assumes "amount \<le> var + 1" shows "(\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. length xs = var + 1 \<and> eval (step_augment luckyFind' IdentityHeuristic amount var L F) (xs @ \<Gamma>))"
field_divide_inverse: fixes a :: "'d"   and b :: "'d" shows "a / b = a * inverse b"
eq_linorder_on_def: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res" shows "eq_linorder_on D cmp \<equiv> linorder_on D cmp \<and> eq_linorder_on_axioms D cmp"
qSwap_involutive: fixes zs :: "'c"   and x :: "'d"   and y :: "'d"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm" shows "X #[[x \<and> y]]_zs #[[x \<and> y]]_zs = X"
simps_5: fixes I :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and b :: "bexp"   and C :: "acom"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "vc ({I} WHILE b DO C) Q = ((\<forall>s. (I s \<longrightarrow> vars b \<subseteq> dom (fst s)) \<and> ((case s of (s, n) \<Rightarrow> I (s, n) \<and> lmaps_to_axpr b True s) \<longrightarrow> pre C (I \<and>* $ 1) s) \<and> ((case s of (s, n) \<Rightarrow> I (s, n) \<and> lmaps_to_axpr b False s) \<longrightarrow> Q s)) \<and> vc C (I \<and>* $ 1))"
simps_1: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signalsInit :: "signal \<Rightarrow> nat"   and signalsResp :: "signal \<Rightarrow> nat"   and more :: "'a" shows "progress \<lparr>ik = ik, secret = secret, progress = progress, signalsInit = signalsInit, signalsResp = signalsResp, \<dots> = more\<rparr> = progress"
m_assoc: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "x \<in> carrier (poly_ring R)"   and "y \<in> carrier (poly_ring R)"   and "z \<in> carrier (poly_ring R)" shows "x \<otimes>\<^bsub>poly_ring R\<^esub> y \<otimes>\<^bsub>poly_ring R\<^esub> z = x \<otimes>\<^bsub>poly_ring R\<^esub> (y \<otimes>\<^bsub>poly_ring R\<^esub> z)"
hrabs_less_Infinitesimal: fixes e :: "real star"   and x :: "real star" assumes "e \<in> Infinitesimal"   and "\<bar>x\<bar> < e" shows "x \<in> Infinitesimal"
finite_product_sigma_finite_axioms_def: fixes I :: "'b set" shows "finite_product_sigma_finite_axioms I \<equiv> finite I"
fBall_def: shows "fBall \<equiv> map_fun fset id Ball"
top: fixes \<Omega> :: "'b set"   and M :: "'b set set" assumes "algebra \<Omega> M" shows "\<Omega> \<in> M"
ord_adds: fixes s :: "'x \<Rightarrow>\<^sub>0 nat"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "s adds t" shows "s \<preceq> t"
surjective: fixes r :: "('m, 'n, 'o, 'p) RShadowRoot_scheme" shows "r = \<lparr>RObject.nothing = RObject.nothing r, RDocument.nothing = RDocument.nothing r, doctype = doctype r, document_element = document_element r, disconnected_nodes = disconnected_nodes r, RShadowRoot.nothing = RShadowRoot.nothing r, mode = mode r, child_nodes = RShadowRoot.child_nodes r, \<dots> = RShadowRoot.more r\<rparr>"
word_gallery_to_0: fixes S :: "'b set"   and ss :: "'b list" assumes "CoxeterComplex S"   and "ss \<noteq> []"   and "ss \<in> lists S" shows "\<exists>xs. SimplicialComplex.maxsimpchain (CoxeterComplex.TheComplex S) (dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) {sum_list ss} # xs @ [dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) 0])"
lhr_fps_def: fixes as :: "'b list"   and fs :: "'b list" shows "lhr_fps as fs = (let m = length fs + 1 - length as; p = lhr_fps_numerator m as ((!) fs); q = lr_fps_denominator as in ratfps_of_poly p / ratfps_of_poly q)"
csqrt_principal: fixes z :: "complex" shows "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)"
stops\<^sub>C_def: fixes c :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)" shows "stops\<^sub>C c \<equiv> \<forall>c'. (c, c') \<notin> eval\<^sub>C"
extract_bind: fixes x :: "'d Monomorphic_Monad.id"   and f :: "'d \<Rightarrow> 'd Monomorphic_Monad.id" shows "extract (bind_id x f) = extract (f (extract x))"
equiv_imp_le: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and u :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "equiv_trans t u" shows "le_trans t u"
b: fixes a :: "'a extended \<Rightarrow> 'a sort_key extended" assumes "a_conditions (insert_messages (M \<union> {Insert m})) a" shows "sorted_wrt (<) (map a (ext_ids t))"
complex_factorization_exists: fixes p :: "complex poly" shows "\<exists>fctrs. is_factorization_of fctrs p"
bounded_clinear_left: fixes b :: "'b" shows "bounded_clinear (\<lambda>a. a ** b)"