hcju/isabelleps huggingface dataset:
cod_tensor: fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a"   and d :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : c \<rightarrow> d\<guillemotright>" shows "cod (f \<otimes> g) = b \<otimes> d"
sp_append: fixes F :: "'n \<Rightarrow> 'o fundef option"   and xs :: "('p, 'q, 'n, 'e, 'f, 'g, 'r, 's) instr list"   and ys :: "('p, 'q, 'n, 'e, 'f, 'g, 'r, 's) instr list"   and \<Sigma> :: "Unboxed.type option list" shows "sp F (xs @ ys) \<Sigma> = sp F xs \<Sigma> \<bind> sp F ys"
simps: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and l :: "('b \<times> nat) tree"   and a :: "'b"   and uu :: "nat"   and r :: "('b \<times> nat) tree" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2" shows "WBT.del_max \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 \<langle>l, (a, uu), r\<rangle> = (if r = \<langle>\<rangle> then (a, l) else let (x, r') = WBT.del_max \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 r in (x, WBT.balanceR \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 l a r'))"
simps_2: shows "Inv \<^bold>\<I> = \<^bold>\<I>"
While_def: fixes t :: "'b"   and c :: "'b" shows "While t Do c End = (t \<cdot> c)\<^sup>\<star> \<cdot> ! t"
bind_def: fixes xs :: "'f list"   and f :: "'f \<Rightarrow> 'e list" shows "List.bind xs f = concat (map f xs)"
characteristic_iso_3: fixes f :: "'a" assumes "Right.ide f" shows "Right.in_hom (Right.comp (f \<star> \<i>[src f]) \<a>[f, src f, src f]) (Right.R (Right.R f)) (Right.R f)"
has_field_sees: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty" assumes "P \<turnstile> C has F,b:T in C" shows "P \<turnstile> C sees F,b:T in C"
read_buffer\<^sub>C_def: shows "C.read_buffer\<^sub>C \<equiv> (Skip@[control_var\<^sub>C +=\<^sub>m AsmNoWrite]) ;; (Skip@[temp\<^sub>C +=\<^sub>m AsmNoReadOrWrite]) ;; (temp\<^sub>C \<leftarrow> aexp\<^sub>C.Load buffer\<^sub>C) ;; (reg\<^sub>C \<leftarrow> aexp\<^sub>C.Load control_var\<^sub>C) ;; Stmt.If (bexp\<^sub>C.Eq reg\<^sub>C 0) (low_var\<^sub>C \<leftarrow> aexp\<^sub>C.Load temp\<^sub>C) (high_var\<^sub>C \<leftarrow> aexp\<^sub>C.Load temp\<^sub>C) ;; (temp\<^sub>C \<leftarrow> aexp\<^sub>C.Const 0) ;; Skip@[temp\<^sub>C -=\<^sub>m AsmNoReadOrWrite]"
irreducible\<^sub>d_square_free: fixes p :: "'a poly" assumes "irreducible\<^sub>d p" shows "square_free p"
Amortized2_def: fixes arity :: "'a \<Rightarrow> nat"   and exec :: "'a \<Rightarrow> 'b list \<Rightarrow> 'b"   and inv :: "'b \<Rightarrow> bool"   and cost :: "'a \<Rightarrow> 'b list \<Rightarrow> nat"   and \<Phi> :: "'b \<Rightarrow> real"   and U :: "'a \<Rightarrow> 'b list \<Rightarrow> real"   and exec' :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and inv' :: "'c \<Rightarrow> bool"   and cost' :: "'a \<Rightarrow> 'c list \<Rightarrow> nat"   and U' :: "'a \<Rightarrow> 'c list \<Rightarrow> real"   and hom :: "'c \<Rightarrow> 'b" shows "Amortized2 arity exec inv cost \<Phi> U exec' inv' cost' U' hom \<equiv> Amortized arity exec inv cost \<Phi> U \<and> Amortized2_axioms arity exec inv cost U exec' inv' cost' U' hom"
Red_I_\<G>_q_def: fixes q :: "'q"   and N :: "('f \<times> 'l) set" shows "Red_I_\<G>_q q N = {\<iota> \<in> Inf_FL. \<G>_I_L_q q \<iota> \<noteq> None \<and> the (\<G>_I_L_q q \<iota>) \<subseteq> Red_I_q q (\<Union> (\<G>_F_L_q q ` N)) \<or> \<G>_I_L_q q \<iota> = None \<and> \<G>_F_L_q q (concl_of \<iota>) \<subseteq> \<Union> (\<G>_F_L_q q ` N) \<union> Red_F_q q (\<Union> (\<G>_F_L_q q ` N))}"
proj_active: fixes i :: "nat"   and t :: "cnf llist"   and c :: "'id" assumes "enat i < llength t"   and "\<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>" shows "\<pi>\<^bsub>c\<^esub>ltake (enat (Suc i)) t = \<pi>\<^bsub>c\<^esub>ltake (enat i) t @\<^sub>l (\<sigma>\<^bsub>c\<^esub>lnth t i #\<^sub>l []\<^sub>l)"
Diff_weaken_left: fixes A :: "'b pl set"   and C :: "'b pl set"   and B :: "'b pl set"   and p :: "'b pl" assumes "A \<subseteq> C"   and "A - B |- p" shows "C - B |- p"
ipset_from_cidr_eq_ip_cidr_set: fixes base :: "'a word"   and len :: "nat" shows "ipset_from_cidr base len = ip_cidr_set base len"
nonmeets2: fixes u :: "'a"   and r :: "'a" assumes "\<I> u"   and "\<I> r" shows "\<not> (u \<parallel> r \<and> u = r)"
not_final_thread_cases2: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x1 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and x :: "'x1"   and ln :: "'a \<Rightarrow>f nat" assumes "r1.not_final_thread s t"   and "thr s t = \<lfloor>(x, ln)\<rfloor>" obtains "\<not> final1 x"   | "ln \<noteq> no_wait_locks"   | w :: "'d wait_set_status" where "wset s t = \<lfloor>w\<rfloor>"
image_mset_eq_permutation: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and g :: "'d \<Rightarrow> 'c" assumes "image_mset f (mset_set A) = image_mset g (mset_set A)"   and "finite A" obtains \<pi> :: "'d \<Rightarrow> 'd" where "\<pi> permutes A" and "\<And>x. x \<in> A \<Longrightarrow> g (\<pi> x) = f x"
equivp_transp: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "equivp R"   and "R x y"   and "R y z" shows "R x z"
ex_min_if_finite: fixes S :: "'a set" assumes "finite S"   and "S \<noteq> {}" shows "\<exists>m\<in>S. \<not> (\<exists>x\<in>S. x < m)"
nn_integral_count_space': fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "finite A"   and "\<And>x. \<lbrakk>x \<in> B; x \<notin> A\<rbrakk> \<Longrightarrow> f x = 0"   and "A \<subseteq> B" shows "integral\<^sup>N (count_space B) f = sum f A"
list_all2_same: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "list_all2 P xs xs = (\<forall>x\<in>set xs. P x x)"
le_divide_eq_numeral_1: fixes w :: "num"   and b :: "'a"   and c :: "'a" shows "(numeral w \<le> b / c) = (if (0::'a) < c then numeral w * c \<le> b else if c < (0::'a) then b \<le> numeral w * c else numeral w \<le> (0::'a))"
Collect_cong: fixes P :: "'m \<Rightarrow> bool"   and Q :: "'m \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "{x. P x} = {x. Q x}"
l2_step2_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and KE :: "msg" shows "l2_step2 Rb A B KE \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> Rb \<notin> dom (progress s) \<and> guessed_frame Rb xpkE = Some KE \<and> Auth A B \<langle>Number 0, KE\<rangle> \<in> chan s \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xpkE, xsk}), chan := {Auth B A (Aenc (NonceF (Rb $ sk)) KE)} \<union> chan s, l1_state.signals := if can_signal s A B then addSignal (l1_state.signals s) (Running A B \<langle>KE, NonceF (Rb $ sk)\<rangle>) else l1_state.signals s, secret := {x. x = NonceF (Rb $ sk) \<and> Rb = test} \<union> secret s\<rparr>}"
digraph_dg_0: shows "digraph \<alpha> dg_0"
assms: shows "t \<oplus> u adds\<^sub>t v"
take_bit_eq_0_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "(take_bit n a = zero) = dvd.dvd times (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) a"
lr_of_tran_fbs_acceptI: fixes rt :: "'b routing_rule list"   and fw :: "'b simple_rule list"   and p :: "('b, 'c) simple_packet_scheme"   and oif :: "char list"   and ifs :: "'d" assumes "valid_prefixes rt"   and "has_default_route rt"   and "no_oif_match fw"   and "has_default_policy fw"   and "simple_linux_router_nol12 rt fw p = Some (p\<lparr>p_oiface := oif\<rparr>)" shows "\<exists>r. generalized_sfw (lr_of_tran_fbs rt fw ifs) p = Some (r, oif, Accept)"
add_mset_commute: fixes x :: "'a"   and y :: "'a"   and M :: "'a multiset" shows "add_mset x (add_mset y M) = add_mset y (add_mset x M)"
interval_eq_empty_cart_2: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" shows "(cbox a b = {}) = (\<exists>i. b $ i < a $ i)"
invar: fixes l :: "'a list \<times> 'a list" shows "fifo.invar l"
d_IN_h_le_value': fixes i :: "nat"   and x :: "'v" shows "d_IN (h_diff i) x \<le> value_flow \<Delta> (h_plus i)"
abs_eq: shows "X = abs_var (BPS (binsert (True, undefined) (binsert (False, undefined) bempty)))"
post_meas1_index_0_alice: fixes \<phi> :: "complex Matrix.mat"   and \<alpha> :: "complex"   and \<beta> :: "complex" assumes "state 1 \<phi>"   and "\<alpha> = \<phi> $$ (0, 0)"   and "\<beta> = \<phi> $$ (1, 0)" shows "post_meas1 3 (alice \<phi>) 0 = Tensor.mat_of_cols_list 8 [[0, 0, 0, 0, \<alpha> / complex_of_real (sqrt 2), - \<beta> / complex_of_real (sqrt 2), - \<beta> / complex_of_real (sqrt 2), \<alpha> / complex_of_real (sqrt 2)]]"
countable_funpow: fixes f :: "'c set \<Rightarrow> 'c set"   and A :: "'c set"   and n :: "nat" assumes "\<And>A. countable A \<Longrightarrow> countable (f A)"   and "countable A" shows "countable ((f ^^ n) A)"
add_\<O>: fixes a :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and b :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "a \<in> \<O> U"   and "b \<in> \<O> U" shows "add_sheaf_spec U a b \<in> \<O> U"
Ball_set: fixes xs :: "'e list"   and P :: "'e \<Rightarrow> bool" shows "Ball (set xs) P = list_all P xs"
match_appE_split: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and t\<^sub>1 :: "term"   and t\<^sub>2 :: "term"   and u :: "'c"   and env :: "(name, 'c) fmap" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "term.match unapp unconst (t\<^sub>1 $ t\<^sub>2) u = Some env" obtains u\<^sub>1 :: "'c" and u\<^sub>2 :: "'c" and env\<^sub>1 :: "(name, 'c) fmap" and env\<^sub>2 :: "(name, 'c) fmap" where "u = app u\<^sub>1 u\<^sub>2" and "term.match unapp unconst t\<^sub>1 u\<^sub>1 = Some env\<^sub>1" and "term.match unapp unconst t\<^sub>2 u\<^sub>2 = Some env\<^sub>2" and "env = env\<^sub>1 ++\<^sub>f env\<^sub>2"
neg_notin_Interp_not_produce: fixes A :: "'a"   and C :: "'a literal multiset"   and D :: "'a literal multiset"   and D'' :: "'a literal multiset" assumes "Neg A \<in># C"   and "A \<notin> Interp D"   and "C \<le> D" shows "production D'' \<noteq> {A}"
simps_1: fixes valuation :: "Literal list" shows "formulaTrue [] valuation = True"
matrix_match_condn_1: fixes A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list"   and i :: "nat"   and j :: "nat" assumes "matrix_match A1 A2 B1 B2 \<and> i < row_length A1 * row_length B1 \<and> j < length A2 * length B2" shows "(A1 \<circ> A2 \<otimes> B1 \<circ> B2) ! j ! i = scalar_product (row A1 (i div row_length B1)) (col A2 (j div length B2)) * scalar_product (row B1 (i mod row_length B1)) (col B2 (j mod length B2))"
minus_add: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> carrier (R Quot I)"   and "y \<in> carrier (R Quot I)" shows "\<ominus>\<^bsub>R Quot I\<^esub> (x \<oplus>\<^bsub>R Quot I\<^esub> y) = \<ominus>\<^bsub>R Quot I\<^esub> x \<oplus>\<^bsub>R Quot I\<^esub> \<ominus>\<^bsub>R Quot I\<^esub> y"
strict_order_iff: fixes a :: "'a"   and b :: "'a" shows "lt a b = (a = local.min a b \<and> a \<noteq> b)"
all_fields_in_has_fields: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> ty) list"   and D :: "char list"   and D' :: "char list"   and fs :: "(char list \<times> ty) list"   and ms :: "(char list \<times> ty list \<times> ty \<times> 'b) list"   and F :: "char list"   and T :: "ty" assumes "P \<turnstile> C has_fields FDTs"   and "P \<turnstile> C \<preceq>\<^sup>* D"   and "class P D = \<lfloor>(D', fs, ms)\<rfloor>"   and "(F, T) \<in> set fs" shows "((F, D), T) \<in> set FDTs"
elims_1: fixes x :: "'b ext"   and xa :: "'b ext"   and y :: "bool" assumes "(x \<le> xa) = y" obtains uu_ :: "'b ext" where "xa__ = ext.Bot" and "xa = uu_" and "ya__"   | uv_ :: "'b" where "xa__ = ext.Val uv_" and "xa = ext.Bot" and "\<not> ya__"   | x :: "'b" and y :: "'b" where "xa__ = ext.Val x" and "xa = ext.Val y" and "ya__ = (x \<le> y)"   | uw_ :: "'b" where "xa__ = ext.Val uw_" and "xa = ext.Top" and "ya__"   | "xa__ = ext.Top" and "xa = ext.Bot" and "\<not> ya__"   | ux_ :: "'b" where "xa__ = ext.Top" and "xa = ext.Val ux_" and "\<not> ya__"   | "xa__ = ext.Top" and "xa = ext.Top" and "ya__"
comp_assoc: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and h :: "'b"   and g :: "'b"   and f :: "'b" assumes "category C" shows "C (C h g) f = C h (C g f)"
planeN4: fixes g :: "graph"   and f :: "face" assumes "Invariants.inv g"   and "f \<in> \<F> g" shows "3 \<le> |vertices f|"
mem_add: fixes x :: "'a"   and y :: "'a" assumes "x \<in> S"   and "y \<in> S" shows "x + y \<in> S"
lim: fixes J :: "nat set"   and X :: "(nat \<Rightarrow> 'a) set" assumes "finite J"   and "X \<in> sets (Pi\<^sub>M J M)" shows "emeasure PF.lim (PF.emb UNIV J X) = emeasure (CI J) X"
transE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "transE \<xi> \<equiv> \<lambda>x. proj (tpOfV x) (\<xi> x)"
args_to_set_def: fixes x :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<times> ((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list" shows "args_to_set x = fst ` (set (fst x) \<union> set (fst (snd x)) \<union> fst ` set (snd (snd x)) \<union> snd ` set (snd (snd x)))"
that: fixes p :: "'a"   and q :: "'a" assumes "p * a + q * b = c" shows "thesis"
deutsch_transform_coeff_2: shows "U\<^sub>f $$ (1, 0) = of_nat (f 0)"
prts: shows "(c - y * A) * x \<le> (let s1 = c1 - y * A2; s2 = c2 - y * A1 in pprt s2 * pprt r2 + pprt s1 * nprt r2 + nprt s2 * pprt r1 + nprt s1 * nprt r1)"
insert_fset_case: fixes x :: "'a"   and S' :: "'a fset" assumes "S = insert_fset x S'" shows "P"
distr_singleton: fixes i :: "'b" shows "distr (Pi\<^sub>M {i} (\<lambda>_. interval_measure (\<lambda>x. x))) (interval_measure (\<lambda>x. x)) (\<lambda>x. x i) = interval_measure (\<lambda>x. x)"
intFI_def: shows "intFI \<equiv> snd3 MI"
kleisli_def: fixes f :: "'c \<Rightarrow> 'c option"   and g :: "'c \<Rightarrow> 'c option" shows "f \<rhd> g \<equiv> \<lambda>x. f x \<bind> g"
array_length_list: fixes a :: "'a array" shows "array_length a = length (list_of_array a)"
coarse_correct: fixes crho :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" assumes "CSHORun A crho HOs SHOs coords"   and "crho 0 = state (rho 0)" shows "P crho"
MatchF_5: shows "pred_mapping (\<lambda>x. x \<le> plen \<pi>) P"
flat_ordI: fixes x :: "'f"   and a :: "'f"   and y :: "'f" assumes "x \<noteq> a \<Longrightarrow> x = y" shows "flat_ord a x y"
simps: fixes v :: "'form \<times> nat list"   and p :: "('form, 'var) antecedent" shows "valid_in_port (v, p) = (v |\<in>| vertices \<and> p |\<in>| inPorts (nodeOf v))"
assms_1: shows "cri.InvMsgInGlob (exchange_config c0)"
bigger_subgroup_is_group: fixes H :: "'a set" assumes "subgroup H G"   and "order G \<le> card H" shows "H = carrier G"
gen_dfs_post_imp_reachable: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and S :: "'a set"   and V0 :: "'a set"   and u0 :: "'a"   and V :: "'a set"   and brk :: "bool" assumes "gen_dfs_pre E U S V0 u0"   and "gen_dfs_post E U S V0 u0 V brk" shows "V \<subseteq> V0 \<union> E\<^sup>* `` {u0}"
lookup_good: fixes \<rho>' :: "(nat \<times> val) list"   and x :: "nat"   and A :: "val"   and \<rho> :: "(nat \<times> bval) list" assumes "lookup \<rho>' x = Some A"   and "good_env \<rho> \<rho>'" shows "\<exists>v. lookup \<rho> x = Some v \<and> v \<in> good A"
finite_induct: fixes P :: "'a\<cdot>('b, 'c) stateT \<Rightarrow> bool"   and n :: "nat"   and x :: "'a\<cdot>('b, 'c) stateT" assumes "P \<bottom>"   and "\<And>cfun. cfun \<noteq> \<bottom> \<Longrightarrow> P (StateT\<cdot>cfun)" shows "P (stateT_take n\<cdot>x)"
mod_ring_mod_integer: shows "(mod_ring_rel_integer ===> mod_ring_rel_integer ===> mod_ring_rel_integer) (\<lambda>x y. if y = 0 then x else 0) (mod)"
cnj_cnj_matrix: fixes A :: "(('b, 'c) vec, 'd) vec" shows "(\<chi>i j. cnj_class.cnj ((\<chi>i j. cnj_class.cnj (A $ i $ j)) $ i $ j)) = A"
i2: shows "ideal_generated {a, b, c} = ideal_generated {d}"
drop_bit_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit 0 = id"
induct: fixes P :: "'c set \<Rightarrow> ('d, 'c) strand_step list \<Rightarrow> bool"   and a0 :: "'c set"   and a1 :: "('d, 'c) strand_step list" assumes "\<And>V. P V []"   and "\<And>V t S. P V S \<Longrightarrow> P V (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>V t S. P (V \<union> fv t) S \<Longrightarrow> P V (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>V s t S. P (V \<union> fv s) S \<Longrightarrow> P V (\<langle>assign: s \<doteq> t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>V s t S. P V S \<Longrightarrow> P V (\<langle>check: s \<doteq> t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>V uu_ uv_ S. P V S \<Longrightarrow> P V (\<forall>uu_\<langle>\<or>\<noteq>: uv_\<rangle>\<^sub>s\<^sub>t # S)" shows "P a0 a1"
sset_atoms: shows "i.R atoms = {\<phi>. max_depth \<phi> < 1}"
unfoldr_filter_generator: fixes P :: "'b \<Rightarrow> bool"   and g :: "('b, 'c) generator"   and s :: "'c" shows "list.unfoldr (filter_generator P g) s = filter P (list.unfoldr g s)"
hconfD: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a :: "nat"   and obj :: "char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set" assumes "P \<turnstile> h \<surd>"   and "h a = \<lfloor>obj\<rfloor>" shows "P,h \<turnstile> obj \<surd>"
canonical_exists: fixes A :: "('b \<Rightarrow> bool) set"   and \<sigma> :: "nat \<Rightarrow> 'b" obtains \<tau> :: "nat \<Rightarrow> 'b" where "\<tau> \<simeq>A\<simeq> \<sigma>" and "canonical_sequence A \<tau>"
simps_8: fixes \<phi> :: "'b ltlr"   and \<psi> :: "'b ltlr" shows "ltlr_to_ltln (\<phi> U\<^sub>r \<psi>) = ltlr_to_ltln \<phi> U\<^sub>n ltlr_to_ltln \<psi>"
norm0_2: shows "\<parallel>fs.gs.gso (i - 1)\<parallel>\<^sup>2 \<noteq> 0"
\<pi>_def: shows "primes_pi = prime_sum_upto (\<lambda>p. 1)"
show5: fixes A :: "real"   and d :: "nat \<Rightarrow> real"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "1 < A"   and "\<forall>n. 1 < d n"   and "\<forall>n. 0 < a n"   and "\<forall>n. 0 < b n"   and "(\<lambda>n. real_of_int (a n) powr (1 / real_of_int (2 ^ n))) \<longlonglongrightarrow> A"   and "LIM n sequentially. d n ^ 2 ^ n / real_of_int (b n) :> at_top"   and "convergent_prod d" shows "\<forall>\<^sub>F n in sequentially. (\<Sum>j. real_of_int (b (n + j)) / real_of_int (a (n + j))) \<le> real_of_int (2 * b n) / real_of_int (a n)"
not_less: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" shows "(\<not> less x y) = less_eq y x"
f_Exec_Stream_nth: fixes n :: "nat"   and xs :: "'a list"   and trans_fun :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b" assumes "n < length xs" shows "f_Exec_Comp_Stream trans_fun xs c ! n = f_Exec_Comp trans_fun (xs \<down> Suc n) c"
fw_invariant_aux_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and j'' :: "nat"   and j :: "nat"   and i :: "nat"   and n :: "nat"   and k :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i' :: "nat"   and j' :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral"   and "j'' \<le> j"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "less_eq (linordered_ab_monoid_add.fw plus less_eq m n k i j i' j') (linordered_ab_monoid_add.fw plus less_eq m n k i j'' i' j')"
components_are_iso: fixes a :: "'c" assumes "C.ide a" shows "D.iso (F a)"
sim_config_emptyI: fixes ts\<^sub>s\<^sub>b :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b :: "nat \<Rightarrow> bool option" assumes "\<forall>i p is xs sb \<D> \<O> \<R>. i < length ts\<^sub>s\<^sub>b \<longrightarrow> ts\<^sub>s\<^sub>b ! i = (p, is, xs, sb, \<D>, \<O>, \<R>) \<longrightarrow> sb = []"   and "length ts = length ts\<^sub>s\<^sub>b"   and "\<forall>i<length ts\<^sub>s\<^sub>b. let (p, is, \<theta>, sb, \<D>, \<O>, \<R>) = ts\<^sub>s\<^sub>b ! i in ts ! i = (p, is, \<theta>, (), \<D>, \<O>, \<R>)" shows "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<sim> (ts, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b)"
a: shows "x \<sharp> (\<theta>_n, \<theta>_c)"
boundedI: fixes b :: "'c"   and a :: "'c"   and c :: "'c" assumes "b \<le> a"   and "c \<le> a" shows "b \<squnion> c \<le> a"
scale_left_imp_eq: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and x :: "'b"   and y :: "'b" assumes "vector_space scale"   and "a \<noteq> (0::'a)"   and "scale a x = scale a y" shows "x = y"
mult_column_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and n :: "'b"   and q :: "'a" shows "mult_column A n q = (\<chi>i j. if j = n then A $ i $ j * q else A $ i $ j)"
hp_conv: fixes h :: "'a"   and l :: "'b" shows "hp (h, l) = h"
2: shows "b2 \<in># remove1_mset b1 \<B>"
s: shows "mk_all_list (vs @ [v]) B = mk_all idn ty (mk_all_list vs B)"
G_Rel_OR: fixes x0 :: "'g"   and x1 :: "'a"   and w :: "'h + 'b" assumes "((x0, x1), w) \<in> set_spmf G" shows "((x0, x1), w) \<in> Rel_OR"
jordan_matrix_Cons: fixes n :: "nat"   and a :: "'b"   and n_as :: "(nat \<times> 'b) list" shows "jordan_matrix ((n, a) # n_as) = four_block_mat (jordan_block n a) (0\<^sub>m n (sum_list (map fst n_as))) (0\<^sub>m (sum_list (map fst n_as)) n) (jordan_matrix n_as)"
filtered_nonempty: fixes X :: "'a set" assumes "filtered X" shows "X \<noteq> {}"
lexord_def: fixes r :: "('d \<times> 'd) set" shows "lexord r = {(x, y). \<exists>a v. y = x @ a # v \<or> (\<exists>u a b v w. (a, b) \<in> r \<and> x = u @ a # v \<and> y = u @ b # w)}"