hcju/isabelleps huggingface dataset:
s_is_partial_execution: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and i :: "nat" shows "s c t i = run_trace (c, take i t)"
assms: fixes i :: "nat" assumes "i \<le> j" shows "fs ! i \<in> carrier_vec n"
RedInstanceOf: fixes s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and v :: "'addr val"   and U :: "ty"   and b :: "bool"   and P :: "'addr J_prog"   and T :: "ty"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id" assumes "typeof\<^bsub>hp s\<^esub> v = \<lfloor>U\<rfloor>"   and "b = (v \<noteq> Null \<and> P \<turnstile> U \<le> T)" shows "extTA,P,t \<turnstile> \<langle>Val v instanceof T,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Val (Bool b),s\<rangle>"
less_imp_Suc_mult_pred_less: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "m < n"   and "0 < k" shows "Suc m * k - Suc 0 < n * k"
sums_snoc: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and xs :: "'b list"   and x :: "'b" assumes "class.monoid_add plus zero" shows "monoid_add.sums plus zero (xs @ [x]) = monoid_add.sums plus zero xs @ [monoid_add.sum_list plus zero (xs @ [x])]"
zexplode_zimplode: fixes A :: "ZF zet" shows "zexplode (zimplode A) = A"
apply_inverse_eq_apply_power_order_minus_one: fixes f :: "'a perm"   and a :: "'a" shows "inverse f \<langle>$\<rangle> a = (f ^ (order f a - 1)) \<langle>$\<rangle> a"
\<Lambda>_pr1_def: shows "\<Lambda>_pr1 = local.comp (\<Lambda> a b c g) (pr1 (local.dom (\<Lambda> a b c g)) (local.dom b))"
Max_wrt_preorder: shows "Max_wrt (R i) = {x \<in> alts. \<forall>y\<in>alts. R i x y \<longrightarrow> R i y x}"
scale_left_distrib: fixes a :: "'c"   and b :: "'c"   and x :: "'c" shows "(a + b) * x = a * x + b * x"
add_mono_thms_linordered_semiring_3: fixes i :: "'a"   and j :: "'a"   and k :: "'a"   and l :: "'a" assumes "i \<le> j \<and> k = l" shows "i + k \<le> j + l"
fimage_converges: fixes x :: "'a"   and F :: "'a set set" assumes "x \<in> carrier"   and "F \<longlongrightarrow> x" shows "fimage F \<longlongrightarrow>\<^bsub>T\<^esub> f x"
leq_antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
preserves_arr: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "hseq (fst f) (snd f)"
f3: fixes A :: "'a set"   and Aa :: "'a set"   and a :: "'a"   and Ab :: "'a set"   and Ac :: "'a set" shows "(A \<subseteq> Aa \<or> aa A Aa \<in> A) \<and> (aa A Aa \<notin> Aa \<or> A \<subseteq> Aa) \<and> (a \<notin> Ab \<or> \<not> Ab \<subseteq> Ac \<or> a \<in> Ac)"
del_list_sorted: fixes xs :: "'b list"   and a :: "'b"   and ys :: "'b list"   and x :: "'b" assumes "sorted (xs @ a # ys)" shows "del_list x (xs @ a # ys) = (if x < a then del_list x xs @ a # ys else xs @ del_list x (a # ys))"
ind_is_open: shows "ind_topology.ind_is_open X is_open X = is_open"
horizontal_identity_is_weak_unit: fixes \<mu> :: "'a" assumes "H.ide \<mu>" shows "weak_unit \<mu>"
continuous: shows "continuous_on (T \<times> X) (\<lambda>(t, x). f t x)"
has_absolute_integral_change_of_variables_compact_family: fixes F :: "nat \<Rightarrow> (real, 'a) vec set"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec"   and b :: "(real, 'b) vec" assumes "\<And>n. compact (F n)"   and "\<And>x. x \<in> \<Union> (range F) \<Longrightarrow> (g has_derivative g' x) (at x within \<Union> (range F))"   and "inj_on g (\<Union> (range F))" shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) absolutely_integrable_on \<Union> (range F) \<and> integral (\<Union> (range F)) (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` \<Union> (range F) \<and> integral (g ` \<Union> (range F)) f = b)"
degree_map_poly: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c poly" assumes "(f (lead_coeff p) = (0::'b)) = (p = 0)" shows "degree (map_poly f p) = degree p"
seq_no_leq_num: fixes i :: "nat" shows "ptoy i \<TTurnstile> onl \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (\<lambda>(\<xi>, l). l \<in> {PToy-:7..PToy-:8} \<longrightarrow> no \<xi> \<le> num \<xi>)"
finprod_def: fixes G :: "('h, 'i) monoid_scheme"   and f :: "'j \<Rightarrow> 'h"   and A :: "'j set" shows "finprod G f A = (if finite A then foldD (carrier G) ((\<otimes>\<^bsub>G\<^esub>) \<circ> f) \<one>\<^bsub>G\<^esub> A else \<one>\<^bsub>G\<^esub>)"
gtrp_preserve_s: fixes s :: "'f"   and c :: "'f multiset"   and e :: "'g el_step"   and s' :: "'f"   and c' :: "'f multiset"   and T :: "(('f \<times> 'f multiset) \<times> 'g \<times> 'f \<times> 'f multiset) set"   and P :: "'f multiset \<Rightarrow> bool" assumes "((s, c), e, s', c') \<in> gtrp T"   and "P (add_mset s c)"   and "\<And>s c s' c' e. \<lbrakk>P (add_mset s c); ((s, c), e, s', c') \<in> T\<rbrakk> \<Longrightarrow> P (add_mset s' c')" shows "P (add_mset s' c')"
stop_cxt: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and E :: "('a, 'c, 'd) Stmt list"   and c :: "('a, 'c, 'd) Stmt" assumes "sifum_lang eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "Stop = sifum_lang.cxt_to_stmt E c" shows "c = Stop"
uwalks_def: fixes G :: "nat set \<times> nat set set" shows "uwalks G \<equiv> {p. set p \<subseteq> uverts G \<and> set (uwalk_edges p) \<subseteq> uedges G \<and> p \<noteq> []}"
simps: fixes a :: "('a \<Rightarrow> 'b) set"   and \<R> :: "('a \<Rightarrow> 'b) set set"   and R :: "('a \<Rightarrow> 'b) set" shows "(a \<in> Succ \<R> R) = (\<exists>u R' t. a = R' \<and> u \<in> R \<and> R \<in> \<R> \<and> R' \<in> \<R> \<and> (0::'b) \<le> t \<and> R' = [u \<oplus> t]\<^sub>\<R>)"
sum: fixes g :: "'d \<Rightarrow> 'a"   and S :: "'d set" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
simps_1: fixes uu :: "('c \<times> 'c) set"   and uv :: "'d \<Rightarrow> 'c"   and U :: "'c set" shows "sources_aux uu uv U [] = U"
"case": fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "(case node_ptr.Ext x of node_ptr.Ext x \<Rightarrow> f x) = f x"
path_singleton: fixes v :: "'a" assumes "v \<in> V\<^bsub>remove_vertex x\<^esub>" shows "H.path [v]"
sym: fixes M :: "'b dfa"   and N :: "'b dfa"   and h :: "hf \<Rightarrow> hf" assumes "dfa_isomorphism M N h" shows "dfa_isomorphism N M (inv_into (dfa.states M) h)"
wf_mult_rel_multpw: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set" assumes "wf s"   and "s O ns \<subseteq> s"   and "refl ns" shows "wf ((multpw ns)\<^sup>* O mult s O (multpw ns)\<^sup>*)"
pth_9_1: fixes c :: "real"   and x :: "'a"   and z :: "'a"   and d :: "real" shows "c *\<^sub>R x + z + d *\<^sub>R x \<equiv> (c + d) *\<^sub>R x + z"
assms_4: shows "D E F G H I SumA D' E' F'"
cring_pow_deg: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P"   and "p (deg R p) [^] n \<noteq> \<zero>" shows "deg R (p [^]\<^bsub>P\<^esub> n) = n * deg R p"
EX_MR: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and p :: "((int \<times> int) set set, int) Combinators list"   and ps :: "((int \<times> int) set set, int) Combinators list" assumes "applied_rule_rev C x p \<noteq> \<lfloor>DenyAll\<rfloor>"   and "p = DenyAll # ps" shows "applied_rule_rev C x p = applied_rule_rev C x ps"
monotone_lappend2: fixes xs :: "'c llist" shows "monotone (\<sqsubseteq>) (\<sqsubseteq>) (lappend xs)"
fact_dvd_pochhammer: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n + 1" shows "fact m dvd pochhammer (int n - int m + 1) m"
simps: fixes a1 :: "'c"   and a2 :: "'c" shows "rel_12_23 a1 a2 = (a1 = \<^bold>1 \<and> a2 = \<^bold>2 \<or> a1 = \<^bold>2 \<and> a2 = \<^bold>3)"
aext_lit: fixes v :: "'c"   and a :: "'e \<Longrightarrow> 'd" shows "\<guillemotleft>v\<guillemotright> \<oplus>\<^sub>p a = \<guillemotleft>v\<guillemotright>"
mod_subset_add: fixes c1 :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and c2 :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and p :: "nat"   and X :: "'c topology"   and d1 :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and d2 :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "(c1, c2) \<in> mod_subset p X"   and "(d1, d2) \<in> mod_subset p X" shows "(c1 + d1, c2 + d2) \<in> mod_subset p X"
summable_add: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "summable f"   and "summable g" shows "summable (\<lambda>n. f n + g n)"
simps_1: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" shows "fold f [] = id"
div_le_mult: fixes i :: "'b word"   and k :: "'b word"   and x :: "'b word" assumes "i \<le> k div x"   and "0 < x" shows "i * x \<le> k"
simps_5: fixes x :: "vrs"   and y :: "vrs"   and t' :: "trm"   and t :: "trm"   and T :: "ty" assumes "x \<sharp> (y, t')" shows "(\<lambda>x:T. t)[y \<mapsto> t'] = (\<lambda>x:T. t[y \<mapsto> t'])"
encode2_def: fixes i :: "nat" shows "encode2 i = digit_decode dims2 (nths (digit_encode dims i) (- vars))"
fam_ext: fixes U :: "'a set" assumes "is_open U" shows "fam_morphisms' U \<in> extensional (\<FF> U)"
distinct_1: fixes x1 :: "name"   and x2 :: "name" shows "InputS x1 \<noteq> BoundOutputS x2"
assms_2: shows "y \<in> space\<^sub>N N"
cSup_atMost: fixes x :: "'d multiset" shows "Sup (subset_mset.atMost x) = x"
map_of_concat: assumes "k \<in> dom (map_of (concat l))" shows "\<exists>i. k \<in> dom (map_of (l ! i)) \<and> i < length l"
refine: fixes G :: "('a, 'b, 'c) igba_rec_scheme"   and S :: "('d, 'b, 'e) sa_rec_scheme"   and Gi :: "('f \<Rightarrow> bool, 'f \<Rightarrow> 'f list, 'f list, 'f \<Rightarrow> integer, 'f \<Rightarrow> 'g \<Rightarrow> bool, 'h) gen_igba_impl_scheme"   and Re :: "('h \<times> 'c) set"   and Rq :: "('f \<times> 'a) set"   and Rl :: "('g \<times> 'b) set"   and Si :: "('i \<Rightarrow> bool, 'i \<Rightarrow> 'i list, 'i list, 'i \<Rightarrow> 'g, 'j) gen_sa_impl_scheme"   and Re2 :: "('j \<times> 'e) set"   and Rs :: "('i \<times> 'd) set" assumes "igba_sys_prod_precond G S"   and "(Gi, G) \<in> igba_impl_rel_ext Re Rq Rl"   and "(Si, S) \<in> sa_impl_rel_ext Re2 Rs Rl" shows "(igba_sys_prod_impl Gi Si, igba_sys_prod_precond.prod G S) \<in> igbg_impl_rel_ext unit_rel (Rq \<times>\<^sub>r Rs)"
open_segment_idem: fixes a :: "'b" shows "{a<--<a} = {}"
conj_eqvt: fixes p :: "perm"   and A :: "bool"   and B :: "bool" shows "p \<bullet> (A \<and> B) = (p \<bullet> A \<and> p \<bullet> B)"
rank_drop_cols: shows "vec_space.rank (length signs) (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))) = dim_col (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))"
ball_trans: fixes y :: "'c"   and z :: "'c"   and q :: "real"   and r :: "real"   and s :: "real" assumes "y \<in> ball z q"   and "r + q \<le> s" shows "ball y r \<subseteq> ball z s"
algebraic_add_cancel_left: fixes x :: "'b"   and y :: "'b" assumes "algebraic x" shows "algebraic (x + y) = algebraic y"
comp_funpow: fixes n :: "nat"   and f :: "'h \<Rightarrow> 'h" shows "(\<circ>) f ^^ n = (\<circ>) (f ^^ n)"
finite_mrexps: fixes r :: "'b rexp" shows "finite (mrexps r)"
induct: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'b"   and x2 :: "'b"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* x1 x2"   and "\<And>a. P a a"   and "\<And>a b c. \<lbrakk>r\<^sup>*\<^sup>* a b; P a b; r b c\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
Exit_source: fixes a :: "'f" assumes "valid_edge a"   and "sourcenode a = Exit" shows "False"
SV_inv4_event_pres: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and D :: "process \<Rightarrow> val option" shows "{SV_inv4 \<inter> (Vinv1 \<inter> Vinv2)} sv_round r S v D {> SV_inv4}"
trancl_rtrancl_absorb: fixes R :: "('c \<times> 'c) set" shows "(R\<^sup>+)\<^sup>* = R\<^sup>*"
mat_incr_dim: fixes n :: "nat" shows "mat_incr n \<in> carrier_mat n n"
cupcake_match_result_eq: fixes pes :: "(pat \<times> exp0) list"   and env :: "v sem_env"   and s :: "'a state"   and v :: "v"   and err_v :: "v" assumes "cupcake_clauses pes" shows "match_result env s v pes err_v = map_result (\<lambda>(e, uu_, env'). (e, env')) id (cupcake_match_result (c env) v pes err_v)"
rightUniqueInjectiveOnFirst: fixes P :: "('a \<times> 'b) set" shows "runiq P = inj_on fst P"
least_def: fixes L :: "('c, 'd) gorder_scheme"   and l :: "'c"   and A :: "'c set" shows "least L l A = (A \<subseteq> carrier L \<and> l \<in> A \<and> (\<forall>x\<in>A. l \<sqsubseteq>\<^bsub>L\<^esub> x))"
cod_cod: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.cod (hom'.cod f) = hom'.cod f"
chart_basis_inverse: fixes x :: "'a proj_space" assumes "chart_basis_domainP b x" shows "local.chart_basis_inv (chart_basis b x) = x"
left_right: fixes m :: "real"   and a :: "real"   and b :: "real" assumes "m \<in> {a..b}"   and "{a..b} \<subseteq> {0..1}" shows "a \<le> leftcut a b m \<and> rightcut a b m \<le> b"
elem_mset_set: fixes A :: "'k set"   and x :: "'k" assumes "finite A" shows "(x \<in># mset_set A) = (x \<in> A)"
h: shows "h \<in> ring_hom (R\<lparr>carrier := K\<rparr>) (S\<lparr>carrier := h ` K\<rparr>)"
cod_MkArr: fixes A :: "'a E.term"   and B :: "'a E.term"   and F :: "'a" assumes "arr (MkArr A B F)" shows "cod (MkArr A B F) = MkIde B"
MaxR_def: fixes r :: "('b \<times> 'b) set" shows "MaxR r \<equiv> Linear_order r"
assms_7: shows "\<forall>m<k + 3. req (t + m) \<noteq> [send]"
mod_poly_less: fixes x :: "'c poly"   and y :: "'c poly" assumes "degree x < degree y" shows "x mod y = x"
\<beta>: shows "\<guillemotleft>\<a>[g, f\<^sub>a, f] \<cdot> (\<nu> \<star> f) \<cdot> \<rho> \<cdot> \<r>[g] : g \<star> src g \<Rightarrow> g \<star> f\<^sub>a \<star> f\<guillemotright> \<and> \<guillemotleft>\<a>[g, f\<^sub>a, f] \<cdot> (\<nu> \<star> f) \<cdot> \<rho> \<cdot> \<r>[g] : src f \<rightarrow> trg g\<guillemotright> \<and> src (\<a>[g, f\<^sub>a, f] \<cdot> (\<nu> \<star> f) \<cdot> \<rho> \<cdot> \<r>[g]) = src g \<and> trg (\<a>[g, f\<^sub>a, f] \<cdot> (\<nu> \<star> f) \<cdot> \<rho> \<cdot> \<r>[g]) = trg g \<and> local.dom (\<a>[g, f\<^sub>a, f] \<cdot> (\<nu> \<star> f) \<cdot> \<rho> \<cdot> \<r>[g]) = g \<star> src g \<and> cod (\<a>[g, f\<^sub>a, f] \<cdot> (\<nu> \<star> f) \<cdot> \<rho> \<cdot> \<r>[g]) = g \<star> f\<^sub>a \<star> f"
normB_S: fixes v :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "v \<in> local.S" shows "norm1 (local.B *v v) \<noteq> 0"
lift_CFG: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Exit :: "'h"   and Def :: "'h \<Rightarrow> 'i set"   and Use :: "'h \<Rightarrow> 'i set"   and ParamDefs :: "'h \<Rightarrow> 'i list"   and ParamUses :: "'h \<Rightarrow> 'i set list" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit" shows "CFG src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_get_proc get_proc Main) (lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind) procs Main"
plus_eq_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and s :: "'a"   and t :: "'a" assumes "class.ninv_comm_monoid_add plus zero"   and "plus s t = zero" shows "s = zero"
extremum_unique: fixes a :: "'e list" shows "prefix a [] = (a = [])"
ide_compE: fixes g :: "'a"   and f :: "'a" assumes "Left_a.ide (Left_a.comp g f)" obtains "Left_a.seq g f" and "Left_a.seq f g" and "Left_a.comp g f = Left_a.dom f" and "Left_a.comp g f = Left_a.cod g"
WS_weak_sim: fixes ms\<^sub>1 :: "'node list"   and s\<^sub>1 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and ms\<^sub>2 :: "'node list"   and s\<^sub>2 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and S :: "'node SDG_node set"   and as :: "'edge list"   and ms\<^sub>1' :: "'node list"   and s\<^sub>1' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "((ms\<^sub>1, s\<^sub>1), ms\<^sub>2, s\<^sub>2) \<in> WS S"   and "S,kind \<turnstile> (ms\<^sub>1,s\<^sub>1) =as\<Rightarrow> (ms\<^sub>1',s\<^sub>1')"   and "s\<^sub>1' \<noteq> []" obtains as' :: "'edge list" where "((ms\<^sub>1', s\<^sub>1'), ms\<^sub>1', transfer (slice_kind S (last as)) s\<^sub>2) \<in> WS S" and "S,slice_kind S \<turnstile> (ms\<^sub>2,s\<^sub>2) =as' @ [last as]\<Rightarrow> (ms\<^sub>1',transfer (slice_kind S (last as)) s\<^sub>2)"
is_canon_rewriterI_eq_sig: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "rb_aux_inv1 bs"   and "b \<in> set bs" shows "is_canon_rewriter rword (set bs) (lt b) b"
inclusion: fixes f :: "'a" assumes "right \<omega> f" shows "arr f"
scalar_prod_cons: fixes ze :: "'c"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and ti :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and as :: "'c list"   and b :: "'c"   and bs :: "'c list" shows "scalar_prodI ze pl ti (a # as) (b # bs) = pl (ti a b) (scalar_prodI ze pl ti as bs)"
iD_isD: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "p \<noteq> []"   and "x \<notin> dom (C (list2FWpolicy p))" shows "C (DenyAll \<oplus> list2FWpolicy (insertDenies p)) x = C DenyAll x"
subset_closed_segment: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "({a--b} \<subseteq> {c--d}) = (a \<in> {c--d} \<and> b \<in> {c--d})"
ide_cod: fixes f :: "(bool, bool list) concrete_category.arr" assumes "J.arr f" shows "J.ide (J.cod f)"
funpow_Suc_right: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b" shows "f ^^ Suc n = f ^^ n \<circ> f"
yes: fixes xs :: "'a list"   and x :: "'a" assumes "index xs x < length xs" shows "xs ! index xs x = x"
dedekind: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ; y \<cdot> z \<le> (x \<cdot> z ; y\<^sup>T) ; (y \<cdot> x\<^sup>T ; z)"
factor_unit: shows "additive.Ker = additive.kernel.Left_Coset \<zero> additive.Ker"
separating_of_bipartite_aux: fixes S :: "('v + 'v) set"   and x :: "'v"   and p :: "'v list"   and y :: "'v" assumes "separating (bipartite_web_of \<Gamma>) S"   and "path \<Gamma> x p y"   and "y \<in> B \<Gamma>"   and "x \<in> A \<Gamma> \<or> Inr x \<in> S" shows "(\<exists>z\<in>set p. z \<in> separating_of_bipartite S) \<or> x \<in> separating_of_bipartite S"
select_convs_3: fixes xpos :: "nat"   and ypos :: "nat"   and more :: "'a" shows "point.more \<lparr>xpos = xpos, ypos = ypos, \<dots> = more\<rparr> = more"
step3: fixes r :: "('a \<times> 'a) set"   and \<tau> :: "'a list"   and \<sigma> :: "'a list"   and \<sigma>' :: "'a list"   and \<tau>' :: "'a list" assumes "trans r"   and "Decreasing_Diagrams.D r \<tau> \<sigma> \<sigma>' \<tau>'" shows "r \<down>m (r|\<sigma>'| + r|\<tau>|) \<subseteq> r \<down>l (\<sigma> @ \<tau>)"
R2: shows "\<forall>x y. R2 x y \<longrightarrow> (\<forall>a b. R2 a b \<longrightarrow> S' x a = T' y b)"
vbrelation_axioms: shows "vbrelation (T\<lparr>ArrVal\<rparr>)"
cm_neg: fixes m :: "nat" shows "continuous_map (nsphere m) (nsphere m) neg"
size_splay: fixes a :: "'b"   and t :: "'b tree" shows "size (splay a t) = size t"
simps: fixes p :: "'a \<times> 'b"   and a :: "'b" shows "sndsp p a = (a = snd p)"
suffix_ConsI: fixes xs :: "'b list"   and ys :: "'b list"   and y :: "'b" assumes "suffix xs ys" shows "suffix xs (y # ys)"
repeat_mset_size: fixes n :: "nat"   and A :: "'a multiset" shows "size (repeat_mset n A) = n * size A"
cases: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and S :: "'a set" assumes "orthogonal \<Delta> f S" obtains "\<And>x y. \<lbrakk>edge \<Delta> x y; x \<in> S; y \<notin> S\<rbrakk> \<Longrightarrow> f (x, y) = capacity \<Delta> (x, y)" and "\<And>x y. \<lbrakk>edge \<Delta> x y; x \<notin> S; y \<in> S\<rbrakk> \<Longrightarrow> f (x, y) = 0"
cmp_def: fixes \<mu>\<nu> :: "'a \<times> 'a" shows "cmp \<mu>\<nu> = (if S.VV.arr \<mu>\<nu> then fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu> else B.null)"
length_word_rsplit_even_size: fixes n :: "nat"   and w :: "'d word"   and m :: "nat" assumes "n = LENGTH('c)"   and "size w = m * n" shows "length (word_rsplit w) = m"
FAssRedsThrow1: 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>throw e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D} := e\<^sub>2,s,b\<rangle> \<rightarrow>* \<langle>throw e',s',b'\<rangle>"
simps: fixes t\<^sub>1 :: "'b"   and t\<^sub>2 :: "'b" shows "\<tau>\<^sub>c\<^sub>s\<^sub>t t\<^sub>1 - \<tau>\<^sub>c\<^sub>s\<^sub>t t\<^sub>2 = \<tau>\<^sub>c\<^sub>s\<^sub>t (t\<^sub>1 - t\<^sub>2)"
minus_dvd_iff: fixes x :: "'a"   and y :: "'a" shows "(- x dvd y) = (x dvd y)"
identify_block_le_rev: fixes j :: "nat"   and A :: "'b mat"   and i :: "nat" assumes "j = ??.Jordan_Normal_Form_Existence.identify_block A i" shows "j \<le> i"
final_iff_Nil: fixes s :: "'a rexp" shows "nullable s = ([] \<in> lang s)"
finite_dimensional_vector_space_pair_1_ow_def: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f"   and scale\<^sub>1 :: "'g \<Rightarrow> 'f \<Rightarrow> 'f"   and basis\<^sub>1 :: "'f set"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h"   and scale\<^sub>2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'h" shows "finite_dimensional_vector_space_pair_1_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 \<equiv> finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 \<and> vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2"
derivation_shift_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and left :: "nat"   and right :: "nat" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "derivation_shift D left right = map (\<lambda>d. (fst d - left + right, snd d)) D"