hcju/isabelleps huggingface dataset:
img_subset: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "f \<in> A \<rightarrow> B" shows "f ` A \<subseteq> B"
inf_rep_def: fixes p1 :: "int extended \<times> int extended"   and p2 :: "int extended \<times> int extended" shows "inf_rep p1 p2 = (let (l1, h1) = p1; (l2, h2) = p2 in (max l1 l2, min h1 h2))"
line_in_subsegment2: fixes i2 :: "real"   and i1 :: "real"   and a :: "'b"   and b :: "'b" assumes "0 < i2"   and "0 < i1"   and "a \<noteq> b" shows "({a -- b}\<^bsub>i1\<^esub> \<in> {a<--<{a -- b}\<^bsub>i2\<^esub>}) = (i1 < i2)"
select_convs_1: fixes init :: "'d set"   and trans :: "('d \<times> 'e \<times> 'd) set"   and more :: "'f" shows "init \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = init"
asIMOD_igOpIPresIGWls: fixes Delta :: "('c, 'd, 'e, 'f, 'g) signature"   and SEM :: "('c, 'd, 'e, 'f, 'g, 'h) semDom" assumes "FixSyn TYPE('b) Delta"   and "FixSyn.sOpPrSWls Delta SEM"   and "sWlsNE SEM" shows "FixSyn.igOpIPresIGWls Delta (FixSyn.asIMOD Delta SEM)"
distinguishing_formula_card_aux: fixes P :: "'state"   and Q :: "'state" shows "|{- p \<bullet> (SOME x. supp x \<subseteq> supp (p \<bullet> P) \<and> x distinguishes (p \<bullet> P) from (p \<bullet> Q)) |p. True}| <o natLeq +c |UNIV|"
that: fixes ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "LT = collect_waits ta" shows "thesis"
quasi_isometry_on_compose: fixes lambda :: "real"   and C :: "real"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and mu :: "real"   and D :: "real"   and Y :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "lambda C -quasi_isometry_on X f"   and "mu D -quasi_isometry_on Y g"   and "f ` X \<subseteq> Y" shows "(lambda * mu) (C * mu + D) -quasi_isometry_on X (g \<circ> f)"
st: shows "s \<in> \<S>\<^sub>a \<and> t \<in> \<S>\<^sub>a \<and> t \<le>\<^sub>\<S> \<lambda> (s, A) \<and> t \<circ> B \<noteq> B"
isomorphicE: fixes a :: "'a"   and a' :: "'a" assumes "B.isomorphic a a'" obtains f :: "'a" where "\<guillemotleft>f : a \<rightarrow>\<^sub>B a'\<guillemotright> \<and> B.iso f"
fvi_subset: shows "fv \<phi>'' \<subseteq> fv \<psi>''"
islimpt_def: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and S :: "'b set" assumes "class.topological_space open" shows "topological_space.islimpt open x S = (\<forall>T. x \<in> T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x))"
square_beta_eta: shows "square (\<rightarrow>\<^sub>\<beta>) (\<rightarrow>\<^sub>\<eta>) (\<rightarrow>\<^sub>\<eta>\<^sup>*) (\<rightarrow>\<^sub>\<beta>)\<^sup>=\<^sup>="
c2_def: fixes y :: "'a" shows "c2 y = f y * g (- y + x)"
remdups_sorted_conv_remdups: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "ord.remdups_sorted less xs = remdups xs"
SReal_epsilon_not_mem: shows "\<epsilon> \<notin> \<real>"
minus_eq: fixes x :: "'a list"   and y :: "'a list" shows "x \<ominus>\<^bsub>poly_ring R\<^esub> y = x \<oplus>\<^bsub>poly_ring R\<^esub> \<ominus>\<^bsub>poly_ring R\<^esub> y"
invertible_Gauss_Jordan: fixes A :: "(('b, 'd) vec, 'c) vec" shows "\<exists>P. invertible P \<and> Gauss_Jordan A = P ** A"
longest_common_prefix: fixes xs :: "'a list"   and ys :: "'a list" shows "\<exists>ps xs' ys'. xs = ps @ xs' \<and> ys = ps @ ys' \<and> (xs' = [] \<or> ys' = [] \<or> hd xs' \<noteq> hd ys')"
COD_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "arr f"   and "arr g" shows "COD (f \<otimes> g) = COD f \<^bold>\<otimes> COD g"
bernoulli_pmf_of_set': fixes A :: "'b set"   and P :: "bool" assumes "finite A" shows "map_pmf (\<lambda>b. {x \<in> A. \<not> b x}) (Pi_pmf A P (\<lambda>_. bernoulli_pmf (1 / 2))) = pmf_of_set (Pow A)"
rank_upper_bound: fixes x :: "nat"   and n :: "nat"   and i :: "nat" assumes "rank x n = Some i" shows "i < max_rank"
split_conv: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and a :: "'f"   and b :: "'g" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
left_neutral: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and a :: "'c" assumes "Groups.monoid f z" shows "f z a = a"
oracle_def: fixes \<sigma> :: "'b \<Rightarrow> bool list option"   and x :: "'b" shows "oracle \<sigma> x = (case \<sigma> x of None \<Rightarrow> spmf_of_set (nlists UNIV len) \<bind> (\<lambda>bs. return_spmf (bs, \<sigma>(x \<mapsto> bs))) | Some bs \<Rightarrow> return_spmf (bs, \<sigma>))"
limit_inter_INF: fixes w :: "nat \<Rightarrow> 'b"   and S :: "'b set" assumes "limit w \<inter> S \<noteq> {}" shows "\<exists>\<^sub>\<infinity>n. w n \<in> S"
rotate2_2: fixes x :: "(real, 2) vec" shows "rotate2 x $ 2 = x $ 1"
catch_env_def: fixes catch :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and m1 :: "('e, 'f) envT"   and m2 :: "('e, 'f) envT" shows "catch_env catch m1 m2 = EnvT (\<lambda>r. catch (run_env m1 r) (run_env m2 r))"
path_append_def: fixes \<pi> :: "nat \<Rightarrow> 'n"   and m :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'n" shows "\<pi> @\<^bsup>m\<^esup> \<pi>' = (\<lambda>n. if n \<le> m then \<pi> n else \<pi>' (n - m))"
map_disj_add_eq_dom_right_eq: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and h\<^sub>0' :: "'c \<Rightarrow> 'd option"   and h\<^sub>1' :: "'c \<Rightarrow> 'd option" assumes "h\<^sub>0 ++ h\<^sub>1 = h\<^sub>0' ++ h\<^sub>1'"   and "dom h\<^sub>0 = dom h\<^sub>0'"   and "h\<^sub>0 \<bottom> h\<^sub>1"   and "h\<^sub>0' \<bottom> h\<^sub>1'" shows "h\<^sub>1 = h\<^sub>1'"
respection: shows "rel_respects_binary_pred Rel (\<lambda>a. hasBarb a CWB)"
fds_right_inverse: fixes f :: "'a fds" assumes "fds_nth f 1 \<noteq> (0::'a)" shows "f * inverse f = 1"
Posix_determ: fixes s :: "'b list"   and r :: "'b rexp"   and v1 :: "'b val"   and v2 :: "'b val" assumes "s \<in> r \<rightarrow> v1"   and "s \<in> r \<rightarrow> v2" shows "v1 = v2"
less_eq_bot_Bot_is_Bot: fixes x :: "'b bot" assumes "x \<le> bot.Bot" shows "x = bot.Bot"
\<sigma>_def: fixes p :: "fm"   and k :: "int"   and t :: "MIR.num" shows "\<sigma> p k t \<equiv> And (Dvd k t) (\<sigma>_\<rho> p (t, k))"
vrange_vdoubleton: fixes a :: "V"   and b :: "V"   and c :: "V"   and d :: "V" shows "\<R>\<^sub>\<circ> (set {\<langle>a, b\<rangle>, \<langle>c, d\<rangle>}) = set {b, d}"
universe_intro: fixes z :: "i" assumes "CMU"   and "\<forall>w. P w z" shows "u = z"
homeomorphic_maps_subtopologies_alt: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a"   and S :: "'a set"   and T :: "'b set" assumes "homeomorphic_maps X Y f g"   and "f ` (topspace X \<inter> S) \<subseteq> T"   and "g ` (topspace Y \<inter> T) \<subseteq> S" shows "homeomorphic_maps (subtopology X S) (subtopology Y T) f g"
inorder_insert: fixes t :: "'a bro"   and h :: "nat"   and a :: "'a" assumes "t \<in> T h"   and "sorted (inorder t)" shows "inorder (insert a t) = ins_list a (inorder t)"
Flawlessness: shows "[\<lambda>w. \<forall>x xa. (G xa m\<rightarrow> m\<not> (P x) m\<rightarrow> m\<not> (x xa)) w]"
typ_of1_eta_red_step_pre: fixes t :: "Core.term"   and Ts :: "typ list"   and \<tau> :: "typ"   and \<tau>' :: "typ" assumes "\<not> loose_bvar1 t 0"   and "typ_of1 Ts (Abs \<tau> (t $ Bv 0)) = Some (\<tau> \<rightarrow> \<tau>')" shows "typ_of1 Ts (decr 0 t) = Some (\<tau> \<rightarrow> \<tau>')"
pre_2: shows "dom (tcsigs (translate_osig sa)) = dom (map_of tao)"
prob_space_axioms: fixes M :: "'c pmf" shows "prob_space (measure_pmf M)"
Eq\<^sub>EI: fixes x :: "\<kappa>"   and y :: "\<kappa>"   and v :: "i" assumes "\<exists>o\<^sub>1 o\<^sub>2. Some (\<omega>\<nu> o\<^sub>1) = Semantics.d\<^sub>\<kappa> x \<and> Some (\<omega>\<nu> o\<^sub>2) = Semantics.d\<^sub>\<kappa> y \<and> o\<^sub>1 = o\<^sub>2" shows "[x \<^bold>=\<^sub>E y in v]"
bc_epsilon: fixes \<epsilon> :: "real" assumes "0 < \<epsilon>"   and "\<epsilon> < 1" shows "\<forall>\<^sub>F n in sequentially. (real_of_int (c (n + 1)) - \<epsilon>) / real_of_int (c n) < real_of_int (b (n + 1)) / real_of_int (b n)"
int_imp_algebraic_int: fixes x :: "'b" assumes "x \<in> \<int>" shows "algebraic_int x"
enat_the_enat: fixes n :: "enat" assumes "n \<noteq> \<infinity>" shows "enat (the_enat n) = n"
p2lNmt: fixes p :: "('b, 'c) Combinators" shows "policy2list p \<noteq> []"
fin: fixes A :: "'b llist set"   and N :: "enat" assumes "GaleStewartDeterminedGames.finite_GSgame A N" shows "N \<noteq> \<infinity>"
closure_def: fixes S :: "'b set" shows "closure S = S \<union> {x. x islimpt S}"
less_sets_UN2: fixes A :: "'a set"   and \<B> :: "'a set set" shows "(A \<lless> \<Union> \<B>) = (\<forall>B\<in>\<B>. A \<lless> B)"
analz_subset_cong: fixes G :: "msg set"   and G' :: "msg set"   and H :: "msg set"   and H' :: "msg set" assumes "analz G \<subseteq> analz G'"   and "analz H \<subseteq> analz H'" shows "analz (G \<union> H) \<subseteq> analz (G' \<union> H')"
union_diff2: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "finite B" shows "F g (A \<union> B) = F g (A - B) \<^bold>* F g (B - A) \<^bold>* F g (A \<inter> B)"
get_shadow_root_shadow_root_ptr_in_heap: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and host :: "(_) element_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr" assumes "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs"   and "heap_is_wellformed h"   and "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr" shows "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h"
path0: fixes \<sigma> :: "'var \<Rightarrow> 'val" shows "path \<sigma> 0 = local.entry"
set_offending_flows_simp: fixes G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b" assumes "wf_graph G" shows "set_offending_flows G nP = {F. F \<subseteq> edges G \<and> (\<not> sinvar G nP \<and> sinvar \<lparr>nodes = nodes G, edges = edges G - F\<rparr> nP) \<and> (\<forall>(e1, e2)\<in>F. \<not> sinvar \<lparr>nodes = nodes G, edges = {(e1, e2)} \<union> (edges G - F)\<rparr> nP)}"
abel_semigroup_axioms: shows "abel_semigroup (\<^bold>\<squnion>)"
wave_plus: assumes "wave (quotient_web \<Gamma> f) (g \<upharpoonleft> \<Gamma> / f)" shows "wave \<Gamma> (f \<frown> g)"
F_def: fixes x :: "(real \<times> 'a) stream \<times> (real \<times> 'a) stream"   and n :: "nat" shows "F x n = (case x of (\<omega>, \<omega>') \<Rightarrow> merge_at \<omega> j \<omega>') !! n"
surjective_pairing: fixes t :: "'d \<times> 'e" shows "t = (fst t, snd t)"
assms_1: shows "trms\<^sub>s\<^sub>t S \<subseteq> SMP S'"
echelon_form_Hermite_of_condition1: fixes A :: "(('a, 'b) vec, 'c) vec"   and ass :: "'a \<Rightarrow> 'a"   and i :: "'c"   and ia :: "'c"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and j :: "'c" assumes "echelon_form A"   and "ass_function ass"   and "\<not> is_zero_row i A"   and "is_zero_row ia (Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res)"   and "ia < j" shows "is_zero_row j (Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res)"
nonzero: shows "MPoly_Type.degree p var \<noteq> 0"
valid_unMultigraph_axioms_def: fixes G :: "('a, 'b) graph" shows "valid_unMultigraph_axioms G \<equiv> (\<forall>v w u'. ((v, w, u') \<in> edges G) = ((u', w, v) \<in> edges G)) \<and> (\<forall>v w. (v, w, v) \<notin> edges G)"
widen_unique_def: 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 C :: "char list"   and D :: "char list"   and Cs :: "char list list" shows "widen_unique P C D Cs = (\<forall>Cs'. Subobjs P C Cs' \<longrightarrow> last Cs' = D \<longrightarrow> Cs = Cs')"
dist_triangle_le: fixes x :: "'b"   and z :: "'b"   and y :: "'b"   and e :: "real" assumes "dist x z + dist y z \<le> e" shows "dist x y \<le> e"
surjective: fixes r :: "('b, 'c) Line_scheme" shows "r = \<lparr>basepoint = basepoint r, direction = direction r, \<dots> = Line.more r\<rparr>"
assert_bound_loop_def: fixes ats :: "('i \<times> 'a atom) list"   and s :: "('i, 'a) state" shows "assert_bound_loop ats s \<equiv> foldl (\<lambda>s' a. if \<U> s' then s' else assert_bound a s') s ats"
op_norm_le_sum_column: fixes A :: "((real, 'a) vec, 'b) vec" shows "\<parallel>A\<parallel>\<^sub>o\<^sub>p \<le> (\<Sum>i\<in>UNIV. \<parallel>column i A\<parallel>)"
inv_cancelL_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> Units R"   and "y \<in> carrier R"   and "z \<in> carrier R"   and "y = z \<otimes> x" shows "y \<otimes> inv x = z"
assms_3: shows "e = Binop_i t iop \<or> e = Binop_f t fop \<or> e = Relop_i t irop \<or> e = Relop_f t frop"
nonempty: fixes x :: "real" shows "{lim (\<lambda>k. f (d k) (r n)) |n. x < r n} \<noteq> {}"
residualGraph_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "residualGraph c f \<equiv> \<lambda>(u, v). if (u, v) \<in> Graph.E c then c (u, v) - f (u, v) else if (v, u) \<in> Graph.E c then f (v, u) else (0::'a)"
eq_upred_def: fixes x :: "('d, 'c) uexpr"   and y :: "('d, 'c) uexpr" shows "(x =\<^sub>u y) = bop (=) x y"
hyps_2: fixes b :: "'a action"   and ba :: "('a, 'b, 'c) psi" assumes "\<alpha> \<prec> P' = b \<prec> ba" shows "distinct (bn b)"
oracle_encrypt_def: fixes k :: "'key"   and b :: "bool"   and L :: "'cipher set" shows "oracle_encrypt k b L = (\<lambda>(msg1, msg0). case msg_predicate msg1 \<and> msg_predicate msg0 of True \<Rightarrow> encrypt k (if b then msg1 else msg0) \<bind> (\<lambda>c. return_spmf (Some c, {c} \<union> L)) | False \<Rightarrow> return_spmf (None, L))"
twos_complement: fixes x :: "'a Word.word" shows "- x = word_succ (~~ x)"
scale_right_imp_eq: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<noteq> (0::'b)"   and "a * x = b * x" shows "a = b"
measure_spmf_spmf_of_set: fixes A :: "'c set" shows "measure_spmf (spmf_of_set A) = (if finite' A then measure_pmf (pmf_of_set A) else null_measure (count_space UNIV))"
cont_eff_ParT_1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and i :: "nat"   and s :: "'state" assumes "properL cl"   and "n < length cl"   and "i < brn (cl ! n)" shows "local.cont (ParT cl) s (brnL cl n + i) = ParT (cl[n := local.cont (cl ! n) s i])"
valid_W_inv_colours: fixes x :: "'d"   and s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'd) Model.local_state"   and p :: "'e process_name" assumes "white x s"   and "valid_W_inv s" shows "x \<notin> W (s p)"
replication_numbers_def: shows "replication_numbers \<equiv> {\<B> rep x |x. x \<in> \<V>}"
CountInv_def: shows "CountInv \<equiv> \<lambda>s ind. \<lbrace>\<acute>(<) ind \<and> \<acute>((!) (Ma s)) = Black\<rbrace> \<subseteq> bc s"
r_4: shows "prob {x. infdist x S \<le> r} < prob S + epsilon"
IrrE_4: fixes K :: "'a set"   and x :: "'a" assumes "subfield K R"   and "x \<in> carrier R"   and "(algebraic over K) x" shows "local.eval (Irr K x) x = \<zero>"
outputCases: fixes a :: "name"   and b :: "name"   and P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and Prop :: "freeRes \<Rightarrow> pi \<Rightarrow> bool" assumes "a{b}.P \<longmapsto> \<alpha> \<prec> P'"   and "\<lbrakk>\<alpha> = a[b]; P = P'\<rbrakk> \<Longrightarrow> Prop (a[b]) P" shows "Prop \<alpha> P'"
Generator_inverse: fixes y :: "'e \<Rightarrow> ('f, 'e) step" assumes "y \<in> {g. terminates g}" shows "generator (Generator y) = y"
encode_problem_parallel_correct_v: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat"   and k :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t"   and "k < length (\<Phi>\<inverse> \<Pi> \<A> t)" shows "\<Phi>\<^sub>S\<inverse> \<Pi> \<A> Suc k = execute_parallel_operator (\<Phi>\<^sub>S\<inverse> \<Pi> \<A> k) ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)"
equivclp_induct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "equivclp r a b"   and "P a"   and "\<And>y z. \<lbrakk>equivclp r a y; r y z \<or> r z y; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
simps_4: fixes r :: "real"   and x :: "quat" shows "Im3 (r *\<^sub>R x) = r * Im3 x"
euclerian_rev: fixes G :: "('a, 'b) graph"   and v' :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list"   and v :: "'a" assumes "valid_unMultigraph G" shows "valid_unMultigraph.is_Eulerian_trail G v' (rev_path ps) v = valid_unMultigraph.is_Eulerian_trail G v ps v'"
l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root_safe :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_safe_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_mode :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, shadow_root_mode) prog"   and get_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root_safe get_shadow_root_safe_locs get_shadow_root get_shadow_root_locs get_mode get_mode_locs \<equiv> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<and> l_get_mode type_wf get_mode get_mode_locs \<and> l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf get_shadow_root_safe get_shadow_root_safe_locs get_shadow_root get_shadow_root_locs get_mode get_mode_locs"
satisfy_equation_R_def: fixes p :: "real poly"   and qs :: "real poly list"   and subset_list :: "(nat list \<times> nat list) list"   and sign_list :: "rat list list" shows "satisfy_equation_R p qs subset_list sign_list = (matrix_A_R sign_list subset_list *\<^sub>v construct_lhs_vector_R p qs sign_list = construct_rhs_vector_R p qs subset_list)"
sublinear_iterates: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool"   and i :: "nat" assumes "healthy (wp body)"   and "sublinear (wp body)" shows "sublinear (iterates body G i)"
approx_add_left_iff: fixes a :: "'c star"   and b :: "'c star"   and c :: "'c star" shows "(a + b \<approx> a + c) = (b \<approx> c)"
undead: fixes x :: "'v"   and y :: "'v" assumes "edge \<Delta> x y" shows "(\<exists>z. edge \<Delta> y z) \<or> (\<exists>z. edge \<Delta> z x)"
max_lideal_def: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" shows "max_lideal I R addition multiplication zero unit \<equiv> lideal I R addition multiplication zero unit \<and> max_lideal_axioms I R addition multiplication zero unit"
pullback_def: fixes T :: "('a, 'c) partial_object_scheme"   and f :: "'a \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'b" shows "pullback T f g = compose (carrier T) g f"
minus_one_word: shows "- 1 = 2 ^ LENGTH('b) - 1"
hyps: shows "\<exists>P. invertible P \<and> Gauss_Jordan_upt_k A k = P ** A"
induct: fixes P :: "('b, 'a) Term.term \<Rightarrow> bool"   and "term" :: "('b, 'a) Term.term" assumes "\<And>x. P (Var x)"   and "\<And>x1a x2. (\<And>x2a. x2a \<in> set x2 \<Longrightarrow> P x2a) \<Longrightarrow> P (Fun x1a x2)" shows "P term"
abs_cnt_nb: fixes nb_a :: "'a \<Rightarrow> nat \<Rightarrow> 'a"   and a_initial_contour :: "'a"   and abs_cnt :: "contour \<Rightarrow> 'a"   and b :: "contour"   and lab :: "nat" assumes "class.contour_a nb_a a_initial_contour abs_cnt" shows "abs_cnt (nb b lab) = nb_a (abs_cnt b) lab"
select_convs_3: fixes sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "current \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = current"
h2_def: fixes g :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and fs :: "('c \<times> 'd) \<times> 'e" shows "h2 g fs = (let (a, b) = fst fs in g a b)"
funion_left_commute: fixes A :: "'a fset"   and B :: "'a fset"   and C :: "'a fset" shows "A |\<union>| (B |\<union>| C) = B |\<union>| (A |\<union>| C)"
simps_1: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.rbt_sorted less rbt.Empty = True"
ACI_norm_flatten: fixes r :: "'c rexp" shows "\<guillemotleft>r\<guillemotright> = PLUS (sorted_list_of_set (ACI_norm ` toplevel_summands r))"
indexlt_def: fixes k' :: "nat"   and u' :: "nat"   and k :: "nat"   and u :: "nat" shows "indexlt k' u' k u = (((k', u'), k, u) \<in> indexlt_rel)"
uconnected_sym: fixes E :: "'b uprod set" shows "sym (uconnected E)"
nonzero_imp_inverse_nonzero: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero" shows "inverse a \<noteq> zero"