hcju/isabelleps huggingface dataset:
transform_upper_closure_3: fixes rs :: "'a common_primitive rule list" assumes "simple_ruleset (rs::'a common_primitive rule list)" shows "\<forall>r::'a common_primitive rule\<in>set (upper_closure (rs::'a common_primitive rule list)). normalized_nnf_match (get_match r) \<and> normalized_src_ports (get_match r) \<and> normalized_dst_ports (get_match r) \<and> normalized_src_ips (get_match r) \<and> normalized_dst_ips (get_match r) \<and> \<not> has_disc is_MultiportPorts (get_match r) \<and> \<not> has_disc is_Extra (get_match r)"
ideal_eSum_closed: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "\<forall>j\<le>m. f j \<in> I" shows "\<Sigma>\<^sub>e K f m \<in> I"
ip_aux: fixes V :: "'v set"   and ws :: "'v \<Rightarrow> 'v list option"   and u :: "'v"   and p :: "'v list" assumes "finite (E\<^sup>* `` U0)"   and "inv (None, V, ws)"   and "ws u = Some p"   and "\<not> P u" shows "inv (None, V \<union> E `` {u}, ws |` (- {u}) ++ (\<lambda>v. if (u, v) \<in> E \<and> v \<notin> V then Some (u # p) else None))"
ucarr: shows "u \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"
bound_outside: fixes z :: "complex" assumes "1 < \<bar>Re z\<bar>" shows "cmod (H z) \<le> exp 1 / (exp 1 - 1)"
simps_2: fixes x :: "octo"   and y :: "octo" shows "Im1 (x - y) = Im1 x - Im1 y"
map_graph_mem_equiv: fixes x :: "'c"   and y :: "'d"   and f :: "'c \<Rightarrow> 'd option" shows "((x, y) \<in> map_graph f) = (f x = Some y)"
cond_actions_ok_\<tau>mred1_inv: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list"   and s1 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s1' :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b"   and ct :: "'b conditional_action" assumes "FWdelay_bisimulation_lift_aux final1 r1 final2 r2 \<tau>move1 \<tau>move2"   and "\<tau>trsys.silent_move (multithreaded_base.redT final1 r1 convert_RA) (\<tau>multithreaded.m\<tau>move \<tau>move1) s1 s1'"   and "final_thread.cond_action_ok final1 s1 t ct" shows "final_thread.cond_action_ok final1 s1' t ct"
lsphere_def: shows "lsphere \<equiv> \<lambda>n. subtopology (nsphere n) (lower n)"
h_eq_p: fixes x :: "real" assumes "x \<in> T" shows "h x = p x"
set_mult_subgroup_idem: fixes H :: "'a set"   and N :: "'a set" assumes "subgroup H G"   and "subgroup N (G\<lparr>carrier := H\<rparr>)" shows "H <#> N = H"
cobounded1: fixes a :: "'a DBMEntry"   and b :: "'a DBMEntry" shows "a \<preceq> linordered_monoid.max a b"
legs_form_cospan_1: shows "C.cospan trg.leg0 f.leg1"
intro: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> nat"   and term_of_pair :: "'b \<times> nat \<Rightarrow> 'a"   and cmp_term :: "'a nat_term_order" assumes "gd_nat_term pair_of_term term_of_pair cmp_term" shows "gd_nat_inf_term pair_of_term term_of_pair cmp_term"
inv_closed: fixes x :: "'c" assumes "x \<in> carrier H" shows "\<ominus>\<^bsub>H\<^esub> x \<in> carrier H"
Succ_def: fixes Kl :: "'b list set"   and kl :: "'b list" shows "Succ Kl kl = {k. kl @ [k] \<in> Kl}"
call_effect_compute: fixes pr :: "real"   and matur :: "nat"   and K :: "real" shows "(\<Sum>w\<in>range (pseudo_proj_True matur). prod (prob_component pr w) {0..<matur} * discounted_value r (\<lambda>m. call_option matur K) matur w) = (\<Sum>y\<in>gener_bool_list matur. lprob_comp pr y * det_discount r matur * max (lgeom_proc u d init (take matur y) - K) 0)"
simps_1: fixes x :: "'a"   and bs :: "bool list" shows "bdd_lookup (Leaf x) bs = x"
elims_1: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c \<times> 'd Filters.filter"   and xa :: "'c \<times> 'd Filters.filter"   and y :: "bool" assumes "triple phi"   and "triple.pairs_less_eq x xa = y" obtains xb :: "'c" and ya :: "'d Filters.filter" and z :: "'c" and w :: "'d Filters.filter" where "x = (xb, ya)" and "xa = (z, w)" and "y = (xb \<le> z \<and> w \<le> ya)"
MkIde_Cod: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr a" shows "MkIde (Cod a) = cod a"
arrI: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and a :: "'a"   and b :: "'a" assumes "partial_magma C"   and "partial_magma.in_hom C f a b" shows "partial_magma.arr C f"
mult_mm_non_zero_l: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det A \<noteq> 0"   and "B \<noteq> mat_zero" shows "A *\<^sub>m\<^sub>m B \<noteq> mat_zero"
typing_fstE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and P :: "'a trm"   and \<tau> :: "PreSimplyTyped.type" assumes "\<Gamma> \<turnstile> Fst P : \<tau>" shows "\<exists>\<sigma>. \<Gamma> \<turnstile> P : TPair \<tau> \<sigma>"
poly_coeff_in_carrier: fixes K :: "'a set"   and p :: "'a list"   and i :: "nat" assumes "subring K R"   and "polynomial K p" shows "local.coeff p i \<in> carrier R"
m_def: shows "??.Example_Semantics.m = Match (Src (IpAddr 0))"
conv2: shows "\<I>',\<I>'' \<turnstile>\<^sub>C conv2 \<sim> map_converter id id f' g' 1\<^sub>C"
hyp_edge: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and c :: "'b"   and "is" :: "nat list"   and n :: "nat"   and s :: "'c"   and ants :: "('b, 'd, 'c, 'a) itnode rose_tree list" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "c \<in> set conclusions"   and "is \<in> it_paths (Abstract_Task.globalize freshenLC subst_renameLCs anyP antecedent [fidx (Abstract_Task.conc_forms conclusions) c, 0] (freshenLC v_away) (Abstract_Task.to_it freshenLC renameLCs lconsts subst subst_lconsts anyP antecedent consequent rules assumptions conclusions (Solved_Task.ts freshenLC renameLCs lconsts subst subst_lconsts antecedent consequent rules assumptions c)))"   and "tree_at (Abstract_Task.globalize freshenLC subst_renameLCs anyP antecedent [fidx (Abstract_Task.conc_forms conclusions) c, 0] (freshenLC v_away) (Abstract_Task.to_it freshenLC renameLCs lconsts subst subst_lconsts anyP antecedent consequent rules assumptions conclusions (Solved_Task.ts freshenLC renameLCs lconsts subst subst_lconsts antecedent consequent rules assumptions c))) is = HNode n s ants" shows "Solved_Task.hyp_edge_at freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions c is n s \<in> Solved_Task.edges freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"
comp_cod_arr: fixes f :: "'b"   and b :: "'b" assumes "J.arr f"   and "J.cod f = b" shows "J.comp b f = f"
poly_eval_alt: fixes a :: "'d \<Rightarrow> 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_eval a p = (\<Sum>t\<in>keys p. lookup p t * (\<Prod>x\<in>keys t. a x ^ lookup t x))"
\<epsilon>: shows "\<guillemotleft>\<epsilon> : f \<star> g \<Rightarrow> trg f\<guillemotright> \<and> S.UP \<epsilon> = UP.unit (trg f) \<cdot>\<^sub>S \<xi>' \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, g))"
assms: shows "Case_Labeling.CTXT IC CT OC (ovalidNF P f Q)"
Proc_CFG_Entry: fixes prog :: "cmd" shows "prog \<turnstile> Entry -IEdge (\<lambda>s. True)\<^sub>\<surd>\<rightarrow>\<^sub>p Label 0"
elims: fixes x :: "'c tree"   and y :: "'c list" assumes "inorder x = y" obtains "x = \<langle>\<rangle>" and "y = []"   | l :: "'c tree" and xa :: "'c" and r :: "'c tree" where "x = \<langle>l, xa, r\<rangle>" and "y = inorder l @ [xa] @ inorder r"
space_subgroup_props_1: fixes n :: "nat"   and E :: "'a set" assumes "dimension n K E" shows "E \<subseteq> carrier R"
le_less: fixes x :: "'c multiset"   and y :: "'c multiset" shows "(x \<subseteq># y) = (x \<subset># y \<or> x = y)"
remdups_append2: fixes xs :: "'a list"   and ys :: "'a list" shows "remdups (xs @ remdups ys) = remdups (xs @ ys)"
base_case_sgas_alt_R: fixes qs :: "real poly list"   and p :: "real poly" assumes "length qs = 1"   and "p \<noteq> 0" shows "set (characterize_consistent_signs_at_roots p qs) \<subseteq> {[1], [0], [- 1]}"
is_pure_assn_basic_simps_2: shows "is_pure_assn emp"
assms: fixes y :: "xml" shows "monotone sum_bot.le_fun sum_bot_ord (p1 y)"
dWHILE_def: shows "dWHILE \<equiv> iWHILE (\<bind>) dRETURN"
max_intros_6: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and p :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>s a. a \<in> supp (P s) \<Longrightarrow> maximal (wp (p a))"   and "\<And>s. sum (P s) (supp (P s)) = 1" shows "maximal (wp (SetPC p P))"
term_path_stable: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat"   and j :: "nat" assumes "is_path \<pi>"   and "\<pi> i = local.return"   and "i \<le> j" shows "\<pi> j = local.return"
alphaAbs_weakestAll_imp_strongestEx: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm"   and xs :: "'e"   and x :: "'d"   and xs' :: "'e"   and x' :: "'d" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGood X"   and "FixVars.qGood X'"   and "alphaAbs_all_distinct_qAFresh xs x X xs' x' X'" shows "alphaAbs_ex_distinct_qAFresh xs x X xs' x' X'"
L_def: shows "\<Sigma>1.L \<equiv> {x. \<exists>w. (x, w) \<in> Rel1}"
faccs_alt: fixes \<delta> :: "('a, 'b) ta_rule set"   and t :: "'b tree" shows "faccs \<delta> t = {q. accs \<delta> t q}"
streamValue43: fixes d :: "nat"   and k :: "nat"   and lose :: "nat \<Rightarrow> bool list"   and t :: "nat" assumes "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]" shows "\<forall>j\<le>2 * d. lose (t + 3 + k + j) = [False]"
subseq_seqseq: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and n :: "nat" assumes "Diagonal_Subsequence.subseqs P" shows "strict_mono (subseqs.seqseq P n)"
simulate_plan_correct: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and \<pi>s :: "char list list"   and s :: "nat \<Rightarrow> nat option"   and s' :: "nat \<Rightarrow> nat option" shows "(ast_problem.simulate_plan problem \<pi>s s = Some s') = ast_problem.path_to problem s \<pi>s s'"
yz2: shows "det2 a11 a12 a21 a22 * y2 + det2 a11 a13 a21 a23 * z2 = (0::'a)"
\<U>: shows "Ball \<U> (openin X) \<and> C \<subseteq> \<Union> \<U>"
child_parent_dual: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "child \<in> set children"   and "known_ptrs h"   and "type_wf h" shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some ptr"
bij_minus: fixes y :: "'a" shows "bij (\<lambda>x. x - y)"
pathin_def: fixes X :: "'b topology"   and g :: "real \<Rightarrow> 'b" shows "pathin X g \<equiv> continuous_map (top_of_set {0..1}) X g"
strong_fusion_character: fixes F :: "i \<Rightarrow> bool" assumes "\<exists>x. F x" shows "(\<forall>y. F y \<longrightarrow> P y (\<sigma>x. F x)) \<and> (\<forall>y. P y (\<sigma>x. F x) \<longrightarrow> (\<exists>z. F z \<and> O y z))"
select_convs_2: fixes nodes :: "'d set"   and edges :: "('d \<times> 'e \<times> 'd) set"   and more :: "'f" shows "edges \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = edges"
Card_order_cexp: fixes r1 :: "('e \<times> 'e) set"   and r2 :: "('d \<times> 'd) set" shows "Card_order (r1 ^c r2)"
UNIV_witness: shows "\<exists>x. x \<in> UNIV"
iv_defined_1: fixes T :: "real set"   and t0 :: "real"   and x0 :: "'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set" assumes "self_mapping T t0 x0 f X" shows "t0 \<in> T"
correct_transition_relation_def: fixes SES :: "('a, 'b) SES_rec" shows "correct_transition_relation SES \<equiv> \<forall>x y z. x y\<longrightarrow>\<^bsub>SES\<^esub> z \<longrightarrow> x \<in> S\<^bsub>SES\<^esub> \<and> y \<in> E\<^bsub>SES\<^esub> \<and> z \<in> S\<^bsub>SES\<^esub>"
ss_hd_b_eq_a: shows "ss ! hd b = a ! hd b"
gmftR_gmnl: fixes n :: "nat"   and m :: "('b, 'c) FingerTreeStruc"   and pr :: "('b, 'c) Digit"   and sf :: "('b, 'c) Node list" assumes "is_leveln_ftree (Suc n) m \<and> is_measured_ftree m"   and "is_leveln_digit n pr \<and> is_measured_digit pr"   and "\<forall>x\<in>set sf. (is_measured_node x \<and> is_leveln_node n x) \<and> length sf \<le> 4" shows "gmft (deepR pr m sf) = gmd pr + gmft m + gmnl sf"
snth_sset: fixes s :: "'c stream"   and n :: "nat" shows "s !! n \<in> sset s"
disj_cancel_right: fixes x :: "'a" shows "x OR NOT x = - (1::'a)"
3: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
imp_true: fixes p :: "'a upred" shows "(p \<Rightarrow> true) = true"
intro: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and ask :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad return bind"   and "monad_reader_axioms bind ask" shows "monad_reader return bind ask"
ce_rel_lm_3: fixes A :: "nat set" assumes "A \<in> ce_sets" shows "ce_set_to_rel A \<in> ce_rels"
takefill_minus_simps_2: fixes n :: "nat"   and fill :: "'b"   and x21 :: "'b"   and x22 :: "'b list" assumes "0 < n" shows "takefill fill n (x21 # x22) = x21 # takefill fill (n - 1) x22"
viewR_aux_def: fixes t :: "('a, 'b) FingerTree" shows "FingerTree.viewR_aux t \<equiv> Abs_viewres (FingerTreeStruc.viewR (Rep_FingerTree t))"
opD: fixes T :: "'a set"   and a :: "'a" assumes "openin (top_of_set S) T"   and "a \<in> T" shows "\<exists>z. z \<in> T \<and> P z"
isin_delete: fixes xs :: "bool list"   and t :: "trie"   and ys :: "bool list" shows "isin (delete xs t) ys = (xs \<noteq> ys \<and> isin t ys)"
knC_serv: fixes Rs :: "fid_t"   and runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option" assumes "Rs \<in> dom runz"   and "fst (the (runz Rs)) = Serv" shows "(sesK (Rs $ sk), Sv) \<in> knC runz"
ord_xy: shows "[\<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<lparr>O!,y\<^sup>P\<rparr> in v]"
is_extensional: fixes f :: "'a" assumes "\<not> A.arr f" shows "F f = B.null"
Awhile_4: shows "lesvars upds \<inter> varacom ({A} WHILE b DO C) = {}"
hull_same: fixes S :: "'c set \<Rightarrow> bool"   and s :: "'c set" assumes "S s" shows "S hull s = s"
irbd_irfb_dual: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Sup_pres \<phi>" shows "bd\<^sup>-\<^sub>\<R> \<phi> = (\<smile> \<circ> fb\<^sup>-\<^sub>\<R> \<circ> \<partial>\<^sub>F) \<phi>"
lrbe_def: fixes \<rho> :: "'e V_rec \<Rightarrow> 'e set"   and ur :: "('s \<times> 's) set" shows "lrbe \<rho> ur \<equiv> \<forall>s\<in>S\<^bsub>SES\<^esub>. \<forall>c\<in>C\<^bsub>\<V>\<^esub>. reachable SES s \<and> En \<rho> s c \<longrightarrow> (\<exists>s'\<in>S\<^bsub>SES\<^esub>. s c\<longrightarrow>\<^bsub>SES\<^esub> s' \<and> (s, s') \<in> ur)"
le_oexp: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>"   and "\<beta> \<noteq> 0" shows "\<alpha> \<le> \<alpha> \<up> \<beta>"
special_coset_pseudominimal_in_below_in: fixes w :: "'w"   and T :: "'w set"   and X :: "'w set" assumes "w \<in> W"   and "T \<in> Pow S"   and "dual_order.pseudominimal_in (dual_order.below_in \<P> (w +o \<langle>T\<rangle>)) X" shows "\<exists>s\<in>S - T. X = w +o \<langle>S - {s}\<rangle>"
g2: shows "R_Chop(Abs_real_int (left r1, right r3),r1,r3)"
simps_3: fixes shared :: "bool"   and m1 :: "(nat \<times> nat) convnet"   and m2 :: "(nat \<times> nat) convnet"   and w :: "nat \<Rightarrow> real" shows "insert_weights shared (Pool m1 m2) w = Pool (insert_weights shared m1 w) (insert_weights shared m2 (if shared then w else (\<lambda>i. w (i + count_weights shared m1))))"
that: fixes rb :: "nat \<Rightarrow> 'b" assumes "b.is_run rb"   and "(ra, rb) \<in> run_rel R" shows "thesis"
Pd'_tax_def: shows "Pd'_tax = Pd'(Xd x := Restr (Pd' (Xd x)) (above (Pd' (Xd x)) x))"
distinct_16: fixes x41 :: "trm"   and x42 :: "trm"   and x31 :: "POPLmark.type"   and x32 :: "trm" shows "x41 \<bullet> x42 \<noteq> (\<lambda><:x31. x32)"
Sum_any_when_dependent_prod_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and g :: "'d \<Rightarrow> 'b"   and h :: "'d \<Rightarrow> 'c" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.Sum_any plus zero (\<lambda>(a, b). zero.when zero (g b) (a = h b)) = comm_monoid_add.Sum_any plus zero g"
delete_map_def: fixes k :: "'b"   and M :: "('b, 'c) Mapping_Str.map" shows "delete_map k M = Map (\<lambda>x. if x = k then None else M\<langle>x\<rangle>)"
assms_1: shows "type_update ts (to_ct_list t_in) (TopType []) = ts'"
simps_1: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'b) Protocol"   and s :: "'s"   and a :: "'a" shows "runJP jp (tInit s) a = pInit (jp a) (envObs a s)"
R_abs_R2: shows "R \<circ> R2 = R"
basic_subtype_x_Real: fixes \<tau> :: "'a basic_type" assumes "\<tau> \<sqsubset>\<^sub>B Real" obtains "\<tau> = Integer"
while_termiE2: fixes b :: "state \<Rightarrow> bool"   and c :: "com"   and s :: "state"   and t :: "state" assumes "WHILE b DO c \<down> s"   and "b s"   and "s -c\<rightarrow> t" shows "WHILE b DO c \<down> t"
Shuffle_Plus_ShuffleLR_One: fixes r :: "'a language" shows "Times_Plus (ShuffleLR r One) = r"
list_of_dnelist_inject: fixes x :: "'a dnelist"   and y :: "'a dnelist" shows "(list_of_dnelist x = list_of_dnelist y) = (x = y)"
wt_subst_fv_termtype_subterm: fixes x :: "('fun, 'atom) Term.term \<times> nat"   and \<theta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and y :: "('fun, 'atom) Term.term \<times> nat" assumes "x \<in> fv (\<theta> y)"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "wf\<^sub>t\<^sub>r\<^sub>m (\<theta> y)" shows "\<Gamma> (Var x) \<sqsubseteq> \<Gamma> (Var y)"
strict_mono_on_def: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "strict_mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r < s \<longrightarrow> f r < f s"
succ_notin_self: fixes i :: "V" shows "ZFC_in_HOL.succ i \<notin>\<^sub>\<circ> i"
rcos_sum: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "H #> x <#> (H #> y) = H #> x \<otimes> y"
take_bv_prepend: fixes n :: "nat"   and b :: "'a"   and x :: "'a list" shows "take n (bv_prepend n b x) = bv_prepend n b []"
root_int_main_1: fixes n :: "int"   and p :: "nat"   and y :: "int"   and b :: "bool" assumes "0 \<le> n"   and "root_int_main p n = (y, b)" shows "0 \<le> y"
WhileFalse: fixes s :: "'a"   and b :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com" assumes "s \<notin> b" shows "\<Gamma>\<turnstile> (While b c, Normal s) \<rightarrow> (Skip, Normal s)"
trancl_idemp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "r\<^sup>+\<^sup>+\<^sup>+\<^sup>+ x y = r\<^sup>+\<^sup>+ x y"
cases: fixes \<Gamma> :: "('a, 'b) web_scheme" assumes "hindered \<Gamma>" obtains f :: "'a \<times> 'a \<Rightarrow> ennreal" where "hindrance \<Gamma> f" and "current \<Gamma> f" and "wave \<Gamma> f"
mult2_alt_nsE: fixes X :: "'a multiset"   and Y :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "(X, Y) \<in> mult2_alt_ns ns s" obtains X1 :: "'a multiset" and X2 :: "'a multiset" and Y1 :: "'a multiset" and Y2 :: "'a multiset" where "X = X1 + X2" and "Y = Y1 + Y2" and "(X1, Y1) \<in> multpw ns" and "\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s)"
CQ_sound: fixes var :: "'sz"   and \<theta> :: "('sf, 'sz) trm"   and \<theta>' :: "('sf, 'sz) trm" shows "CQ_holds var \<theta> \<theta>'"
below_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
Cauchy_theorem_singularities: fixes s :: "complex set"   and pts :: "complex set"   and f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and h :: "complex \<Rightarrow> real" assumes "open s"   and "connected s"   and "finite pts"   and "f holomorphic_on s - pts"   and "valid_path g"   and "pathfinish g = pathstart g"   and "path_image g \<subseteq> s - pts"   and "\<forall>z. z \<notin> s \<longrightarrow> winding_number g z = 0"   and "\<forall>p\<in>s. 0 < h p \<and> (\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pts))" shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
absorb_comp_converter: fixes d :: "('k, 'l) distinguisher"   and conv :: "('k, 'l, 'm, 'n) converter"   and conv' :: "('m, 'n, 'i, 'j) converter" shows "absorb d (conv \<odot> conv') = absorb (absorb d conv) conv'"
restr_rel_def: fixes R :: "'e set" shows "restr_rel R \<equiv> {(s, s'). (ss_stack s, ss_stack s') \<in> \<langle>Id \<times>\<^sub>r {(U, U'). U - R = U'}\<rangle>list_rel \<and> on_stack s = on_stack s' \<and> visited s = visited s' \<union> R \<and> visited s' \<inter> R = {} \<and> simple_state.more s = simple_state.more s'}"
carrier_def: shows "manifold_eucl.carrier = \<Union> (domain ` charts_eucl)"
R_lin_independent_imp_same_scalars: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and rs :: "'a list"   and ss :: "'a list"   and ms :: "'b list" assumes "RModule R smult M"   and "length rs = length ss"   and "length rs \<le> length ms"   and "set rs \<subseteq> R"   and "set ss \<subseteq> R"   and "set ms \<subseteq> M"   and "R_scalar_mult.R_lin_independent R smult ms"   and "scalar_mult.lincomb smult rs ms = scalar_mult.lincomb smult ss ms" shows "rs = ss"
shift_left_inj: fixes xs :: "'a list"   and s1 :: "'a stream"   and s2 :: "'a stream" shows "(xs @- s1 = xs @- s2) = (s1 = s2)"
best_degenerate_in_best_overall: shows "\<exists>x\<in>local.degenerate_lotteries. \<forall>y\<in>local.\<P>. x \<succeq>[\<R>] y"
c: shows "(x, T\<^sub>2) \<in> set \<Gamma>"
pure_app_eq: fixes R :: "('b \<times> 'c) set"   and a :: "'c"   and c :: "'b" shows "pure R a c = \<up> ((c, a) \<in> R)"
fps_divide_shift_denom_conv_times_fps_X_power: fixes n :: "nat"   and g :: "'a fps"   and f :: "'a fps" assumes "n \<le> subdegree g"   and "subdegree g \<le> subdegree f" shows "f / fps_shift n g = f / g * fps_X ^ n"
dsg4: fixes x :: "'a"   and y :: "'a" shows "ad (ad x) \<cdot> ad (ad y) = ad (ad y) \<cdot> ad (ad x)"
standard_decomp_SucE: fixes X :: "'x set"   and U :: "'x set"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite X"   and "U \<subseteq> X"   and "h \<in> P[X]"   and "h \<noteq> 0" obtains ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" where "valid_decomp X ps" and "cone_decomp (cone (h, U)) ps" and "standard_decomp (Suc (poly_deg h)) ps" and "\<lbrakk>is_monomial h; lcf h = (1::'b)\<rbrakk> \<Longrightarrow> monomial_decomp ps" and "homogeneous h \<Longrightarrow> hom_decomp ps"