hcju/isabelleps huggingface dataset:
convex_polychainI: fixes xs :: "((real \<times> real) \<times> real \<times> real) list" assumes "polychain xs"   and "\<And>i. Suc i < length xs \<Longrightarrow> 0 < det3 (fst (xs ! i)) (snd (xs ! i)) (snd (xs ! Suc i))" shows "convex_polychain xs"
map\<^sub>1_in_hom_1: fixes f :: "'a" assumes "C.ide f" shows "\<guillemotleft>\<tau>\<^sub>1 f : F.map\<^sub>0 (src\<^sub>C f) \<rightarrow>\<^sub>D G.map\<^sub>0 (trg\<^sub>C f)\<guillemotright>"
ind_in_set_inj: fixes a :: "nat"   and A :: "nat set"   and b :: "nat" assumes "a \<in> A"   and "b \<in> A"   and "ind_in_set A a = ind_in_set A b" shows "a = b"
rep_uminus: fixes r :: "'c \<Rightarrow>\<^sub>0 'c" shows "ideal.rep (- r) = - ideal.rep r"
Fadmit_And_simps: fixes \<sigma> :: "('a, 'b, 'c) subst"   and p :: "('a, 'b, 'c) formula"   and q :: "('a, 'b, 'c) formula" shows "Fadmit \<sigma> (p && q) = (Fadmit \<sigma> p \<and> Fadmit \<sigma> q)"
distinct_227: fixes x91 :: "floatarith"   and x92 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Min x91 x92 \<noteq> x151 ^\<^sub>e x152"
not_contains_insertion: fixes var :: "nat"   and p :: "real mpoly"   and L :: "real list"   and x :: "real"   and val :: "real"   and y :: "real" assumes "var \<notin> vars p"   and "insertion (nth_default 0 (L[var := x])) p = val" shows "insertion (nth_default 0 (L[var := y])) p = val"
step_sb: shows "(ts, m, \<S>) \<Rightarrow>\<^sub>s\<^sub>b\<^sup>* (ts', m', \<S>')"
inf_left_idem: fixes x :: "'d upred"   and y :: "'d upred" shows "(x \<and> x \<and> y) = (x \<and> y)"
lock_thread_okD: fixes ls :: "('d, 'e) locks"   and ts :: "'e \<Rightarrow> ('f \<times> 'd \<Rightarrow>f nat) option"   and l :: "'d"   and t :: "'e" assumes "lock_thread_ok ls ts"   and "has_lock (ls $ l) t" shows "\<exists>xw. ts t = \<lfloor>xw\<rfloor>"
lookup_pair_eq_0: fixes xs :: "('a \<times> 'd) list"   and k :: "'a" assumes "oalist_inv_raw xs" shows "(lookup_pair xs k = (0::'d)) = (k \<notin> fst ` set xs)"
simps_5: fixes ypos' :: "nat \<Rightarrow> nat"   and xpos :: "nat"   and ypos :: "nat"   and more :: "'a" shows "ypos_update ypos' \<lparr>xpos = xpos, ypos = ypos, \<dots> = more\<rparr> = \<lparr>xpos = xpos, ypos = ypos' ypos, \<dots> = more\<rparr>"
r_prefixes_Suc: fixes n :: "nat"   and j :: "nat" shows "eval r_prefixes [Suc n, j] \<down>= (let b = the (eval r_prefixes [n, j]) in e_snoc (e_append_zeros b (the (eval r_find_k [b]))) 1)"
integral_norm_bound_integral: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> real" assumes "integrable M f"   and "integrable M g"   and "\<And>x. x \<in> space M \<Longrightarrow> norm (f x) \<le> g x" shows "norm (integral\<^sup>L M f) \<le> integral\<^sup>L M g"
finite_s: fixes s :: "'a set" assumes "s \<in> simplices" shows "finite s"
that: assumes "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"   and "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "\<CC>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = a"   and "\<CC>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b" shows "thesis"
filtermap_linear_at_right: fixes c :: "'a"   and b :: "'a"   and x :: "'a" assumes "c \<noteq> (0::'a)" shows "filtermap (\<lambda>x. c * x + b) (at_right x) = (if (0::'a) < c then at_right (c * x + b) else at_left (c * x + b))"
verts_add_arc_conv: fixes a :: "'b" shows "verts (add_arc a) = verts G \<union> {tail G a, head G a}"
LI_pos: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set" assumes "valuation K v"   and "ideal (Vr K v) I" shows "0 \<le> LI K v I"
cos_ms_aux': fixes xs :: "('a \<times> real) msllist"   and basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real" assumes "ms_exp_gt 0 (ms_aux_hd_exp xs)"   and "basis_wf basis"   and "is_expansion_aux xs f basis" shows "is_expansion_aux (cos_ms_aux' xs) (\<lambda>x. cos (f x)) basis"
res_def: fixes side_oracle :: "'b cstate \<Rightarrow> 'a \<Rightarrow> ('c option \<times> 'b cstate) spmf" shows "channel.res side_oracle \<equiv> RES (side_oracle \<oplus>\<^sub>O A_CHAN.send_oracle \<oplus>\<^sub>O A_CHAN.recv_oracle) Void"
eq_iff: fixes a :: "'c list"   and b :: "'c list" shows "(a = b) = (prefix b a \<and> prefix a b)"
coeffs_sdiv: fixes p :: "'b poly"   and a :: "'b" shows "coeffs (sdiv_poly p a) = strip_while ((=) (0::'b)) (map (\<lambda>x. x div a) (coeffs p))"
visits_via_trivial: fixes W :: "'a set"   and A :: "'a set" assumes "v \<in> W" shows "visits_via P A W"
sim: shows "similar_mat (step_2 (step_1 A)) A"
simps_1: fixes x :: "'a" shows "remove1 x [] = []"
true_clss_mono: fixes DD :: "'b literal multiset set"   and CC :: "'b literal multiset set"   and I :: "'b set" assumes "DD \<subseteq> CC"   and "I \<TTurnstile>s CC" shows "I \<TTurnstile>s DD"
fundamental_property: fixes n :: "'node"   and c :: "'com"   and s :: "'state"   and c' :: "'com"   and s' :: "'state" assumes "n \<triangleq> c"   and "\<langle>c,s\<rangle> \<Rightarrow> \<langle>c',s'\<rangle>" shows "\<exists>n' as. n -as\<rightarrow>* n' \<and> transfers (kinds as) s = s' \<and> preds (kinds as) s \<and> n' \<triangleq> c'"
Sbis_matchT_T: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>s d" shows "matchT_T c d"
map_map: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "map f (map g xs) = map (f \<circ> g) xs"
assms: fixes x :: "'a"   and y1 :: "'a"   and y2 :: "'a" assumes "r\<^sup>*\<^sup>* x y1"   and "r\<^sup>*\<^sup>* x y2" shows "\<exists>z. r\<^sup>*\<^sup>* y1 z \<and> r\<^sup>*\<^sup>* y2 z"
add_4: shows "add_mset x' X' = add_mset x X"
converse_ordermap_mono: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and y :: "'a"   and x :: "'a" assumes "ordermap A r y \<in> elts (ordermap A r x)"   and "wf r"   and "total_on A r"   and "x \<in> A"   and "y \<in> A"   and "small A" shows "(y, x) \<in> r"
ord_isom_inj: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and a :: "'a"   and b :: "'a" assumes "Order E"   and "ord_isom D E f"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "(a = b) = (f a = f b)"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "design point_set block_collection"   and "proper_design_axioms block_collection" shows "proper_design point_set block_collection"
less_eq_cblinfun_def_heterogenous: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c"   and B :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "(A \<le> B) = (if heterogenous_same_type_cblinfun TYPE('b) TYPE('c) then \<forall>\<psi>. 0 \<le> \<psi> \<bullet>\<^sub>C ((B - A) *\<^sub>V heterogenous_cblinfun_id *\<^sub>V \<psi>) else A = B)"
eval_monom_expr: fixes a :: "'a" assumes "a \<in> carrier R" shows "eval R R id a (monom (UP R) \<one> 1 \<ominus>\<^bsub>UP R\<^esub> monom (UP R) a 0) = \<zero>"
hyps: fixes kv :: "'a \<times> 'b" assumes "sorted_wrt lt (map fst xs)" shows "sorted_wrt lt (map fst (update_by_pair kv xs))"
unat_arith_simps_6: fixes x :: "'a word"   and y :: "'a word" shows "unat (x div y) = unat x div unat y"
map_lc_cong: fixes f :: "form"   and f1 :: "nat \<Rightarrow> nat"   and f2 :: "nat \<Rightarrow> nat" assumes "\<And>x. x \<in> lc f \<Longrightarrow> f1 x = f2 x" shows "map_lc f1 f = map_lc f2 f"
MGU_Var1: fixes v :: "'b"   and t :: "('a, 'b) Term.term" assumes "\<not> Var v \<sqsubset> t" shows "MGU (Var(v := t)) (Var v) t"
agree: shows "Vagree \<nu> \<nu>' (FVDiff ($f var args))"
strategyproofness: shows "pmf (sds R15) c + pmf (sds R15) a \<le> pmf (sds R19) c + pmf (sds R19) a"
inj_gen: shows "inj_on ((*\<^sub>v) B) gen"
list_all2_in_set_ExD: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a list"   and q :: "'b list"   and x :: "'a" assumes "list_all2 R p q"   and "x \<in> set p" shows "\<exists>y\<in>set q. R x y"
norm_f: fixes i :: "nat" shows "set_integrable M (A i) (\<lambda>x. norm (f x))"
xs'_def: shows "xs' = xs \<restriction>\<^sup>l\<^sub>\<circ> vcard xs'"
hyps_6: fixes b :: "(W.tvar \<times> ty) list" shows " b<(x, close \<Gamma> T1) # \<Gamma>> \<turnstile> t2 : b<T2> "
quotientE: fixes X :: "'a set"   and A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "X \<in> A // r" obtains x :: "'a" where "X = r `` {x}" and "x \<in> A"
sin_of_real_pi: shows "sin (of_real pi) = (0::'a)"
tab_simps_5: shows "cod \<rho> = r \<star> r\<^sub>0"
gmw_asym_def: fixes S1_14_OT :: "nat \<Rightarrow> bool \<times> bool \<times> bool \<times> bool \<Rightarrow> unit \<Rightarrow> 'a spmf"   and R1_14_OT :: "nat \<Rightarrow> bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> 'a spmf"   and S2_14_OT :: "nat \<Rightarrow> bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'b spmf"   and R2_14_OT :: "nat \<Rightarrow> bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> 'b spmf"   and protocol_14_OT :: "nat \<Rightarrow> bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> (unit \<times> bool) spmf"   and adv_14_OT :: "nat \<Rightarrow> real" shows "gmw_asym S1_14_OT R1_14_OT S2_14_OT R2_14_OT protocol_14_OT adv_14_OT \<equiv> \<forall>n. gmw_base (S1_14_OT n) (R1_14_OT n) (S2_14_OT n) (R2_14_OT n) (protocol_14_OT n) (adv_14_OT n)"
length_nat_to_bv_lower_limit: fixes k :: "nat"   and n :: "nat" assumes "2 ^ k \<le> n" shows "k < length (nat_to_bv n)"
nodes_target_elim: fixes q :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" assumes "q \<in> gnodes A w V" obtains r :: "(nat \<times> 'a) list" and p :: "nat \<times> 'a" where "p \<in> ginitial A \<inter> V" and "gpath A w V r p" and "q = gtarget r p"
rel_refl: fixes Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a le_constraint" assumes "\<And>x. Ra x x" shows "rel_le_constraint Ra x x"
plus_compose: fixes x :: "'e \<Rightarrow> 'd"   and y :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" shows "x + y \<circ> f = (x \<circ> f) + (y \<circ> f)"
insert_no_failure: fixes i :: "'c option"   and xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool" assumes "i = None \<or> (\<exists>i'. i = Some i' \<and> i' \<in> fst ` set xs)" shows "\<exists>xs'. Ordered_List.insert xs e i = Some xs'"
i: shows "set (divisors_nat 75) = {1, 3, 5, 15, 25, 75}"
correct_shortest_path: fixes v :: "'a" assumes "v \<in> verts G" shows "dist v = \<mu> c s v"
rep_eq: fixes x :: "'d"   and xa :: "'c"   and xb :: "('a, 'b, 'c, 'd) formula" shows "Rep_formula (Act x xa xb) = Act\<^sub>\<alpha> x xa (Rep_formula xb)"
are_all_effects_consistent_tail: fixes op :: "('a, 'b) strips_operator_scheme"   and ops :: "('a, 'b) strips_operator_scheme list" assumes "are_all_operator_effects_consistent (op # ops)" shows "are_all_operator_effects_consistent ops"
sum_list_posI: fixes xs :: "'a list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> (0::'a) < x"   and "xs \<noteq> []" shows "(0::'a) < sum_list xs"
obj_def: fixes a :: "'a arrow_of_spans_data" shows "obj a \<equiv> V.arr a \<and> src a = a"
esize_atMost: fixes n :: "enat" shows "esize {..n} = eSuc n"
set_map: fixes f :: "'e \<Rightarrow> 'd"   and v :: "'e multiset" shows "set_mset (image_mset f v) = f ` set_mset v"
ltake_mono: fixes nxs :: "enat \<times> 'c llist" shows "mono_llist (\<lambda>ltake. case nxs of (n, LNil) \<Rightarrow> LNil | (0, LCons x xsa) \<Rightarrow> LNil | (eSuc na, LCons x xsa) \<Rightarrow> LCons x (curry ltake na xsa))"
length_set_of_appr: fixes xrs :: "real list"   and xs :: "'b list" assumes "xrs \<in> set_of_appr xs" shows "length xrs = length xs"
connected_Union: fixes S :: "'b set set" assumes "\<And>s. s \<in> S \<Longrightarrow> connected s"   and "\<Inter> S \<noteq> {}" shows "connected (\<Union> S)"
domain_eq: shows "Domainp pcr_mod_ring = (\<lambda>x. x \<in> {0..<int CARD('b)})"
ccINF_sup: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and a :: "'c" assumes "class.countable_complete_distrib_lattice Inf Sup inf less_eq less sup bot top"   and "countable B" shows "sup (Inf (f ` B)) a = Inf ((\<lambda>b. sup (f b) a) ` B)"
3: shows "{(x, y) |y. x < y in xs \<and> y < x in ys} \<subseteq> Inv xs ys"
seq_append_def: fixes k :: "hf"   and f :: "hf"   and g :: "hf" shows "seq_append k f g \<equiv> hrestrict f k \<squnion> shift g k"
div_unit_dvd_iff: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "is_unit b" shows "(a div b dvd c) = (a dvd c)"
\<tau>moves_False: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" shows "\<tau>trsys.silent_move r (\<lambda>s ta s'. False) = (\<lambda>s s'. False)"
AbsI: fixes x :: "\<kappa>"   and v :: "i" assumes "\<exists>o\<^sub>1 y. Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 = \<alpha>\<nu> y" shows "[\<lparr>A!,x\<rparr> in v]"
set_category_axioms: shows "set_category (\<cdot>) \<AA>"
a_in_o: shows "a \<in> orbit G.face_cycle_succ a'"
IH: fixes P :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w1 :: "'a \<Rightarrow> 'b"   and w2 :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<forall>y. ListMem y l1 \<longrightarrow> P y"   and "P x" shows "wlp R (\<lambda>y. if P y then w1 y else w2 y) g f x l1 = wlp R w1 g f x l1"
N_def: shows "N \<equiv> distr M (stock_measure INTEG) (lift_IntVal f)"
adjuster_wit': fixes n_gs :: "('a vec \<times> 'a) list"   and gs :: "'a vec list"   and f :: "'a vec"   and mus_acc :: "'a list" assumes "n_gs = map (\<lambda>x. (x, \<parallel>x\<parallel>\<^sup>2)) gs"   and "f \<in> carrier_vec n"   and "set gs \<subseteq> carrier_vec n" shows "mus_adjuster f n_gs mus_acc (0\<^sub>v n) = adjuster_wit mus_acc f gs"
appl_methds_Base_foo: fixes S :: "qtname" shows "appl_methds tprg S (ClassT Base) \<lparr>name = foo, parTs = [NT]\<rparr> = {((ClassT Base, \<lparr>access = Public, static = False, pars = [z], resT = Class Base\<rparr>), [Class Base])}"
hyperb_ineq_4_points: fixes e :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and t :: "'a" shows "Min {Gromov_product_at e x y, Gromov_product_at e y z, Gromov_product_at e z t} - 2 * deltaG TYPE('a) \<le> Gromov_product_at e x t"
Always_Int_I: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set" assumes "F \<in> Always A"   and "F \<in> Always B" shows "F \<in> Always (A \<inter> B)"
explode: fixes x :: "literal" shows "explode x \<in> {cs. \<forall>c\<in>set cs. \<not> digit7 c}"
only_vertical_division_1: shows "only_vertical_division one_chain_typeI two_chain_typeI"
cobounded_unbounded_components: fixes S :: "'a set" assumes "bounded (- S)" shows "\<exists>c. c \<in> components S \<and> \<not> bounded c"
build_ip_partition_distinct: fixes c :: "parts_connection"   and rs :: "'b simple_rule list" shows "distinct (map wordinterval_to_set (build_ip_partition c rs))"
simps: fixes g :: "'a"   and v :: "'var"   and unvisited :: "'node list"   and n :: "'node" shows "i.phiDefNodes_aux g v unvisited n = (if n \<notin> set unvisited \<or> v \<in> defs2 g n then {} else fold (\<union>) (map (i.phiDefNodes_aux g v (removeAll n unvisited)) (i.predecessors g n)) (if length (i.predecessors g n) \<noteq> 1 then {n} else {}))"
uminus_add_conv_diff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ab_group_add plus zero minus uminus" shows "plus (uminus a) b = minus b a"
adjust_scope_partial: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "HT_partial \<pi> P c Q" shows "HT_partial \<pi> (\<lambda>s. P <<>|s>) (SCOPE c) (\<lambda>s\<^sub>0 s. \<exists>l. Q <<>|s\<^sub>0> <l|s>)"
jumpNestingOk_stmt_simp1: fixes jmps :: "jump set"   and s :: "stmt" shows "jumpNestingOk jmps \<langle>s\<rangle> = jumpNestingOkS jmps s"
degenerate_has_min_preferred: shows "{x \<in> local.degenerate_lotteries. \<forall>y\<in>local.degenerate_lotteries. y \<succeq>[\<R>] x} \<noteq> {}"
Cons_1: fixes M :: "('a \<times> bool) list" assumes "\<exists>s. p' @ s = M"   and "e \<in> set (elements p')" shows "prefixToElement e M = prefixToElement e p'"
Rep_formula': fixes x :: "('a, 'b, 'c, 'd) formula" shows "hereditarily_fs (Rep_formula x)"
a_kernel_defs_1: fixes G :: "('e, 'f) ring_scheme"   and H :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" shows "a_kernel G H h = kernel (add_monoid G) (add_monoid H) h"
x: shows "\<forall>\<^sub>F x in F. l \<le> f x"
run_eval_list_def: shows "run_eval_list \<equiv> SOME f. \<forall>env es s. evaluate_list True env s es (f env es s)"
assms_2: shows "t_covering_design V B k t \<Lambda>\<^sub>t"
splitset_rearrange: fixes F :: "'d list \<Rightarrow> 'd list \<Rightarrow> 'd list \<Rightarrow> 'c"   and x :: "'d list" shows "\<Sum> {\<Sum> {F (fst p) (fst q) (snd q) |q. q \<in> splitset (snd p)} |p. p \<in> splitset x} = \<Sum> {\<Sum> {F (fst q) (snd q) (snd p) |q. q \<in> splitset (fst p)} |p. p \<in> splitset x}"
cases: fixes r :: "thread_t" obtains ev_counter :: "nat" where "r = \<lparr>ev_counter = ev_counter\<rparr>"
theNFTBoth_def: fixes cl :: "('d, 'e, 'f) cmd list"   and dl :: "('g, 'h, 'i) cmd list" shows "theNFTBoth cl dl \<equiv> theNFT cl \<inter> theNFT dl"
measure_eqI_lessThan: fixes M :: "real measure"   and N :: "real measure" assumes "sets M = sets borel"   and "sets N = sets borel"   and "\<And>x. emeasure M {x<..} < \<infinity>"   and "\<And>x. emeasure M {x<..} = emeasure N {x<..}" shows "M = N"
inj_on_contraD: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c"   and y :: "'c" assumes "inj_on f A"   and "x \<noteq> y"   and "x \<in> A"   and "y \<in> A" shows "f x \<noteq> f y"
list_set_autoref_isEmpty: fixes R :: "('e \<times> 'f) set" shows "(is_Nil, op_set_isEmpty) \<in> \<langle>R\<rangle>list_set_rel \<rightarrow> bool_rel"
2: shows "- v \<squnion> - x \<sqinter> - - z = - - (- v \<squnion> - x \<sqinter> - - z)"
assms_3: shows "\<FF>' = \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>"
invariant_nsc_def: shows "invariant_nsc = Incremental_Atom_Ops_For_NS_Constraint_Ops.invariant_nsc invariant_s preprocess"
connected_component_idemp: fixes S :: "'a set"   and x :: "'a" shows "connected_component_set (connected_component_set S x) x = connected_component_set S x"
equal_if_bisimilar: fixes K :: "'a list set"   and L :: "'a list set" assumes "K \<subseteq> lists S"   and "L \<subseteq> lists S"   and "local.bisimilar K L" shows "K = L"
l11_41_aux: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Col A B C"   and "Bet B A D"   and "A \<noteq> D" shows "A C B LtA C A D"
encode_injective: fixes f :: "recf"   and g :: "recf" assumes "encode f = encode g" shows "f = g"
distinct_55: fixes x71 :: "'b poly"   and x72 :: "nat"   and x81 :: "'b poly"   and x82 :: "nat"   and x83 :: "'b poly" shows "Pw x71 x72 \<noteq> CN x81 x82 x83"
proper_part_irreflexivity: assumes "PP x y" shows "\<not> PP y x"
subgame_path_conforms_with_strategy: fixes V' :: "'a set"   and p :: "Player"   and P :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "V' \<subseteq> V\<^bsub>subgame V'\<^esub>"   and "G'.path_conforms_with_strategy p P \<sigma>"   and "lset P \<subseteq> V'" shows "ParityGame.path_conforms_with_strategy (G'.subgame V') p P \<sigma>"
simps_1: fixes S :: "char list \<Rightarrow> 'av"   and X :: "char list set" shows "m\<^sub>o (Some S) X = m\<^sub>s S X"
app_hyps_1: shows "size t1 = Suc n div 2 \<and> heap t1 \<and> braun t1 \<and> mset_tree t1 = mset (take (Suc n div 2) xs)"
assms_4: shows "\<forall>i. lo \<le> i \<and> i < lo' \<longrightarrow> xs' ! i = xs ! i"
finite_number_segment: fixes n :: "nat" shows "card {m. 0 < m \<and> m < n} = n - 1"
waveD_separating: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "wave \<Gamma> f" shows "separating \<Gamma> (TER\<^bsub>\<Gamma>\<^esub> f)"
traces_refinement: fixes sched\<^sub>C :: "nat list"   and gc\<^sub>C :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)"   and gc\<^sub>C' :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)"   and gc\<^sub>A :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" assumes "conc.meval_sched_abv gc\<^sub>C sched\<^sub>C gc\<^sub>C'"   and "length (fst gc\<^sub>A) = length cms"   and "length (fst gc\<^sub>C) = length cms"   and "\<And>i. i < length cms \<Longrightarrow> ((fst gc\<^sub>A ! i, snd gc\<^sub>A), fst gc\<^sub>C ! i, snd gc\<^sub>C) \<in> \<R>_rel (cms ! i)"   and "abs.sound_mode_use gc\<^sub>A"   and "modes_respect_priv (map snd (fst gc\<^sub>C))" shows "\<exists>sched\<^sub>A gc\<^sub>A''. abs.meval_sched_abv gc\<^sub>A sched\<^sub>A gc\<^sub>A'' \<and> (\<forall>i<length cms. ((fst gc\<^sub>A'' ! i, snd gc\<^sub>A''), fst gc\<^sub>C' ! i, snd gc\<^sub>C') \<in> \<R>_rel (cms ! i)) \<and> abs.sound_mode_use gc\<^sub>A''"
split_beta: fixes f :: "'i \<Rightarrow> 'j \<Rightarrow> 'h"   and prod :: "'i \<times> 'j" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
canonical_alt_def: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b" shows "canonical M n = canonical_subs n {0..n} M"
select_convs_1: fixes init :: "'c set"   and trans :: "('c \<times> 'd \<times> 'c) set"   and more :: "'e" shows "init \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = init"
reachable_in_verts_1: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {v. u \<rightarrow>\<^sup>* v}\<^esub> v" shows "u \<in> verts (G \<restriction> {v. u \<rightarrow>\<^sup>* v})"
assms_1: shows "P \<Longrightarrow>\<^sup>^a<b> \<prec> P'"
proots_count_times: fixes p :: "'b poly"   and q :: "'b poly"   and s :: "'b set" assumes "p * q \<noteq> 0" shows "proots_count (p * q) s = proots_count p s + proots_count q s"
fls_eq0_below_subdegree: fixes n :: "int"   and f :: "'a fls" assumes "n < fls_subdegree f" shows "fls_nth f n = (0::'a)"
exhaust: fixes y :: "g_state" obtains "y = Even"   | "y = Odd"
Group_Im: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b" assumes "GroupHom G T" shows "Group (T ` G)"
bij_betw_imp_inj_on: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "bij_betw f A B" shows "inj_on f A"
useful_tautologies_2: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<^bold>\<not>\<^bold>\<not>\<phi> in v]"
not_eq_diff_nonzero: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<noteq> b"   and "a \<in> carrier Zp"   and "b \<in> carrier Zp" shows "a \<ominus> b \<in> nonzero Zp"
subgroup_kernel: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and h :: "'d \<Rightarrow> 'f" assumes "group_hom G H h" shows "subgroup (kernel G H h) G"
set_ev_sd: shows "set (pawalk_verts u (sd_path e w p)) \<subseteq> set (pawalk_verts u p) \<union> {w}"
Rep_uint32_inject: fixes x :: "uint32"   and y :: "uint32" shows "(Rep_uint32 x = Rep_uint32 y) = (x = y)"
m_closed: fixes H :: "'f set"   and G :: "('f, 'g) monoid_scheme"   and x :: "'f"   and y :: "'f" assumes "subgroup H G"   and "x \<in> H"   and "y \<in> H" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> H"