hcju/isabelleps huggingface dataset:
mbufnt_take_eqD_2: fixes f :: "event_data option list set list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and buf :: "event_data option list set list list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "event_data option list set list list"   and nts' :: "nat list"   and i :: "nat"   and js :: "nat list"   and Ps :: "(nat \<Rightarrow> event_data option list set \<Rightarrow> bool) list"   and R :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and j :: "nat"   and k :: "nat" assumes "mbufnt_take f z buf nts = (z', buf', nts')"   and "wf_mbufn i js Ps buf"   and "list_all2 R [i..<j] nts"   and "\<And>k. k \<in> set js \<Longrightarrow> k \<le> j"   and "k = Mini (i + length nts) js" shows "list_all2 R [k..<j] nts'"
assms_1: shows "rel_gpv'' A C (R OO R') gpv gpv'"
stateinterpol_left: fixes x :: "variable"   and S :: "variable set"   and \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real" assumes "x \<in> S" shows "stateinterpol \<nu> \<omega> S x = \<nu> x"
cring_simprules_5: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier K"   and "y \<in> carrier K" shows "x \<otimes>\<^bsub>K\<^esub> y \<in> carrier K"
abs_eq: fixes xa :: "ereal"   and x :: "ereal" assumes "eq_onp ((\<le>) 0) xa xa"   and "eq_onp ((\<le>) 0) x x" shows "e2ennreal xa * e2ennreal x = e2ennreal (xa * x)"
inf_wf_to_inf: fixes order :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'b"   and x :: "'c" assumes "well_founded order"   and "inf_wf r order n x" shows "Inf.inf r x"
IH: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "higher_differentiable_on S f n"   and "higher_differentiable_on S g n"   and "open S" shows "higher_differentiable_on S (\<lambda>x. f x + g x) n"
rPar2: fixes xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> !P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "\<And>C. Prop C \<Psi> (!P) M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' P')"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* C"   and "distinct xvec" shows "Prop C \<Psi> (P \<parallel> !P) M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' (P \<parallel> P'))"
subset_contr: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subset> B"   and "B \<subseteq> A" shows "False"
im_sheaf_def: fixes V :: "'c set" shows "local.im_sheaf V \<equiv> \<FF> (f \<^sup>\<inverse> S V)"
rel_option_restrict_relp_iff: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option" shows "rel_option (R \<upharpoonleft> P \<otimes> Q) x y = (rel_option R x y \<and> pred_option P x \<and> pred_option Q y)"
a: shows "cs\<^bsup>\<pi>\<^esup> (\<pi> \<exclamdown> xs) = xs"
ab_group_add_ow_def: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and mns :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and um :: "'b \<Rightarrow> 'b" shows "ab_group_add_ow S pls z mns um \<equiv> comm_monoid_add_ow S pls z \<and> ab_group_add_ow_axioms S pls z mns um"
le_supI1: fixes x :: "'d multiset"   and a :: "'d multiset"   and b :: "'d multiset" assumes "x \<subseteq># a" shows "x \<subseteq># a \<union># b"
h4: shows "\<turnstile>\<^sub>p {adjoint (exexH_k (n - 1)) * (adjoint (tensor_P mat_Ph (1\<^sub>m K)) * (adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1)) * tensor_P mat_Ph (1\<^sub>m K)) * exexH_k (n - 1)} hadamard_n n;; Utrans_P vars1 mat_Ph {adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1)}"
scaleRat_zero: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "rat" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat" shows "scaleRat a zero = zero"
assms_2: fixes x :: "'b" assumes "ys = sum_bot.Right x" shows "f x = g x"
between_exchange3: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Bet A B C"   and "Bet A C D" shows "Bet B C D"
univ_poly_not_field: fixes K :: "'a set" assumes "subring K R" shows "\<not> field (K [X])"
NS_refl: fixes s :: "('f, 'a) Term.term" shows "NS s s"
gen_length_def: fixes n :: "nat"   and xs :: "'m list" shows "List.gen_length n xs = n + length xs"
dims0_def: shows "st1.dims0 = nths dims st1.vars0"
bounded_plus_comp: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "bounded (f ` S)"   and "bounded (g ` S)" shows "bounded ((\<lambda>x. f x + g x) ` S)"
prems_1: shows "H \<turnstile> PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F)"
stable_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and \<tau>s :: "'c list"   and p :: "nat" shows "stable r step \<tau>s p = (\<forall>(q, \<tau>)\<in>set (step p (\<tau>s ! p)). \<tau> \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! q)"
contractible_eq_simply_connected_2d: fixes S :: "complex set" assumes "open S" shows "contractible S = simply_connected S"
pwd_spy_ii_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_ii n)"
ennreal_cases: fixes x :: "ennreal" obtains r :: "real" where "0 \<le> r" and "x = ennreal r"   | "x = \<top>"
function_zero_def: fixes S :: "'a set"   and R :: "('b, 'c) ring_scheme" shows "function_zero S R = (\<lambda>x\<in>S. \<zero>\<^bsub>R\<^esub>)"
continuous_on_TimesI: assumes "\<And>x. x \<in> X \<Longrightarrow> continuous_on T (\<lambda>t. f t x)" shows "continuous_on (T \<times> X) (\<lambda>(t, x). f t x)"
iso_is_arr: fixes f :: "'b" assumes "D.iso f" shows "D.arr f"
list_all2_dropI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list"   and n :: "nat" assumes "list_all2 P xs ys" shows "list_all2 P (drop n xs) (drop n ys)"
mor_image2: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and B1' :: "'g set"   and B2' :: "'h set"   and s1' :: "'g \<Rightarrow> ('f, 'g, 'h) F1"   and s2' :: "'h \<Rightarrow> ('f, 'g, 'h) F2"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h" assumes "mor B1 B2 s1 s2 B1' B2' s1' s2' f g" shows "g ` B2 \<subseteq> B2'"
ord_dvd_group_order: fixes a :: "'a" assumes "a \<in> carrier (mult_of R)" shows "G.ord a dvd order (mult_of R)"
healthy_nnegD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "healthy t"   and "sound P" shows "nneg (t P)"
simps_4: fixes P :: "('a \<times> 'a) set"   and Pw :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set"   and Rw :: "('a \<times> 'a) set" shows "SN_rel_ext_step P Pw R Rw normal_ns = Rw"
cis_times_2pi: fixes n :: "nat" shows "cis (real n * 2 * pi) = 1"
less_add_iff2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and e :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "class.ordered_ring plus minus zero less_eq less times uminus" shows "less (plus (times a e) c) (plus (times b e) d) = less c (plus (times (minus b a) e) d)"
NewC: fixes Env :: "env"   and B :: "lname set"   and C :: "qtname" shows "Env\<turnstile> B \<guillemotright>\<langle>NewC C\<rangle>\<guillemotright> \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
axioms_2: fixes default :: "'b"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "one_mono_ordered_semiring_1 default gt" shows "one_mono_ordered_semiring_1_axioms default gt"
length_filter_le_axis_median: fixes ps :: "(real, 'a) vec list"   and k :: "'a" assumes "0 < length ps"   and "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)" shows "length (filter (\<lambda>p. p $ k \<le> axis_median k ps) ps) = (length ps - 1) div 2 + 1"
fract_poly_is_unit: fixes p :: "'a poly" assumes "p dvd 1" shows "is_unit (map_poly to_fract p)"
sat_limit_iff: fixes Ns :: "'a literal multiset set llist" assumes "chain sr_ext.derive Ns" shows "satisfiable (Liminf_llist Ns) = satisfiable (lhd Ns)"
assms: fixes bl :: "'a set" assumes "bl \<in># \<B>"   and "p \<in> bl" shows "2 \<le> card bl"
prime_dvd_mult_nat: fixes p :: "nat"   and m :: "nat"   and n :: "nat" assumes "prime p"   and "p dvd m * n" shows "p dvd m \<or> p dvd n"
less_irrefl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.preorder less_eq less" shows "\<not> less x x"
J_eq: shows "J = {i. i < dim_col A \<and> i \<in> J}"
connectedin_def: fixes X :: "'b topology"   and S :: "'b set" shows "connectedin X S \<equiv> S \<subseteq> topspace X \<and> connected_space (subtopology X S)"
identify_blocksD_5: fixes k :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and A :: "'b mat" assumes "k < n"   and "(i, j) \<in> set (??.Jordan_Normal_Form_Existence.identify_blocks A k)"   and "0 < i" shows "A $$ (i - 1, i - 1) \<noteq> A $$ (k, k) \<or> A $$ (i - 1, i) \<noteq> (1::'b)"
simple_path_join_loop: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "arc g1"   and "arc g2"   and "pathfinish g1 = pathstart g2"   and "pathfinish g2 = pathstart g1"   and "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}" shows "simple_path (g1 +++ g2)"
c_gt_0: fixes A :: "real mat"   and n :: "nat"   and m :: "nat"   and lam :: "complex"   and n_as :: "(nat \<times> complex) list" assumes "spectral_radius_1_jnf_max A n m lam n_as" shows "0 < spectral_radius_1_jnf_max.c m"
proj_psi'_dim: fixes l :: "nat" shows "proj_psi'_l l \<in> carrier_mat N N"
distinct_list_bind: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b list" assumes "distinct xs"   and "\<And>x. x \<in> set xs \<Longrightarrow> distinct (f x)"   and "disjoint_family_on (set \<circ> f) (set xs)" shows "distinct (List.bind xs f)"
starset_n_Compl: fixes A :: "nat \<Rightarrow> 'a set" shows "*sn* (\<lambda>n. - A n) = - (*sn* A)"
discI_1: fixes tm :: "('a, 'b) tm"   and x1 :: "('a, 'b) hd" assumes "tm = Hd x1" shows "is_Hd tm"
listrel_sym: fixes r :: "('g \<times> 'g) set" assumes "sym r" shows "sym (listrel r)"
HEndPhase2_HInv5_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase2 s s' p" shows "HInv5_inner s' p"
coeff_0_normalize_quot_nonzero: fixes x :: "'b poly \<times> 'b poly" assumes "constant_term (snd x) \<noteq> (0::'b)" shows "constant_term (snd (normalize_quot x)) \<noteq> (0::'b)"
le_supI: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and x :: "'a"   and b :: "'a" assumes "class.semilattice_sup sup less_eq less"   and "less_eq a x"   and "less_eq b x" shows "less_eq (sup a b) x"
\<phi>_simps_1: shows "arr f\<^sub>0g\<^sub>1.\<phi>"
rem_implicit_pres_ops_numVals: fixes prob :: "(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 x :: "nat" shows "ast_problem.numVals (rem_implicit_pres_ops prob) x = ast_problem.numVals prob x"
add_pred_def: fixes P :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set"   and e :: "'BExp" shows "P +\<^sub>\<S> e \<equiv> if pred_stable \<S> e then P \<union> {e} else P"
reduced_GB_nonzero_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "0 \<notin> punit.reduced_GB F"
induct: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and Pa :: "'b \<Rightarrow> bool" assumes "accessible_on P A x"   and "\<And>x. \<lbrakk>x \<in> A; \<And>y. \<lbrakk>y \<in> A; P y x\<rbrakk> \<Longrightarrow> accessible_on P A y; \<And>y. \<lbrakk>y \<in> A; P y x\<rbrakk> \<Longrightarrow> Pa y\<rbrakk> \<Longrightarrow> Pa x" shows "Pa x"
length_takeWhile_le: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "length (takeWhile P xs) \<le> length xs"
State_Idle_append_replicate_NoMsg_output_last_message: fixes localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'d message_af list"   and c :: "'a"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun xs c))" shows "last_message (map output_fun (f_Exec_Comp_Stream trans_fun (xs @ NoMsg\<^bsup>n\<^esup>) c)) = last_message (map output_fun (f_Exec_Comp_Stream trans_fun xs c))"
assms: shows "\<forall>a. lossless_spmf (D2 a)"
make_lit_def: fixes v :: "bool"   and a :: "'b" shows "make_lit v a \<equiv> case v of True \<Rightarrow> a\<^sup>+ | False \<Rightarrow> a\<inverse>"
inter_union_distrib_right: fixes C :: "'c multiset"   and A :: "'c multiset"   and B :: "'c multiset" shows "C + A \<inter># B = (C + A) \<inter># (C + B)"
LT1: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> F \<leadsto> F"
row_add_code: fixes A :: "(('b, 'a) vec, 'c) vec"   and a :: "'c"   and b :: "'c"   and q :: "'b"   and i :: "'c" shows "($) (row_add_row A a b q i) = (\<lambda>j. if i = a then A $ a $ j + q * A $ b $ j else A $ i $ j)"
map_converter_id_move_left: fixes f :: "'a \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'b"   and f' :: "'e \<Rightarrow> 'c"   and g' :: "'d \<Rightarrow> 'f" shows "map_converter f g f' g' 1\<^sub>C = map_converter (f' \<circ> f) (g \<circ> g') id id 1\<^sub>C"
irreducible\<^sub>dD_1: fixes p :: "'b poly" assumes "irreducible\<^sub>d p" shows "0 < degree p"
Spec: fixes s :: "'d"   and t :: "'d"   and r :: "('d \<times> 'd) set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and n :: "nat" assumes "(s, t) \<in> r" shows "\<Gamma>\<turnstile> \<langle>Spec r,Normal s\<rangle> =n\<Rightarrow> Normal t"
execute_heap: fixes f :: "heap \<Rightarrow> 'a \<times> heap" shows "execute (heap f) = Some \<circ> f"
\<tau>move2_CallObjD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and M :: "String.literal"   and es :: "'addr expr1 list"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (e\<bullet>M(es)) pc xcp"   and "pc < length (compE2 e)" shows "\<tau>move2 P h stk e pc xcp"
sum_upto_ind: fixes P :: "nat \<Rightarrow> bool"   and x :: "real" shows "sum_upto (ind P) x = of_nat (card {n. 0 < n \<and> real n \<le> x \<and> P n})"
simps_28: fixes f1 :: "'a"   and f2 :: "bool \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "real \<Rightarrow> 'a"   and f5 :: "val \<Rightarrow> val \<Rightarrow> 'a"   and x4 :: "real" shows "(case RealVal x4 of UnitVal \<Rightarrow> f1 | BoolVal x \<Rightarrow> f2 x | IntVal x \<Rightarrow> f3 x | RealVal x \<Rightarrow> f4 x | <|x, xa|> \<Rightarrow> f5 x xa) = f4 x4"
put_put: fixes \<sigma> :: "'b"   and v :: "'a"   and u :: "'a" shows "put (put \<sigma> v) u = put \<sigma> u"
bind_advantage: fixes \<A> :: "'g \<times> 'a \<Rightarrow> (('j \<times> 'd) \<times> 'e \<times> ('k \<times> 'f) \<times> 'e \<times> 'k \<times> 'f) spmf" shows "AND_\<Sigma>_commit.abstract_com.bind_advantage \<A> \<le> AND_\<Sigma>_commit.rel_advantage (AND_\<Sigma>_commit.adversary \<A>)"
analz_insert_pan: fixes evs :: "event list"   and K :: "nat"   and P :: "nat" assumes "evs \<in> set_cr"   and "K \<notin> invKey ` pubEK ` range CA" shows "(Pan P \<in> analz (insert (Key K) (knows Spy evs))) = (Pan P \<in> analz (knows Spy evs))"
suminf_diff: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "summable f"   and "summable g" shows "suminf f - suminf g = (\<Sum>n. f n - g n)"
add_Sup_distrib_id: fixes A :: "V"   and x :: "V" assumes "A \<noteq> 0" shows "x + \<Squnion> (elts A) = \<Squnion> ((+) x ` elts A)"
inject: fixes x1 :: "'a"   and y1 :: "'a" shows "(Action x1 = Action y1) = (x1 = y1)"
assms_2: shows "A = [a, a', a'']\<^sub>\<circ>"
cap_non_negative: shows "\<forall>u v. 0 \<le> c (u, v)"
A: shows "ipurge_tr_rev_aux I D U xs = ws @ [x] \<longrightarrow> (\<exists>ys zs. xs = ys @ x # zs \<and> ipurge_tr_rev_aux I D U zs = [])"
ProcModifyReturn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'d"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and ModifAbr :: "'e \<Rightarrow> 'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P call init p return' c Q,A"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/UNIV \<^esub>{\<sigma>} Call p (Modif \<sigma>),(ModifAbr \<sigma>)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P call init p return c Q,A"
wf_wellorderI: assumes "wf {(x, y). x < y}"   and "OFCLASS('b, linorder_class)" shows "OFCLASS('b, wellorder_class)"
poly_add_consistent: fixes K :: "'a set" assumes "subring K R" shows "ring.poly_add (R\<lparr>carrier := K\<rparr>) = poly_add"
zero_order_2: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and n :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq n zero = (n = zero)"
incl: shows "H1 <#> H \<inter> K1 \<subseteq> H1 <#> H \<inter> K"
evenE: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and a :: "'k" assumes "class.semiring_parity plus minus zero times one divide modulo"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" obtains b :: "'k" where "a = times (numeral.numeral one plus (num.Bit0 num.One)) b"
subgraphI_max_subgraph: fixes P :: "('a, 'b) pre_digraph \<Rightarrow> bool"   and x :: "('a, 'b) pre_digraph" assumes "max_subgraph P x" shows "subgraph x G"
valid_strategy_only_in_V: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "strategy p \<sigma>"   and "\<And>v. v \<in> V \<Longrightarrow> \<sigma> v = \<sigma>' v" shows "strategy p \<sigma>'"
simps_1: fixes e :: "'v \<times> 'v" shows "h_minus 0 e = 0"
leg0_simps_1: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c" assumes "tabulation_data V H \<a> \<i> src trg r \<rho> f g" shows "partial_magma.ide V f"
HPhase1or2ReadElse_Inv2c: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "Inv2c s"   and "HPhase1or2ReadElse s s' p d q"   and "Inv2a s" shows "Inv2c s'"
compl_pmdl_def: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" shows "compl_pmdl compl = (\<forall>gs bs ps sps data. is_Groebner_basis (fst ` set gs) \<longrightarrow> sps \<noteq> [] \<longrightarrow> set sps \<subseteq> set ps \<longrightarrow> unique_idx (gs @ bs) data \<longrightarrow> fst ` set (fst (compl gs bs (ps -- sps) sps data)) \<subseteq> pmdl (args_to_set (gs, bs, ps)))"
iterate_add_empty: fixes n :: "nat" shows "iterate_add n (0::'c) = (0::'c)"
ClassicMark_def: shows "ClassicMark = (\<lambda>(i, j). case (i, j) of (init, loop) \<Rightarrow> QQ1_a \<sqinter> QQ2_a | (init, _) \<Rightarrow> \<top> | (loop, init) \<Rightarrow> \<top> | (loop, loop) \<Rightarrow> QQ3_a \<sqinter> QQ4_a \<sqinter> (QQ5_a \<sqinter> QQ6_a \<sqinter> QQ7_a) | (loop, final) \<Rightarrow> QQ8_a | (final, b) \<Rightarrow> \<top>)"
ord_sym: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ord a b c" shows "ord c b a"
admissible_eq_mcontI: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lub :: "'d set \<Rightarrow> 'd"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "mcont luba orda lub ord f"   and "mcont luba orda lub ord g" shows "ccpo.admissible luba orda (\<lambda>x. f x = g x)"
cod_is_obj: fixes Obj :: "'a set"   and Arr :: "'b set"   and Dom :: "'b \<Rightarrow> 'a"   and Cod :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "graph Obj Arr Dom Cod"   and "x \<in> Arr" shows "Cod x \<in> Obj"
map_raw_def: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> ('g, 'f) step"   and s :: "'f" shows "map_raw f g s = (case g s of Done \<Rightarrow> Done | Skip s' \<Rightarrow> Skip s' | Yield a s' \<Rightarrow> Yield (f a) s')"
ik0_def: shows "ik0 \<equiv> range aAgt \<union> range aNum \<union> aKey ` corrKey"
holds_aand: fixes P :: "'a \<Rightarrow> bool"   and steps :: "'a stream"   and Q :: "'a \<Rightarrow> bool" shows "(holds P steps \<and> holds Q steps) = holds (\<lambda>step. P step \<and> Q step) steps"
mcont_def: fixes luba :: "'f set \<Rightarrow> 'f"   and orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lubb :: "'g set \<Rightarrow> 'g"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g" shows "mcont luba orda lubb ordb f = (monotone orda ordb f \<and> cont luba orda lubb ordb f)"
lnth_lmap_ldropn: fixes n :: "nat"   and xs :: "'c llist"   and f :: "'c \<Rightarrow> 'd" assumes "enat n < llength xs" shows "lmap f (ldropn n xs) $ 0 = lmap f xs $ n"
orderE: fixes b :: "'d"   and a :: "'d" assumes "b \<le> a" obtains "a = a \<squnion> b"
imp_step: fixes l\<^sub>1 :: "lit"   and l\<^sub>2 :: "lit" assumes "sat_lit \<sigma> l\<^sub>1"   and "(l\<^sub>1, l\<^sub>2) \<in> imp_graph cnf" shows "sat_lit \<sigma> l\<^sub>2"
all_formulas_entailed: fixes Bot :: "'b set"   and entails :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and N2 :: "'b set"   and N1 :: "'b set" assumes "consequence_relation Bot entails"   and "\<forall>C\<in>N2. entails N1 {C}" shows "entails N1 N2"
coprime_iff_coprime: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one" shows "comm_monoid_mult.coprime times one = algebraic_semidom.coprime times one"