hcju/isabelleps huggingface dataset:
weaken_otherwith: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "nat set"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'a"   and \<sigma>' :: "nat \<Rightarrow> 'a"   and a :: "'b seq_action"   and P' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q' :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> bool" assumes "otherwith P I (orecvmsg Q) \<sigma> \<sigma>' a"   and "\<And>\<sigma> m. P \<sigma> m \<Longrightarrow> P' \<sigma> m"   and "\<And>\<sigma> m. Q \<sigma> m \<Longrightarrow> Q' \<sigma> m" shows "otherwith P' I (orecvmsg Q') \<sigma> \<sigma>' a"
prv_imp_sdsj_in: fixes \<phi> :: "'fmla"   and F :: "'fmla set" assumes "\<phi> \<in> fmla"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> F" shows "prv (imp \<phi> (sdsj F))"
invariant_steps: fixes a :: "'a"   and as :: "'a list" assumes "steps (a # as)"   and "P a" shows "list_all P as"
equivclp_trans: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "equivclp r a b"   and "equivclp r b c" shows "equivclp r a c"
progress_le: fixes \<sigma> :: "(char list \<times> 'b list) trace"   and \<phi> :: "'b MFOTL.formula"   and j :: "nat" shows "Monitor.progress \<sigma> \<phi> j \<le> j"
path_Exit_source: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit.path n as n'"   and "n = NewExit" shows "n' = NewExit \<and> as = []"
mult_cancel_right1: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and c :: "'e"   and b :: "'e" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(c = times b c) = (c = zero \<or> b = one)"
A: fixes ys' :: "'a list"   and k' :: "nat" shows "xs = ys' @ x # zs \<longrightarrow> offset k' x (ys' @ x # zs) \<noteq> None"
get_child_nodes_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "type_wf h"   and "known_ptr ptr"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_child_nodes ptr"
tensor_add_0_left: fixes A :: "'b tensor" shows "tensor0 (dims A) + A = A"
invar: fixes s :: "('a, unit) HashMap.hashmap" shows "hs.invar s"
LstSeqP_graph_aux_def: shows "LstSeqP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>LstSeqP_aux s k y. x1 = (s, k, y) \<and> x2 = OrdP k AND HDomain_Incl s (SUCC k) AND HFun_Sigma s AND HPair k y IN s)"
finite_alts: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "pref_profile_wf agents alts R" shows "finite alts"
assms: shows "a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)"
take_update_cancel: fixes n :: "nat"   and m :: "nat"   and xs :: "'c list"   and y :: "'c" assumes "n \<le> m" shows "take n (xs[m := y]) = take n xs"
assms_2: fixes \<psi> :: "'a ltl" assumes "G \<psi> \<in> \<G>" shows "S \<Turnstile>\<^sub>P G \<psi>"
to_Gromov_completion: shows "x = to_Gromov_completion xi"
map_poly_div: fixes p :: "real poly"   and q :: "real poly" shows "map_poly of_real (p div q) = map_poly of_real p div map_poly of_real q"
freshStarPair_2: fixes xvec :: "name list"   and x :: "'a"   and y :: "'b" shows "xvec \<sharp>* (x, y) = (xvec \<sharp>* x \<and> xvec \<sharp>* y)"
cCons_def: fixes x :: "'b"   and xs :: "'b list" shows "x ## xs = (if xs = [] \<and> x = (0::'b) then [] else x # xs)"
plus_adds_pp: fixes s :: "'a"   and t :: "'a"   and v :: "'t" shows "(s + t adds\<^sub>p v) = (t adds\<^sub>p v \<and> s adds\<^sub>p v \<ominus> t)"
abs_eq: fixes xa :: "nat \<Rightarrow> rat"   and x :: "nat \<Rightarrow> rat" assumes "realrel xa xa"   and "realrel x x" shows "Real xa + Real x = Real (\<lambda>n. xa n + x n)"
hby: shows "\<forall>l\<in>C. \<forall>m\<in>C. \<bar>f l - f m\<bar> \<le> x"
Alap: shows "A \<inter> \<Union> S \<noteq> {}"
merge_tailrec_listappend: fixes as :: "'a list"   and bs :: "'a list"   and accs1 :: "'a list"   and accs2 :: "'a list" shows "merge_tailrec as bs (accs1 @ accs2) = rev accs2 @ merge_tailrec as bs accs1"
integral_by_parts: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and F :: "real \<Rightarrow> real"   and G :: "real \<Rightarrow> real" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> isCont f x"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> isCont g x"   and "\<And>x. (F has_real_derivative f x) (at x)"   and "\<And>x. (G has_real_derivative g x) (at x)" shows "LBINT x. F x * g x * indicat_real {a..b} x = F b * G b - F a * G a - (LBINT x. f x * G x * indicat_real {a..b} x)"
SD_singleton: fixes x :: "'alt"   and y :: "'alt" assumes "x \<in> alts"   and "y \<in> alts" shows "SD (Pareto R) (return_pmf x) (return_pmf y) = Pareto R x y"
insertI1: fixes a :: "'b"   and B :: "'b set" shows "a \<in> Set.insert a B"
quad_equality_helper: fixes var :: "nat"   and L :: "real list"   and Cv :: "real"   and C :: "real mpoly"   and A :: "real mpoly"   and Av :: "real"   and B :: "real mpoly"   and Bv :: "real" assumes "var < length L"   and "0 \<le> Cv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) C = Cv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) A = Av"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) B = Bv" shows "aEval (Eq (A + B * Var var)) (L[var := sqrt Cv]) = eval (And (fm.Atom (Leq (A * B))) (fm.Atom (Eq (A\<^sup>2 - B\<^sup>2 * C)))) (L[var := sqrt Cv])"
inf_sup_ord_4: fixes y :: "'a"   and x :: "'a" shows "y \<le> x \<sqinter> y"
assms_2: shows "w \<le>p v\<^sup>\<omega>"
f1: shows "pderiv (p div q) * (q * q) div (q * q) = pderiv (p div q)"
path2_strict_prefix_ex: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and m' :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "m' \<in> set (butlast ns)" obtains ns' :: "'node list" where "g \<turnstile> n-ns'\<rightarrow>m'" and "strict_prefix ns' ns" and "m' \<notin> set (butlast ns')"
Sup_4: fixes x :: "'a" assumes "x \<in> M" shows "x = a \<or> x \<in> iterates_above (f a)"
enumerate_aexp_inputs_list: fixes a :: "vname aexp" shows "\<exists>l. enumerate_aexp_inputs a = set l"
rotate_gal: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'b list" shows "(rotater n xs = ys) = (rotate n ys = xs)"
simps_2: fixes I :: "'at set"   and A :: "'at" shows "I \<Turnstile> Neg A = (A \<notin> I)"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "in_middle (as, lm) (s, l, r) ires = (\<exists>lm1 lm2 tn m ml mr rn. lm @ 0 \<up> tn = lm1 @ [m] @ lm2 \<and> length lm1 = s \<and> m + 1 = ml + mr \<and> ml \<noteq> 0 \<and> tn = s + 1 - length lm \<and> (if lm1 = [] then l = Oc \<up> ml @ Bk # Bk # ires else l = Oc \<up> ml @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> (r = Oc \<up> mr @ [Bk] @ <lm2> @ Bk \<up> rn \<or> lm2 = [] \<and> r = Oc \<up> mr))"
resolution_classD1: fixes S :: "'a set set" assumes "resolution_class S" shows "partition_on \<V> S"
insert_def: fixes x :: "'b"   and xs :: "'b list" shows "List.insert x xs = (if x \<in> set xs then xs else x # xs)"
ket_zero_k_dim: fixes k :: "nat" assumes "k < n" shows "ket_zero_k k \<in> carrier_vec (2 ^ Suc k)"
nres_rel_def: fixes R :: "('e \<times> 'f) set" shows "\<langle>R\<rangle>nres_rel \<equiv> {(c, a). c \<le> \<Down> R a}"
sum: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "Modules.additive f" shows "f (sum g A) = (\<Sum>x\<in>A. f (g x))"
simps: fixes stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status" shows "curr_method (stk, loc, C, M, pc, ics) = M"
intro: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "linear s1 s2 f"   and "invertible_lf_axioms f" shows "invertible_lf s1 s2 f"
exhaust_sel: fixes sem_env :: "'b sem_env" obtains "sem_env = make_sem_env (sem_env.v sem_env) (c sem_env)"
IH: assumes "\<And>m n. \<lbrakk>m < length xs; n < m\<rbrakk> \<Longrightarrow> xs ! n \<sqsubset>\<^sup>i xs ! m" shows "hb.hb_consistent (node_deliver_messages xs)"
lim2: shows "LIM k sequentially. k - 1 :> sequentially"
eq: fixes s :: "complex" assumes "0 < Re s" shows "Dirichlet_L n \<chi> s = eval_fds (fds \<chi>) s"
index_mat_multrow_2: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and mul :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "i < dim_row A"   and "j < dim_col A" shows "mat_multrow_gen mul i a A $$ (i, j) = mul a (A $$ (i, j))"
edges_face_eq: fixes a :: "nat"   and b :: "nat"   and f :: "face" shows "((a, b) \<in> \<E> f) = (f \<bullet> a = b \<and> a \<in> \<V> f)"
f2_nth_simps_3: fixes n :: "nat" assumes "2 < n" shows "f\<^sup>2 $ n = (0::'a)"
results'_rpv_def: fixes rpv :: "('d, 'f, 'e) rpv" shows "results'_rpv rpv = range rpv \<bind> results'_gpv"
leg1_is_map: shows "is_left_adjoint p\<^sub>1"
in_doubt_deny_denies_DropReject: fixes a :: "action"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and m :: "'c match_expr"   and p :: "'d" assumes "a = action.Drop \<or> a = Reject"   and "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "Semantics.matches \<gamma> m p" shows "Matching_Ternary.matches (\<beta>, in_doubt_deny) m a p"
continuous_isCont: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "isCont f x0 = isCont f x0"
pWls: shows "presWlsAll (rec MOD) (recAbs MOD) MOD"
takeWhile_raw_def: fixes P :: "'c \<Rightarrow> bool"   and g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" shows "takeWhile_raw P g s = (case g s of Done \<Rightarrow> Done | Skip s' \<Rightarrow> Skip s' | Yield a s' \<Rightarrow> if P a then Yield a s' else Done)"
finite_Diff2: fixes B :: "'d set"   and A :: "'d set" assumes "finite B" shows "finite (A - B) = finite A"
reduced_run_words_fin: fixes q :: "'state"   and v\<^sub>1 :: "'action list"   and v\<^sub>2 :: "'action stream"   and l :: "'action list"   and w :: "'action list"   and w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and u :: "'action stream" assumes "reduced_run q v\<^sub>1 v\<^sub>2 l w w\<^sub>1 w\<^sub>2 u" shows "R.path w q"
queue_invar_def: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.queue_invar q \<equiv> \<forall>t\<in>set q. SkewBinomialHeapStruc.tree_invar t"
continuous_closed_preimage: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "continuous_on S f"   and "closed S"   and "closed T" shows "closed (S \<inter> f -` T)"
assms_4: shows "set_mset (measure_conv r \<delta>1) \<subseteq> r \<down>m M"
BIT_special_simps_1: shows "0 BIT False = 0"
FORWEAK_def: fixes X :: "'f set"   and d :: "'e nres"   and f :: "'f \<Rightarrow> 'e nres"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> 'e nres" shows "FORWEAK X d f c = (if X = {} then d else op_set_npick_remove X \<bind> (\<lambda>(a, X). f a \<bind> (\<lambda>b. WHILE (\<lambda>(uu_, X). \<not> op_set_isEmpty X) (\<lambda>(b, X). ASSERT (X \<noteq> {}) \<bind> (\<lambda>_. op_set_npick_remove X \<bind> (\<lambda>(a, X). f a \<bind> (\<lambda>b'. c b b' \<bind> (\<lambda>b. RETURN (b, X)))))) (b, X) \<bind> (\<lambda>(b, uu_). RETURN b))))"
FOREACHci_rule': fixes S :: "'c set"   and I :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd nres"   and P :: "'d \<Rightarrow> bool" assumes "finite S"   and "I S \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> it; it \<subseteq> S; I it \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (it - {x}))"   and "\<And>\<sigma>. \<lbrakk>I {} \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>"   and "\<And>it \<sigma>. \<lbrakk>it \<subseteq> S; I it \<sigma>; \<not> c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "FOREACH\<^sub>C\<^bsup>I\<^esup> S c f \<sigma>0 \<le> SPEC P"
games_gfp_def: shows "games_gfp \<equiv> gfp fixgames"
AE_measure_spmf_iff: fixes P :: "'a \<Rightarrow> bool"   and p :: "'a spmf" shows "almost_everywhere (measure_spmf p) P = (\<forall>x\<in>set_spmf p. P x)"
sup_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "Err.sup f = lift2 (\<lambda>x y. OK (x \<squnion>\<^bsub>f\<^esub> y))"
terminates_list_update_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and b :: "'f" assumes "terminates g" shows "terminates (list_update_raw g b)"
vdisjnt_iff: fixes A :: "V"   and B :: "V" shows "(A \<inter>\<^sub>\<circ> B = []\<^sub>\<circ>) = (\<forall>x. \<not> (x \<in>\<^sub>\<circ> A \<and> x \<in>\<^sub>\<circ> B))"
plus_real_alg: fixes x :: "real_alg"   and y :: "real_alg" shows "real_of x + real_of y = real_of (x + y)"
fmrestrict_set_empty: fixes A :: "'a set" shows "fmrestrict_set A fmempty = fmempty"
rep_eq: fixes x :: "'c stone_phi_pair"   and xa :: "'c stone_phi_pair" shows "(x < xa) = stone_phi.pairs_less (Rep_stone_phi_pair x) (Rep_stone_phi_pair xa)"
intro: fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'i spmf"   and S1 :: "'g \<Rightarrow> 'j \<Rightarrow> 'i spmf"   and R2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'k spmf"   and S2 :: "'h \<Rightarrow> 'l \<Rightarrow> 'k spmf"   and funct :: "'g \<Rightarrow> 'h \<Rightarrow> ('j \<times> 'l) spmf" assumes "\<And>m1 m2. lossless_spmf (R1 m1 m2)"   and "\<And>m1 out1. lossless_spmf (S1 m1 out1)"   and "\<And>m1 m2. lossless_spmf (R2 m1 m2)"   and "\<And>m2 out2. lossless_spmf (S2 m2 out2)"   and "\<And>m1 m2. lossless_spmf (funct m1 m2)" shows "sim_det_def R1 S1 R2 S2 funct"
measure_empty: fixes M :: "'d measure" shows "Sigma_Algebra.measure M {} = 0"
path_edges_remove_prefix: fixes xs :: "'a list"   and x :: "'a"   and xs' :: "'a list" assumes "H.path (xs @ x # xs')" shows "H.edges_of_walk (xs @ [x]) = H.edges_of_walk (xs @ x # xs') - H.edges_of_walk (x # xs')"
intros_2: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> parts H" shows "X \<in> parts H"
TR1_mem: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme"   and m1 :: "'f"   and m2 :: "'f"   and n :: "'h" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N"   and "m1 \<in> carrier M"   and "m2 \<in> carrier M"   and "n \<in> carrier N" shows "(m1 \<plusminus>\<^bsub>M\<^esub> m2, n) \<plusminus>\<^bsub>P\<^esub> -\<^sub>a\<^bsub>P\<^esub> ((m1, n) \<plusminus>\<^bsub>P\<^esub> (m2, n)) \<in> TR\<^bsub>R\<^esub> M N P"
d_2: fixes f :: "'b poly"   and g :: "'b poly" shows "divide_poly_list f g = (let cg = coeffs g in if cg = [] then g else let cf = coeffs f; cgr = rev cg in poly_of_list (divide_poly_main_list (hd cgr) [] (rev cf) cgr (1 + length cf - length cg)))"
direct_decomp_hom_deg_set: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and ss :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set list"   and z :: "nat" assumes "direct_decomp A ss"   and "\<And>s. s \<in> set ss \<Longrightarrow> homogeneous_set s" shows "direct_decomp (hom_deg_set z A) (map (hom_deg_set z) ss)"
prefixes_fin: fixes x :: "'a list" shows "let prefixes = {t. \<exists>t2. x = t @ t2} in finite prefixes \<and> card prefixes = length x + 1"
cr_def: shows "cr \<equiv> \<lambda>s m. if m < ord then interpret_floatarith (deriv_rec f m) ((map real_of_float xs)[0 := real_of_float (xs ! 0)]) / fact m - real_of_float (mid (z ! m)) else interpret_floatarith (deriv_rec f m) ((map real_of_float xs)[0 := s]) / fact ord - real_of_float (mid (z ! ord))"
coeff_int: fixes p :: "'b poly"   and n :: "nat" shows "coeff_int p (int n) = coeff p n"
assms_2: shows "approx_form prec f (replicate n None) ss"
CCs_imageCC: fixes F :: "'edge set" shows "CCs F = CC F ` V"
eval_CompDiag: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag t"   and "Diag u"   and "Seq t u" shows "\<lbrace>t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u\<rbrace> = \<lbrace>t\<rbrace> \<cdot> \<lbrace>u\<rbrace>"
mkArr_eqI': fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's"   and F' :: "'s \<Rightarrow> 's" assumes "S.arr (S.mkArr A B F)"   and "\<And>x. x \<in> A \<Longrightarrow> F x = F' x" shows "S.mkArr A B F = S.mkArr A B F'"
fold_stores_def: fixes ws :: "('e, 'g, 'h) mem_store_action list" shows "fold_stores ws = fold (\<lambda>w. (\<circ>) (do_store_action w)) ws id"
cases: fixes t :: "'t"   and a1 :: "(status \<times> 'x1) \<times> 'm1"   and a2 :: "(status \<times> 'x2) \<times> 'm2" assumes "t \<turnstile> a1 \<approx>i a2" obtains x1 :: "'x1" and m1 :: "'m1" and x2 :: "'x2" and m2 :: "'m2" where "a1 = ((PreStart, x1__), m1__)" and "a2 = ((PreStart, x2__), m2__)" and "t \<turnstile> (x1__, m1__) \<approx> (x2__, m2__)"   | x1 :: "'x1" and m1 :: "'m1" and x2 :: "'x2" and m2 :: "'m2" where "a1 = ((Running, x1a__), m1a__)" and "a2 = ((Running, x2a__), m2a__)" and "t \<turnstile> (x1a__, m1a__) \<approx> (x2a__, m2a__)"   | x1 :: "'x1" and m1 :: "'m1" and x2 :: "'x2" and m2 :: "'m2" where "a1 = ((Finished, x1), m1)" and "a2 = ((Finished, x2), m2)" and "t \<turnstile> (x1, m1) \<approx> (x2, m2)" and "final1 x1" and "final2 x2"
Roots'_def: shows "Roots' = \<Union> (Roots ` P)"
eqvt_def: fixes f :: "'e" shows "eqvt f \<equiv> \<forall>p. p \<bullet> f = f"
scaleC_minus1_left: fixes x :: "'a" shows "- 1 *\<^sub>C x = - x"
action_needed_asses_def: fixes a :: "('a, 'b) fmap \<times> 'c"   and s :: "('a, 'b) fmap" shows "action_needed_asses a s \<equiv> fmrestrict_set (action_needed_vars a s) s"
rpr_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "rpr A r \<equiv> complete A r \<and> refl_on A r \<and> trans r"
adds_antisym: fixes s :: "'a"   and t :: "'a" assumes "s adds t"   and "t adds s" shows "s = t"
a_subset: fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme" assumes "additive_subgroup H G" shows "H \<subseteq> carrier G"
refine_IdD: fixes c :: "'g nres"   and a :: "'g nres" assumes "c \<le> \<Down> Id a" shows "c \<le> a"
univ_second_countable_sequence: obtains B :: "nat \<Rightarrow> 'b set" where "inj B" and "\<And>n. open (B n)" and "\<And>S. open S \<Longrightarrow> \<exists>k. S = \<Union> {B n |n. n \<in> k}"
continuous_on_prod_compactE: fixes U :: "'a set"   and C :: "'b set"   and fx :: "'a \<times> 'b \<Rightarrow> 'c"   and x0 :: "'a"   and e :: "real" assumes "continuous_on (U \<times> C) fx"   and "compact C"   and "x0 \<in> U"   and "0 < e" obtains X0 :: "'a set" where "x0 \<in> X0" and "open X0" and "\<forall>x\<in>X0 \<inter> U. \<forall>t\<in>C. dist (fx (x, t)) (fx (x0, t)) \<le> e"
borel_measurable_if_D: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "(\<lambda>x. if x \<in> S then f x else (0::'b)) \<in> borel_measurable lebesgue" shows "f \<in> borel_measurable (lebesgue_on S)"
approx_SComplex_mult_cancel: fixes a :: "complex star"   and w :: "complex star"   and z :: "complex star" assumes "a \<in> SComplex"   and "a \<noteq> 0"   and "a * w \<approx> a * z" shows "w \<approx> z"
matrix_to_iarray_eq_of_fun: fixes A :: "(('c, 'd) vec, 'b) vec"   and f :: "nat \<Rightarrow> 'c iarray"   and n :: "nat" assumes "\<forall>i. vec_to_iarray (A $ i) = f (mod_type_class.to_nat i)"   and "n = IArray.length (matrix_to_iarray A)" shows "matrix_to_iarray A = IArray.of_fun f n"
shiftl_eq_push_bit: fixes 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"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.semiring_bit_syntax plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_bit_syntax.shiftl push_bit a n = push_bit n a"
one_poly_eq_simps_2: shows "([:1::'a:] = 1) = True"
elims_3: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and xa :: "'b equation" assumes "\<not> validate_ground_eq x xa" obtains I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and t :: "'b trm" and s :: "'b trm" where "x = I" and "xa = Eq t s" and "\<not> I t s"
graph_rule_translation: fixes X :: "'b allegorical_term"   and Y :: "'b allegorical_term" shows "graph_rule (translation X, translation (A_Int X Y))"
smn_lemma: fixes n :: "nat"   and cs :: "nat list"   and m :: "nat"   and xs :: "nat list"   and p :: "nat" assumes "0 < n"   and "length cs = m"   and "length xs = n" shows "eval (r_universal (m + n)) (p # cs @ xs) = eval (r_universal n) (the (eval (r_smn n m) (p # cs)) # xs)"
set: fixes x1 :: "'a match_expr"   and x2 :: "action" shows "set_rule (Rule (x1::'a match_expr) (x2::action)) = set_match_expr x1"
induct: fixes P :: "bexp \<Rightarrow> bool"   and a0 :: "bexp" assumes "P (Bc True)"   and "P (Bc False)"   and "\<And>v. P (bexp.Not v)"   and "\<And>v va. P (And v va)"   and "\<And>v va. P (Less v va)" shows "P a0"
elims: fixes x :: "'b option"   and xa :: "'b option"   and y :: "'b option" assumes "x + xa = y" obtains uu_ :: "'b option" where "x = None" and "xa = uu_" and "y = None"   | v :: "'b" where "x = Some v" and "xa = None" and "y = None"   | a :: "'b" and b :: "'b" where "x = Some a" and "xa = Some b" and "y = Some (a + b)"
inside_frontier_eq_interior: fixes S :: "'b set" assumes "bounded S"   and "convex S" shows "inside (frontier S) = interior S"
swap_def: fixes p :: "'d \<times> 'c" shows "prod.swap p = (snd p, fst p)"
forward_lockstep_simulation: fixes s1 :: "('fenv_std, 'henv, ('fun, 'dyn) frame) state"   and s1' :: "('fenv_std, 'henv, ('fun, 'dyn) frame) state"   and s2 :: "('fenv_inca, 'henv, ('fun, 'dyn) frame) state" assumes "Sstd.step s1 s1'"   and "s1 \<sim> s2" shows "\<exists>s2'. Sinca.step s2 s2' \<and> s1' \<sim> s2'"
pivotandupdate_lvars: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "PivotAndUpdate eq_idx_for_lvar pivot_and_update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (EqForLVar.eq_for_lvar eq_idx_for_lvar (\<T> s) x\<^sub>i)" shows "lvars (\<T> (pivot_and_update x\<^sub>i x\<^sub>j c s)) = lvars (\<T> s) - {x\<^sub>i} \<union> {x\<^sub>j}"
A2: shows "n_leaves (bt_map f t2) = n_leaves t2"
mat_max_comm: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat" assumes "class.non_strict_order less_eq"   and "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "ord.mat_max less_eq A B = ord.mat_max less_eq B A"
sip_in_kD: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PAodv-:7} \<union> {PAodv-:5} \<union> {PRrep-:0..PRrep-:1} \<union> {PRreq-:0..PRreq-:3} \<longrightarrow> sip \<xi> \<in> kD (rt \<xi>))"
full_vars_assn_def: shows "full_vars_assn = hr_comp (hs.assn string_assn) (\<langle>var_rel\<rangle>set_rel)"
latestAct_prop_1: fixes n :: "nat"   and nid :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'. latestAct_cond nid t n n'" shows "\<parallel>nid\<parallel>\<^bsub>t \<langle>nid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>\<^esub>"
simps: fixes a1 :: "'c"   and a2 :: "'c" shows "a1 -\<tau>2\<rightarrow> a2 = (\<exists>s s' tl. a1 = s \<and> a2 = s' \<and> trsys2 s tl s' \<and> False)"