hcju/isabelleps huggingface dataset:
adopt_node_children_subset: fixes h :: "(_) heap"   and owner_document :: "(_) document_ptr"   and node :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and children' :: "(_) node_ptr list" assumes "h \<turnstile> Shadow_DOM.adopt_node owner_document node \<rightarrow>\<^sub>h h'"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h' \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children'"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "set children' \<subseteq> set children"
starfun2_star_n: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and X :: "nat \<Rightarrow> 'b"   and Y :: "nat \<Rightarrow> 'c" shows "(*f2* f) (star_n X) (star_n Y) = star_n (\<lambda>n. f (X n) (Y n))"
preserves_dom: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr f" shows "Span.dom (Span.trg f) = Span.trg (Span.dom f)"
aff_dim_unique: fixes B :: "'a set"   and V :: "'a set" assumes "affine hull B = affine hull V \<and> \<not> affine_dependent B" shows "int (card B) = aff_dim V + 1"
nonzero_of_rat_inverse: fixes a :: "rat" assumes "a \<noteq> 0" shows "of_rat (inverse a) = inverse (of_rat a)"
square_free_factorization_order_root_no_mem: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list"   and x :: "'a" assumes "square_free_factorization p (c, bs)"   and "p \<noteq> 0"   and "\<And>a i. (a, i) \<in> set bs \<Longrightarrow> poly a x \<noteq> (0::'a)" shows "order x p = 0"
syncHdAdd: fixes a :: "'a event list"   and t :: "'a event list"   and A :: "'a event set"   and u :: "'a event list" assumes "tl a setinterleaves ((t, u), A) \<and> hd a \<notin> A \<and> hd t \<in> A \<and> a \<noteq> []" shows "a setinterleaves ((t, hd a # u), A)"
mult_commute: fixes a :: "'ag"   and b :: "'ag" assumes "a \<in> U"   and "b \<in> U" shows "a *\<^sub>o\<^sub>w b = b *\<^sub>o\<^sub>w a"
less_irrefl: fixes x :: "'a list" shows "\<not> strict_border x x"
same_prefix_prefix: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs \<cdot> ys \<le>p xs \<cdot> zs) = (ys \<le>p zs)"
siso0_Sbis: fixes c :: "('test, 'atom) com" assumes "siso0 c" shows "c \<approx>T c"
numeral_inc: fixes x :: "num" shows "numeral (Num.inc x) = numeral x + (1::'a)"
b: shows "steps0 (1, l, r) ap stpa = (sa, la, ra)"
ennreal_indicator: fixes A :: "'c set"   and x :: "'c" shows "ennreal (indicat_real A x) = indicator A x"
vset_neq_2: fixes b :: "V"   and A :: "V"   and a :: "V" assumes "b \<in>\<^sub>\<circ> A"   and "a \<notin>\<^sub>\<circ> A" shows "b \<noteq> a"
mult_imp_msetext_huet: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "irreflp gt"   and "transp gt"   and "(mset xs, mset ys) \<in> mult {(x, y). gt y x}" shows "msetext_huet gt ys xs"
sum_wt_ParT_sub_notWtFT_notFinished_shift: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and cl :: "('c, 'd, 'a) cmd list"   and n :: "nat"   and I :: "nat set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b" assumes "PL_Indis cval indis"   and "properL cl"   and "WtFT cl \<noteq> 1"   and "n < length cl"   and "\<not> finished (cl ! n)"   and "I \<subseteq> {..<PL.brn (cl ! n)}" shows "sum (PL.wt aval tval cval (ParT cl) s) (PL_Indis.shift cl n I) = 1 / real (length cl) / (1 - WtFT cl) * sum (PL.wt aval tval cval (cl ! n) s) I"
tranclp_rtranclp_tranclp: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "r\<^sup>+\<^sup>+ a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
Prim1_def: shows "Prim1 \<equiv> {\<lambda>x. eval r [x] |r. prim_recfn 1 r}"
O_assoc: fixes R :: "('d \<times> 'g) set"   and S :: "('g \<times> 'f) set"   and T :: "('f \<times> 'e) set" shows "(R O S) O T = R O S O T"
UnitCategory_def: shows "UnitCategory = MakeCat \<lparr>Obj = {()}, Mor = {()}, Dom = \<lambda>f. (), Cod = \<lambda>f. (), Id = \<lambda>f. (), Comp = \<lambda>f g. ()\<rparr>"
res_seq_res: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "res_seq s k \<in> carrier Zp\<^bsup>\<omega>\<^esup>"
rbt_sorted_fold_insert: fixes t :: "('a, 'c) rbt"   and t' :: "('a, 'c) rbt" assumes "rbt_sorted t" shows "rbt_sorted (RBT_Impl.fold rbt_insert t' t)"
support_wpw: fixes b :: "bexp"   and c :: "com"   and x :: "char list"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "support (\<lambda>l. wpw b c (l x) Q l) \<subseteq> insert x (support Q)"
split_sel: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f"   and option :: "'g option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (option = Some (the option) \<longrightarrow> P (f2 (the option))))"
pred_envir_def: fixes p :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c" shows "pred_envir p f = all_envir (pure p \<diamondop> f)"
Res: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and x :: "name" assumes "(\<Psi>, P, Q) \<in> {(\<Psi>, \<lparr>\<nu>*xvec\<rparr>P \<parallel> R, \<lparr>\<nu>*xvec\<rparr>Q \<parallel> R) |xvec \<Psi> P Q R. xvec \<sharp>* \<Psi> \<and> (\<forall>A\<^sub>R \<Psi>\<^sub>R. extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle> \<and> A\<^sub>R \<sharp>* \<Psi> \<and> A\<^sub>R \<sharp>* P \<and> A\<^sub>R \<sharp>* Q \<longrightarrow> \<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<sim> Q)} \<union> bisim"   and "x \<sharp> \<Psi>" shows "(\<Psi>, \<lparr>\<nu>x\<rparr>P, \<lparr>\<nu>x\<rparr>Q) \<in> {(\<Psi>, \<lparr>\<nu>*xvec\<rparr>P \<parallel> R, \<lparr>\<nu>*xvec\<rparr>Q \<parallel> R) |xvec \<Psi> P Q R. xvec \<sharp>* \<Psi> \<and> (\<forall>A\<^sub>R \<Psi>\<^sub>R. extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle> \<and> A\<^sub>R \<sharp>* \<Psi> \<and> A\<^sub>R \<sharp>* P \<and> A\<^sub>R \<sharp>* Q \<longrightarrow> \<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<sim> Q)} \<union> bisim"
proper_interval_Some2: fixes ao :: "'a option"   and x :: "'a" shows "proper_interval ao (Some x) = (\<exists>z\<in>above ao. z < x)"
exhaust: fixes y :: "msg" obtains x1 :: "agent" where "y = Agent x1"   | x2 :: "nat" where "y = Number x2"   | x3 :: "nat" where "y = Nonce x3"   | x4 :: "nat" where "y = Key x4"   | x5 :: "msg" where "y = Hash x5"   | x6 :: "nat" where "y = Passwd x6"   | x7 :: "rat" where "y = Pri_AgrK x7"   | x8 :: "int" where "y = Pub_AgrK x8"   | x91 :: "rat" and x92 :: "rat" where "y = Auth_Data x91 x92"   | x101 :: "nat" and x102 :: "msg" where "y = Crypt x101 x102"   | x111 :: "msg" and x112 :: "msg" where "y = \<lbrace>x111, x112\<rbrace>"
S: fixes x :: "'n"   and d :: "real" assumes "x \<in> S"   and "0 < d" shows "\<exists>i. i \<in> K \<and> x \<in> cball (a i) (r i) \<and> r i < d"
contour_integral_part_circlepath_reverse': fixes b :: "real"   and a :: "real"   and c :: "complex"   and r :: "real"   and f :: "complex \<Rightarrow> complex" assumes "b < a" shows "contour_integral (part_circlepath c r a b) f = - contour_integral (part_circlepath c r b a) f"
reflexive: fixes P :: "pi" shows "P \<sim>\<^sup>s P"
simps: fixes rh :: "'e \<Rightarrow> 'f"   and bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "('e, 'f) T\<^sub>m"   and a2 :: "('e, 'f) T\<^sub>m" shows "blinding_of_T rh bo a1 a2 = (\<exists>x y. a1 = T\<^sub>m x \<and> a2 = T\<^sub>m y \<and> blinding_of_F rh bo (root_hash_T rh) (blinding_of_T rh bo) x y)"
l_span_closed2Tr: fixes A :: "'b set"   and H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "H \<subseteq> carrier M" shows "s \<in> {j. j \<le> n} \<rightarrow> A \<and> f \<in> {j. j \<le> n} \<rightarrow> linear_span R M (carrier R) H \<longrightarrow> l_comb R M n s f \<in> linear_span R M A H"
select_convs_3: fixes c_records :: "'d zmultiset"   and c_temp :: "'c \<Rightarrow> 'd zmultiset"   and c_msg :: "'c \<Rightarrow> 'c \<Rightarrow> 'd zmultiset list"   and c_glob :: "'c \<Rightarrow> 'd zmultiset"   and more :: "'e" shows "c_msg \<lparr>c_records = c_records, c_temp = c_temp, c_msg = c_msg, c_glob = c_glob, \<dots> = more\<rparr> = c_msg"
subcocycle_integrable_1: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "subcocycle u" shows "integrable M (u n)"
simps_1: fixes c :: "'i"   and \<sigma> :: "'j \<Rightarrow> ('h, 'i) trm" shows "OsubstFO (OVar c) \<sigma> = OVar c"
assms_3: shows "II \<subseteq> {..<brnL cl (length cl)}"
f_def: shows "f = (\<lambda>(g, h) ab. case ab of Inl a \<Rightarrow> g a | Inr b \<Rightarrow> h b)"
S: fixes x :: "real" assumes "x \<in> {0..1} - S" shows "g1 +++ g2 differentiable at x within {0..1}"
Fun_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ" shows "Fun (mkIde A) = (\<lambda>x\<in>A. x)"
map_graph_def: fixes f :: "'a \<Rightarrow> 'b option" shows "map_graph f = {(x, y) |x y. f x = Some y}"
step: shows "\<theta>\<turnstile> (s, t) \<rightarrow>\<^sub>s ((s', t'), is')"
bind_le_nofailI: fixes m :: "'e nres"   and f :: "'e \<Rightarrow> 'f nres"   and m' :: "'f nres" assumes "nofail m"   and "\<And>x. RETURN x \<le> m \<Longrightarrow> f x \<le> m'" shows "m \<bind> f \<le> m'"
complex_roots_of_int_poly_2: fixes p :: "int poly" shows "distinct (complex_roots_of_int_poly p)"
Least_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" shows "ord.Least less_eq P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> less_eq x y))"
comp_Suc: fixes n :: "nat"   and p :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "swapidseq n p"   and "a \<noteq> b" shows "swapidseq (Suc n) (Fun.swap a b id \<circ> p)"
valofn_def: fixes x :: "('a, 'b) IEEE.float" shows "valofn x = 2 ^ IEEE.exponent x / 2 ^ bias TYPE(('a, 'b) IEEE.float) * (1 + real (fraction x) / 2 ^ LENGTH('b))"
HPT_Conf_Target_Source: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d" assumes "TS \<in> HPT ST"   and "S \<in> Conf ST"   and "S \<in> Target TS" shows "S \<in> Source TS"
lesspoll_trans1: fixes X :: "'d set"   and Y :: "'e set"   and Z :: "'f set" assumes "X \<lesssim> Y"   and "Y \<prec> Z" shows "X \<prec> Z"
mstep_def: fixes tdb :: "(char list \<times> 'a list) set \<times> nat"   and st :: "'a mstate" shows "mstep tdb st = (let (xs, m) = meval (mstate_n st) (snd tdb) (fst tdb) (mstate_m st) in (\<Union> (set (map (\<lambda>(i, X). Pair i ` X) (List.enumerate (mstate_i st) xs))), \<lparr>mstate_i = mstate_i st + length xs, mstate_m = m, mstate_n = mstate_n st\<rparr>))"
basis_orthogonal: fixes B :: "'b set" assumes "finite B" shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
sum_list_prod_map_smult_distrib: fixes m :: "'v"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> ('f, 'g) aezfun"   and xys :: "('a \<times> 'b) list" assumes "m \<in> V"   and "set (map (\<lambda>(x, y). f x y) xys) \<subseteq> FG" shows "(\<Sum>(x, y)\<leftarrow>xys. f x y) \<cdot> m = (\<Sum>(x, y)\<leftarrow>xys. f x y \<cdot> m)"
A_n_A: fixes n :: "nat" shows "A_n n \<subseteq> A \<Gamma>"
mult_Real: fixes X :: "nat \<Rightarrow> rat"   and Y :: "nat \<Rightarrow> rat" assumes "Real.cauchy X"   and "Real.cauchy Y" shows "Real.Real X * Real.Real Y = Real.Real (\<lambda>n. X n * Y n)"
mset_factors_def: fixes F :: "'b multiset"   and p :: "'b" shows "mset_factors F p \<equiv> F \<noteq> {#} \<and> (\<forall>f. f \<in># F \<longrightarrow> irreducible f) \<and> p = \<Prod>\<^sub># F"
C2O_cons: fixes n :: "C"   and ns :: "C list" shows "C2O (C (n # ns)) = (if n = C [] then S (C2O (C ns)) else L (\<lambda>i. C2O (C (funC n i @ ns))))"
swapInp_preserves_wls: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "wlsInp delta inp" shows "wlsInp delta (inp %[z1 \<and> z2]_zs)"
get_disconnected_nodes_def: shows "get_disconnected_nodes = (\<lambda>document_ptr. get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes)"
swaprows_mat_inv: fixes k :: "nat"   and n :: "nat"   and l :: "nat" assumes "k < n"   and "l < n" shows "swaprows_mat n k l * swaprows_mat n k l = 1\<^sub>m n"
of_real_def: fixes r :: "real" shows "of_real r = r *\<^sub>R (1::'c)"
is_extensional: fixes f :: "'s" assumes "\<not> S.arr f" shows "\<Phi>\<Psi>.map f = S.null"
succ_rel_functional: fixes a :: "'b"   and b1 :: "'b option"   and xs :: "'b list"   and b2 :: "'b option" assumes "(a, b1) \<in> succ_rel xs"   and "(a, b2) \<in> succ_rel xs"   and "distinct xs" shows "b1 = b2"
iTILL_def: fixes n :: "nat" shows "[\<dots>n] \<equiv> {..n}"
OK_le_err_OK: fixes x :: "'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c" shows "(OK x \<sqsubseteq>\<^bsub>Err.le r\<^esub> OK y) = (x \<sqsubseteq>\<^bsub>r\<^esub> y)"
resolve_rename_either_way_eq_congls_of_inferences_between: fixes C :: "'a literal list"   and Q :: "'a literal list set" shows "mset ` set (resolve_rename C C) \<union> (\<Union>D\<in>Q. mset ` set (resolve_rename_either_way C D)) = concls_of (inference_system.inferences_between (ord_FO_\<Gamma> S) (mset ` Q) (mset C))"
mult_imp_less_div_pos: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and y :: "'d"   and z :: "'d"   and x :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less (times z y) x" shows "less z (divide x y)"
eqvt_def: fixes f :: "'b" shows "eqvt f \<equiv> \<forall>p. p \<bullet> f = f"
finalConflictingState: fixes state :: "State" assumes "InvariantUniq (getM state)"   and "InvariantReasonClauses (getF state) (getM state)"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "\<not> applicableExplain state"   and "\<not> applicableBackjump state"   and "getConflictFlag state" shows "getC state = []"
urev_def: fixes m :: "nat"   and n :: "nat" shows "urev m n = u n m"
assm_2: fixes x :: "'a"   and K :: "'a set" assumes "(x, K) \<in> s" shows "x \<in> K"
step: fixes n :: "nat"   and x :: "(nat \<times> form) list"   and s :: "(nat \<times> form) list"   and y :: "(nat \<times> form) list" assumes "(n, x) \<in> deriv s"   and "y \<in> set (subs x)" shows "(Suc n, y) \<in> deriv s"
vsubst_subst_compose2: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and z1 :: "'var" assumes "good X"   and "good Y"   and "ys \<noteq> zs \<or> y \<notin> {z, z1}" shows "X #[Y / y]_ys #[z1 // z]_zs = X #[z1 // z]_zs #[Y #[z1 // z]_zs / y]_ys"
simps_4: fixes c :: "int"   and e :: "MIR.num" shows "iszlfm (NEq (CN 0 c e)) = (\<lambda>bs. 0 < c \<and> numbound0 e \<and> isint e bs)"
wf_measure: fixes f :: "'a \<Rightarrow> nat" shows "wf (measure f)"
left_zero: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and a :: "'c" assumes "ring R addition multiplication zero unit"   and "a \<in> R" shows "multiplication zero a = zero"
dest_IO_eq_None_iff: fixes generat :: "('i, 'g, 'h) generat" shows "(dest_IO generat = None) = is_Pure generat"
ub1: shows "\<forall>x. poly p x = 0 \<longrightarrow> x < ub1"
sstart_in_streams: fixes xs :: "'b list"   and S :: "'b set" assumes "xs \<in> lists S" shows "sstart S xs \<subseteq> streams S"
p: shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' Z) Call p (Q' Z),(A' Z)"
fB: shows "\<forall>ps n. I (ps, n) \<and> vars b \<subseteq> dom ps \<and> \<not> pbval b ps \<longrightarrow> Q (ps, n)"
AAs0'_AAs: shows "AAs0' \<cdot>\<cdot>aml \<eta>s0' = AAs"
refl_on_converse: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "refl_on A (r\<inverse>) = refl_on A r"
trans: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "c2 x y = Lt"   and "c2 y z = Lt" shows "c2 x z = Lt"
weight_map_spmf: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d spmf" shows "weight_spmf (map_spmf f p) = weight_spmf p"
power_inductl: fixes z :: "'a list set"   and x :: "'a list set"   and y :: "'a list set"   and n :: "nat" assumes "z \<union> p_prod x y \<subseteq> y" shows "p_prod (path_dioid.power x n) z \<subseteq> y"
sup_action: fixes P :: "('a, 'b) action"   and Q :: "('a, 'b) action" shows "sup P Q \<equiv> action_of (action.relation_of P \<and> action.relation_of Q)"
cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def: fixes node :: "'m RNode_scheme" shows "cast node = RObject.extend (RObject.truncate node) (Inl (RObject.more node))"
pickedOneOfThem: fixes n :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and var :: "nat"   and step :: "nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm" assumes "heuristicPicker n L F = Some (var, step)" shows "step = qe_eq_repeat \<or> step = gen_qe \<or> step = luckyFind'"
f_eq: shows "AE x in M. \<forall>i. (if x \<in> space M - N then f i x else 0) = f i x"
mult_one_div_unit_factor: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "times a (divide one (unit_factor b)) = divide a (unit_factor b)"
v_of_merge: fixes env2 :: "v sem_env"   and env1 :: "v sem_env" shows "sem_env.v (extend_dec_env env2 env1) = nsAppend (sem_env.v env2) (sem_env.v env1)"
right_closed: fixes a :: "'a set \<times> 'b"   and b :: "'a set \<times> 'b" assumes "(a, b) \<in> {(x, y). stk2.rel x y}" shows "b \<in> Sigma I \<FF>2"
eq_im_imp_diff_in_Ker: fixes g :: "'g"   and h :: "'g" assumes "g \<in> G"   and "h \<in> G"   and "T g = T h" shows "g - h \<in> Ker"
cobounded2: fixes b :: "'a"   and a :: "'a" shows "b \<le> sup a b"
strict_trans1: fixes b :: "'k multiset"   and a :: "'k multiset"   and c :: "'k multiset" assumes "b \<subseteq># a"   and "c \<subset># b" shows "c \<subset># a"
rel_chain_Nil: fixes r :: "('b \<times> 'b) set" shows "rel_chain [] r"
set_map: fixes f :: "'e \<Rightarrow> 'd"   and v :: "'e list" shows "set (map f v) = f ` set v"
simps: fixes ws :: "'a list"   and Z :: "'a set" shows "local.rtrancl_while_step (ws, Z) = (let x = hd ws; new = remdups (filter (\<lambda>y. y \<notin> Z) (f x)) in (new @ tl ws, set new \<union> Z))"
d_equal_def: fixes DA :: "'c \<Rightarrow> 'd"   and d :: "'d"   and m :: "'c \<Rightarrow> 'e"   and m' :: "'c \<Rightarrow> 'e" shows "WHATWHERE.d_equal DA d m m' \<equiv> \<forall>x. DA x \<le> d \<longrightarrow> m x = m' x"
dbl_inc_simps_4: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one (uminus one) = uminus one"
update_put: fixes f :: "('v \<Rightarrow> int option) \<Rightarrow> 'v \<Rightarrow> int option"   and s :: "'v \<Rightarrow> int option"   and m :: "'m" shows "update f (put s m) = put s m"
prime_def: fixes p :: "nat" shows "prime p = (1 < p \<and> (\<forall>m. m dvd p \<longrightarrow> m = 1 \<or> m = p))"
A_dom: fixes x :: "'a"   and y :: "'a" shows "d (A x y) = A x y"
root: shows "takeWhile (\<lambda>x. x = a) w \<in> [a]*"
is_lasso_pre_empty: fixes L :: "'Q lasso" assumes "V0 = {}" shows "\<not> is_lasso_pre L"
interior_def: fixes S :: "'a set" shows "interior S = \<Union> {T. open T \<and> T \<subseteq> S}"
row_carrier: fixes A :: "'b mat"   and i :: "nat" shows "Matrix.row A i \<in> carrier_vec (dim_col A)"
congs_distinct: fixes F\<^sub>1 :: "'a list"   and F\<^sub>2 :: "'a list" assumes "F\<^sub>1 \<cong> F\<^sub>2" shows "distinct F\<^sub>2 = distinct F\<^sub>1"
intersects_x_axis_mk_circline: fixes A :: "complex"   and B :: "complex" assumes "is_real A"   and "A \<noteq> 0 \<or> B \<noteq> 0" shows "intersects_x_axis (mk_circline A B (cnj B) A) = (A = 0 \<or> (Re A)\<^sup>2 < (Re B)\<^sup>2)"
rec2_def: fixes s1 :: "('d, 'd IF1 \<times> 'f, 'd IF2 \<times> 'e) F1 \<Rightarrow> 'f"   and s2 :: "('d, 'd IF1 \<times> 'f, 'd IF2 \<times> 'e) F2 \<Rightarrow> 'e" shows "rec2 s1 s2 = snd \<circ> fold2 <ctor1 \<circ> F1map id fst fst , s1> <ctor2 \<circ> F2map id fst fst , s2>"
token_succeeds_alt_def: fixes x :: "nat" shows "token_succeeds x = (\<forall>\<^sub>\<infinity>n. token_run x n \<in> F)"
lea_asym: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LeA D E F"   and "D E F LeA A B C" shows "A B C CongA D E F"
elims: fixes x :: "'b"   and xa :: "nat"   and y :: "nat set" assumes "configuration x xa = y" obtains q :: "'b" and n :: "nat" where "x = q" and "xa = n" and "y = {x. x \<le> n \<and> token_run x n = q}"
eigen_value_def: fixes A :: "(('c, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and k :: "'c" shows "eigen_value A k = (\<exists>v. eigen_vector A v k)"
ty_of_htype_eq_Integer: fixes hT :: "htype" shows "ty_of_htype hT \<noteq> Integer"
source_in_SPRAY: fixes x :: "'a" assumes "SPRAY x \<noteq> {}" shows "\<exists>P\<in>SPRAY x. x \<in> P"
restrict_gpv_def: fixes \<I> :: "('h, 'i) \<I>" shows "restrict_gpv \<I> \<equiv> corec_gpv (\<lambda>gpv. map_pmf (\<lambda>uu. case uu of None \<Rightarrow> map_option (map_generat id id ((\<circ>) Inl)) None | Some (Pure uua) \<Rightarrow> map_option (map_generat id id ((\<circ>) Inl)) ((Some \<circ> Pure) uua) | Some (IO out c) \<Rightarrow> if out \<in> outs_\<I> \<I> then Some (IO (id out) (\<lambda>input. if input \<in> responses_\<I> \<I> out then Inr (c input) else Inl Fail)) else map_option (map_generat id id ((\<circ>) Inl)) None) (the_gpv gpv))"
1: shows "pfp_wn (local.step' \<top>) c = Some c'"
extremum_uniqueI: fixes a :: "'a list" assumes "prefix a []" shows "a = []"
ccw'_not_coll_6: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "ccw' a b c" shows "det3 c b a \<noteq> 0"
FW''_def: fixes n :: "nat"   and M :: "nat \<times> nat \<Rightarrow> 'b" shows "FW'' n M = FW' M n"
not_less_simps_1: fixes n :: "nat"   and m :: "nat" assumes "\<not> n < m" shows "(n < Suc m) = (n = m)"
iteration_sep4: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "y \<cdot> (0::'a) = (0::'a)"   and "z \<cdot> x = (0::'a)"   and "y \<cdot> x \<le> (x + z) \<cdot> y\<^sup>\<star>" shows "(x + y + z)\<^sup>\<infinity> = x\<^sup>\<infinity> \<cdot> (y + z)\<^sup>\<infinity>"
zoom_def: fixes l :: "'d \<Longrightarrow> 'b"   and m :: "('d, 'c) state" shows "l \<rhd> m = State (\<lambda>b. case run_state m (get\<^bsub>l\<^esub> b) of (c, a) \<Rightarrow> (c, put\<^bsub>l\<^esub> b a))"
assms_1: shows "prim_invar2_ctd Q \<pi>"
diff_add_inverse: fixes n :: "nat"   and m :: "nat" shows "n + m - n = m"