hcju/isabelleps huggingface dataset:
C_merge_all': fixes xss :: "'b list list" shows "int (C_merge_all xss) \<le> int (length (concat xss)) * \<lceil>log 2 (real (length xss))\<rceil>"
AE_measure_pmf_iff: fixes P :: "'d \<Rightarrow> bool"   and M :: "'d pmf" shows "almost_everywhere (measure_pmf M) P = (\<forall>y\<in>set_pmf M. P y)"
set_mset_A: shows "set_mset A = {x \<in> S. 0 < f x}"
tensor_mat_def: fixes dims :: "nat list"   and vars :: "nat set"   and m1 :: "'a mat"   and m2 :: "'a mat" shows "tensor_mat dims vars m1 m2 = mat (state_sig.d dims) (state_sig.d dims) (\<lambda>(i, j). m1 $$ (partial_state.encode1 dims vars i, partial_state.encode1 dims vars j) * m2 $$ (partial_state.encode2 dims vars i, partial_state.encode2 dims vars j))"
size_split_min: fixes t :: "'b tree"   and a :: "'b"   and t' :: "'b tree" assumes "Root_Balanced_Tree.split_min t = (a, t')"   and "t \<noteq> \<langle>\<rangle>" shows "size t' = size t - 1"
cases: fixes aa :: "'b"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "aa \<in> segment f a b" obtains "aa = f a" and "f a \<noteq> b"   | x :: "'b" where "aa = f x" and "x \<in> segment f a b" and "f x \<noteq> b"
varFIXvar: fixes b :: "BExpr"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and \<beta> :: "(Location \<times> Location) set" shows "PhiWhile b \<Phi> (s, t, \<beta>) = ((b, \<Phi>, \<beta>, s, t) \<in> var)"
sdiv_int_0_div: fixes x :: "int" shows "0 sdiv x = 0"
finite_Lim_measure_incseq: fixes A :: "nat \<Rightarrow> 'a set"   and M :: "'a pmf" assumes "range A \<subseteq> measure_pmf.events M"   and "incseq A" shows "(\<lambda>i. measure_pmf.prob M (A i)) \<longlonglongrightarrow> measure_pmf.prob M (\<Union> (range A))"
a_coset_add_zero: fixes M :: "'a set" assumes "M \<subseteq> carrier R" shows "M +> \<zero> = M"
rbt_insert: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "rbt t" shows "rbt (RBT_Set.insert x t)"
rep_eq: fixes x :: "'b fset" shows "sorted_list_of_fset x = sorted_list_of_set (fset x)"
that: fixes e :: "'b" assumes "e \<in> arcs G"   and "tail G e = u"   and "head G e = v" shows "thesis"
Rep_fs_set: fixes x :: "'b fs_set" shows "Rep_fs_set x \<in> {x. finite (supp x)}"
list_less_def: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs < ys) = ((xs, ys) \<in> lenlex {(u, v). u < v})"
TensorDiag_preserves_Can: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t"   and "monoidal_language.Can C u"   and "monoidal_language.Diag C t"   and "monoidal_language.Diag C u" shows "monoidal_language.Can C (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u)"
cases: fixes x :: "J.Arr" obtains "x = J.AA"   | "x = J.BB"   | "x = J.TT"   | "x = J.AT"   | "x = J.BT"   | "x = J.Null"
eucl_Inf: fixes X :: "'a set" shows "Inf X = (\<Sum>i\<in>Basis. (INF x\<in>X. x \<bullet> i) *\<^sub>R i)"
h3: shows "list_constr (a # I) (length qs) \<longrightarrow> sign ! a = (if 0 < poly (qs ! a) x then 1 else - 1)"
co_path_induct: fixes P :: "'b \<times> 'b \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'b) list \<Rightarrow> bool"   and e :: "'b \<times> 'b"   and w :: "'b"   and p :: "('b \<times> 'b) list" assumes "P e w []"   and "\<And>e'. P e w [e']"   and "\<And>e1 e2 es. \<lbrakk>e1 = (fst e, w) \<and> e2 = (w, snd e); P e w es\<rbrakk> \<Longrightarrow> P e w (e1 # e2 # es)"   and "\<And>e1 e2 es. \<lbrakk>\<not> (e1 = (fst e, w) \<and> e2 = (w, snd e)) \<and> e1 = (snd e, w) \<and> e2 = (w, fst e); P e w es\<rbrakk> \<Longrightarrow> P e w (e1 # e2 # es)"   and "\<And>e1 e2 es. \<lbrakk>\<not> (e1 = (fst e, w) \<and> e2 = (w, snd e)); \<not> (e1 = (snd e, w) \<and> e2 = (w, fst e)); P e w (e2 # es)\<rbrakk> \<Longrightarrow> P e w (e1 # e2 # es)" shows "P e w p"
trace_induct2: fixes t :: "'b Trace"   and t' :: "'c Trace"   and P :: "'b Trace \<Rightarrow> 'c Trace \<Rightarrow> bool" assumes "tLength t = tLength t'"   and "\<And>s s'. P (tInit s) (tInit s')"   and "\<And>s s' t t'. \<lbrakk>tLength t = tLength t'; P t t'\<rbrakk> \<Longrightarrow> P (t \<leadsto> s) (t' \<leadsto> s')" shows "P t t'"
da_elim_cases_35: fixes Env :: "env"   and B :: "lname set"   and binop :: "binop"   and e1 :: "expr"   and e2 :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>BinOp binop e1 e2\<rangle>\<guillemotright> A" obtains E1 :: "assigned" and E2 :: "assigned" where "binop = CondAnd" and "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1__" and "Env\<turnstile> B \<union> assigns_if True e1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2__" and "nrm A = B \<union> assigns_if True (BinOp CondAnd e1 e2) \<inter> assigns_if False (BinOp CondAnd e1 e2)" and "brk A = (\<lambda>l. UNIV)"   | E1 :: "assigned" and E2 :: "assigned" where "binop = CondOr" and "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1a__" and "Env\<turnstile> B \<union> assigns_if False e1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2" and "nrm A = B \<union> assigns_if True (BinOp CondOr e1 e2) \<inter> assigns_if False (BinOp CondOr e1 e2)" and "brk A = (\<lambda>l. UNIV)"   | E1 :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1" and "Env\<turnstile> nrm E1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> A" and "binop \<noteq> CondAnd" and "binop \<noteq> CondOr"
dual_downset_set: shows "\<Down> = ord.upset_set (\<lambda>x y. y \<le> x)"
trans_greater_equiv: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "cmp a b = Greater"   and "cmp b c = Equiv" shows "cmp a c = Greater"
simps_9: fixes x31 :: "'b trm"   and x32 :: "'b trm"   and x2 :: "'b" shows "x31 \<cdot> x32 \<noteq> Const x2"
Gramian_determinant_1: fixes k :: "nat" assumes "k \<le> length fs" shows "rat_of_int (gs.Gramian_determinant fs k) = (\<Prod>j<k. \<parallel>gs.gso j\<parallel>\<^sup>2)"
Suc_2: fixes n :: "nat" assumes "\<And>l. n \<le> l \<Longrightarrow> card (level A w (2 * Suc k) l) \<le> card (nodes A) - Suc k" shows "thesis"
inject: fixes x :: "'b list"   and ya :: "'b list" shows "(IArray x = IArray ya) = (x = ya)"
list_all2'': shows "list_all2 (\<lambda>x y. triple_eq_pair x y (\<lambda>tp'. filter_a1_map pos tp' a1_map) (\<lambda>ts' tp'. filter_a2_map I ts' tp' a2_map'')) tl_aux (zip (linearize (tl_queue tss')) [tp - (len - 1)..<tp])"
perp_perp_in: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C A" shows "Tarski_neutral_dimensionless.PerpAt Bet Cong A A B C A"
that_1: assumes "list_all (\<lambda>seg. 0 \<le> det3 (fst seg) (snd seg) p) [x, y, (snd y, fst x)]" shows "thesis"
semiring_norm_50: fixes a :: "'a" shows "(0::'a) + a = a"
abs_interface_def: shows "abs_interface \<equiv> quot_type.abs (\<simeq>) Abs_interface"
SD_efficient_no_pareto_loser: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b pmf" assumes "pref_profile_wf agents alts R"   and "SD_efficient R p"   and "p \<in> lotteries_on alts" shows "set_pmf p \<inter> pareto_losers R = {}"
g_image_filter_impl: shows "map_image_filter m1.\<alpha> m1.invar m2.\<alpha> m2.invar g_image_filter"
mk_finite_char_def: fixes C :: "'b set set" shows "mk_finite_char C = {S. \<forall>S'\<subseteq>S. finite S' \<longrightarrow> S' \<in> C}"
scast_of_nat: fixes x :: "nat" shows "SCAST('b signed \<rightarrow> 'b) (word_of_nat x) = word_of_nat x"
assms: shows "\<exists>k\<le>n. c k \<noteq> (0::'a)"
intro: fixes \<Delta> :: "('c, 'd) network_scheme" assumes "countable \<^bold>E\<^bsub>\<Delta>\<^esub>"   and "source \<Delta> \<noteq> sink \<Delta>"   and "\<And>e. e \<notin> \<^bold>E\<^bsub>\<Delta>\<^esub> \<Longrightarrow> capacity \<Delta> e = 0"   and "\<And>e. capacity \<Delta> e \<noteq> \<top>" shows "countable_network \<Delta>"
PLam_constrains: fixes i :: "nat"   and I :: "nat set"   and F :: "nat \<Rightarrow> ('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program"   and A :: "'c set"   and B :: "'c set" assumes "i \<in> I"   and "\<forall>j. F j \<in> preserves snd" shows "(PLam I F \<in> lift_set i (A \<times> UNIV) co lift_set i (B \<times> UNIV)) = (F i \<in> A \<times> UNIV co B \<times> UNIV)"
parent_child_rel_acyclic: fixes h :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h" shows "acyclic (Shadow_DOM.parent_child_rel h)"
coeff_in_carrier: fixes p :: "'a list"   and i :: "nat" assumes "set p \<subseteq> carrier R" shows "local.coeff p i \<in> carrier R"
elem_out_ideal_nonzero: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and x :: "'c" assumes "Ring R"   and "ideal R I"   and "x \<in> carrier R"   and "x \<notin> I" shows "x \<noteq> \<zero>\<^bsub>R\<^esub>"
norm_inverse_le_norm: fixes r :: "real"   and x :: "'a" assumes "r \<le> norm x"   and "0 < r" shows "norm (inverse x) \<le> inverse r"
decomps\<^sub>e\<^sub>s\<^sub>t_pts_symbolic_c: fixes D :: "('fun, 'var) extstrand_step list"   and A :: "('fun, 'var) extstrand_step list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list set" assumes "D \<in> decomps\<^sub>e\<^sub>s\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t A) (assignment_rhs\<^sub>e\<^sub>s\<^sub>t A) \<I>" shows "(S, A) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (S, A @ D)"
AsBasicSkip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and Q :: "'e set"   and A :: "'e set"   and p :: "'e set"   and f :: "'e \<Rightarrow> 'e" shows "assertionsR \<Gamma> \<Theta> Q A (AnnExpr p) (Basic f) Q"
letter_eq: fixes a :: "'a1"   and b :: "'a2" assumes "letter_eq a b" shows "(a \<in> set alphabet1) = (b \<in> set alphabet2)"
abs_reach_def: fixes \<Phi> :: "'b ltl_prop_equiv_quotient" shows "af_abs_opt.abs_reach \<Phi> = {foldl af_abs_opt.f_abs \<Phi> w |w. True}"
sum_conjugate: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite X" shows "conjugate (sum f X) = (\<Sum>x\<in>X. conjugate (f x))"
strongest_intuitionistic: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "\<nexists>Q. (\<forall>h. Q h \<longrightarrow> sep_algebra.sep_conj plus sep_disj P (\<lambda>s. True) h) \<and> sep_algebra.intuitionistic plus sep_disj Q \<and> Q \<noteq> sep_algebra.sep_conj plus sep_disj P (\<lambda>s. True) \<and> (\<forall>h. P h \<longrightarrow> Q h)"
WCFG_EntryD: fixes prog :: "cmd"   and et :: "(char list \<Rightarrow> val option) edge_kind"   and n' :: "w_node" assumes "prog \<turnstile> (_Entry_) -et\<rightarrow> n'" shows "n' = (_Exit_) \<and> et = (\<lambda>s. False)\<^sub>\<surd> \<or> n' = (_ 0 _) \<and> et = (\<lambda>s. True)\<^sub>\<surd>"
Ej_def: fixes a :: "nat list"   and b :: "nat list"   and j :: "nat"   and x :: "nat list" shows "hlde_ops.Ej a b j x = {eij a b i j - 1 |i. i < length x \<and> dij a b i j \<le> x ! i}"
l9_39: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and P :: "'a"   and R :: "'a"   and D :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TSP Bet A B C P R"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D"   and "Tarski_neutral_dimensionless.Out Bet D P Q" shows "Tarski_neutral_dimensionless.TSP Bet A B C Q R"
lift_preserves_sub: fixes F :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program"   and i :: "nat"   and v :: "'c \<Rightarrow> 'e"   and j :: "nat" assumes "F \<in> preserves snd" shows "(lift i F \<in> preserves (v \<circ> sub j \<circ> fst)) = (if i = j then F \<in> preserves (v \<circ> fst) else True)"
simps_1: fixes a :: "'a"   and b :: "'a" shows "a r-0\<rightarrow> b = b"
fldclass_simp: fixes n :: "vname"   and c :: "qtname" shows "fldclass (n, c) = c"
is_unit_pCons_ex_iff: fixes p :: "'a poly" shows "is_unit p = (\<exists>a. a \<noteq> (0::'a) \<and> p = [:a:])"
assms_3: shows "continuous_on D (\<lambda>x. V' x (f x))"
cp_thm: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and d :: "int" assumes "iszlfm p (a # bs)"   and "d_\<beta> p 1"   and "d_\<delta> p d"   and "0 < d" shows "(\<exists>x. Ifm (real_of_int x # bs) p) = (\<exists>j\<in>{1..d}. Ifm (real_of_int j # bs) (minusinf p) \<or> (\<exists>b\<in>set (\<beta> p). Ifm ((Inum (a # bs) b + real_of_int j) # bs) p))"
assms: fixes h :: "heap \<times> nat set" assumes "h \<Turnstile> \<Gamma>" shows "hn_refine \<Gamma> c \<Gamma>' R a"
preList_Seq: fixes upds :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set) list"   and C1 :: "acom"   and C2 :: "acom"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "preList upds (C1;; C2) l s = preList (map (\<lambda>(x, e, v). (x, preT C2 e, fune C2 v)) upds) C1 l s"
not_principal_aux: fixes c :: "('a, 'b) sequent"   and T :: "'b"   and Ts :: "('a, 'b) form list"   and M :: "'b"   and \<Gamma> :: "('a, 'b) form multiset"   and N :: "'b"   and \<Delta> :: "('a, 'b) form multiset"   and F :: "'a"   and Fs :: "('a, 'b) form list" assumes "ModalSequents.mset c = \<LM> Modal T Ts  \<RM>"   and "M \<cdot> \<Gamma> + succ c = N \<cdot> \<Delta> \<oplus> Compound F Fs" shows "False"
gState_progress_rel_gState_invI1: fixes g :: "'b gState_scheme"   and g' :: "'b gState_scheme"   and prog :: "program" assumes "(g, g') \<in> gState_progress_rel prog" shows "gState_inv prog g"
in_homI: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d"   and a :: "'d"   and b :: "'d" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.dom C f = a"   and "partial_magma.cod C f = b" shows "partial_magma.in_hom C f a b"
map_pmf_const: fixes c :: "'b"   and M :: "'c pmf" shows "map_pmf (\<lambda>_. c) M = return_pmf c"
ltln_WeakUntil_to_Release: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n \<phi> W\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<psi> R\<^sub>n (\<phi> or\<^sub>n \<psi>)"
simps_2: fixes \<eta> :: "nat"   and vkey :: "'vkey"   and sigkey :: "'sigkey"   and log :: "('message \<times> 'signature) list"   and uv :: "unit" shows "vkey_oracle \<eta> (Some (vkey, sigkey, log)) uv = return_spmf (vkey, Some (vkey, sigkey, log))"
trancl_id: fixes r :: "('c \<times> 'c) set" assumes "trans r" shows "r\<^sup>+ = r"
pairwise_finiteI: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite {b. \<exists>a. P a b}"   and "finite {a. \<exists>b. P a b}" shows "finite {(a, b). P a b}"
T_def: fixes P :: "'a process" shows "T P = TRACES (Rep_process P)"
finitesum_sprod: fixes f :: "'f \<Rightarrow> 'a set"   and I :: "'f set"   and r :: "'b"   and g :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> I \<rightarrow> {X. submodule R M X}"   and "I \<noteq> {}"   and "r \<in> carrier R"   and "g \<in> {j. j \<le> n} \<rightarrow> finitesum_base M I f" shows "r \<cdot>\<^sub>s \<Sigma>\<^sub>e M g n = \<Sigma>\<^sub>e M (\<lambda>x. r \<cdot>\<^sub>s g x) n"
cBex_triv_one_point2: fixes A :: "'f cset"   and a :: "'f" shows "cBex A ((=) a) = cin a A"
create_character_data_impl: shows "create_character_data = l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_create_character_data set_val get_disconnected_nodes set_disconnected_nodes"
limit_g: shows "(g \<longlongrightarrow> C) at_top"
strand_subst_comp: fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and S :: "('b, 'a) strand_step list"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "range_vars \<delta> \<inter> bvars\<^sub>s\<^sub>t S = {}" shows "S \<cdot>\<^sub>s\<^sub>t \<delta> \<circ>\<^sub>s \<theta> = (S \<cdot>\<^sub>s\<^sub>t \<delta>) \<cdot>\<^sub>s\<^sub>t \<theta>"
characterize_consistent_signs_at_roots_copr_def: fixes p :: "real poly"   and qss :: "real poly list" shows "characterize_consistent_signs_at_roots_copr p qss = remdups (map (consistent_sign_vec_copr qss) (characterize_root_list_p p))"
fundamental_property: fixes prog :: "cmd"   and n :: "w_node"   and c :: "cmd"   and s :: "char list \<Rightarrow> val option"   and c' :: "cmd"   and s' :: "char list \<Rightarrow> val option" assumes "labels_nodes prog n c"   and "\<langle>c,s\<rangle> \<rightarrow>* \<langle>c',s'\<rangle>" shows "\<exists>n' as. While_CFG.path prog n as n' \<and> transfers (While_CFG.kinds as) s = s' \<and> preds (While_CFG.kinds as) s \<and> labels_nodes prog n' c'"
ent_disjE: fixes A :: "assn"   and C :: "assn"   and B :: "assn" assumes "A \<Longrightarrow>\<^sub>A C"   and "B \<Longrightarrow>\<^sub>A C" shows "A \<or>\<^sub>A B \<Longrightarrow>\<^sub>A C"
surjective: fixes r :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme" shows "r = \<lparr>ISignature = iS\<^bsub>r\<^esub>, ICategory = iC\<^bsub>r\<^esub>, ITypes = ITypes r, IFunctions = IFunctions r, \<dots> = Interpretation.more r\<rparr>"
simps: fixes \<delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>0 :: "'a"   and w :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "run\<^sub>t \<delta> q\<^sub>0 w i = (run \<delta> q\<^sub>0 w i, w i, run \<delta> q\<^sub>0 w (Suc i))"
empty_Collect_eq: fixes P :: "'b \<Rightarrow> bool" shows "({} = Collect (P::'b \<Rightarrow> bool)) = (\<forall>x::'b. \<not> P x)"
mod_int_unique: fixes k :: "int"   and l :: "int"   and q :: "int"   and r :: "int" assumes "eucl_rel_int k l (q, r)" shows "k mod l = r"
VDMIff: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and d :: "OBJ"   and B :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and b :: "BExpr" assumes "G \<rhd> c : A"   and "G \<rhd> d : B" shows "G \<rhd> Iff b c d : \<lambda>s t. (evalB b (fst s) \<longrightarrow> A s t) \<and> (\<not> evalB b (fst s) \<longrightarrow> B s t)"
coeff_f_0_code: fixes f :: "'a poly" shows "poly.coeff f 0 = (case coeffs f of [] \<Rightarrow> 0::'a | x # xa \<Rightarrow> x)"
HC_intros_13: fixes F :: "'b formula" shows "AX10 \<turnstile>\<^sub>H (\<^bold>\<not> (F::'b formula) \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> F"
modes_respect_priv: shows "modes_respect_priv (map snd cms\<^sub>C)"
atLeastLessThan_less_eq: fixes m :: "nat"   and n :: "nat" shows "({0..<m} \<subseteq> {0..<n}) = (m \<le> n)"
wf_trm_subst_range_iff: fixes \<theta> :: "'b \<Rightarrow> ('fun, 'b) Term.term" shows "(\<forall>x. wf\<^sub>t\<^sub>r\<^sub>m (\<theta> x)) = wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"
dvd_prodI: fixes A :: "'a set"   and x :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "x \<in> A" shows "f x dvd prod f A"
monom_Suc_1: fixes a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "a \<in> carrier Zp" shows "monom (UP Zp) a (Suc n) = monom (UP Zp) \<one> 1 \<otimes>\<^bsub>UP Zp\<^esub> monom (UP Zp) a n"
drest_smap_drest_smap_drest: fixes vs :: "'d set"   and x :: "('d, 'e) fmap"   and s :: "('d, 'e) fmap" shows "(fmrestrict_set vs x \<subseteq>\<^sub>f s) = (fmrestrict_set vs x \<subseteq>\<^sub>f fmrestrict_set vs s)"
image_mset_empty: fixes f :: "'b \<Rightarrow> 'a" shows "f `# {#} = {#}"
herbrand_evalt_2: fixes ts :: "'a FOL_Fitting.term list"   and e :: "nat \<Rightarrow> 'a hterm" assumes "closedts 0 ts" shows "terms_of_hterms (evalts e HApp ts) = ts"
simps_1: fixes p :: "('a, 'b) form" shows "sub_consts [] p = p"
le_err_refl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e :: "'b err" assumes "\<forall>x. x \<sqsubseteq>\<^bsub>r\<^esub> x" shows "e \<sqsubseteq>\<^bsub>Err.le r\<^esub> e"
DiffI: fixes c :: "'b::type"   and A :: "'b::type set"   and B :: "'b::type set" assumes "(c::'b::type) \<in> (A::'b::type set)"   and "(c::'b::type) \<notin> (B::'b::type set)" shows "(c::'b::type) \<in> (A::'b::type set) - (B::'b::type set)"
AND_upper2: fixes y :: "int"   and x :: "int" assumes "0 \<le> y" shows "x AND y \<le> y"
ASuc: fixes n :: "nat" shows "A (Suc n) = (if \<exists>U\<subseteq>A n. closed U \<and> \<phi> U \<and> U \<inter> B n = {} then SOME U. U \<subseteq> A n \<and> closed U \<and> \<phi> U \<and> U \<inter> B n = {} else A n)"
mult_neg_pos: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a zero"   and "less zero b" shows "less (times a b) zero"
reachable_def: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" shows "reachable s \<equiv> \<exists>e s'. execution e s' s"
sort_of_Rep: fixes a :: "'a" shows "sort_of (Rep a) \<in> range sort_fun"
ex_un_null: shows "\<exists>!n. \<forall>f. Left.comp n f = n \<and> Left.comp f n = n"
mul_ext_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and xs :: "'b list"   and ys :: "'b list" shows "mul_ext f xs ys \<equiv> let s = {(x, y). fst (f x y)}; ns = {(x, y). snd (f x y)} in ((mset xs, mset ys) \<in> s_mul_ext ns s, (mset xs, mset ys) \<in> ns_mul_ext ns s)"
update_convs_3: fixes document_element' :: "(_) element_ptr option \<Rightarrow> (_) element_ptr option"   and nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p" shows "document_element_update document_element' \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element' document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr>"
assms_3: shows "semiring_of_sets \<Omega> (G ` I)"
measure_pmf_UNIV: fixes p :: "'c pmf" shows "measure_pmf.prob p UNIV = 1"
local_ring_diff: fixes mx :: "'a set" assumes "\<not> zeroring R"   and "ideal R mx"   and "mx \<noteq> carrier R"   and "\<forall>a\<in>carrier R - mx. Unit R a" shows "local_ring R \<and> maximal_ideal R mx"
less_one: fixes n :: "nat" shows "(n < 1) = (n = 0)"
fMin_le: fixes x :: "'a"   and A :: "'a fset" assumes "x |\<in>| A" shows "fMin A \<le> x"
ncoplanar_perm_7: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Coplanar A B C D" shows "\<not> Coplanar B A D C"
insert_index_1: fixes i' :: "nat"   and i :: "nat" assumes "i' < i" shows "insert_index i i' = i'"