hcju/isabelleps huggingface dataset:
eventually_ball_finite: fixes A :: "'b set"   and P :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and net :: "'c filter" assumes "finite A"   and "\<forall>y\<in>A. \<forall>\<^sub>F x in net. P x y" shows "\<forall>\<^sub>F x in net. \<forall>y\<in>A. P x y"
CptnModCondF: fixes P1 :: "'b com"   and s :: "'b"   and ys :: "('b com option \<times> 'b) list"   and b :: "'b set"   and P0 :: "'b com" assumes "(Some P1, s) # ys \<in> cptn_mod"   and "s \<notin> b" shows "(Some (Cond b P0 P1), s) # (Some P1, s) # ys \<in> cptn_mod"
size_add_hi: fixes t :: "'b tree"   and m :: "nat"   and x :: "'b" assumes "braun t"   and "m = size t" shows "size (update1 (Suc m) x t) = size t + 1"
integral_exp_minus_squared_real: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "((\<lambda>t. exp (- t\<^sup>2)) has_integral sqrt pi / 2 * (erf b - erf a)) {a..b}"
nLt_le_conv: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" assumes "comparator comp" shows "(comp x y \<noteq> Lt) = le_of_comp comp y x"
Lxx_def: fixes x :: "nat"   and y :: "nat" shows "Lxx x y = lang (L_lasthasxx x y)"
seg_def: fixes i :: "nat" shows "seg i \<equiv> {S ! j |j. seg_start i \<le> j \<and> j < seg_end i}"
spec\<^sub>7_def: fixes p :: "prog" shows "spec\<^sub>7 p \<equiv> p = p\<^sub>0"
int_distrib_2: fixes w :: "int"   and z1 :: "int"   and z2 :: "int" shows "w * (z1 + z2) = w * z1 + w * z2"
sound_reduction: fixes \<Gamma>\<^sub>1 :: "'c constr list"   and n\<^sub>1 :: "nat"   and \<Psi>\<^sub>1 :: "'c TESL_atomic list"   and \<Phi>\<^sub>1 :: "'c TESL_atomic list"   and \<Gamma>\<^sub>2 :: "'c constr list"   and n\<^sub>2 :: "nat"   and \<Psi>\<^sub>2 :: "'c TESL_atomic list"   and \<Phi>\<^sub>2 :: "'c TESL_atomic list" assumes "\<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow> \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2" shows "\<lbrakk>\<lbrakk> \<Gamma>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk>\<lbrakk> \<Psi>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^sub>2\<^esup> \<inter> \<lbrakk>\<lbrakk> \<Phi>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> Suc n\<^sub>2\<^esup> \<subseteq> \<lbrakk>\<lbrakk> \<Gamma>\<^sub>1 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk>\<lbrakk> \<Psi>\<^sub>1 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^sub>1\<^esup> \<inter> \<lbrakk>\<lbrakk> \<Phi>\<^sub>1 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> Suc n\<^sub>1\<^esup>"
InvariantReasonClausesAfterBackjump: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and p :: "(Literal \<times> bool) list"   and bClause :: "Literal list"   and bLiteral :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantReasonClauses F M"   and "InvariantUniq M"   and "isPrefix p M"   and "isUnitClause bClause bLiteral (elements p)"   and "formulaEntailsClause F bClause"   and "M' = p @ [(bLiteral, False)]" shows "InvariantReasonClauses F M'"
CSP4_imp_right_Skip: fixes P :: "('c, 'd) action" assumes "action.relation_of P is CSP4 healthy" shows "(P `;` Skip) = P"
lex_ccw_left: fixes x :: "real \<times> real"   and y :: "real \<times> real"   and r :: "real" assumes "Counterclockwise_2D_Arbitrary.lex x y"   and "0 < r" shows "ccw y (y + (0, r)) x"
finprod_singleton: fixes i :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "i \<in> A"   and "finite A"   and "f \<in> A \<rightarrow> carrier R" shows "(\<Oplus>j\<in>A. if i = j then f j else \<zero>) = f i"
that: fixes r :: "'c list" assumes "path M (w || r) q"   and "length w = length r" shows "thesis"
is_cond_exp: fixes X :: "'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'c"   and N :: "'c measure" assumes "integrable M X"   and "disc_fct Y"   and "point_measurable M (space N) Y" shows "\<forall>A\<in>sets N. LINT w|M. X w * indicat_real (Y -` A \<inter> space M) w = LINT w|M. expl_cond_expect M Y X w * indicat_real (Y -` A \<inter> space M) w"
of_nat_Suc: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and m :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (Suc m) = plus one (semiring_1.of_nat one plus zero m)"
BK3_imp_Gets: fixes A :: "agent"   and B :: "agent"   and Ticket :: "msg"   and K :: "nat"   and Ta :: "nat"   and evs :: "event list" assumes "Says A B \<lbrace>Ticket, Crypt K \<lbrace>Agent A, Number Ta\<rbrace>\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "evs \<in> bankerb_gets" shows "\<exists>Tk. Gets A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) \<in> set evs"
mem_simps_8: fixes b :: "'b"   and B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "(b \<in> \<Union> (B ` A)) = (\<exists>x\<in>A. b \<in> B x)"
component_of_imp_component: fixes F :: "'c program"   and H :: "'c program" assumes "F component_of H" shows "F \<le> H"
state_col_ket_vec: fixes v :: "complex Matrix.mat" assumes "state 1 v" shows "state 1 |Matrix.col v 0\<rangle>"
append_assoc: fixes xs :: "['b]"   and ys :: "['b]"   and zs :: "['b]" shows "(xs ++ ys) ++ zs = xs ++ ys ++ zs"
card_product_dependent: fixes S :: "'c set"   and T :: "'c \<Rightarrow> 'd set" assumes "finite S"   and "\<forall>x\<in>S. finite (T x)" shows "card {(x, y). x \<in> S \<and> y \<in> T x} = (\<Sum>x\<in>S. card (T x))"
strict_trans1: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "prefix a b"   and "strict_prefix b c" shows "strict_prefix a c"
vec_simps_22: fixes ad :: "'a"   and bd :: "'a"   and cc :: "'a" shows "ad - bd + cc = ad + cc - bd"
list_rev_decomp: fixes l :: "'f list" assumes "l \<noteq> []" shows "\<exists>ll e. l = ll @ [e]"
entailment_def: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" shows "\<Gamma> \<TTurnstile> F \<equiv> \<forall>\<A>. (\<forall>G\<in>\<Gamma>. \<A> \<Turnstile> G) \<longrightarrow> \<A> \<Turnstile> F"
valuation_equivTr2: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "valuation K v'"   and "carrier (Vr K v) \<subseteq> carrier (Vr K v')"   and "vp K v = carrier (Vr K v) \<inter> vp K v'" shows "carrier (Vr K v') \<subseteq> carrier (Vr K v)"
NDVD_l: fixes i :: "int"   and a :: "MIR.num" shows "isrlfm (rsplit (NDVD i) a)"
smult_zero: fixes r :: "('f, 'g) aezfun" assumes "r \<in> ActingGroup.group_ring" shows "r \<cdot> (0::'v) = (0::'v)"
simps_10: fixes opt :: "atom fm \<Rightarrow> atom fm"   and step :: "nat \<Rightarrow> nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm"   and \<phi> :: "atom fm" shows "QE_dnf opt step (AllN 0 \<phi>) = QE_dnf opt step \<phi>"
suminf_sum: fixes I :: "'b set"   and f :: "'b \<Rightarrow> nat \<Rightarrow> 'c" assumes "\<And>i. i \<in> I \<Longrightarrow> summable (f i)" shows "(\<Sum>n. \<Sum>i\<in>I. f i n) = (\<Sum>i\<in>I. \<Sum>n. f i n)"
symmetric_monoidal_category_axioms: shows "symmetric_monoidal_category (\<cdot>) T \<alpha> \<iota> \<sigma>"
distinct_2: fixes x21 :: "secprop set"   and x22 :: "key"   and x23 :: "payload"   and x11 :: "secprop set"   and x12 :: "agent"   and x13 :: "agent"   and x14 :: "payload" shows "DynCh x21 x22 x23 \<noteq> StatCh x11 x12 x13 x14"
o_apply: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "(f \<circ> g) x = f (g x)"
sets_stream_filteration_le_stream_space: fixes M :: "'b measure"   and n :: "enat" shows "sets (stream_filtration M n) \<subseteq> sets (stream_space M)"
subspace_diff_fun_space: shows "subspace sub.diff_fun_space"
lm082: fixes X :: "'c set"   and Y :: "'c set"   and P :: "('c \<times> 'd) set" assumes "X \<inter> Y = {}" shows "(P outside Y) `` X = P `` X"
rep_inj: fixes x :: "'a"   and y :: "'a" assumes "rep_nat x = rep_nat y" shows "x = y"
Cd_greatest: fixes d :: "'b"   and X :: "'a set" shows "ManiPref'.Cd d X = greatest (TruePref.Pd_shuffle_to_top ds' (ManiPref.cop ds) d) (X \<inter> Field (TruePref.Pd_shuffle_to_top ds' (ManiPref.cop ds) d))"
Cons_1: fixes vs :: "'a list" assumes "v \<in> set vs" shows "v \<notin> f ` set (remdups_wrt_rev f xs vs)"
ordinal_div_less: fixes x :: "ordinal"   and y :: "ordinal"   and z :: "ordinal" assumes "x < y * z" shows "x div y < z"
add_vec_span: fixes us :: "rat vec list"   and j :: "nat"   and i :: "nat"   and c :: "rat" assumes "set us \<subseteq> Rn"   and "j < length us"   and "i < length us"   and "i \<noteq> j" shows "gs.span (set us) = gs.span (set (us[i := us ! i + c \<cdot>\<^sub>v us ! j]))"
assms_1: shows "arb_pf = (\<lambda>x n w. 0)(x := a, y := b)"
zassenhaus_reconstruction_irreducible\<^sub>d: fixes p :: "int"   and f :: "int poly"   and n :: "nat"   and hs :: "int poly list"   and fs :: "int poly list" assumes "prime p"   and "comm_monoid_mult_class.coprime (lead_coeff f) p"   and "poly_mod.square_free_m p f"   and "0 < degree f"   and "berlekamp_hensel p n f = hs"   and "2 * \<bar>lead_coeff f\<bar> * factor_bound f (degree_bound hs) < p ^ n"   and "zassenhaus_reconstruction hs p n f = fs" shows "f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible\<^sub>d fi)"
integral_measure_spmf: fixes p :: "'c spmf"   and f :: "'c \<Rightarrow> real" assumes "integrable (measure_spmf p) f" shows "integral\<^sup>L (measure_spmf p) f = LINT x|count_space UNIV. spmf p x * f x"
preserves_ide: fixes a :: "'a" assumes "C.ide a" shows "D.ide (F a)"
not_Leaf_if_not_single: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree" assumes "\<not> single l r" shows "l \<noteq> \<langle>\<rangle>"
theorem24xxiv_pp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x * y) \<squnion> - - (x * z) = - (x * (y \<sqinter> - z)) \<squnion> - - (x * z)"
of_nat_simps_2: shows "of_nat 1 = (1::'d)"
det_state_cover_finite: fixes M :: "('d, 'e, 'f) FSM"   and V :: "'d list set" assumes "is_det_state_cover M V"   and "finite (nodes M)" shows "finite V"
mult_norm_matrix_sgn_eq: fixes A :: "(('b, 'd) vec, 'c) vec"   and x :: "('b, 'd) vec" shows "(\<parallel>A *v sgn x\<parallel>) * (\<parallel>x\<parallel>) = \<parallel>A *v x\<parallel>"
simply_connected_sphere_eq: fixes a :: "'a"   and r :: "real" shows "simply_connected (sphere a r) = (3 \<le> DIM('a) \<or> r \<le> 0)"
rel_list_all2_Cons: fixes S :: "('a \<times> 'b) set"   and x :: "'a"   and xs :: "'a list"   and y :: "'b"   and ys :: "'b list" shows "list_all2 (fun_of S) (x # xs) (y # ys) = ((x, y) \<in> S \<and> list_all2 (fun_of S) xs ys)"
DF_COPY: shows "DF (range left \<union> range right) \<sqsubseteq>\<^sub>F\<^sub>D COPY"
iT_Plus_neg_Int: fixes A :: "nat set"   and B :: "nat set"   and k :: "nat" shows "A \<inter> B \<oplus>- k = (A \<oplus>- k) \<inter> (B \<oplus>- k)"
nrows: fixes A :: "'a matrix"   and m :: "nat"   and n :: "nat" assumes "nrows A \<le> m" shows "Rep_matrix A m n = (0::'a)"
var_swap_ref1: fixes a :: "'b"   and b :: "'b" shows "(''z'' ::= (\<lambda>s. s ''x'')) ; rel_R \<lceil>\<lambda>s. s ''z'' = a \<and> s ''y'' = b\<rceil> \<lceil>\<lambda>s. s ''x'' = b \<and> s ''y'' = a\<rceil> \<subseteq> rel_R \<lceil>\<lambda>s. s ''x'' = a \<and> s ''y'' = b\<rceil> \<lceil>\<lambda>s. s ''x'' = b \<and> s ''y'' = a\<rceil>"
inv_solve_left': fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group.group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(inv\<^bsub>G\<^esub> b \<otimes>\<^bsub>G\<^esub> c = a) = (c = b \<otimes>\<^bsub>G\<^esub> a)"
RedNewFail: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and C :: "char list"   and l :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "new_Addr h = None" shows "P \<turnstile> \<langle>new C,(h, l)\<rangle> \<rightarrow> \<langle>THROW OutOfMemory,(h, l)\<rangle>"
assms: fixes e :: "real" assumes "0 < e" shows "AE x in M. F x \<le> G x + e"
simps: fixes n :: "nat" shows "\<MM>.\<S>\<^sub>R n = {q. \<G> \<Turnstile>\<^sub>P Rep q} \<union> {q. \<exists>j\<ge>the \<MM>.smallest_accepting_rank\<^sub>R. \<MM>.state_rank q n = Some j}"
src_def: fixes \<mu> :: "'c" shows "C'.src \<mu> = (if C'.arr \<mu> then src\<^sub>C \<mu> else C'.null)"
robbins: fixes x :: "'a"   and y :: "'a" shows "- (- (x \<squnion> y) \<squnion> - (x \<squnion> - y)) = x"
fv_subst_subset: fixes x :: "'a"   and t :: "('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "x \<in> fv t" shows "fv (\<theta> x) \<subseteq> fv (t \<cdot> \<theta>)"
R1: shows "X Midpoint P P1 \<and> Col A B X"
OP_def: fixes x :: "'c" shows "OP x \<equiv> x"
int_of_integer_0_iff: fixes n :: "integer" shows "(int_of_integer n = 0) = (n = 0)"
intro: fixes r1 :: "('d, 'a, 'b, 'c, 'e, 'f) semantics"   and r2 :: "('d, 'a, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'a, 'b, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'a, 'g, 'h, 'e, 'f) \<tau>moves" assumes "\<And>t. \<tau>inv (r1 t) (r2 t) (bisim t) (ta_bisim bisim) \<tau>move1 \<tau>move2" shows "FWdelay_bisimulation_lift_axioms r1 r2 bisim \<tau>move1 \<tau>move2"
nn_integral_fst: fixes f :: "'d \<times> 'e \<Rightarrow> ennreal"   and M1 :: "'d measure"   and M :: "'e pmf" assumes "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M measure_pmf M)" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>measure_pmf M \<partial>M1 = integral\<^sup>N (M1 \<Otimes>\<^sub>M measure_pmf M) f"
toEnum_def: fixes n :: "nat" shows "toEnum n \<equiv> if n < length enum_class.enum then enum_class.enum ! n else the None"
lspasl_d: fixes Gamma :: "bool"   and h2 :: "'a"   and h1 :: "'a"   and Delta :: "bool" assumes "Gamma \<and> (0::'a,0::'a\<triangleright>h2) \<and> h1 = (0::'a) \<longrightarrow> Delta" shows "Gamma \<and> (h1,h1\<triangleright>h2) \<longrightarrow> Delta"
simps_10: fixes v :: "linear_poly"   and va :: "linear_poly" shows "nonstrict_constr (GTPP v va) = False"
simps_9: fixes V :: "'a set"   and uy :: "('b, 'a) Term.term"   and uz :: "('b, 'a) Term.term"   and S :: "('b, 'a) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (\<langle>uy in uz\<rangle> # S) = wf'\<^sub>s\<^sub>s\<^sub>t V S"
elims_2: fixes x :: "'a option"   and xa :: "'a option" assumes "x \<le> xa" obtains x :: "'a" and y :: "'a" where "xa__ = Some x" and "xa = Some y__" and "x \<le> y__"   | y :: "'a option" where "xa__ = None" and "xa = y"
notin_fset: fixes x :: "'b"   and S :: "'b fset" shows "(x |\<notin>| S) = (x \<notin> fset S)"
x_less_i_eq_x_i: fixes i :: "nat" assumes "i < n" shows "x mod base ^ n div base ^ i mod base = x div base ^ i mod base"
l_remove_child_axioms: shows "l_remove_child DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs remove_child remove_child_locs get_owner_document get_child_nodes get_disconnected_nodes"
k: shows "\<forall>n\<ge>k. \<bar>X n - X k\<bar> < s"
eq2: shows "lt (monom_mult c s p) = s \<oplus> lt p"
asym: fixes x :: "'a"   and y :: "'a" assumes "x \<bowtie> y"   and "y \<bowtie> x"   and "x \<in> A"   and "y \<in> A" shows "False"
le_fin_concat_end: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w :: "'item list" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2" shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2 @ w"
abs_sum_abs: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "\<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar> = (\<Sum>a\<in>A. \<bar>f a\<bar>)"
bind_gpv_try_gpv_map_Some: fixes gpv :: "('k, 'i, 'j) gpv"   and f :: "'k \<Rightarrow> ('h, 'i, 'j) gpv" shows "TRY map_gpv Some id gpv ELSE Done None \<bind> (\<lambda>x. case x of None \<Rightarrow> Fail | Some y \<Rightarrow> f y) = gpv \<bind> f"
mod_cut_greater_aux_le_x: fixes r :: "nat"   and t :: "nat"   and x :: "nat"   and m :: "nat" assumes "r \<le> t"   and "t < x"   and "x mod m = r mod m" shows "t + m - (t - r) mod m \<le> x"
direct_decompD_3: fixes A :: "'a set"   and ss :: "'a set list" assumes "direct_decomp A ss" shows "sum_list ` listset ss = A"
snd_convol': fixes f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a"   and x :: "'c" shows "snd (BNF_Def.convol f g x) = g x"
count_greater_eq_one_iff: fixes M :: "'k multiset"   and x :: "'k" shows "(1 \<le> count M x) = (x \<in># M)"
HSem_ignores_fresh_restr'': fixes \<Gamma> :: "('b \<times> 'a) list"   and S :: "'b set"   and \<rho> :: "'b \<Rightarrow> 'c" assumes "fv \<Gamma> \<subseteq> S" shows "(\<lbrace> \<Gamma> \<rbrace>\<rho>) f|` S = \<lbrace> \<Gamma> \<rbrace>\<rho> f|` S"
tarski_first3_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "tarski_first3 C \<equiv> (\<forall>a b. C a b b a) \<and> (\<forall>a b p q r s. C a b p q \<and> C a b r s \<longrightarrow> C p q r s) \<and> (\<forall>a b c. C a b c c \<longrightarrow> a = b)"
order_iff_strict: fixes b :: "'g"   and a :: "'g" shows "(b \<le> a) = (b < a \<or> a = b)"
separate_othogonal: fixes P :: "'a \<times> 'b \<Rightarrow> bool"   and Q :: "'a \<times> 'b \<Rightarrow> bool"   and ps :: "'a"   and n :: "'b" assumes "\<And>ps n. P (ps, n) \<Longrightarrow> n = (0::'b)"   and "\<And>ps n. Q (ps, n) \<Longrightarrow> ps = (0::'a)" shows "(P \<and>* Q) (ps, n) = (P (ps, 0::'b) \<and> Q (0::'a, n))"
guarded_def: fixes sys :: "'c \<Rightarrow> ('d, 'e) process_term" shows "guarded sys \<equiv> \<forall>X Y. sys X \<noteq> VAR Y"
\<Psi>_simp: fixes yx :: "'d \<times> 'c" assumes "\<Phi>\<Psi>.DopxC.ide yx" shows "\<phi>\<psi>.\<Psi> yx = \<phi>\<psi>.\<Psi>o yx"
add_smult_distrib_left_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat"   and k :: "'c" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "k \<cdot>\<^sub>m (A + B) = k \<cdot>\<^sub>m A + k \<cdot>\<^sub>m B"
Rep_token_run_af: fixes \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set"   and x :: "nat"   and n :: "nat" shows "Rep (semi_mojmir_def.token_run \<up>af\<^sub>G (Abs \<phi>) w x n) \<equiv>\<^sub>P af\<^sub>G \<phi> (w [x \<rightarrow> n])"
fin_digraphI_app_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "fin_digraph (app_iso hom G)"
div_ge: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k \<le> m div n" shows "n * k \<le> m"
ennreal_spmf_lub_spmf: fixes x :: "'a" assumes "Y \<noteq> {}" shows "ennreal (spmf local.lub_spmf x) = (\<Squnion>p\<in>Y. ennreal (spmf p x))"
refl_NS: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" assumes "pre_order_pair S NS" shows "refl NS"
leneq: shows "llength (ltake (llength us) xs) = llength (ltake (llength us) ys)"
31: shows "bprv (all yy' (imp (NN \<langle>\<phi>\<rangle> \<langle>neg \<phi>\<rangle>) (imp (NN \<langle>\<phi>\<rangle> (Var yy')) (eql \<langle>neg \<phi>\<rangle> (Var yy')))))"
algebra_Pow: fixes sp :: "'b set" shows "algebra sp (Pow sp)"
is_extensional: fixes f :: "'d" assumes "\<not> D.arr f" shows "src\<^sub>D f = D.null"
hyps: assumes "\<forall>x\<in>set xs. dim_vec x = n"   and "i < n" shows "M.sumlist xs $ i = (\<Sum>j = 0..<length xs. xs ! j $ i)"
slp_get_proc: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^bsub>sl\<^esub>* n'" shows "get_proc n = get_proc n'"
theI': fixes P :: "'h \<Rightarrow> bool" assumes "\<exists>!x. P x" shows "P (THE x. P x)"
rep_eq: fixes x :: "nat set"   and xa :: "nat" shows "mapping_of (powsum_mpoly x xa) = powsum_mpoly_aux x xa"
disc_16: fixes x2 :: "('d, 'e) Term.term" shows "\<not> stateful_strand_step.is_Equality (receive\<langle>x2\<rangle>)"
of_nat_of_bool: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and P :: "bool" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (of_bool P) = zero_neq_one.of_bool one zero P"
has_vector_derivative_def: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and net :: "real filter" shows "(f has_vector_derivative f') net = (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
bounded_lattice_def: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" shows "class.bounded_lattice inf less_eq less sup bot top \<equiv> class.lattice inf less_eq less sup \<and> class.order_bot bot less_eq less \<and> class.order_top less_eq less top"
orientable_11_implies_12: fixes x :: "'a" assumes "x \<sqinter> id x = top \<and> x \<squnion> id x = top" shows "x \<sqinter> - top \<sqinter> id (x \<sqinter> - top) = bot \<and> x \<sqinter> - top \<squnion> id (x \<sqinter> - top) = - top"
Bernstein_Poly_rescale: fixes a :: "real"   and b :: "real"   and j :: "nat"   and p :: "nat"   and c :: "real"   and d :: "real" assumes "a \<noteq> b" shows "Bernstein_Poly j p c d \<circ>\<^sub>p [:a, 1:] \<circ>\<^sub>p [:0, b - a:] = Bernstein_Poly j p ((c - a) / (b - a)) ((d - a) / (b - a))"
map_fun_def: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c"   and h :: "'d \<Rightarrow> 'e" shows "(f ---> g) h = g \<circ> h \<circ> f"
return_slice2: fixes n'' :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s-p\<rightarrow>\<^bsub>ret\<^esub> n'"   and "n' \<in> sum_SDG_slice2 n" shows "n'' \<in> sum_SDG_slice2 n"
tseytin_assmt_distinct: fixes F :: "'a formula" shows "distinct (map fst (tseytin_assmt F))"
i_take_prefix_le_conv: fixes a :: "nat"   and f :: "nat \<Rightarrow> 'b"   and b :: "nat" shows "prefix (f \<Down> a) (f \<Down> b) = (a \<le> b)"
assms_6: shows "\<forall>v k. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<longrightarrow> \<A> (State k (index (\<Pi>\<^sub>\<V>) v)) = ((trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi> ! k) v = Some True)"
of_nat_eq: fixes n :: "nat"   and w :: "'b word" shows "(word_of_nat n = w) = (\<exists>q. n = unat w + q * 2 ^ LENGTH('b))"
\<beta>2_le_\<beta>1': shows "zhmset_of \<beta>2 \<le> zhmset_of \<beta>1"
init: shows "\<psi> j \<triangleright> (e_length b - 1) = b"
det_fst_Gauss_Jordan_column_k_PA_eq_fst_Gauss_Jordan_column_k_det_P: fixes P :: "(('b, 'c) vec, 'c) vec"   and i :: "nat"   and A :: "(('b, 'c) vec, 'c) vec"   and k :: "nat" shows "fst (Gauss_Jordan_column_k_det_P (det P, i, A) k) = det (fst (Gauss_Jordan_column_k_PA (P, i, A) k))"
unit_disc_fix_moebius_preserve_poincare_between': fixes M :: "moebius"   and u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "unit_disc_fix M"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "w \<in> unit_disc" shows "poincare_between (moebius_pt M u) (moebius_pt M v) (moebius_pt M w) = poincare_between u v w"
monom_mult_sum_right: fixes c :: "'c"   and t :: "'b"   and f :: "'d \<Rightarrow> 'b \<Rightarrow>\<^sub>0 'c"   and P :: "'d set" shows "MPoly_Type_Class.punit.monom_mult c t (sum f P) = (\<Sum>p\<in>P. MPoly_Type_Class.punit.monom_mult c t (f p))"
Stable_eq_stable: fixes F :: "'d program"   and A :: "'d set" shows "(F \<in> Stable A) = (F \<in> stable (reachable F \<inter> A))"
nat_mirror_le_conv: fixes a :: "nat"   and l :: "nat"   and r :: "nat"   and b :: "nat" assumes "a \<le> l + r" shows "(nat_mirror b l r \<le> nat_mirror a l r) = (a \<le> b)"
map_snd_zip: fixes xs :: "'g list"   and ys :: "'h list" assumes "length xs = length ys" shows "map snd (zip xs ys) = ys"
inj_on_diff: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d set" assumes "inj_on f A" shows "inj_on f (A - B)"
cod_hcomp: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "cod (\<nu> \<star> \<mu>) = cod \<nu> \<star> cod \<mu>"