hcju/isabelleps huggingface dataset:
union_disjoint: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "sum_0 g (A \<union> B) = sum_0 g A + sum_0 g B"
Id_compose: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "f \<in> A \<rightarrow> B"   and "f \<in> extensional A" shows "identity B \<circ> f \<down> A = f"
pow_eq_if_list': fixes p :: "'b list"   and m :: "nat" shows "p \<^sup>@ m = (if m = 0 then \<epsilon> else p \<^sup>@ (m - 1) \<cdot> p)"
lfp_def: fixes f :: "'c \<Rightarrow> 'c" shows "lfp f = Inf {u. f u \<le> u}"
ordermap: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and a :: "'a" assumes "wf r" shows "ordermap A r a = set (ordermap A r ` {y \<in> A. (y, a) \<in> r})"
Pair_bottom_iff: fixes x :: "'a"   and y :: "'b" shows "((x, y) = \<bottom>) = (x = \<bottom> \<and> y = \<bottom>)"
remove1_insort: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "remove1 x (linorder.insort_key less_eq (\<lambda>x. x) x xs) = xs"
TESL_interp_unfold_stepwise_timedelayed: fixes master :: "clock"   and \<delta>\<tau> :: "'b tag_const"   and measuring :: "clock"   and slave :: "clock" shows "\<lbrakk> master time-delayed by \<delta>\<tau> on measuring implies slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<Inter> {Y. \<exists>n. Y = \<lbrakk> master time-delayed by \<delta>\<tau> on measuring implies slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup>}"
bounded_variation_absolutely_integrable_interval: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c"   and B :: "real" assumes "f integrable_on cbox a b"   and "\<And>d. d division_of cbox a b \<Longrightarrow> (\<Sum>K\<in>d. norm (integral K f)) \<le> B" shows "f absolutely_integrable_on cbox a b"
rel_fun_Grp: fixes h :: "'j \<Rightarrow> 'h"   and A :: "'i set"   and g :: "'i \<Rightarrow> 'k" shows "rel_fun (BNF_Def.Grp UNIV h)\<inverse>\<inverse> (BNF_Def.Grp A g) = BNF_Def.Grp {f. f ` range h \<subseteq> A} (map_fun h g)"
lift2_eq_ErrD: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and y :: "'b"   and A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "x \<squnion>\<^bsub>f\<^esub> y = Err"   and "semilat (err A, Err.le r, lift2 f)"   and "x \<in> A"   and "y \<in> A" shows "\<not> (\<exists>u\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> u \<and> y \<sqsubseteq>\<^bsub>r\<^esub> u)"
diag_def: fixes \<phi> :: "'fmla" shows "diag \<phi> \<equiv> SOME \<psi>. \<psi> \<in> fmla \<and> Fvars \<psi> = {} \<and> bprv (eqv \<psi> (subst \<phi> \<langle>\<psi>\<rangle> xx))"
length_sequence_of: fixes enum :: "nat \<Rightarrow> 'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "bij_betw enum {0..<card A} A"   and "f \<in> A \<rightarrow>\<^sub>E B" shows "length (sequence_of A enum f) = card A"
inheritedFounded: fixes subs :: "'a \<Rightarrow> 'a set"   and P :: "'a \<Rightarrow> bool" shows "inherited subs (founded subs P)"
Ana_m: shows "Ana m = (K \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t (!) T, map ((!) T) N)"
assms_4: shows "exception_\<I> \<I> \<turnstile>g \<A> \<surd>"
intersect'_nodes_card: fixes A :: "('e, 'd) dbta"   and B :: "('e, 'f) dbta" assumes "finite (DBTA.nodes A)"   and "finite (DBTA.nodes B)" shows "card (DGBTA.nodes (intersect' A B)) \<le> card (DBTA.nodes A) * card (DBTA.nodes B)"
prod_cases5: fixes y :: "'a \<times> 'b \<times> 'd \<times> 'e \<times> 'f" obtains a :: "'a" and b :: "'b" and c :: "'d" and d :: "'e" and e :: "'f" where "y = (a, b, c, d, e)"
eq_Max_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and m :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(m = linorder.Max less_eq A) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq a m))"
gseq_iff: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "((f, g) \<in> gseq) = (f \<in> SEQ A \<and> g \<in> SEQ A \<and> (\<exists>i. size (g i) < size (f i) \<and> (\<forall>j<i. f j = g j)))"
axioms_2: fixes \<eta> :: "'c \<Rightarrow> 'd"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and M' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d" assumes "monoid_homomorphism \<eta> M composition unit M' composition' unit'" shows "monoid M composition unit"
nonneg_coeffs: fixes n :: "nat" shows "fds_nth f n \<in> \<real>\<^sub>\<ge>\<^sub>0"
2: shows "Maps.commutative_square \<lbrakk>\<lbrakk>tab\<^sub>0 g\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>tab\<^sub>1 h\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>TTfgh.p\<^sub>0\<rbrakk>\<rbrakk>"
normal_def2: fixes s :: "state" shows "normal s = (fst s = None)"
fimageI: fixes x :: "'d"   and A :: "'d fset"   and f :: "'d \<Rightarrow> 'e" assumes "x |\<in>| A" shows "f x |\<in>| f |`| A"
sub: fixes a :: "'a \<Rightarrow> 'a" assumes "a \<in> translation ` G" shows "a \<in> G \<rightarrow>\<^sub>E G"
trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_funs_term_cases: fixes t :: "('c, 'd) Term.term"   and F :: "(('c, 'e) Term.term \<times> ('c, 'e) Term.term) list"   and \<theta> :: "'e \<Rightarrow> ('c, 'd) Term.term"   and f :: "'c" assumes "t \<in> trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>)"   and "f \<in> funs_term t" shows "(\<exists>u\<in>trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F. f \<in> funs_term u) \<or> (\<exists>x\<in>fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F. f \<in> funs_term (\<theta> x))"
distinct_1: fixes x1 :: "'c"   and x21 :: "'c"   and x22 :: "'d"   and x23 :: "('c, 'd) chain" shows "\<lbrace> x1 \<rbrace> \<noteq> \<lbrace> x21 \<rbrace> \<cdot> x22 \<cdot> x23"
of_nat_stream: fixes n :: "nat" shows "of_nat n = pure (of_nat n)"
subdistl_eq: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z \<cdot> x \<squnion> z \<cdot> (x \<squnion> y) = z \<cdot> (x \<squnion> y)"
ex_project: fixes x :: "nat" shows "\<exists>v. project_Nat_nat v = x"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "incidence_system point_set block_collection"   and "finite_incidence_system_axioms point_set" shows "finite_incidence_system point_set block_collection"
vfiniteE2: fixes A :: "V" assumes "vfinite A" obtains n :: "V" where "n \<in>\<^sub>\<circ> \<omega>" and "n \<approx>\<^sub>\<circ> A"
inner_eq_zero_iff: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and x :: "'e" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(inner x x = 0) = (x = zero)"
zero_eq_HMSet: fixes m :: "hmultiset multiset" shows "(0 = HMSet m) = (m = {#})"
wt_subst_trm'': fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "typed_model arity public Ana \<Gamma>"   and "typed_model.wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<Gamma> \<sigma>" shows "\<Gamma> t = \<Gamma> (t \<cdot> \<sigma>)"
continuous_within_topological: fixes x :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'd" shows "continuous (at x within s) f = (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
h_plus: fixes x :: "'a"   and y :: "'a" shows "h (x + y) = h x + h y"
seq_char: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" shows "V.seq \<nu> \<mu> = (arrow_of_spans (\<cdot>) \<mu> \<and> arrow_of_spans (\<cdot>) \<nu> \<and> Dom \<nu> = Cod \<mu>)"
synth_analz_mono: fixes G :: "msg set"   and H :: "msg set" assumes "G \<subseteq> H" shows "synth (analz G) \<subseteq> synth (analz H)"
elims: fixes x :: "'b \<Rightarrow> 'a"   and xa :: "'b \<times> 'b"   and y :: "'a \<times> 'a" assumes "vec_map x xa = y" obtains f :: "'b \<Rightarrow> 'a" and xb :: "'b" and ya :: "'b" where "x = f" and "xa = (xb, ya)" and "y = (f xb, f ya)"
length_splice: fixes xs :: "'b list"   and ys :: "'b list" shows "length (splice xs ys) = length xs + length ys"
less: shows "s\<^sub>2 \<sqsubseteq>\<^bsub>r\<^esub> s\<^sub>1"
nat_power_le_imp_le_base: fixes n :: "nat"   and a :: "nat"   and b :: "nat" assumes "n \<noteq> 0"   and "a ^ n \<le> b ^ n" shows "a \<le> b"
ex_final1_conv_ex_final2: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves" assumes "FWdelay_bisimulation_obs final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2" shows "(\<exists>x1. final1 x1) = (\<exists>x2. final2 x2)"
intro: fixes M :: "'d \<Rightarrow> 'e measure"   and I :: "'d set" assumes "product_sigma_finite M"   and "finite_product_sigma_finite_axioms I" shows "finite_product_sigma_finite M I"
lcsTr8: fixes H :: "'a set"   and a :: "'a"   and h :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "h \<in> H" shows "a \<cdot> h \<in> a \<diamondsuit> H"
simps_1: fixes K\<^sub>1 :: "clock"   and \<tau> :: "'b tag_const"   and K\<^sub>2 :: "clock"   and i :: "nat" shows "\<lbrakk> K\<^sub>1 sporadic \<tau> on K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> i\<^esup> = {\<rho>. \<exists>n\<ge>i. hamlet (Rep_run \<rho> n K\<^sub>1) \<and> time (Rep_run \<rho> n K\<^sub>2) = \<tau>}"
negligible_finite: fixes s :: "'b set" assumes "finite s" shows "negligible s"
gfs_Suc_eq: fixes n :: "nat" shows "p x x (Suc n) *\<^sub>R z ^ Suc n = (\<Sum>i\<le>n. u x x i *\<^sub>R z ^ Suc i * p x x (n - i) *\<^sub>R z ^ (n - i))"
brc_step_abs: fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule set"   and \<Sigma> :: "('a, unit) HashMap.hashmap \<times> 'a list \<times> (('a, 'b) ta_rule, nat) HashMap.hashmap" assumes "rqrm_invar rqrm"   and "rqrm_prop \<delta> rqrm"   and "\<Sigma> \<in> brc_invar \<delta>"   and "brc_cond \<Sigma>" shows "(brc_\<alpha> \<Sigma>, brc_\<alpha> (brc_step rqrm \<Sigma>)) \<in> br'_step \<delta>"
mgp_pre_iso_test: fixes g :: "graph" assumes "minGraphProps g" shows "pre_iso_test (fgraph g)"
numeral_lt_\<omega>\<^sub>z: fixes n :: "num" shows "numeral n < \<omega>\<^sub>z"
fun_type_length_eq: fixes f :: "'fun"   and X :: "('fun, 'var) Term.term list"   and g :: "'fun"   and Y :: "('fun, 'atom) Term.term list" assumes "\<Gamma> (Fun f X) = Fun g Y" shows "length X = length Y"
proper_Un: fixes A :: "'b set"   and B :: "'b set" assumes "proper A"   and "proper B" shows "proper (A \<union> B)"
left_inverse: fixes a :: "'b" assumes "a \<noteq> (0::'b)" shows "inverse a * a = (1::'b)"
is_empty_empty: fixes t :: "('b, 'c) RBT.rbt" shows "RBT.is_empty t = (t = RBT.empty)"
Diff_infinite_finite: fixes T :: "'a set"   and S :: "'a set" assumes "finite T"   and "infinite S" shows "infinite (S - T)"
sqr_of_cmod_of_prod: fixes z1 :: "complex"   and z2 :: "complex" shows "(cmod (z1 * z2))\<^sup>2 = (cmod z1)\<^sup>2 * (cmod z2)\<^sup>2"
exec_gpv_plus_oracle_left: fixes oracle1 :: "'i \<Rightarrow> 'j \<Rightarrow> ('l \<times> 'i) spmf"   and oracle2 :: "'i \<Rightarrow> 'k \<Rightarrow> ('m \<times> 'i) spmf"   and gpv :: "('h, 'j, 'l) gpv"   and s :: "'i" shows "exec_gpv (oracle1 \<oplus>\<^sub>O oracle2) (left_gpv gpv) s = exec_gpv oracle1 gpv s"
HFail_HInv4a: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p"   and "HInv4a s q" shows "HInv4a s' q"
kruskal_injective_inv: fixes f :: "'a"   and q :: "'a"   and e :: "'a" assumes "injective f"   and "covector q"   and "q * f\<^sup>T \<le> q"   and "e \<le> q"   and "q * f\<^sup>T \<le> - e"   and "injective e"   and "coreflexive (q\<^sup>T * q \<sqinter> f\<^sup>T * f)" shows "injective (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e)"
accept_def: fixes w :: "nat \<Rightarrow> 'l" shows "a.accept w \<equiv> \<exists>r. a.is_run r \<and> w = a.L \<circ> r"
integrable_fst: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "integrable (lborel \<Otimes>\<^sub>M lborel) (\<lambda>(x, y). f x y)" shows "integrable lborel (\<lambda>x. integral\<^sup>L lborel (f x))"
A_trusts_OR1: fixes A :: "agent"   and Na :: "nat"   and M :: "nat"   and B :: "agent"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Nonce Na, Nonce M, Agent A, Agent B\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> orb" shows "Says A B \<lbrace>Nonce M, Agent A, Agent B, Crypt (shrK A) \<lbrace>Nonce Na, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs"
induce_eq_iff_induced: fixes H :: "('a, 'b) pre_digraph" assumes "induced_subgraph H H" shows "H \<restriction> verts H = H"
R_def: shows "R = {(V, V'). V = V' \<and> (V = [] \<or> V = [skip\<^bsub>\<iota>\<^esub>])}"
par_comp_def: fixes s :: "('d, 'f seq_action) automaton"   and t :: "('e, 'f seq_action) automaton" shows "s \<langle>\<langle> t \<equiv> \<lparr>init = init s \<times> init t, trans = parp_sos (automaton.trans s) (automaton.trans t)\<rparr>"
ln_exp_cf5_lower_bound_neg: fixes x :: "real" assumes "x \<le> 0"   and "0 < numer_cf5 x" shows "ln (exp_cf5 x) \<le> x"
degree_m_eq_monic: fixes f :: "int poly" assumes "monic f"   and "1 < p" shows "degree_m f = degree f"
dvd_add_times_triv_right_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus b (times c a)) = dvd.dvd times a b"
transfer: shows "pcr_ccsubspace (=) {0::'d} \<bottom>"
facetrel_card: fixes y :: "'b set"   and x :: "'b set" assumes "y \<lhd> x" shows "card (x - y) = 1"
elims: fixes x :: "('e, 'f, 'g) FSM"   and xa :: "('e, 'f, 'h) FSM"   and xb :: "('e, 'f) ATC set"   and xc :: "'e list set"   and xd :: "nat"   and xe :: "nat"   and y :: "'e list set" assumes "C x xa xb xc xd xe = y" obtains M2 :: "('e, 'f, 'g) FSM" and M1 :: "('e, 'f, 'h) FSM" and \<Omega> :: "('e, 'f) ATC set" and V :: "'e list set" and m :: "nat" where "x = M2a__" and "xa = M1a__" and "xb = \<Omega>'__" and "xc = Va__" and "xd = ma__" and "xe = 0" and "y = {}"   | M2 :: "('e, 'f, 'g) FSM" and M1 :: "('e, 'f, 'h) FSM" and \<Omega> :: "('e, 'f) ATC set" and V :: "'e list set" and m :: "nat" where "x = M2b__" and "xa = M1b__" and "xb = \<Omega>''__" and "xc = Vb__" and "xd = mb__" and "xe = Suc 0" and "y = Vb__"   | M2 :: "('e, 'f, 'g) FSM" and M1 :: "('e, 'f, 'h) FSM" and \<Omega> :: "('e, 'f) ATC set" and V :: "'e list set" and m :: "nat" and v :: "nat" where "x = M2" and "xa = M1" and "xb = \<Omega>" and "xc = V" and "xd = m" and "xe = Suc (Suc v)" and "y = append_set (C M2 M1 \<Omega> V m (Suc v) - RM M2 M1 \<Omega> V m (Suc v)) (inputs M2) - TS M2 M1 \<Omega> V m (Suc v)"
Polygamma_approx_ge_2_complex: fixes x :: "complex"   and m :: "nat"   and j :: "nat" assumes "0 < Re x"   and "0 < m" shows "Polygamma_approx (Suc (Suc j)) m x = stirling_sum' (Suc j) m x"
constrains_Int_distrib: fixes C :: "'b set"   and A :: "'b set"   and B :: "'b set" shows "C co A \<inter> B = (C co A) \<inter> (C co B)"
sum_eq_exp: shows "(\<Sum>k = 0..n. (- 1) ^ k / fact k) = exp (- 1) - exp t / fact (n + 1) * (- 1) ^ (n + 1)"
sel_17: fixes x31 :: "nat"   and x32 :: "'b"   and x33 :: "'b"   and x34 :: "'c"   and x35 :: "'c"   and x36 :: "'a" shows "getMsg (Recv x31 x32 x33 x34 x35 x36) = x36"
analytic_on_compose: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "complex \<Rightarrow> complex" assumes "f analytic_on S"   and "g analytic_on f ` S" shows "g \<circ> f analytic_on S"
add_arc_simps_4: fixes G :: "('d, 'c) pre_digraph"   and a :: "'c" shows "head (pre_digraph.add_arc G a) = head G"
pax4: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool"   and v :: "'d \<Rightarrow> 'e" shows "|~ \<box>[P]_v = (_actrans P v \<and> \<circle>\<box>[P]_v)"
sameFw_behave_sets: fixes A :: "'a word set list"   and c :: "'b parts_connection_scheme"   and rs :: "'a simple_rule list" assumes "\<forall>w\<in>set A. \<forall>a1\<in>w. \<forall>a2\<in>w. same_fw_behaviour_one a1 a2 c rs"   and "\<forall>w1\<in>set A. \<forall>w2\<in>set A. \<exists>a1\<in>w1. \<exists>a2\<in>w2. same_fw_behaviour_one a1 a2 c rs" shows "\<forall>w1\<in>set A. \<forall>w2\<in>set A. \<forall>a1\<in>w1. \<forall>a2\<in>w2. same_fw_behaviour_one a1 a2 c rs"
log: shows "log base (real (D fs)) \<le> log base (real N ^ (m * m))"
extend_indets_monomial: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and c :: "'c" shows "extend_indets (monomial c t) = MPoly_Type_Class.punit.monom_mult c 0 (subst_pp extend_indets_subst t)"
some_elem_def: fixes s :: "'b set" shows "some_elem s = (SOME x. x \<in> s)"
tr: shows "(s, i:deliver(d), s') \<in> automaton.trans (pnet np (p1 \<parallel> p2))"
allow_all_fun_def: fixes f :: "'d \<Rightarrow> 'e" shows "A\<^sub>f f = A\<^sub>p (Some \<circ> f)"
upd_incoming__ident_node: fixes nd :: "('b, 'c) node_scheme"   and n :: "('b, 'd) node_scheme"   and ns :: "('b, 'c) node_scheme set" assumes "nd \<in> upd_incoming n ns"   and "nd \<in> ns" shows "incoming n \<subseteq> incoming nd \<or> \<not> (old nd = old n \<and> next nd = next n)"
i: fixes x :: "nat" shows "eval r_phi [i, x] = eval g [x]"
semiring_of_sets_def: fixes \<Omega> :: "'b set"   and M :: "'b set set" shows "semiring_of_sets \<Omega> M \<equiv> subset_class \<Omega> M \<and> semiring_of_sets_axioms M"
prems_2: fixes op2 :: "'b"   and r :: "'a" assumes "op2 \<in> snd ` set (ys @ [y])"   and "r \<in> deps op2" shows "r \<noteq> oid"
exec1_end: fixes P :: "instr list"   and c :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and c' :: "int \<times> (char list \<Rightarrow> int) \<times> int list" assumes "size P \<le> fst c" shows "\<not> P \<turnstile> c \<rightarrow> c'"
perfect_sec_P2_def: fixes R2 :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd \<times> 'e) spmf"   and S2 :: "'b \<Rightarrow> 'e \<Rightarrow> 'c spmf"   and Out2 :: "'b \<Rightarrow> 'a \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) spmf"   and funct :: "'a \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'e) spmf"   and m1 :: "'a"   and m2 :: "'b" shows "sim_non_det_def.perfect_sec_P2 R2 S2 Out2 funct m1 m2 \<equiv> R2 m1 m2 = funct m1 m2 \<bind> (\<lambda>(s1, s2). sim_non_det_def.Ideal2 S2 Out2 m2 m1 s2)"
ncols_not_0: fixes A :: "(('a, 'b) vec, 'c) vec" shows "0 \<noteq> ncols A"
orthogonal_in_span_eq_0: fixes v :: "(real, 'a) vec"   and S :: "(real, 'a) vec set" assumes "v \<in> span S"   and "\<And>w. w \<in> S \<Longrightarrow> orthogonal v w" shows "v = 0"
\<delta>_prod_sng_alt_2: fixes \<delta>1 :: "('a, 'c) ta_rule set"   and r :: "('b, 'c) ta_rule" shows "\<delta>_prod_sng2 \<delta>1 r = \<delta>_prod \<delta>1 {r}"
simps_2: fixes t :: "Core.term"   and u :: "Core.term"   and k :: "nat" shows "loose_bvar1 (t $ u) k = (loose_bvar1 t k \<or> loose_bvar1 u k)"
surjective: fixes r :: "('a, 'b, 'c) pre_digraph_scheme" shows "r = \<lparr>verts = verts r, arcs = arcs r, tail = tail r, head = head r, \<dots> = pre_digraph.more r\<rparr>"
coeff_max_bddTr: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c" shows "c_max S c \<le> fst c"
Acc: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and va :: "Term.var"   and Q :: "'b assn" assumes "G,A\<turnstile>{Normal P} va=\<succ> {\<lambda>Var:(v, f):. Q\<leftarrow>In1 v}" shows "G,A\<turnstile>{Normal P} Acc va-\<succ> {Q}"
berlekamp_monic_factorization_i_def: fixes p :: "int"   and ff_ops :: "'a arith_ops_record"   and d :: "nat"   and f :: "'a list" shows "berlekamp_monic_factorization_i p ff_ops d f = (let vs = berlekamp_basis_i p ff_ops f in berlekamp_factorization_main_i p ff_ops (arith_ops_record.zero ff_ops) (arith_ops_record.one ff_ops) d [f] vs (length vs))"
inject_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(x2\<inverse> = y2\<inverse>) = (x2 = y2)"
multiplicity_gauss_int_of_nat: fixes a :: "nat"   and b :: "nat" shows "multiplicity (of_nat a) (of_nat b) = multiplicity a b"
short_ch_card_2: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and X :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "MinkowskiBetweenness.ch_by_ord \<P> betw f X" shows "MinkowskiBetweenness.short_ch \<P> X = (card X = 2)"
assms_4: shows "proj2_abs (i *\<^sub>R a + r) = proj2_abs (j *\<^sub>R a + r)"
Crypt_notin_image_Key: fixes K :: "nat"   and X :: "msg"   and KK :: "nat set" shows "Crypt K X \<notin> Key ` KK"
liftAll2_True: fixes inp :: "'c \<Rightarrow> 'a option"   and inp' :: "'c \<Rightarrow> 'b option" shows "liftAll2 (\<lambda>v v'. True) inp inp'"
rep_eq: fixes x :: "'b"   and xa :: "'c" shows "lookup (single x xa) = (\<lambda>k'. xa when x = k')"
dom_dom: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.VV.dom (B.VV.dom f) = B.VV.dom f"
a_gla: fixes y :: "'a"   and x :: "'a" shows "(y \<cdot> ar x = (zero_class.zero::'a)) = (ar x \<le> ar y)"
match_eval_trm: fixes ts :: "Formula.trm list"   and n :: "nat"   and f :: "nat \<Rightarrow> event_data option"   and xs :: "event_data list" assumes "\<forall>t\<in>set ts. \<forall>i\<in>fv_trm t. i < n"   and "Some f = match ts xs" shows "map (Formula.eval_trm (tabulate (\<lambda>i. the (f i)) 0 n)) ts = xs"
Image_singleton: fixes r :: "('b \<times> 'a) set"   and a :: "'b" shows "r `` {a} = {b. (a, b) \<in> r}"
simps_23: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b"   and f5 :: "char list \<Rightarrow> 'b" shows "(case Boolean of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f2"
fmap_rel_alt_def: fixes K :: "('a \<times> 'c) set"   and V :: "('b \<times> 'd) set" shows "\<langle>K, V\<rangle>fmap_rel \<equiv> {(m1, m2). (\<forall>i j. i \<in># dom_m m2 \<longrightarrow> (j, i) \<in> K \<longrightarrow> (the (fmlookup m1 j), the (fmlookup m2 i)) \<in> V) \<and> fset (fmdom m1) \<subseteq> Domain K \<and> fset (fmdom m2) \<subseteq> Range K \<and> (\<forall>i j. (i, j) \<in> K \<longrightarrow> (j \<in># dom_m m2) = (i \<in># dom_m m1))}"
cbox_borel: fixes a :: "'a"   and b :: "'a" shows "cbox a b \<in> sets borel"
assms: shows "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> m"
configuration_Min_in: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n \<noteq> {}" shows "Min (semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n) \<in> semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n"
1: shows "i < length (map (\<lambda>A. local.tensor_prod_otimes A B) As)"
stails_bottom_iff: fixes xs :: "[:'a:]" shows "(stails\<cdot>xs = \<bottom>) = (xs = \<bottom>)"