hcju/isabelleps huggingface dataset:
closure_of_closed: fixes S :: "('a, 'b) eq_object_scheme"   and A :: "'a set" shows "closure_of\<^bsub>S\<^esub> A \<subseteq> carrier S"
domA_fv_subset: fixes \<Gamma> :: "('d \<times> 'e) list" shows "domA \<Gamma> \<subseteq> fv \<Gamma>"
calculus_axioms: shows "calculus Bot_G Inf_G (\<Turnstile>G) Red_I_G Red_F_G"
equiv_int_cong_hcomp: fixes P :: "conc_interface"   and Q :: "conc_interface"   and P' :: "conc_interface"   and Q' :: "conc_interface" assumes "P \<simeq> Q"   and "P' \<simeq> Q'" shows "P \<otimes>\<^sub>c P' \<simeq> Q \<otimes>\<^sub>c Q'"
map2set_to_list: fixes Rk :: "('g \<times> 'h) set"   and R :: "('g \<times> 'h) set \<Rightarrow> (unit \<times> unit) set \<Rightarrow> ('i \<times> ('h \<Rightarrow> unit option)) set"   and it :: "'i \<Rightarrow> ('g \<times> unit) list" assumes "GEN_ALGO_tag (is_map_to_list Rk unit_rel R it)" shows "is_set_to_list Rk (map2set_rel R) (to_list ((map_iterator_dom \<circ>\<circ>\<circ> (\<circ>)) foldli it))"
graphI: fixes G :: "('c, 'd) pre_digraph" assumes "digraph G"   and "symmetric G" shows "graph G"
hyps_2: fixes r'' :: "dB" assumes "t \<rightarrow>\<^sub>\<beta> r''" shows "r[s/0] \<degree>\<degree> ss \<rightarrow>\<^sub>s r''"
divide_less_0_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide a b) zero = (less zero a \<and> less b zero \<or> less a zero \<and> less zero b)"
add_diff_cancel_left: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus c a) (plus c b) = minus a b"
ma_uminus: fixes x :: "mini_alg" shows "- real_of x = real_of (ma_uminus x)"
SUP_mult_left_ennreal: fixes c :: "ennreal"   and f :: "'b \<Rightarrow> ennreal"   and I :: "'b set" shows "c * Sup (f ` I) = (SUP i\<in>I. c * f i)"
bind_def: fixes m :: "'a + 'c"   and f :: "'c \<Rightarrow> 'a + 'b" shows "m \<bind> f = (case m of Inl e \<Rightarrow> Inl e | Inr x \<Rightarrow> f x)"
simps_2: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and k :: "nat"   and u :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k (Suc u) = limit (LocalLexing.Append \<NN> \<TT> \<RR> \<SS> (LocalLexing.\<Z> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k (Suc u)) k) (LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u)"
P9: shows "Bet X1 Q C \<and> Bet A1 Q B1"
errMOD_igAbsRenSTR: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igVarIPresIGWls Delta MOD"   and "FixSyn.igSubstIPresIGWls Delta MOD"   and "igWlsDisj MOD"   and "FixSyn.igAbsRen Delta MOD" shows "igAbsRenSTR (FixSyn.errMOD Delta MOD)"
verit_sum_simplify: fixes a :: "'c" shows "a + (0::'c) = a"
card_has: fixes S :: "'b set"   and n :: "nat" assumes "card S = n" shows "has n S"
surjective: fixes r :: "('c, 'd) edge_scheme" shows "r = \<lparr>src = src r, tgt = tgt r, \<dots> = edge.more r\<rparr>"
fresh_13: fixes a :: "name"   and x2 :: "name"   and x3 :: "trm"   and x1 :: "name" shows "a \<sharp> AndL2 x2.x3 x1 = (a \<sharp> [x2].x3 \<and> a \<sharp> x1)"
Vinv1_def: shows "Vinv1 = {s. \<forall>r\<ge>next_round s. votes s r = Map.empty}"
pprogress_eq: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and \<sigma> :: "(char list \<times> 'a list) trace"   and \<phi> :: "'a MFOTL.formula" assumes "prefix_of \<pi> \<sigma>" shows "pprogress \<phi> \<pi> = Monitor.progress \<sigma> \<phi> (plen \<pi>)"
prod_mset_empty: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one {#} = one"
document_ptr_document_ptr_cast: fixes document_ptr :: "(_) document_ptr" shows "is_document_ptr_kind (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)"
branchTerminalPropagates: fixes subs :: "'b \<Rightarrow> 'b set"   and Gamma :: "'b"   and f :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" assumes "branch subs Gamma f"   and "terminal subs (f m)" shows "terminal subs (f (m + n))"
list_or_upt: fixes lI :: "'a list"   and l :: "'a attree list"   and s :: "'a set" assumes "lI \<noteq> []"   and "length l = length lI"   and "nodup_all lI"   and "\<And>i. i < length lI \<Longrightarrow> \<turnstile>l ! i \<and> attack (l ! i) = ({lI ! i}, s)" shows "\<turnstile>l \<oplus>\<^sub>\<or>\<^bsup>(set lI, s)\<^esup>"
int_step_then_rat_poly_step: fixes a1 :: "int poly"   and a2 :: "int poly"   and q :: "int poly"   and b :: "int"   and c1 :: "int poly"   and c2 :: "int poly" assumes "Some (a1, a2) = div_int_poly_step q b (Some (c1, c2))" shows "(of_int_poly a1, of_int_poly a2) = (div_rat_poly_step (of_int_poly q) \<circ> rat_of_int) b (of_int_poly c1, of_int_poly c2)"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category A"   and "category B" shows "functor_category A B"
connected_finite_iff_sing: fixes S :: "'a set" assumes "connected S" shows "finite S = (S = {} \<or> (\<exists>a. S = {a}))"
update_convs_3: fixes routing_action' :: "'b routing_action \<Rightarrow> 'b routing_action"   and routing_match :: "'b prefix_match"   and metric :: "nat"   and routing_action :: "'b routing_action"   and more :: "'c" shows "routing_action_update routing_action' \<lparr>routing_match = routing_match, metric = metric, routing_action = routing_action, \<dots> = more\<rparr> = \<lparr>routing_match = routing_match, metric = metric, routing_action = routing_action' routing_action, \<dots> = more\<rparr>"
map_add_def: fixes m1 :: "'d \<Rightarrow> 'e option"   and m2 :: "'d \<Rightarrow> 'e option" shows "m2 \<Oplus> m1 = (\<lambda>x. case m2 x of \<bottom> \<Rightarrow> m1 x | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>y\<rfloor>)"
map_1: fixes f :: "'b \<Rightarrow> 'a" shows "map f [] = []"
that: fixes D' :: "'a set set" assumes "D' \<subseteq> \<D>"   and "finite D'"   and "Sigma_Algebra.measure lebesgue (\<Union> \<D>) - e < Sigma_Algebra.measure lebesgue (\<Union> D')" shows "thesis"
sum_pos2: fixes I :: "'c set"   and i :: "'c"   and f :: "'c \<Rightarrow> 'd multiset" assumes "finite I"   and "i \<in> I"   and "{#} \<subset># f i"   and "\<And>i. i \<in> I \<Longrightarrow> {#} \<subseteq># f i" shows "{#} \<subset># comm_monoid_add.sum (+) {#} f I"
rules: shows "R1 \<subseteq> upRules \<and> R2 \<subseteq> nprovRules \<and> R3 \<subseteq> provRules"
vec_of_list_coeffs_replicate_nth: fixes i :: "nat"   and u :: "'b poly"   and h :: "'c poly" assumes "i \<in> {..<degree u}"   and "degree h < degree u" shows "vec_of_list (coeffs h @ replicate (degree u - length (coeffs h)) (0::'c)) $ i = coeff h i"
nonempty_ex: fixes A :: "'e set" assumes "A \<noteq> {}" shows "\<exists>x. x \<in> A"
NSDeriv_unique: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and D :: "'a"   and E :: "'a" assumes "NSDERIV f x :> D"   and "NSDERIV f x :> E"   and "NSDERIV f x :> D"   and "NSDERIV f x :> E" shows "D = E"
PData2Data_def: fixes D :: "'b pdata" shows "PData2Data D = (let (OL, DP) = Rep_pdata D; L = map the OL in Abs_data (L, DP))"
Subspace_FGModuleHomSet: fixes smult' :: "('f, 'g) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and W :: "'a set" assumes "FGModule G smult' W" shows "RModule UNIV (\<lambda>a T v. aezfun_scalar_mult.fsmult smult' a (T v)) (FGModuleHomSet G (\<cdot>) V smult' W) \<and> FGModuleHomSet G (\<cdot>) V smult' W \<subseteq> VectorSpaceHomSet (\<sharp>\<cdot>) V (aezfun_scalar_mult.fsmult smult') W"
ae: shows "AE x in M. \<not> (x \<in> A \<and> x \<in> B)"
assms_2: shows "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs"
space_K: fixes x :: "'a" assumes "x \<in> space M" shows "space (K x) = space M"
homeomorphic_convex_compact: fixes S :: "'c set"   and T :: "'c set" assumes "convex S"   and "compact S"   and "interior S \<noteq> {}"   and "convex T"   and "compact T"   and "interior T \<noteq> {}" shows "S homeomorphic T"
vector_derivative_at_within_non_trivial_limit: fixes x :: "real"   and s :: "real set"   and f :: "real \<Rightarrow> 'b"   and f' :: "'b" assumes "at x within s \<noteq> bot \<and> (f has_vector_derivative f') (at x)" shows "vector_derivative f (at x within s) = f'"
mem_correct_tracing_empty: shows "mem_correct_empty (state_mem_defs.lookup_trace (\<lambda>k. State_Monad.get \<bind> (\<lambda>m. State_Monad.return (Mapping.lookup m k)))) (state_mem_defs.update_trace (\<lambda>k v. State_Monad.get \<bind> (\<lambda>m. State_Monad.set (Mapping.update k v m)))) ((\<lambda>_. True) \<circ> snd) ([], Mapping.empty)"
plus_ms_aux_MSLNil_2: fixes G :: "('a \<times> real) msllist" shows "plus_ms_aux MSLNil G = G"
simps: fixes bnd :: "nat"   and dd :: "nat"   and c :: "int"   and g :: "int poly"   and mm :: "int"   and k :: "nat"   and kk :: "nat" shows "Mahler_Measure.mahler_approximation_main bnd dd c g mm k kk = (let mmm = mahler_landau_graeffe_approximation kk dd g; new_mm = if k = 0 then mmm else min mm mmm in if bnd \<le> k then new_mm else Mahler_Measure.mahler_approximation_main bnd (dd * dd) c (graeffe_one_step c g) new_mm (Suc k) (2 * kk))"
signof_def: fixes p :: "nat \<Rightarrow> nat" shows "signof p = (if sign p = 1 then 1::'b else - (1::'b))"
invim_def: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" shows "invim f A B = {x \<in> A. f x \<in> B}"
is_projection_on_cadjoint: fixes \<pi> :: "'b \<Rightarrow> 'b"   and M :: "'b set" assumes "is_projection_on \<pi> M"   and "closed_csubspace M" shows "\<pi>\<^sup>\<dagger> = \<pi>"
udvd_imp_mod_eq_0: fixes v :: "'b word"   and w :: "'b word" assumes "v udvd w" shows "w mod v = 0"
app_mono: fixes p :: "'c prog \<Rightarrow> String.literal \<Rightarrow> 'c mdecl \<Rightarrow> bool"   and P :: "'c prog"   and \<tau> :: "(ty list \<times> ty err list) option"   and \<tau>' :: "(ty list \<times> ty err list) option"   and i :: "'d instr"   and m :: "nat"   and rT :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list" assumes "wf_prog p P"   and "P \<turnstile> \<tau> \<le>' \<tau>'"   and "app i P m rT pc mpc xt \<tau>'" shows "app i P m rT pc mpc xt \<tau>"
le_iff_diff_le_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a b = less_eq (minus a b) zero"
Re_power_real: fixes x :: "complex"   and n :: "nat" assumes "Im x = 0" shows "Re (x ^ n) = Re x ^ n"
differentiable_at_fst: fixes f :: "'b \<Rightarrow> 'c \<times> 'd"   and x :: "'b"   and X :: "'b set" assumes "f differentiable at x within X" shows "(\<lambda>x. fst (f x)) differentiable at x within X"
cbg: shows "cball z e \<subseteq> - path_image \<gamma>"
disj_subst: fixes P :: "'a upred"   and R :: "'a upred"   and Q :: "'a upred" assumes "P = R" shows "(P \<or> Q) = (R \<or> Q)"
poincare_collinear_zero_polar_form: fixes x :: "complex"   and y :: "complex" assumes "poincare_collinear {0\<^sub>h, of_complex x, of_complex y}"   and "x \<noteq> 0"   and "y \<noteq> 0"   and "of_complex x \<in> unit_disc"   and "of_complex y \<in> unit_disc" shows "\<exists>\<phi> rx ry. x = cor rx * cis \<phi> \<and> y = cor ry * cis \<phi> \<and> rx \<noteq> 0 \<and> ry \<noteq> 0"
hyps: shows "rat_poly.row_length (blockmat ls) = 2 ^ nat (domain_block ls)"
simps_2: fixes env :: "pvar \<Rightarrow> nat" shows "sem Fls env = fls"
T_ndet: fixes P :: "'a process"   and Q :: "'a process" shows "T (P |-| Q) = T P \<union> T Q"
intro: fixes x :: "'b \<Longrightarrow> 'a" assumes "\<And>\<sigma> \<rho>. put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<rho>) = \<rho>" shows "bij_lens_axioms x"
zero_in_group_closure: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "class.ab_group_add plus zero minus uminus" shows "zero \<in> ab_group_add.group_closure zero minus S"
Dynkin_systemI': fixes M :: "'c set set"   and \<Omega> :: "'c set" assumes "\<And>A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"   and "{} \<in> M"   and "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"   and "\<And>A. \<lbrakk>disjoint_family A; range A \<subseteq> M\<rbrakk> \<Longrightarrow> \<Union> (range A) \<in> M" shows "Dynkin_system \<Omega> M"
nextMax_max_1: fixes x :: "nat set"   and v :: "nat" assumes "finite x"   and "v \<in> x" shows "v < nextMax x"
suminf_eq_zero_iff: assumes "summable f"   and "\<And>n. (0::'a) \<le> f n" shows "(suminf f = (0::'a)) = (\<forall>n. f n = (0::'a))"
subset_W: fixes y :: "'a"   and x :: "'a" assumes "y \<in> I x"   and "x \<in> carrier" shows "{j \<in> N. y \<in> W j} \<subseteq> {j \<in> N. W j \<inter> I x \<noteq> {}}"
runWriterT_unitWT: fixes x :: "'e" shows "runWriterT\<cdot>(unitWT\<cdot>x) = return\<cdot>(Writer\<cdot>mempty\<cdot>x)"
if_SE_execE': fixes \<sigma> :: "'d"   and P :: "'d \<Rightarrow> bool"   and B\<^sub>1 :: "'d \<Rightarrow> ('e \<times> 'd) option"   and B\<^sub>2 :: "'d \<Rightarrow> ('e \<times> 'd) option"   and M :: "'d \<Rightarrow> (bool \<times> 'd) option" assumes "\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2 ;- M" obtains "P \<sigma>" and "\<sigma> \<Turnstile> B\<^sub>1 ;- M"   | "\<not> P \<sigma>" and "\<sigma> \<Turnstile> B\<^sub>2 ;- M"
hb: shows "2 ^ length (tl b) \<le> bv_to_nat b"
Prod2Rg_def: fixes A1 :: "('d, 'e) Ring_scheme"   and A2 :: "('d, 'e) Ring_scheme" shows "A1 \<Oplus>\<^sub>r A2 = r\<Pi>\<^bsub>{0, Suc 0}\<^esub> prodB1 A1 A2"
CTermP_sf: fixes vf :: "bool"   and t :: "tm" shows "Sigma_fm (CTermP vf t)"
dvd_diff: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "dvd.dvd times x y"   and "dvd.dvd times x z" shows "dvd.dvd times x (minus y z)"
buffered_val_witness: fixes sb :: "'b memref list"   and a :: "nat"   and v :: "nat" assumes "buffered_val sb a = Some v" shows "\<exists>volatile sop A L R W. Write\<^sub>s\<^sub>b volatile a sop v A L R W \<in> set sb"
urel32_def: fixes p :: "int"   and pp :: "uint32"   and x :: "uint32"   and y :: "int" assumes "mod_ring_locale TYPE('b) p"   and "p = int_of_uint32 pp"   and "p \<le> 65535" shows "mod_ring_locale.urel32 p x y = (y = int_of_uint32 x \<and> y < p)"
SentP_graph_aux_def: shows "SentP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SentP_aux y z w x. x1 = x \<and> x2 = SyntaxN.Ex y (SyntaxN.Ex z (SyntaxN.Ex w (FormP (Var y) AND FormP (Var z) AND FormP (Var w) AND (x EQ Q_Imp (Var y) (Var y) OR x EQ Q_Imp (Var y) (Q_Disj (Var y) (Var z)) OR x EQ Q_Imp (Q_Disj (Var y) (Var y)) (Var y) OR x EQ Q_Imp (Q_Disj (Var y) (Q_Disj (Var z) (Var w))) (Q_Disj (Q_Disj (Var y) (Var z)) (Var w)) OR x EQ Q_Imp (Q_Disj (Var y) (Var z)) (Q_Imp (Q_Imp (Var y) (Var w)) (Q_Disj (Var z) (Var w))))))) \<and> atom y \<sharp> (z, w, x) \<and> atom z \<sharp> (w, x) \<and> atom w \<sharp> x)"
pequiv_def: fixes prog :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and prog' :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "prog \<simeq> prog' \<equiv> \<forall>P. sound P \<longrightarrow> wp prog P = wp prog' P"
le_ncols: fixes n :: "nat"   and A :: "'b matrix" shows "(n \<le> ncols A) = (\<forall>m. (\<forall>j i. m \<le> i \<longrightarrow> Rep_matrix A j i = (0::'b)) \<longrightarrow> n \<le> m)"
real_power_nonzero: fixes d :: "real"   and a :: "'a" assumes "0 < d" shows "real_power d a \<noteq> (0::'a)"
snoc: fixes xs :: "'a llist"   and x :: "'a" assumes "lfinite xs"   and "P xs" shows "P (lappend xs (LCons x LNil))"
float_neg_sign1: fixes a :: "('a, 'b) IEEE.float"   and b :: "('c, 'd) IEEE.float" shows "(sign a = sign (- b)) = (sign a \<noteq> sign b)"
PiM_empty: fixes M :: "'a \<Rightarrow> 'b measure" shows "Pi\<^sub>M {} M = count_space {\<lambda>_. undefined}"
pref_pow_ext': fixes u :: "'b list"   and r :: "'b list"   and k :: "nat" assumes "u \<le>p r \<^sup>@ k" shows "u \<le>p r \<cdot> r \<^sup>@ k"
Int_Diff: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and B :: "'c set" assumes "finite A" shows "sum g A = sum g (A \<inter> B) + sum g (A - B)"
shift_bounds_fact: shows "(\<Prod>i = Suc 0..Suc d. nth_digit_general n i p choose nth_digit_general k i p) = (\<Prod>i = 0..d. nth_digit_general n (Suc i) p choose nth_digit_general k (Suc i) p)"
is_lrun_def: fixes l :: "nat \<Rightarrow> 'l"   and s :: "nat \<Rightarrow> 'a" shows "A.is_lrun l s \<equiv> s 0 = as\<^sub>0 \<and> (\<forall>i. alstep (l i) (s i) (s (Suc i)))"
a_ptr_disconnected_node_rel_def: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap" shows "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_ptr_disconnected_node_rel get_disconnected_nodes h = (\<lambda>(x, y). (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r x, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r y)) ` {(document_ptr, disconnected_node). document_ptr |\<in>| document_ptr_kinds h \<and> disconnected_node \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r}"
assms: shows "eSubstAbs MOD ys eY y eA = OK Z"
distinct_15: fixes x31 :: "'a"   and x32 :: "('a, 'b) com"   and x33 :: "('a, 'b) com"   and x41 :: "'a"   and x42 :: "('a, 'b) com" shows "if x31 then x32 else x33 \<noteq> while x41 do x42"
sub_Union: fixes C :: "'c set set"   and A :: "'c set" assumes "\<exists>X\<in>C. A \<subseteq> X" shows "A \<subseteq> \<Union> C"
simps_2: fixes x :: "brick list"   and ys :: "wall" shows "codomain_wall (x * ys) = codomain_wall ys"
sim_action_simps_4: fixes ad :: "'a"   and al :: "addr_loc"   and v :: "'a val"   and obs :: "('a, 'b) obs_event action" shows "NormalAction (ReadMem ad al v) \<approx> obs = (\<exists>v'. obs = NormalAction (ReadMem ad al v'))"
Builds_def: fixes B :: "hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and l :: "hf" shows "Builds B C s l \<equiv> B (app s l) \<or> (\<exists>m\<^bold>\<in>l. \<exists>n\<^bold>\<in>l. C (app s l) (app s m) (app s n))"
less_imp_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.preorder less_eq less"   and "less x y" shows "less_eq x y"
lin_dep_def: fixes S :: "rat vec set" shows "gs.lin_dep S = (\<exists>A a v. finite A \<and> A \<subseteq> S \<and> True \<and> gs.lincomb a A = 0\<^sub>v n \<and> v \<in> A \<and> a v \<noteq> 0)"
eqvt_at_apply: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and p :: "perm" assumes "eqvt_at f x" shows "(p \<bullet> f) x = f x"
supp_Some: fixes x :: "'a" shows "supp (Some x) = supp x"
assms_4: fixes x :: "real" assumes "x \<in> {a..b}" shows "g x \<in> S"
SUP_UNION: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "Sup (f ` \<Union> (g ` A)) = (SUP y\<in>A. Sup (f ` g y))"
basic_trans_rules_18: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b"   and "a \<noteq> b" shows "a < b"
nth_prefixes_is_prefix_tl: fixes n :: "nat"   and p :: "'c list" assumes "Suc n < length p" shows "prefix (tl (prefixes p) ! n) (tl (prefixes p) ! Suc n)"
nb\<^sub>1\<^sub>3: fixes v :: "'a"   and a' :: "'b" assumes "(v, a') \<in> dom (map_of (effect_to_assignments op'))"   and "(v, a') \<in> (\<Union>(v, a)\<in>set (effect_of op). {(v, a') |a'. a' \<in> \<R>\<^sub>+ \<Psi> v \<and> a' \<noteq> a})" shows "(\<phi>\<^sub>S \<Psi> s ++ map_of (effect_of (\<phi>\<^sub>O\<inverse> \<Psi> op'))) (v, a') = Some False"
inj_map_strong: fixes x :: "'b list"   and xa :: "'b list"   and f :: "'b \<Rightarrow> 'c"   and fa :: "'b \<Rightarrow> 'c" assumes "\<And>z za. \<lbrakk>z \<in> set x; za \<in> set xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map f x = map fa xa" shows "x = xa"
vD_iD_gives_kD_1: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "ip \<in> vD rt" shows "ip \<in> kD rt"
that: fixes q :: "real \<Rightarrow> 'a" assumes "arc q"   and "path_image q \<subseteq> path_image p"   and "pathstart q = a"   and "pathfinish q = b" shows "thesis"
simps: fixes f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and uu :: "int iarray"   and i :: "nat"   and j :: "nat" shows "dmu_ij_state (f, mu, uu) i j = mu !! i !! j"
bdd_aboveI: fixes A :: "'c multiset set"   and M :: "'c multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<subseteq># M" shows "subset_mset.bdd_above A"
assms_6: shows "d \<in> {4 * delta + 2 * C..dist (p a) (p b)}"
irreflp_on_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "irreflp_on P A = (\<forall>a\<in>A. \<not> P a a)"
supset_pseudominimal_ne_bottom: fixes P :: "'b set set"   and x :: "'b set" assumes "dual_order.pseudominimal_in P x" shows "x \<noteq> dual_order.bottom P"
push_down_oldest_token_token_run: 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"   and x :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w q n = Some x" shows "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n = q"
vwalk_imp_ex_vpath: fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph"   and u :: "'a"   and v :: "'a" assumes "vwalk p G"   and "hd p = u"   and "last p = v" shows "\<exists>q. vpath q G \<and> hd q = u \<and> last q = v"
am_is_in_V_def: fixes am :: "nat \<Rightarrow> nat list"   and u :: "nat" shows "am_is_in_V am u \<equiv> return (am u \<noteq> [])"
mv_boxes_correct2: fixes n :: "nat"   and aa :: "nat"   and ba :: "nat"   and lm1 :: "nat list"   and lm2 :: "nat list"   and lm3 :: "nat list"   and lm4 :: "nat list" assumes "n \<le> aa - ba"   and "ba < aa"   and "length lm1 = ba"   and "length lm2 = aa - ba - n"   and "length lm3 = n" shows "{\<lambda>nl. nl = lm1 @ 0 \<up> n @ lm2 @ lm3 @ lm4} mv_boxes aa ba n {\<lambda>nl. nl = lm1 @ lm3 @ lm2 @ 0 \<up> n @ lm4}"
l2_inv7_aux: fixes R :: "fid_t"   and s :: "l2_state" assumes "NonceF (R $ sk) \<in> analz (ik s)"   and "s \<in> l2_inv7" shows "R \<noteq> test \<and> R \<notin> partners"