hcju/isabelleps huggingface dataset:
perp_up_at_end: fixes a :: "proj2"   and b :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "proj2_incident b l" shows "B\<^sub>\<real> cart2_pt (perp_foot a l) cart2_pt a cart2_pt (perp_up a l)"
stepsClosure_refl: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" shows "refl (stepsClosure Rel Cal)"
thread_start_actions_ok_prefix: fixes E' :: "('a \<times> ('b, 'a) obs_event action) llist"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "thread_start_actions_ok E'"   and "lprefix E E'" shows "thread_start_actions_ok E"
id_comp: fixes g :: "'k \<Rightarrow> 'l" shows "id \<circ> g = g"
well_formed_productF: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'g) FSM"   and FAIL :: "'g \<times> 'f"   and PM :: "('d, 'e, 'g \<times> 'f) FSM" assumes "well_formed M1"   and "well_formed M2"   and "productF M2 M1 FAIL PM" shows "well_formed PM"
iMODb_mod_0: fixes r :: "nat"   and c :: "nat" shows "[ r, mod 0, c ] = [r\<dots>,0]"
append_eq_Cons_conv: fixes ys :: "'b list"   and zs :: "'b list"   and x :: "'b"   and xs :: "'b list" shows "(ys @ zs = x # xs) = (ys = [] \<and> zs = x # xs \<or> (\<exists>ys'. ys = x # ys' \<and> ys' @ zs = xs))"
simps_2: fixes l :: "('a, 'b) rbt"   and c :: "color"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "bd_inv (Node l c k v r) = (bd_inv l \<and> bd_inv r \<and> black_depth l = black_depth r)"
ttree_contI3: fixes f :: "'b \<Rightarrow> 'c ttree" assumes "cont (\<lambda>x. paths (f x))" shows "cont f"
plus_cont: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> bcontfun"   and "g \<in> bcontfun" shows "(\<lambda>x. f x + g x) \<in> bcontfun"
mult_mult_of: fixes R :: "('b, 'c) ring_scheme" shows "(\<otimes>\<^bsub>mult_of R\<^esub>) = (\<otimes>\<^bsub>R\<^esub>)"
\<eta>_in_terms_of_\<phi>: fixes y :: "'b" assumes "B.ide y" shows "\<eta> y = \<eta>\<epsilon>.\<phi> y (F y)"
mksynlr_leq: fixes x :: "('a \<times> 'b) set"   and y :: "('a \<times> 'b) set" assumes "x \<in> synlr"   and "y \<in> synlr" shows "(mksynlr x \<le> mksynlr y) = (x \<subseteq> y)"
filtermap_nhds_minus: fixes a :: "'a" shows "filtermap uminus (nhds a) = nhds (- a)"
carrier_matD_1: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "dim_row A = nr"
rsp: shows "eq_onp (\<lambda>x. x \<in> {0..1}) 0 0"
sat_push_alt: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and e :: "nat \<times> nat" assumes "sat_push_precond f l e" shows "push_effect f e = augment_edge f e (cf_of f e)"
SourcesS5_L2: shows "Sources level2 sS5 = {sS2, sS4}"
measurable_bind: fixes f :: "'b \<Rightarrow> 'c measure"   and M :: "'b measure"   and N :: "'c measure"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd measure"   and R :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "(\<lambda>x. g (fst x) (snd x)) \<in> M \<Otimes>\<^sub>M N \<rightarrow>\<^sub>M subprob_algebra R" shows "(\<lambda>x. f x \<bind> g x) \<in> M \<rightarrow>\<^sub>M subprob_algebra R"
normalize_discover: fixes l' :: "(bool \<times> 'c) list"   and l :: "(bool \<times> 'c) list" assumes "canceled l'"   and "cancels_to l l'" shows "Cancelation.normalize l = l'"
unfold: shows "root2 = (\<Lambda> pat. rep2\<cdot>pat\<cdot>([::], pat))"
assms_3: fixes P :: "ccs"   and Q :: "ccs" assumes "Prop P Q" shows "F P \<leadsto><weakBisimulation> F Q"
sing_pow_exp: fixes w :: "'b list"   and a :: "'b" assumes "w \<in> [a]*" shows "w = [a] \<^sup>@ \<^bold>|w\<^bold>|"
parcs_xss: shows "parcs (list_digraph G_list) = \<Union> (sset xss)"
in_numbers_of_marks_eq: fixes m :: "nat"   and n :: "nat"   and bs :: "bool list" shows "(m \<in> numbers_of_marks n bs) = (m \<in> {n..<n + length bs} \<and> bs ! (m - n))"
funion_fminus_finter: fixes A :: "'b fset"   and B :: "'b fset" shows "A |-| B |\<union>| (A |\<inter>| B) = A"
pmf_cond: fixes p :: "'h pmf"   and s :: "'h set"   and x :: "'h" assumes "set_pmf p \<inter> s \<noteq> {}" shows "pmf (cond_pmf p s) x = (if x \<in> s then pmf p x / measure_pmf.prob p s else 0)"
map_default_def: fixes k :: "'b"   and v :: "'c"   and f :: "'c \<Rightarrow> 'c"   and m :: "('b, 'c) mapping" shows "map_default k v f m = map_entry k f (Mapping.default k v m)"
intro: fixes n :: "nat"   and fs :: "'a vec list" assumes "gram_schmidt_fs_lin_indpt n fs"   and "gram_schmidt_fs_int_axioms n fs" shows "gram_schmidt_fs_int n fs"
INF_insert: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and 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"   and f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (f ` insert a A) = inf (f a) (Inf (f ` A))"
starset_finite: fixes A :: "'b set" assumes "finite A" shows "*s* A = star_of ` A"
cones_map_mapsto: fixes f :: "'c" assumes "C.arr f" shows "D.cones_map f \<in> extensional (D.cones (C.cod f)) \<inter> (D.cones (C.cod f) \<rightarrow> D.cones (C.dom f))"
pick_in_set_inf: fixes S :: "nat set"   and n :: "nat" assumes "infinite S" shows "pick S n \<in> S"
cut_less_CommuteWithIntervalCut4: fixes t :: "'a" shows "CommuteWithIntervalCut4 (\<lambda>I. I \<down>< t)"
octo_of_real_of_nat_eq: fixes x :: "nat" shows "octo_of_real (real x) = octo_of_nat x"
size_less_than_median: fixes xs :: "'b list" shows "size {#y \<in># mset xs. y < median xs#} \<le> (length xs - 1) div 2"
finite_finfun_default_aux: fixes f :: "'a \<Rightarrow> 'b" assumes "f \<in> finfun" shows "finite {a. f a \<noteq> finfun_default_aux f}"
assms: fixes n :: "nat" assumes "m < n" shows "f n = (0::'a)"
simps_5: fixes i :: "'lbl strand_label"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" shows "tr\<^sub>p\<^sub>c ((i, insert\<langle>t,s\<rangle>) # A) D = tr\<^sub>p\<^sub>c A (List.insert (i, t, s) D)"
opnet_net_ips_net_tree_ips_init: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and ns :: "'d net_state"   and np :: "nat \<Rightarrow> ((nat \<Rightarrow> 'c) \<times> 'd, 'e seq_action) automaton"   and p :: "net_tree" assumes "(\<sigma>, ns) \<in> init (opnet np p)" shows "net_ips ns = net_tree_ips p"
bind_optionT_def: shows "Monad_Overloading.bind = bind_option Monad_Overloading.return Monad_Overloading.bind"
finite_subset: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "finite B" shows "finite A"
cod_simps_2: shows "trg_trg.cod.leg0 = trg_trg.leg0"
simps_5: fixes f :: "('d, 'c) trm"   and g :: "('d, 'c) trm" shows "FVT (Times f g) = FVT f \<union> FVT g"
init: fixes s :: "'a \<times> 'b" assumes "s \<in> init A" shows "P s"
chain_tSup_tllist_ord: fixes A :: "('c, 'b) tllist set"   and xs :: "('c, 'b) tllist" assumes "Complete_Partial_Order.chain local.tllist_ord A"   and "\<And>xs'. xs' \<in> A \<Longrightarrow> local.tllist_ord xs' xs" shows "local.tllist_ord (local.tSup A) xs"
R_lemma: fixes x :: "'a" assumes "x \<cdot> x = x" shows "x\<^sup>\<star> = (1::'a) + x"
elims: fixes x :: "('b, 'a, 'c) unboxed"   and y :: "'a option" assumes "cast_Ubx1 x = y" obtains xa :: "'a" where "x = OpUbx1 xa" and "y = Some xa"   | v :: "'b" where "x = OpDyn v__" and "y = None"   | v :: "'c" where "x = OpUbx2 v" and "y = None"
km_3: shows "(\<pi> @\<^bsup>n\<^esup> \<pi>') m = ipd (\<pi> k)"
distinct_1: fixes x2 :: "('a, 'b) Node" shows "Empty \<noteq> Single x2"
simps_3: fixes V :: "'b set"   and N :: "'b set"   and C :: "'b set"   and more :: "'c" shows "C \<lparr>V = V, N = N, C = C, \<dots> = more\<rparr> = C"
fff: shows "finite_field_factorization_int p f = (c, gs)"
face_cong_if_norm_eq: fixes xs :: "nat list"   and ys :: "nat list" assumes "rotate_min xs = rotate_min ys"   and "xs \<noteq> []"   and "ys \<noteq> []" shows "xs \<cong> ys"
many_strongly_connected_iff_11: fixes x :: "'a" assumes "is_inj x" shows "many_strongly_connected x = (x\<^sup>\<star> ; x\<^sup>T ; x \<le> x\<^sup>+)"
simulation_and_inverse_contrasimulation_impl_steps_closure_is_contrasimulation: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_simulation Rel Cal"   and "weak_reduction_contrasimulation (Rel\<inverse>) Cal" shows "weak_reduction_contrasimulation (stepsClosure Rel Cal) Cal"
nf_inres_RES: fixes X :: "'b set"   and x :: "'b" shows "nf_inres (RES X) x = (x \<in> X)"
arg_le_sf: fixes x :: "nat" shows "x \<le> sf x"
row_four_block_mat_2: fixes A :: "'a mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'a mat"   and nc2 :: "nat"   and C :: "'a mat"   and nr2 :: "nat"   and D :: "'a mat"   and i :: "nat" assumes "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2"   and "\<not> i < nr1"   and "i < nr1 + nr2" shows "row (four_block_mat A B C D) i = row C (i - nr1) @\<^sub>v row D (i - nr1)"
space_lebesgue_on: fixes S :: "'c set" shows "space (lebesgue_on S) = S"
i_def: shows "i \<equiv> \<lambda>x. ((1 - x) ^ (n - 1) / fact (n - 1)) *\<^sub>R Df (X + x *\<^sub>R H) n H"
ncc: fixes c :: "char list"   and rs :: "'a rule list" shows "no_call_to c rs = (c \<notin> called_c rs)"
in_bigo_zero_iff: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "(f \<in> O[F](\<lambda>_. 0::'c)) = (\<forall>\<^sub>F x in F. f x = (0::'c))"
h_2: shows "evalUni (elimVarUni_atom L2' f') x"
Inf_reductive: fixes \<iota> :: "'f inference" assumes "\<iota> \<in> Inf" shows "concl_of \<iota> < main_prem_of \<iota>"
body_generalize_fundef: fixes fd :: "('dyn, 'd, 'e, 'f, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" shows "body (Subx.generalize_fundef fd) = map Subx.generalize_instr (body fd)"
not_coll_ordered_lexI: fixes l :: "real \<times> real"   and x0 :: "real \<times> real"   and x1 :: "real \<times> real"   and r :: "real \<times> real" assumes "l \<noteq> x0"   and "Counterclockwise_2D_Arbitrary.lex x1 r"   and "Counterclockwise_2D_Arbitrary.lex x1 l"   and "Counterclockwise_2D_Arbitrary.lex r x0"   and "Counterclockwise_2D_Arbitrary.lex l x0"   and "ccw' x0 l x1"   and "ccw' x0 x1 r" shows "det3 x0 l r \<noteq> 0"
AtCatchExpr1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and a1 :: "('e, 'd, 'f) ann"   and c1 :: "('e, 'd, 'f) com"   and a :: "'e set \<times> ('e, 'd, 'f) com"   and a2 :: "('e, 'd, 'f) ann"   and c2 :: "('e, 'd, 'f) com" assumes "atomicsR \<Gamma> \<Theta> a1 c1 a" shows "atomicsR \<Gamma> \<Theta> (AnnComp a1 a2) (Catch c1 c2) a"
huntington_is_boolean: shows "class.boolean_algebra (-) uminus (\<sqinter>) (\<sqsubseteq>) (\<sqsubset>) (\<squnion>) \<bottom> \<top>"
trascl_less: fixes x :: "'a"   and y :: "'a"   and a :: "'a"   and R :: "('a \<times> 'a) set" assumes "x \<noteq> y"   and "(a, x) \<in> R\<^sup>*" shows "(a, x) \<in> (Restr R (- {y}))\<^sup>* \<or> (y, x) \<in> R O (Restr R (- {y}))\<^sup>*"
d1: shows "adjoint (mat (dim_vec v) 1 (\<lambda>(i, j). v $ i)) \<in> carrier_mat 1 n"
domintros_2: fixes c :: "nat"   and ns :: "C list" shows "stepC_dom (c, C (C [] # ns))"
comp2eq_def: fixes cmp :: "'a \<Rightarrow> 'b \<Rightarrow> comp_res"   and a :: "'a"   and b :: "'b" shows "comp2eq cmp a b \<equiv> case cmp a b of EQUAL \<Rightarrow> True | _ \<Rightarrow> False"
sum_eq: fixes s :: "real" shows "(\<Sum>x. indicat_real {0<..} s *\<^sub>R (complex_of_real s powr (z - 1) / complex_of_real (exp ((real x + a) * s)))) = indicat_real {0<..} s *\<^sub>R (complex_of_real s powr (z - 1) * complex_of_real (exp (- a * s) / (1 - exp (- s))))"
LLL_invD_7: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" assumes "LLL_invariant upw i fs" shows "lattice_of fs = L"
in_basis_finmapI: fixes I :: "'d set"   and S :: "'d \<Rightarrow> 'e set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> S i \<in> basis_proj" shows "Pi' I S \<in> basis_finmap"
linear_strict_order_split: fixes x :: "'a" shows "linear_orderable_2 x = linear_orderable_3 x"
two_comparisons_into_case_order_9: fixes x :: "'a"   and y :: "'a"   and P :: "'e"   and R :: "'e"   and Q :: "'e" shows "(if x = y then P else if lt y x then R else Q) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
even_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) zero"
ordering_axioms: shows "ordering (\<^bold>\<le>) (\<^bold><)"
sq_norm_vec_def: fixes v :: "'b vec" shows "\<parallel>v\<parallel>\<^sup>2 \<equiv> sum_list (map sq_norm (list_of_vec v))"
list_of_vec_append: fixes v :: "'b vec"   and w :: "'b vec" shows "list_of_vec (v @\<^sub>v w) = list_of_vec v @ list_of_vec w"
vimage_Collect_eq: fixes f :: "'c \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool" shows "f -` Collect P = {y. P (f y)}"
sums_offset: fixes f :: "nat \<Rightarrow> 'a"   and i :: "nat"   and l :: "'a" assumes "(\<lambda>n. f (n + i)) sums l" shows "f sums (l + sum f {..<i})"
extreme_bound_singleton: fixes x :: "'a" assumes "x \<in> A" shows "extreme_bound A (sympartp (\<sqsubseteq>)) {x} x"
dord_fun_zero_min: fixes ord :: "('a \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'c) \<Rightarrow> bool"   and s :: "'a \<Rightarrow> 'c" assumes "\<And>t. ord t t"   and "finite (supp_fun s)" shows "dord_fun ord 0 s"
IH: fixes vs :: "'a set"   and s :: "('a, 'b) fmap"   and s' :: "('a, 'b) fmap"   and ss :: "('a, 'b) fmap list" assumes "vars_change as vs s = s' # ss" shows "\<exists>as1 act as2. as = as1 @ act # as2 \<and> vars_change as1 vs s = [] \<and> state_succ (exec_plan s as1) act = s' \<and> vars_change as2 vs (state_succ (exec_plan s as1) act) = ss"
get_parent_reads: fixes node_ptr :: "(_) node_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "reads get_parent_locs (get_parent node_ptr) h h'"
use_spec_rule: fixes m :: "'b nres"   and \<Psi> :: "'b \<Rightarrow> bool"   and \<Phi> :: "'b \<Rightarrow> bool" assumes "m \<le> SPEC \<Psi>"   and "m \<le> SPEC (\<lambda>s. \<Psi> s \<longrightarrow> \<Phi> s)" shows "m \<le> SPEC \<Phi>"
gc_W_empty_locs_def: shows "gc_W_empty_locs = idle_locs \<union> init_locs \<union> sweep_locs \<union> {mark_load_fM, mark_store_fA, mark_end} \<union> prefixed ''mark_noop'' \<union> prefixed ''mark_loop_get_roots'' \<union> prefixed ''mark_loop_get_work''"
mag_inverse: fixes x :: "('e, 'f, 'g) Quantity_scheme" shows "mag (inverse x) = inverse (mag x)"
substL_congL: fixes l :: "('c, 'd) lit"   and \<pi>1 :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<pi>2 :: "Preliminaries.var \<Rightarrow> 'c trm" assumes "\<And>x. x \<in> varsL l \<Longrightarrow> \<pi>1 x = \<pi>2 x" shows "substL \<pi>1 l = substL \<pi>2 l"
Object_Thread_Throwable_neq_1: shows "Thread \<noteq> Object"
prv_eqvI: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (imp \<phi> \<chi>)"   and "prv (imp \<chi> \<phi>)" shows "prv (eqv \<phi> \<chi>)"
closed_seqs_memI: fixes s :: "nat \<Rightarrow> 'a"   and R :: "('a, 'b) ring_scheme" assumes "\<And>k. s k \<in> carrier R" shows "s \<in> carrier R\<^bsup>\<omega>\<^esup>"
has_integral_implies_lebesgue_measurable: fixes f :: "'b \<Rightarrow> 'c"   and I :: "'c"   and \<Omega> :: "'b set" assumes "(f has_integral I) \<Omega>" shows "(\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x) \<in> borel_measurable lebesgue"
flowsto_def: fixes X0 :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set"   and T :: "real set"   and CX :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set"   and X1 :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set" shows "Y.flowsto X0 T CX X1 = (\<forall>(x0, d0)\<in>X0. \<exists>h\<in>T. h \<in> Y.existence_ivl0 x0 \<and> (Y.flow0 x0 h, Y.Dflow x0 h o\<^sub>L d0) \<in> X1 \<and> (\<forall>h'\<in>{0<--<h}. (Y.flow0 x0 h', Y.Dflow x0 h' o\<^sub>L d0) \<in> CX))"
VA: shows "Vagree \<nu> \<omega> (\<Union>y\<in>{y. Inl (Inr y) \<in> SIGO (OProd ODE1 ODE2)}. FVT (\<sigma> y))"
transaction_views_for_same_hash: fixes t\<^sub>m :: "transaction\<^sub>m"   and participant :: "participant"   and t :: "transaction" assumes "t\<^sub>m \<in> set (Canton_Transaction_Tree.transaction_views_for participant t)" shows "hash_transaction t\<^sub>m = hash_transaction (embed_transaction t)"
Mem_Eats_I2: fixes H :: "fm set"   and u :: "tm"   and z :: "tm"   and t :: "tm" assumes "H \<turnstile> u EQ z" shows "H \<turnstile> u IN Eats t z"
proj2_Col_coeff_abs: fixes p :: "proj2"   and q :: "proj2"   and j :: "real"   and i :: "real" assumes "p \<noteq> q"   and "j \<noteq> 0" shows "proj2_Col_coeff p q (proj2_abs (i *\<^sub>R proj2_rep p + j *\<^sub>R proj2_rep q)) = i / j"
Int_commute: fixes A :: "'b set"   and B :: "'b set" shows "A \<inter> B = B \<inter> A"
select_convs_2: fixes heap :: "heap"   and locals :: "locals"   and more :: "'a" shows "locals \<lparr>heap = heap, locals = locals, \<dots> = more\<rparr> = locals"
funion_assoc: fixes A :: "'b fset"   and B :: "'b fset"   and C :: "'b fset" shows "A |\<union>| B |\<union>| C = A |\<union>| (B |\<union>| C)"
gcsort_in_def: fixes xs :: "'c list" shows "gcsort_in xs \<equiv> (0, [length xs], xs)"
FalseD: assumes "False \<in> set_pmf (primality_test P n)" shows "\<not> prime n"
value_less_eq: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "y \<in> carrier K"   and "v x < v y" shows "v x = v (x \<plusminus> y)"
ImagepI: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d" assumes "P x"   and "R x y" shows "Imagep R P y"
slice_edges_Append: fixes S :: "'node SDG_node set"   and cs :: "'edge list"   and as :: "'edge list"   and as' :: "'edge list"   and asx :: "'edge list"   and asx' :: "'edge list" assumes "slice_edges S cs as = as'"   and "slice_edges S (upd_cs cs as) asx = asx'" shows "slice_edges S cs (as @ asx) = as' @ asx'"
index_minus_mat_3: fixes A :: "'a mat"   and B :: "'a mat" shows "dim_col (A - B) = dim_col B"
pw_Sup_nofail: shows "nofail (Sup (f ` X)) = (\<forall>x\<in>f ` X. nofail x)"
tendsto_real_root: fixes f :: "'a \<Rightarrow> real"   and x :: "real"   and F :: "'a filter"   and n :: "nat" assumes "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. root n (f x)) \<longlongrightarrow> root n x) F"
bounded_out: fixes \<Gamma> :: "('a, 'b) web_scheme"   and x :: "'a" assumes "bounded_countable_web \<Gamma>"   and "x \<in> \<^bold>V\<^bsub>\<Gamma>\<^esub> - B \<Gamma>" shows "integral\<^sup>N (count_space (\<^bold>E\<^bsub>\<Gamma>\<^esub> `` {x})) (weight \<Gamma>) < \<top>"
dsjCont: fixes A :: "fmla" shows "A OR A IMP A \<in> boolean_axioms"
nonneg_bounded: shows "\<exists>K\<ge>0. \<forall>x. norm (g x) \<le> norm x * K"
lexordp_eq_simps_4: fixes x :: "'t"   and xs :: "'t list"   and y :: "'t"   and ys :: "'t list" shows "ord_term_lin.lexordp_eq (x # xs) (y # ys) = (x \<prec>\<^sub>t y \<or> \<not> y \<prec>\<^sub>t x \<and> ord_term_lin.lexordp_eq xs ys)"
pivot_tableau: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and v :: "nat \<Rightarrow> 'a" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "let s' = pivot x\<^sub>i x\<^sub>j s in v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> s' \<and> \<triangle> (\<T> s')"
dX_singular: fixes X :: "'x set"   and x :: "'x"   and d :: "'d" assumes "inj_on Xd X"   and "x \<in> X"   and "d = Xd x" shows "dX X d = {x}"
rel_pmf_bind_reflI: fixes p :: "'a pmf"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b pmf"   and g :: "'a \<Rightarrow> 'c pmf" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> rel_pmf R (f x) (g x)" shows "rel_pmf R (p \<bind> f) (p \<bind> g)"
cobounded1: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<preceq> a \<oplus> b"
seq'_simps_2: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and i :: "nat" shows "fx0.seq' f x (Suc i) = f (fx0.seq' f x i)"
has_integral_cmult_real: fixes c :: "real"   and f :: "'a \<Rightarrow> real"   and x :: "real"   and A :: "'a set" assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A" shows "((\<lambda>x. c * f x) has_integral c * x) A"
set: shows "bl2 \<in># remove1_mset bl \<B>"
assms_1: shows "T \<in> M \<rightarrow>\<^sub>M M"
primitive_part_dvd_primitive_partI: fixes p :: "'a poly"   and q :: "'a poly" assumes "p dvd q" shows "primitive_part p dvd primitive_part q"
spmf_eqI: fixes p :: "'a spmf"   and q :: "'a spmf" assumes "\<And>i. spmf p i = spmf q i" shows "p = q"
subst_map: fixes t :: "lam"   and ts :: "lam list"   and x :: "name"   and u :: "lam" shows "(t \<degree>\<degree> ts)[x::=u] = t[x::=u] \<degree>\<degree> map (\<lambda>t. t[x::=u]) ts"
plays_are_positions_conv: fixes p :: "'a list" shows "play (llist_of p) = (position p \<and> enat (length p) = 2 * N)"
H: fixes d :: "real" assumes "0 < d \<and> d \<le> 1" shows "(\<forall>l. 0 < deltaf d l) \<and> deltaf d \<longlonglongrightarrow> 0 \<and> ennreal (1 - d) \<le> emeasure M {x \<in> space M. 1 - d \<le> upper_asymptotic_density {n. \<forall>l\<in>{1..n}. - deltaf d l * real l < u n x - u (n - l) ((T ^^ l) x)}}"
isl_def: fixes sum :: "'c + 'd" shows "isl sum = (\<exists>x1. sum = Inl x1)"
simps_2: fixes m :: "'a SecurityInvariant"   and Ms :: "'a SecurityInvariant list"   and G :: "'a list_graph" shows "implc_get_offending_flows (m # Ms) G = implc_offending_flows m G @ implc_get_offending_flows Ms G"
var_assign_eval: fixes x :: "loc"   and s :: "loc \<Rightarrow> nat"   and n :: "nat" shows "(X x, s(x := n)) -|-> (n, s(x := n))"
vconn_implies_hpar_1: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright>" shows "src\<^sub>C \<mu> = src\<^sub>C f"
prv_eqv_trans: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c"   and \<psi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (Syntax_with_Connectives.eqv cnj imp \<phi> \<chi>)"   and "prv (Syntax_with_Connectives.eqv cnj imp \<chi> \<psi>)" shows "prv (Syntax_with_Connectives.eqv cnj imp \<phi> \<psi>)"
pdivides_iff_shell: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subfield K (A [X])"   and "p \<in> carrier (K [X]\<^bsub>A [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>A [X]\<^esub>)" shows "p pdivides\<^bsub>A [X]\<^esub> q = p divides\<^bsub>K [X]\<^bsub>A [X]\<^esub>\<^esub> q"
hom_mult: fixes x :: "'b poly"   and y :: "'b poly"   and p :: "'b poly" shows "(x * y) \<circ>\<^sub>p p = x \<circ>\<^sub>p p * y \<circ>\<^sub>p p"