hcju/isabelleps huggingface dataset:
measure_of_def: fixes \<Omega> :: "'b set"   and A :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> ennreal" shows "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>}, \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"
lin_vectorspace: fixes x :: "'a" assumes "x \<in> E" shows "vectorspace (lin x)"
infinite: fixes A :: "'a set" assumes "infinite A" shows "Lcm\<^sub>f\<^sub>i\<^sub>n A = (0::'a)"
le_infI1: fixes inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and a :: "'a::type"   and x :: "'a::type"   and b :: "'a::type" assumes "class.semilattice_inf (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool)"   and "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (a::'a::type) (x::'a::type)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) ((inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (a::'a::type) (b::'a::type)) (x::'a::type)"
Hash_imp_parts2: fixes evs :: "event list"   and X :: "msg"   and M :: "nat"   and Y :: "msg"   and N :: "nat" assumes "evs \<in> set_cr"   and "Hash \<lbrace>X, Nonce M, Y, Nonce N\<rbrace> \<in> parts (knows Spy evs)" shows "Nonce M \<in> parts (knows Spy evs) \<and> Nonce N \<in> parts (knows Spy evs)"
decomp: fixes i :: "nat"   and x :: "nat" assumes "i < k"   and "x\<^sub>1 \<le> x" shows "bs ! i * real x + (hs ! i) x = real ((ts ! i) x)"
nan: fixes x :: "('e, 'f) IEEE.float" assumes "is_nan x" shows "P x"
update_last: fixes k :: "'a"   and v :: "'b"   and v' :: "'b"   and al :: "('a \<times> 'b) list" shows "AList.update k v (AList.update k v' al) = AList.update k v al"
fresh_psubst: fixes x :: "Syntax.var"   and e :: "Syntax.term"   and vs :: "(Syntax.var, Syntax.term) fmap" assumes "atom x \<sharp> e"   and "atom x \<sharp> vs" shows "atom x \<sharp> psubst_term e vs"
RSubmodule_RSpan_single: fixes m :: "'c" assumes "m \<in> V" shows "RModule FG (\<cdot>) (GSpan [m]) \<and> GSpan [m] \<subseteq> V"
simps_20: fixes C :: "char list"   and e :: "char list exp"   and Cs :: "char list list"   and e' :: "char list exp" shows "fv (RI (C,e) ; Cs \<leftarrow> e') = fv e \<union> fv e'"
init_empty_stack: shows "gds_init gds \<le>\<^sub>n SPEC (gds_is_empty_stack gds)"
llength_geq_enat_lfiniteD: fixes xs :: "'a llist"   and n :: "nat" assumes "llength xs \<le> enat n" shows "lfinite xs"
distinct_1: fixes x2 :: "'g" shows "None \<noteq> \<lfloor>x2\<rfloor>"
assms_3: shows "f \<in> carrier L \<rightarrow> carrier L"
restrictP_subsume2: fixes B :: "'b set"   and A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "B \<subseteq> A" shows "r |` A |` B = r |` B"
rhs_eq_val_def: fixes v :: "(nat, 'a) mapping"   and x\<^sub>i :: "nat"   and c :: "'a"   and e :: "nat \<times> linear_poly" shows "rhs_eq_val v x\<^sub>i c e \<equiv> let x\<^sub>j = lhs e; a\<^sub>i\<^sub>j = coeff (rhs e) x\<^sub>i in \<langle>v\<rangle> x\<^sub>j + a\<^sub>i\<^sub>j *R (c - \<langle>v\<rangle> x\<^sub>i)"
SC_discr_imp_SC_ZObis: fixes indis :: "('a \<times> 'a) set"   and aval :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('c, 'b) com"   and tval :: "'c \<Rightarrow> 'a \<Rightarrow> bool" assumes "PL_Indis indis"   and "PL_Indis.SC_discr aval indis c" shows "PL_Indis.SC_ZObis tval aval indis c"
Min_le: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "le (local.Min A) x"
Disjunctive_Sup: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and X :: "('e \<Rightarrow> 'f) set" assumes "Disjunctive_fun R" shows "(R .. \<Squnion>X) = \<Squnion>{y. \<exists>x\<in>X. y = (R .. x)}"
fds_abs_converges_moebius_mu: fixes s :: "'a" assumes "1 < s \<bullet> (1::'a)" shows "fds_abs_converges (fds moebius_mu) s"
ideD_1: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" assumes "category C"   and "partial_magma.ide C a" shows "partial_magma.arr C a"
aprimedivisor_dvd: fixes p :: "'a"   and n :: "'a" assumes "prime p"   and "p dvd n" shows "aprimedivisor n dvd n"
ide_is_iso: fixes a :: "'c" assumes "C.ide a" shows "C.iso a"
orthogonal_transformation_inj: fixes f :: "'a \<Rightarrow> 'b" assumes "orthogonal_transformation f" shows "inj f"
sp: shows "supp \<alpha> - {atom i} \<subseteq> atom ` Vs"
subterm_subst_img_subterm: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('a, 'b) Term.term" assumes "t \<sqsubseteq> s \<cdot> \<theta>"   and "\<And>s'. s' \<sqsubseteq> s \<Longrightarrow> t \<noteq> s' \<cdot> \<theta>" shows "\<exists>w\<in>fv s. t \<sqsubset> \<theta> w"
simps: fixes usvs :: "[:'r:] \<times> [:'r:]"   and l :: "([:'r:] \<times> [:'r:]) tree"   and r :: "([:'r:] \<times> [:'r:]) tree" assumes "usvs \<noteq> \<bottom>" shows "abs2\<cdot>(Node\<cdot>usvs\<cdot>l\<cdot>r) = usvs"
min_cong1: fixes x :: "nat" assumes "0 < x"   and "x < p - 1" shows "[a ^ x \<noteq> 1] (mod p)"
match_def: fixes s :: "'b rexp"   and w :: "'a list" shows "match s w = final (fold delta w (init s))"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "\<And>j. \<lbrakk>j + 2 < n \<Longrightarrow> P (j + 1); \<And>x. \<lbrakk>j + 2 < n; x \<in> {j + 3..n - 1}\<rbrakk> \<Longrightarrow> P x\<rbrakk> \<Longrightarrow> P j" shows "P a0"
finite: shows "\<forall>ts v. nP v = ts \<longrightarrow> finite (taints ts)"
simps_5: fixes r :: "'b rexp"   and s :: "'b rexp" shows "\<guillemotleft>Times r s\<guillemotright> = Times \<guillemotleft>r\<guillemotright> \<guillemotleft>s\<guillemotright>"
rb_def: shows "rb = (let ((bs, uu_, uu_), z) = rb_aux (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), 0) in (bs, z))"
set_iterator_map_linord_def: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "('a \<times> 'e) set" shows "set_iterator_map_linord it S0 \<equiv> set_iterator_genord it S0 (\<lambda>(k, uu_) (k', uu_). k \<le> k')"
bet_out__suppa: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and E :: "'p"   and D :: "'p"   and F :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "Bet A B C"   and "E Out D F" shows "A B C SuppA D E F "
iso_bij_betwn_block_sets_inv: shows "bij_betw ((`) (inv_into \<V> \<pi>)) (set_mset \<B>') (set_mset \<B>)"
individually_rational_on_empty: fixes ds :: "'b set" shows "individually_rational_on ds {}"
filter_conv_foldr: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "List.filter P xs = foldr (\<lambda>x xs. if P x then x # xs else xs) xs []"
asset_ii_secret: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and B :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_prikey"   and "(Asset n, \<lbrace>Num 2, PubKey B\<rbrace>) \<in> s" shows "PriKey B \<notin> spied s"
simps_1: fixes t :: "'a bro" shows "tree (N1 t) = t"
those_eq_Some_iff: fixes ys :: "'a option list"   and xs :: "'a list" shows "(those ys = Some xs) = (ys = map Some xs)"
update_convs_1: fixes cpu_reg' :: "(CPU_register \<Rightarrow> word32) \<Rightarrow> CPU_register \<Rightarrow> word32"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "cpu_reg_update cpu_reg' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg' cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
span_union_is_sum: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> carrier V"   and "B \<subseteq> carrier V" shows "span (A \<union> B) = subspace_sum (span A) (span B)"
intro: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis\<^sub>1 :: "'d set"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and basis\<^sub>2 :: "'f set" assumes "finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1"   and "finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 basis\<^sub>2" shows "finite_dimensional_vector_space_pair_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 basis\<^sub>2"
fps_of_poly_power: fixes p :: "'b poly"   and n :: "nat" shows "fps_of_poly (p ^ n) = fps_of_poly p ^ n"
\<Z>_csgrp_commutative: fixes \<alpha> :: "V"   and \<SS> :: "V"   and a :: "V"   and b :: "V" assumes "\<Z>_csgrp \<alpha> \<SS>"   and "a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>"   and "b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>" shows "a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b = b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> a"
g: fixes i :: "nat" assumes "m < i"   and "i < n" shows "g i = f i"
leg0_composite: shows "rf.leg0 = f.leg0 \<cdot> rf.prj\<^sub>0"
simps_2: fixes var :: "nat"   and uu :: "real list" shows "convert_fm var TrueF uu = Some TrueFUni"
until_as_suntil: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "(\<phi> until \<psi>) \<omega> = ((\<phi> suntil \<psi>) \<omega> \<or> alw \<phi> \<omega>)"
add_mask_fold: fixes x :: "'b word"   and n :: "nat" shows "x + 2 ^ n - 1 = x + mask n"
map: fixes f :: "'b \<Rightarrow> 'a"   and x1 :: "'b list"   and x2 :: "'b" shows "map_inference f (Infer x1 x2) = Infer (map f x1) (f x2)"
Inf_pres_Inf: fixes F :: "('c \<Rightarrow> 'd) set" assumes "\<forall>f\<in>F. Inf_pres f" shows "Inf_pres (\<Sqinter> F)"
dominated_convergence_absolutely_integrable: fixes S :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>k. f k absolutely_integrable_on S"   and "h integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> norm (g x) \<le> h x"   and "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x" shows "g absolutely_integrable_on S"
lastMarkedNotInPrefixBeforeLastMarked: fixes M :: "('b \<times> bool) list" assumes "uniq (elements M)"   and "markedElements M \<noteq> []" shows "lastMarked M \<notin> set (elements (prefixBeforeLastMarked M))"
inversion_unit_disc: shows "inversion ` unit_disc = unit_disc_compl"
strategy_attracts_invalid_path: fixes P :: "'c llist"   and v :: "'c"   and w :: "'c"   and P' :: "'c llist"   and A :: "'c set"   and W :: "'c set" assumes "P = LCons v (LCons w P')"   and "v \<in> A - W"   and "w \<notin> A \<union> W" shows "\<not> visits_via P A W"
bounded_pbernpoly: fixes n :: "nat" obtains c :: "real" where "\<And>x. norm (pbernpoly n x) \<le> c"
suffix_snoc: fixes xs :: "'c list"   and ys :: "'c list"   and y :: "'c" shows "suffix xs (ys @ [y]) = (xs = [] \<or> (\<exists>zs. xs = zs @ [y] \<and> suffix zs ys))"
1: fixes A :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) set" assumes "A \<in> set ss" shows "homogeneous_set A"
rep_eq: fixes x :: "'a set" shows "space_as_set (ccspan x) = closure (cspan x)"
safe_set_def: fixes ode_ops :: "ode_ops"   and X :: "'c set" shows "safe_set ode_ops X = approx_form_spec (safe_form ode_ops) (list_of_eucl ` X) \<bind> (\<lambda>b1. isFDERIV_spec (D ode_ops) [0..<D ode_ops] (ode_e ode_ops) (list_of_eucl ` X) \<bind> (\<lambda>b2. RETURN (b1 \<and> b2)))"
DirProds_one'': fixes G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set" shows "\<one>\<^bsub>DirProds G I\<^esub> = (\<lambda>i\<in>I. \<one>\<^bsub>G i\<^esub>)"
empty_forest_refine: shows "((per_init V, {}), {}) \<in> uf_graph_rel"
kripke_rels_trc_worlds: fixes w :: "'e"   and w' :: "'e"   and M :: "('f, 'g, 'e) KripkeStructure"   and W :: "'e set" assumes "(w, w') \<in> (\<Union> (range (relations M)))\<^sup>*"   and "w \<in> worlds M"   and "kripke M"   and "W = worlds M" shows "w' \<in> W"
sorted_list_of_mset_set: fixes A :: "'b set" shows "sorted_list_of_multiset (mset_set A) = sorted_list_of_set A"
mono_restrict_space: fixes M :: "'b measure"   and N :: "'b measure"   and X :: "'b set" assumes "sets M \<subseteq> sets N" shows "sets (restrict_space M X) \<subseteq> sets (restrict_space N X)"
n_circ_def: shows "n_circ \<equiv> \<lambda>n. ((+++) p_circ ^^ n) p_circ_pt"
inf_export: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> - (x \<sqinter> y) = x \<sqinter> - y"
CC: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" assumes "\<^bold>\<not> F \<triangleright> \<Gamma> \<turnstile> \<bottom>" shows "\<Gamma> \<turnstile> F"
edges2: fixes e :: "'a set" assumes "e \<in> E" shows "card e = 2"
\<delta>0s: fixes i :: "'a" shows "0 < (SOME \<delta>. 0 < \<delta> \<and> (\<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < \<delta> \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon> / real CARD('a)))"
special_sg_G: fixes G :: "('e, 'f) Group_scheme" assumes "Group G" shows "G \<guillemotright> carrier G "
conway_assm2_def: fixes x :: "('b, 'c) boffa_mon" shows "conway_assm2 x = (\<forall>i. x\<^bsub>i,i\<^esub>\<^sup>\<star> = x\<^bsub>i,i\<^esub>)"
CONS_def: shows "CONS \<equiv> {U. \<exists>\<psi> s. learn_cons \<psi> U s}"
an_na: fixes a :: "ant" assumes "a \<noteq> \<infinity>"   and "0 \<le> a" shows "an (na a) = a"
elims: fixes x :: "'b heap"   and y :: "'b tree" assumes "hom x = y" obtains "x = heap.Empty" and "y = \<langle>\<rangle>"   | xa :: "'b" and hs :: "'b heap list" where "x = Hp xa hs" and "y = \<langle>homs hs, xa, \<langle>\<rangle>\<rangle>"
rt: shows "returns_to {x \<in> S. s x = 0} x"
step_8: shows "Max (poly_deg ` fst ` set (fst (shift_list (h, U) x qs, B - {(h, U)}))) \<le> Max (poly_deg ` fst ` set (fst (shift1 (shift_list (h, U) x qs, B - {(h, U)}))))"
IH: fixes b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and e :: "real" assumes "basis_wf (b # basis)"   and "e < 0" shows "eval_monom (1, es) basis \<in> \<omega>(\<lambda>x. b x powr e)"
rel_sel: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "('d, 'f, 'h) gpv"   and b :: "('e, 'g, 'h) gpv" shows "rel_gpv R1 R2 a b = rel_spmf (rel_generat R1 R2 (rel_rpv R1 R2)) (the_gpv a) (the_gpv b)"
join_ir_R: fixes i :: "value list"   and r :: "nat \<Rightarrow>f value option"   and n :: "nat" shows "join_ir i r (R n) = r $ n"
weakly_preserves_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "D.isomorphic (F (trg\<^sub>C \<mu>)) (trg\<^sub>D (F \<mu>))"
st_proj2_inv: fixes x :: "'b sphere" assumes "x \<noteq> bot_sphere" shows "st_proj2_inv (st_proj2 x) = x"
polyfun_tensors_from_net: fixes m :: "(nat \<times> nat) convnet"   and "is" :: "nat list"   and j :: "nat"   and s :: "bool" assumes "valid_net m"   and "is \<lhd> input_sizes m"   and "j < output_size m" shows "polyfun {..<count_weights s m} (\<lambda>f. Tensor.lookup (tensors_from_net (insert_weights s m f) $ j) is)"
n_in: shows "\<iota> \<in> Red_I (lnth Ns i0)"
in_set_update_listD: fixes y :: "nat \<times> 'd"   and n :: "nat"   and x :: "'d"   and ys :: "(nat \<times> 'd) list" assumes "y \<in> set (update_list n x ys)" shows "y = (n, x) \<or> y \<in> set ys"
\<F>\<G>_semantics: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<F>\<G> \<phi> w = {\<psi> \<in> subformulas\<^sub>\<nu> \<phi>. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>)}"
simps_3: fixes n :: "nat"   and F :: "'b formula" shows "cut_bound' n (\<^bold>\<not> F) = cut_bound' n F"
assms_2: shows "X \<in> obj\<^bsub>CatDom FId' C\<^esub>"
imirror_finite: fixes I :: "nat set" assumes "finite I" shows "finite (imirror I)"
cc_single_valued': fixes ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option"   and b :: "contour"   and d :: "d"   and ds :: "d list" assumes "\<forall>b'\<in>contours_in_ve ve. b' < b"   and "\<forall>b'\<in>contours_in_d d. b' < b"   and "\<forall>d'\<in>set ds. \<forall>b'\<in>contours_in_d d'. b' < b" shows "single_valued (\<F>\<cdot>(Discr (d, ds, ve, b))) \<and> (\<forall>((lab, \<beta>), t)\<in>\<F>\<cdot>(Discr (d, ds, ve, b)). \<exists>b'. b' \<in> ran \<beta> \<and> b \<le> b')"
substitute_simps_2: fixes x :: "'b"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree" shows "(x # xs \<in> paths (substitute f T t)) = (possible t x \<and> xs \<in> paths (substitute (f_nxt f T x) T (nxt t x \<otimes>\<otimes> f x)))"
eventually_returns_to: fixes S :: "'a set"   and s :: "'a \<Rightarrow> real"   and x :: "'a"   and Ds :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" assumes "returns_to {x \<in> S. s x = 0} x"   and "closed S"   and "\<And>x. (s has_derivative blinfun_apply (Ds x)) (at x)"   and "\<And>x. isCont Ds x"   and "\<forall>\<^sub>F x in at (poincare_map {x \<in> S. s x = 0} x). s x = 0 \<longrightarrow> x \<in> S"   and "blinfun_apply (Ds (poincare_map {x \<in> S. s x = 0} x)) (f (poincare_map {x \<in> S. s x = 0} x)) \<noteq> 0"   and "x \<notin> S \<or> s x \<noteq> 0 \<or> blinfun_apply (Ds x) (f x) \<noteq> 0" shows "eventually (returns_to {x \<in> S. s x = 0}) (at x)"
fsubset_finsertI: fixes B :: "'b fset"   and a :: "'b" shows "B |\<subseteq>| finsert a B"
parAssocLeft: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs"   and Q :: "ccs"   and R :: "ccs" assumes "\<And>S T U. (S \<parallel> T \<parallel> U, S \<parallel> (T \<parallel> U)) \<in> Rel" shows "P \<parallel> Q \<parallel> R \<leadsto>[Rel] P \<parallel> (Q \<parallel> R)"
intP_def: shows "intP \<equiv> ProblemIkTpart_TEModel.intP parOf tpD tpFD (\<lambda>\<sigma>. D) eintF eintP"
inject_4: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) cmd"   and x53 :: "('a, 'b, 'c) cmd"   and y51 :: "'c"   and y52 :: "('a, 'b, 'c) cmd"   and y53 :: "('a, 'b, 'c) cmd" shows "(Ch x51 x52 x53 = Ch y51 y52 y53) = (x51 = y51 \<and> x52 = y52 \<and> x53 = y53)"
minus_power_mult_self: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "times (power.power one times (uminus a) n) (power.power one times (uminus a) n) = power.power one times a (2 * n)"
intro: fixes s\<^sub>0 :: "'a"   and lstep :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'a" assumes "df_system s\<^sub>0 lstep"   and "run_axioms s\<^sub>0 lstep s" shows "DF_System.run s\<^sub>0 lstep s"
simps: fixes bs :: "atom set"   and x :: "'b"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom set"   and y :: "'b" shows "(bs, x) \<approx>set R f p (cs, y) = (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs)"
rpr_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "rpr A r \<equiv> complete A r \<and> refl_on A r \<and> trans r"
not_neutral_contains_not_neutral: fixes g :: "'d \<Rightarrow> 'a"   and A :: "'d set" assumes "sum_0 g A \<noteq> (0::'a)" obtains a :: "'d" where "a \<in> A" and "g a \<noteq> (0::'a)"
cell_subdivision_lemma: fixes \<F> :: "'b set set"   and d :: "int"   and I :: "('b \<times> real) set" assumes "finite \<F>"   and "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"   and "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> d"   and "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X"   and "finite I" shows "\<exists>\<G>. \<Union> \<G> = \<Union> \<F> \<and> finite \<G> \<and> (\<forall>C\<in>\<G>. \<exists>D. D \<in> \<F> \<and> C \<subseteq> D) \<and> (\<forall>C\<in>\<F>. \<forall>x\<in>C. \<exists>D. D \<in> \<G> \<and> x \<in> D \<and> D \<subseteq> C) \<and> (\<forall>X\<in>\<G>. polytope X) \<and> (\<forall>X\<in>\<G>. aff_dim X \<le> d) \<and> (\<forall>X\<in>\<G>. \<forall>Y\<in>\<G>. X \<inter> Y face_of X) \<and> (\<forall>X\<in>\<G>. \<forall>x\<in>X. \<forall>y\<in>X. \<forall>a b. (a, b) \<in> I \<longrightarrow> a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or> b \<le> a \<bullet> x \<and> b \<le> a \<bullet> y)"
fminusD1: fixes c :: "'b"   and A :: "'b fset"   and B :: "'b fset" assumes "c |\<in>| A |-| B" shows "c |\<in>| A"
mem_simps_10: fixes b :: "'a"   and B :: "'b \<Rightarrow> 'a set"   and A :: "'b set" shows "(b \<in> \<Inter> (B ` A)) = (\<forall>x\<in>A. b \<in> B x)"
vector_scaleR_commute: fixes A :: "(('b, 'd) vec, 'c) vec"   and c :: "real"   and x :: "('b, 'd) vec" shows "A *v c *\<^sub>R x = c *\<^sub>R (A *v x)"
ground_wfP: shows "wfP (\<lambda>s t. ground t \<and> t >\<^sub>t\<^sub>w s)"
span_same: fixes S :: "'a Matrix.vec set" assumes "S \<subseteq> mat_kernel (char_matrix A lam)" shows "local.span S = NC.span S"
meet_top: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "bounded_lattice L"   and "x \<in> carrier L" shows "x \<sqinter>\<^bsub>L\<^esub> \<top>\<^bsub>L\<^esub> = x"
genby_lcoset_order2: fixes s :: "'b"   and w :: "'b" assumes "s + s = (0::'b)" shows "w +o \<langle>{s}\<rangle> = {w, w + s}"
invariant_1_2E: fixes rai :: "int poly \<times> rat \<times> rat" assumes "invariant_1_2 rai" obtains "invariant_1 rai" and "1 < degree (poly_real_alg_1 rai)"
vpeq_sym: fixes u :: "partition_id_t"   and s :: "state_t"   and t :: "state_t" assumes "vpeq u s t" shows "vpeq u t s"
sel: fixes out :: "'b"   and c :: "('a, 'b, 'c) rpv" shows "the_gpv (Generative_Probabilistic_Value.Pause out c) = return_spmf (IO out c)"
matrix_def: fixes inner_cols :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec \<Rightarrow> 'a"   and inner_rows :: "('a, 'c) vec \<Rightarrow> ('a, 'c) vec \<Rightarrow> 'a" shows "Generalizations2.matrix inner_cols inner_rows \<equiv> vec_real_inner (\<lambda>x y. x * cnj_class.cnj y) inner_cols \<and> vec_real_inner (\<lambda>x y. x * cnj_class.cnj y) inner_rows"
h3: shows "\<exists>m\<in>monomials (p + q). lookup m var = n"
hyps_1: shows "\<langle>c\<^sub>1,s\<rangle> \<rightarrow> \<langle>c\<^sub>1',s'\<rangle>"
compare_double_greater: fixes x :: "double"   and y :: "double" shows "(0 < compare_double x y) = (\<not> Code_Double.is_nan x \<and> Code_Double.is_nan y \<or> fcompare_double x y = ccode.Gt)"
D0'_D: shows "D0' \<cdot> \<eta>0' = D"