hcju/isabelleps huggingface dataset:
between_distinct_r12: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "distinct vs"   and "ram1 \<noteq> ram2" shows "distinct (ram1 # between vs ram1 ram2 @ [ram2])"
homotopic_on_components_eq: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "locally connected S \<or> compact S"   and "ANR T" shows "homotopic_with_canon (\<lambda>x. True) S T f g = ((continuous_on S f \<and> f ` S \<subseteq> T \<and> continuous_on S g \<and> g ` S \<subseteq> T) \<and> (\<forall>C\<in>components S. homotopic_with_canon (\<lambda>x. True) C T f g))"
chine_simps_1: shows "C.arr \<mu>\<nu>.chine"
normalize_dvd_iff: fixes a :: "'a"   and b :: "'a" shows "(normalize a dvd b) = (a dvd b)"
distinct_set_list: fixes A :: "'a set" assumes "finite A" shows "distinct (list A) \<and> set (list A) = A"
col_swaprows_3: fixes k :: "nat"   and n :: "nat"   and l :: "nat" assumes "k < n"   and "l < n" shows "col (swaprows_mat n k l) l = unit_vec n k"
LIMSEQ_indicator_decseq: fixes A :: "nat \<Rightarrow> 'a set"   and x :: "'a" assumes "decseq A" shows "(\<lambda>i. indicator (A i) x) \<longlonglongrightarrow> indicator (\<Inter> (range A)) x"
limproc_Seq_D1: fixes Y :: "nat \<Rightarrow> 'c process"   and S :: "'c process" assumes "chain Y" shows "D (lim_proc (range Y) `;` S) = D (lim_proc (range (\<lambda>i. Y i `;` S)))"
cases: fixes partial_hd :: "'a list \<Rightarrow> 'a option"   and a1 :: "'a list"   and a2 :: "'a" assumes "A.hd_predicate_in_locale partial_hd a1 a2" obtains xs :: "'a list" and x :: "'a" where "a1 = xs" and "a2 = x" and "partial_hd xs = Some x"
absorb_iff2: fixes a :: "'e multiset"   and b :: "'e multiset" shows "(a \<subseteq># b) = (a \<union># b = b)"
\<tau>Runs_terminate_final1: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and final1 :: "'g \<Rightarrow> bool"   and final2 :: "'i \<Rightarrow> bool"   and s1 :: "'g"   and tls1 :: "('h, 'g option) tllist"   and s2 :: "'i"   and tls2 :: "('j, 'i option) tllist"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and s1' :: "'g" assumes "delay_bisimulation_final_base trsys1 trsys2 bisim \<tau>move1 \<tau>move2 final1 final2"   and "\<tau>trsys.\<tau>Runs trsys1 \<tau>move1 s1 tls1"   and "\<tau>trsys.\<tau>Runs trsys2 \<tau>move2 s2 tls2"   and "tllist_all2 tlsim (rel_option bisim) tls1 tls2"   and "tfinite tls1"   and "terminal tls1 = \<lfloor>s1'\<rfloor>"   and "final1 s1'" shows "\<exists>s2'. tfinite tls2 \<and> terminal tls2 = \<lfloor>s2'\<rfloor> \<and> final2 s2'"
gnz: fixes x :: "'a" assumes "x \<in> sphere (0::'a) 1 \<inter> S" shows "g x \<noteq> (0::'a)"
numsub_nz: fixes a :: "Ferrack.num"   and b :: "Ferrack.num" assumes "nozerocoeff a"   and "nozerocoeff b" shows "nozerocoeff (numsub a b)"
zs_a: shows "set_mset Zs \<subseteq> A"
dagger_of_id_is_id: fixes n :: "nat" shows "1\<^sub>m n\<^sup>\<dagger> = 1\<^sub>m n"
distinct_44: fixes x81 :: "nat"   and x82 :: "msg"   and x4 :: "nat" shows "Crypt x81 x82 \<noteq> Pan x4"
next_recvupd_vacant: fixes s :: "('d, 'c, 'e) configuration_scheme stream"   and t :: "'c" assumes "NrecVacantUpto (shd s) t"   and "next_recvupd s" shows "NrecVacantUpto (shd (stl s)) t"
Fract_raw_def: fixes a :: "int"   and b :: "int" shows "Fract_raw a b = (if b = 0 then (0, 1) else (a, b))"
tso_store_inv_def: shows "tso_store_inv = (\<lambda>s. (\<forall>x. x \<in> set (sys_mem_store_buffers gc s) \<longrightarrow> gc_writes x) \<and> (\<forall>x xa. xa \<in> set (sys_mem_store_buffers (mutator x) s) \<longrightarrow> mut_writes xa))"
ospec_cons: fixes m :: "'d option"   and Q :: "'d \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool" assumes "ospec m Q"   and "\<And>r. Q r \<Longrightarrow> P r" shows "ospec m P"
matrix_to_iarray_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "matrix_to_iarray A = IArray (map (vec_to_iarray \<circ> ($) A \<circ> mod_type_class.from_nat) [0..<CARD('d)])"
lprefix_lhdD: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "xs \<sqsubseteq> ys"   and "\<not> lnull xs" shows "lhd xs = lhd ys"
reduce_column_aux_preserves: fixes H :: "'a mat"   and m :: "nat"   and n :: "nat"   and P_init :: "'a mat"   and K_init :: "'a mat"   and P :: "'a mat"   and K :: "'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "nat list"   and i :: "nat" assumes "H \<in> carrier_mat m n"   and "P_init \<in> carrier_mat m m"   and "K_init \<in> carrier_mat m n"   and "P_init * H = K_init"   and "(P, K) = local.reduce_column_aux div_op xs H (P_init, K_init)"   and "0 < m"   and "invertible_mat P_init"   and "0 \<notin> set xs"   and "i \<notin> set xs"   and "i < m" shows "Matrix.row K i = Matrix.row K_init i"
one: shows "\<forall>i<m. \<exists>no. \<forall>j<n. \<forall>n\<ge>no. P n i j"
Vinv3_v_round: fixes r :: "nat"   and v_f :: "process \<Rightarrow> val option"   and d_f :: "process \<Rightarrow> val option" shows "{Vinv3 \<inter> Vinv1} v_round r v_f d_f {> Vinv3}"
assms_2: assumes "{#a, b#} \<subseteq># A" shows "P"
P': fixes x :: "'a"   and x' :: "'b" shows "P x x' = \<up> (P' x x')"
order_less_irrefl: fixes x :: "'k" shows "\<not> x < x"
IrrE_1: fixes K :: "'a set"   and x :: "'a" assumes "subfield K R"   and "x \<in> carrier R"   and "(algebraic over K) x" shows "Irr K x \<in> carrier (K [X])"
assms: fixes w :: "bool stream" shows "f w = g (stake n w)"
2: shows "Map F = \<lbrakk>REP F\<rbrakk>\<^sub>B"
preserves_cod: fixes f :: "'a" assumes "\<tau>.A.arr f" shows "\<tau>.B.cod (\<nu> f) = H (\<tau>.A.cod f)"
div_smult_right: fixes a :: "'b"   and x :: "'b poly"   and y :: "'b poly" assumes "a \<noteq> (0::'b)" shows "x div smult a y = smult (inverse a) (x div y)"
surjective: fixes r :: "'a env_scheme" shows "r = \<lparr>prg = prg r, cls = cls r, lcl = lcl r, \<dots> = env.more r\<rparr>"
chain_sup_def: fixes S :: "nat \<Rightarrow> ('d, 'e) labeled_graph" shows "chain_sup S \<equiv> LG (\<Union>i. edges (S i)) (\<Union>i. vertices (S i))"
scale_right_diff_distrib: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and x :: "'b"   and y :: "'b" assumes "module scale" shows "scale a (x - y) = scale a x - scale a y"
homogeneous_homogenize: fixes x :: "'c"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "homogeneous (homogenize x p)"
deg_sectI: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and d :: "nat" assumes "t \<in> .[X]"   and "deg_pm t = d" shows "t \<in> deg_sect X d"
mset_neg_zcount_nonneg: fixes M :: "'a zmultiset"   and t :: "'a" assumes "mset_neg M = {#}" shows "0 \<le> zcount M t"
pr0_simps_1: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "arr (hfsetcat.pr0 a b)"
homotopic_with_imp_continuous_maps: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "homotopic_with P X Y f g" shows "continuous_map X Y f \<and> continuous_map X Y g"
refl: fixes a :: "'a multiset" shows "a \<subseteq># a"
gcd_mult_right: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd (times a c) (times b c) = normalize (times (gcd b a) c)"
finset_def: shows "finset \<equiv> map_fun id Abs_fset set"
partitions_of_unityE: fixes X :: "'c \<Rightarrow> 'a set"   and A :: "'c set" assumes "carrier \<subseteq> \<Union> (X ` A)"   and "\<And>i. i \<in> A \<Longrightarrow> open (X i)" obtains \<phi> :: "'c \<Rightarrow> 'a \<Rightarrow> real" where "\<And>i. i \<in> A \<Longrightarrow> diff_fun k charts (\<phi> i)" and "\<And>i x. \<lbrakk>i \<in> A; x \<in> carrier\<rbrakk> \<Longrightarrow> 0 \<le> \<phi> i x" and "\<And>i x. \<lbrakk>i \<in> A; x \<in> carrier\<rbrakk> \<Longrightarrow> \<phi> i x \<le> 1" and "\<And>x. x \<in> carrier \<Longrightarrow> (\<Sum>i | i \<in> A \<and> \<phi> i x \<noteq> 0. \<phi> i x) = 1" and "\<And>i. i \<in> A \<Longrightarrow> csupport_on carrier (\<phi> i) \<inter> carrier \<subseteq> X i" and "locally_finite_on carrier A (\<lambda>i. csupport_on carrier (\<phi> i))"
rep_eq: shows "(*\<^sub>v) 0 = (\<lambda>x. 0::'c)"
SL_def: fixes A :: "'b set" shows "SL A = Pi_pmf A 0 (\<lambda>_. geometric_pmf p)"
mult_scalar_one: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "1 \<odot> p = p"
flatten_sigma_2_monotone: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and M :: "'b ltln set"   and M' :: "'b ltln set" assumes "w \<Turnstile>\<^sub>n \<phi>[M]\<^sub>\<Sigma>\<^sub>2"   and "M \<subseteq> M'" shows "w \<Turnstile>\<^sub>n \<phi>[M']\<^sub>\<Sigma>\<^sub>2"
memo_def: fixes f :: "'v \<Rightarrow> 'm"   and x :: "'v" shows "memo f x = get (\<lambda>table. case table x of None \<Rightarrow> bind (f x) (\<lambda>y. update (\<lambda>m. m(x \<mapsto> y)) (return y)) | Some y \<Rightarrow> return y)"
assoc_p_subid: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> y \<cdot> (z \<cdot> 1\<^sub>\<pi>) = x \<cdot> (y \<cdot> (z \<cdot> 1\<^sub>\<pi>))"
lincomb_Cons: fixes smult :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and r :: "'b"   and rs :: "'b list"   and m :: "'a"   and ms :: "'a list" shows "scalar_mult.lincomb smult (r # rs) (m # ms) = smult r m + scalar_mult.lincomb smult rs ms"
card_Mi_twice_card_Vi: fixes L :: "'b \<Rightarrow> nat"   and E :: "('b \<times> 'b) set"   and V :: "'b set"   and M :: "('b \<times> 'b) set"   and i :: "nat" assumes "OSC L E \<and> matching V E M \<and> 1 < i" shows "2 * card (matching_i i V E M L) = card (V_i i V E M L)"
prv_prfOf: fixes \<phi> :: "'c" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "prv \<phi> = (\<exists>n\<in>num. prfOf n \<phi>)"
simps: fixes q :: "'b"   and n :: "nat" shows "pre_oldest_tokens q n = {x. \<exists>q'. oldest_token q' n = Some x \<and> q = \<delta> q' (w n)} \<union> (if q = q\<^sub>0 then {Suc n} else {})"
exhaust: fixes y :: "'b match_expr" obtains x1 :: "'b" where "y = Match x1"   | x2 :: "'b match_expr" where "y = MatchNot x2"   | x31 :: "'b match_expr" and x32 :: "'b match_expr" where "y = MatchAnd x31 x32"   | "y = MatchAny"
cases: fixes a1 :: "key"   and a2 :: "agent"   and runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option" assumes "(a1, a2) \<in> knC runz" obtains Ra :: "fid_t" and A :: "agent" and B :: "agent" and K :: "key" and al :: "atom list" where "a1 = K__" and "a2 = A__" and "runz Ra = Some (Init, [A__, B__], aKey K__ # al__)"   | Rb :: "fid_t" and A :: "agent" and B :: "agent" and K :: "key" and al :: "atom list" where "a1 = Ka__" and "a2 = B" and "runz Rb = Some (Resp, [Aa__, B], aKey Ka__ # al)"   | Rs :: "fid_t" where "a1 = sesK (Rs $ sk)" and "a2 = Sv" and "Rs \<in> dom runz" and "fst (the (runz Rs)) = Serv"   | K :: "key" and A :: "agent" where "a1 = K" and "a2 = A" and "(K, A) \<in> keySetup"
word_upto_eq_upto: fixes s :: "nat"   and e :: "nat" assumes "s \<le> e"   and "e \<le> unat (- 1)" shows "word_upto (word_of_nat s) (word_of_nat e) = map word_of_nat [s..<Suc e]"
inv_unknown_set: fixes G :: "graph"   and S :: "state" assumes "Dijkstra.inv G S" shows "unknown_set S = verts G - known_set S"
dist_z_iff: fixes v :: "nat"   and v' :: "nat" shows "dist v 0 v' = (v' = v)"
routing_ipassmt_ipassmt_sanity_disjoint: fixes rt :: "'b routing_rule list" assumes "valid_prefixes rt" shows "ipassmt_sanity_disjoint (map_of (routing_ipassmt rt))"
updown: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and lm :: "nat"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" assumes "p \<in> sparsegrid dm lm" shows "updown dm lm \<alpha> p = (\<Sum>p'\<in>sparsegrid dm lm. \<alpha> p' * l2 p' p)"
qFresh_imp_ex_qAFresh2: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and xs :: "'e"   and x :: "'d"   and ys :: "'e"   and y :: "'d" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGood X"   and "qFresh xs x X"   and "qFresh ys y X" shows "\<exists>X'. FixVars.qGood X' \<and> X #= X' \<and> qAFresh xs x X' \<and> qAFresh ys y X'"
rel_gpv''_pos_distr: fixes A :: "'g \<Rightarrow> 'm \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'n \<Rightarrow> bool"   and R :: "'i \<Rightarrow> 'o \<Rightarrow> bool"   and A' :: "'m \<Rightarrow> 'j \<Rightarrow> bool"   and C' :: "'n \<Rightarrow> 'k \<Rightarrow> bool"   and R' :: "'o \<Rightarrow> 'l \<Rightarrow> bool" shows "rel_gpv'' A C R OO rel_gpv'' A' C' R' \<le> rel_gpv'' (A OO A') (C OO C') (R OO R')"
ts_tuple_rel': shows "ts_tuple_rel (set (map (\<lambda>(t, rel). (t, join rel pos X)) auxlist)) = {tas \<in> ts_tuple_rel (set (linearize data_prev) \<union> set (linearize data_in)). valid_tuple tuple_since' tas}"
prod_fun_imageE: fixes c :: "'e \<times> 'f"   and f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and R :: "('g \<times> 'h) set" assumes "c \<in> map_prod f g ` R" obtains x :: "'g" and y :: "'h" where "c = (f x, g y)" and "(x, y) \<in> R"
no_reg_aval_swap_regs: fixes a :: "vname aexp"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option"   and r' :: "nat \<Rightarrow>f value option" assumes "max_reg a = None" shows "aval a (join_ir i r) = aval a (join_ir i r')"
th: fixes a :: "'a fls"   and b :: "'a fls" shows "(dist a b = 0) = (a = b)"
IH2_3: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M2" shows "subterms f \<subseteq> Mapping.keys M2"
assms_2: shows "simplex (x_mat_eq_vec c A\<^sup>T) = Inr X"
assms: shows "change_on_1 (block i t) (gap i t)"
basis_inD_aux: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> \<E>'"   and "\<E>'\<^sub>1.basis_in \<E> X" shows "indep_system.basis \<E> (\<E>'\<^sub>1.indep_in \<E>) X"
intros_13: fixes i :: "nat"   and i' :: "nat"   and i'' :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and R :: "nat set"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set" assumes "i \<noteq> i'"   and "i \<noteq> i''"   and "\<sigma>' i = \<sigma> i" shows "((\<sigma>, NodeS i s R), connect(i', i''), \<sigma>', NodeS i s R) \<in> onode_sos S"
rref_condition1: fixes A :: "(('d, 'e) vec, 'f) vec" assumes "reduced_row_echelon_form A" shows "\<forall>i. is_zero_row i A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row j A)"
super_orientable_irreflexive_inf: fixes x :: "'a" assumes "super_orientable x" shows "irreflexive_inf x"
surj_to_el2: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and b :: "'e" assumes "surj_to f A B"   and "b \<in> B" shows "\<exists>a\<in>A. f a = b"
wtE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "Ik.wtE \<xi> \<equiv> \<forall>x. intT (TE.tpOfV x) (\<xi> x)"
three_neq_aux_3: shows "\<^bold>2 \<noteq> \<^bold>3"
subsetD: fixes A :: "'e set"   and B :: "'e set"   and c :: "'e" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
mono_trans_def: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "mono_trans t \<equiv> \<forall>P Q. sound P \<and> sound Q \<and> P \<tturnstile> Q \<longrightarrow> t P \<tturnstile> t Q"
power_less_top_ennreal: fixes x :: "ennreal"   and n :: "nat" shows "(x ^ n < \<top>) = (x < \<top> \<or> n = 0)"
fmdomE: fixes x :: "'d"   and m :: "'d \<rightharpoonup>\<^sub>f 'e" assumes "x |\<in>| fmdom m" obtains y :: "'e" where "fmlookup m x = Some y"
set_less_eq_aux_code: fixes xs :: "'a list"   and ys :: "'a list" assumes "sorted xs"   and "distinct xs"   and "sorted ys"   and "distinct ys" shows "(set xs \<sqsubseteq>' set ys) = ord.lexordp_eq (\<lambda>x y. y < x) xs ys"
map_index_cong: fixes xs :: "'a list"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<forall>p<length xs. f p (xs ! p) = g p (xs ! p)" shows "map_index f xs = map_index g xs"
IH_1: fixes i :: "int"   and j :: "int" assumes "inorder t1 = [i..j]"   and "i \<le> j + 1" shows "fst ` leaves i t1 = {i..j + 1}"
gen_wf_\<alpha>e_def: shows "gen_wf_\<alpha>e \<equiv> map_fun Rep_gen_cfg_wf id gen_\<alpha>e"
major: shows "\<tau>Exec_movest ci P t es h (stk, loc, pc, xcp) (stk'', loc'', pc'', xcp'')"
preserves_left_adjoint: fixes f :: "'a" assumes "is_left_adjoint f" shows "S.is_left_adjoint (S.UP f)"
preserves_comp: fixes g :: "'b"   and f :: "'b" assumes "Adj.D.seq g f" shows "F (g \<cdot>\<^sub>D f) = F g \<cdot>\<^sub>C F f"
normalized_tableau_def: fixes t :: "(nat \<times> linear_poly) list" shows "\<triangle> t \<equiv> distinct (map lhs t) \<and> lvars t \<inter> rvars t = {} \<and> 0 \<notin> rhs ` set t"
of_nat_CARD_eq_0: shows "of_nat CARD('q) = 0"
dist_j: shows "(\<O>\<^sub>i \<union> all_acquired sb\<^sub>i) \<inter> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb\<^sub>j = {}"
soundI_memv: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" assumes "\<And>I \<nu>. \<lbrakk>is_interp I; \<And>\<phi> \<nu>. List.member SG \<phi> \<Longrightarrow> \<nu> \<in> seq_sem I \<phi>\<rbrakk> \<Longrightarrow> \<nu> \<in> seq_sem I C" shows "sound (SG, C)"
assms_6: fixes e1 :: "pexpr"   and e2 :: "pexpr" assumes "e = PExpr2 (PMul e1 e2)" shows "P"
alphaBoundOutput: fixes x' :: "name"   and P :: "pi"   and a :: "name"   and x :: "name" assumes "x' \<sharp> P" shows "a<\<nu>x> \<prec> P = a<\<nu>x'> \<prec> ([(x, x')] \<bullet> P)"
abs_minus: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and 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" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (uminus a) = abs a"
abs_eq: fixes x :: "'b" shows "return_pmf x = Abs_pmf (return (count_space UNIV) x)"
oalist_inv_raw_sort_oalist_aux: fixes xs :: "('a \<times> 'd) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs" shows "oalist_inv_raw ko (sort_oalist_aux ko xs)"
dom_cod: fixes f :: "'a" assumes "C.arr f" shows "C.dom (C.cod f) = C.cod f"
complete_lattice_partial_function_definitions: fixes Inf :: "'h set \<Rightarrow> 'h"   and Sup :: "'h set \<Rightarrow> 'h"   and inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and bot :: "'h"   and top :: "'h" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "partial_function_definitions less_eq Sup"
list_cdh_def: fixes \<G> :: "'b cyclic_group" shows "list_cdh \<G> \<equiv> cyclic_group \<G>"
while_sum: fixes s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int" assumes "(wsum, s) \<Rightarrow> t" shows "t ''y'' = s ''y'' + sum (s ''x'')"
3: fixes a :: "'a"   and ys :: "['a]"   and Pa :: "'a \<rightarrow> tr"   and xsa :: "['a]" shows "map\<cdot>(nth\<cdot>(a : ys) oo (+1))\<cdot>(findIndices\<cdot>Pa\<cdot>xsa) = map\<cdot>(nth\<cdot>ys)\<cdot>(findIndices\<cdot>Pa\<cdot>xsa)"
goodstein_step: fixes i :: "nat" assumes "0 < local.goodstein i" shows "encode (i + 3) 0 (local.goodstein (i + 1)) < encode (i + 2) 0 (local.goodstein i)"
redT_updWs_def: fixes t :: "'a" shows "redT_updWs t = rtrancl3p (redT_updW t)"
has_left_adjoint_functor: shows "left_adjoint_functor D C F'"
least: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.pcpo below" shows "\<exists>x. \<forall>y. below x y"
ABORT_eq_Done: fixes a :: "'f option" shows "(ABORT = Resumption.resumption.Done a) = (a = None)"
simple_Group_def: fixes G :: "('e, 'f) Group_scheme" shows "simple_Group G = ({N. G \<guillemotright> N } = {carrier G, {\<one>\<^bsub>G\<^esub>}})"
all_is_branch_def: fixes p :: "('b, 'c) fm"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set" shows "all_is_branch p i branch xs \<equiv> \<forall>(v, v')\<in>xs. v < length branch \<longrightarrow> is_at p i branch v v'"
simps_1: fixes u :: "nat"   and v :: "nat" shows "cf.isPath u [] v = (u = v)"
rinit_Val_ext: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and e'' :: "char list exp"   and Cs :: "char list list"   and e\<^sub>0 :: "char list exp"   and s'' :: "heap \<times> sfields \<times> sheap"   and v' :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>RI (C,e'') ; Cs \<leftarrow> e\<^sub>0,s''\<rangle> \<Rightarrow> \<langle>Val v',s\<^sub>1\<rangle>"   and "\<And>e' s'. P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle> \<Longrightarrow> P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>RI (C,e) ; Cs \<leftarrow> e\<^sub>0,s\<rangle> \<Rightarrow> \<langle>Val v',s\<^sub>1\<rangle>"
termi_c2': shows "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1',Normal s\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>c2' \<down> Normal s'"
tdev_def: fixes x :: "'b pdevs" shows "tdev x = (\<Sum>i<degree x. \<bar>pdevs_apply x i\<bar>)"
simps_11: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and f3 :: "'e \<Rightarrow> 'd \<Rightarrow> 'c"   and x31 :: "'e"   and x32 :: "'d" shows "(case Yield x31 x32 of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = f3 x31 x32"
transpose_one: fixes n :: "nat" shows "(1\<^sub>m n)\<^sup>T = 1\<^sub>m n"
redT_updTs_Some: fixes ts :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option"   and t :: "'c"   and xw :: "'a \<times> 'b \<Rightarrow>f nat"   and tas :: "('c, 'a, 'd) new_thread_action list" assumes "ts t = \<lfloor>xw\<rfloor>"   and "thread_oks ts tas" shows "redT_updTs ts tas t = \<lfloor>xw\<rfloor>"
fg_round_between_start_rounds: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> ('g, 'f, 'h) config"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f"   and p :: "'f"   and r :: "nat"   and n :: "nat" assumes "fg_run A rho HOs SHOs coords"   and "fg_start_round rho p r \<le> n"   and "n < fg_start_round rho p (Suc r)" shows "round (rho n) p = r"
empty_replicate: fixes n :: "nat"   and x :: "'k" shows "([] = replicate n x) = (n = 0)"
infinite_atMost_int: fixes n :: "int" shows "infinite {..n}"
that: fixes C :: "'a set" assumes "countable C"   and "C \<subseteq> K"   and "pairwise (\<lambda>i j. disjnt (ball (a i) (r i)) (ball (a j) (r j))) C"   and "S \<subseteq> (\<Union>i\<in>C. ball (a i) (5 * r i))" shows "thesis"
PiNil_def: shows "\<zero> \<equiv> Abs_pi PiNil_Rep"
IH: fixes r :: "'trm"   and s :: "'trm"   and y :: "'var" assumes "r \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm"   and "\<forall>x\<in>snd ` set txs. x \<notin> FvarsT s"   and "\<forall>t\<in>fst ` set txs. y \<notin> FvarsT t"   and "distinct (map snd txs)"   and "s \<in> atrm"   and "y \<in> var"   and "y \<notin> snd ` set txs" shows "substT (rawpsubstT r txs) s y = rawpsubstT r ((s, y) # txs)"
cl10_d: fixes x :: "'a" shows "d (x \<sqinter> nc) = 1\<^sub>\<sigma> \<sqinter> (x \<sqinter> nc) \<cdot> nc"