hcju/isabelleps huggingface dataset:
simps_2: fixes x :: "pdf_type" shows "op_type Mult x = (case x of PRODUCT INTEG INTEG \<Rightarrow> Some INTEG | PRODUCT INTEG (PRODUCT pdf_type1 x) \<Rightarrow> Map.empty x | PRODUCT INTEG _ \<Rightarrow> None | PRODUCT REAL REAL \<Rightarrow> Some REAL | PRODUCT REAL (PRODUCT pdf_type1 x) \<Rightarrow> Map.empty x | PRODUCT REAL _ \<Rightarrow> None | PRODUCT (PRODUCT pdf_type1a x) pdf_type2 \<Rightarrow> Map.empty x | PRODUCT _ pdf_type2 \<Rightarrow> None | _ \<Rightarrow> None)"
fi_map_def: fixes G :: "('a, 'b) graph_rec_scheme"   and f :: "'a \<Rightarrow> 'c"   and x :: "'c" assumes "g_rename_precond G f" shows "g_rename_precond.fi_map G f x = (if x \<in> f ` g_V G then Some (g_rename_precond.fi G f x) else None)"
greatestPD3: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "greatestP P x"   and "y \<in> Collect P" shows "y \<le> x"
length_mat_to_cols_list: fixes A :: "complex Matrix.mat" shows "length (mat_to_cols_list A) = dim_col A"
transpose_infmatrix_closed: fixes x :: "'b matrix" shows "Rep_matrix (Abs_matrix (transpose_infmatrix (Rep_matrix x))) = transpose_infmatrix (Rep_matrix x)"
wellFoundedEmbed: fixes rel :: "('a \<times> 'a) set"   and rel' :: "('a \<times> 'a) set" assumes "\<forall>x y. (x, y) \<in> rel \<longrightarrow> (x, y) \<in> rel'"   and "wf rel'" shows "wf rel"
high_proc_def: fixes p :: "prog" shows "high_proc p \<equiv> \<forall>\<sigma>\<in>states p. \<forall>\<sigma>'. body p \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<longrightarrow> the (\<sigma> ''lowIn'') + the (\<sigma> ''highIn'') \<le> the (\<sigma>' ''highOut'')"
inv_ide: fixes a :: "'a" assumes "A.ide a" shows "A.inv a = a"
continuous_map_image_subset_topspace: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e" assumes "continuous_map X Y f" shows "f ` topspace X \<subseteq> topspace Y"
lambda_var: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and S :: "stack_elem list" assumes "map_of \<Gamma> x = Some e"   and "isVal e" shows "(\<Gamma>, Var x, S) \<Rightarrow>\<^sup>* ((x, e) # delete x \<Gamma>, e, S)"
clo_cT: shows "closed (cbox (- c) c \<inter> T)"
exec: shows "\<Gamma>\<turnstile> \<langle>Language.normalize c,s\<rangle> \<Rightarrow> t"
assms: shows "(\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<N>\<lbrace>\<Gamma>\<rbrace>\<^esub>)\<cdot>r \<noteq> \<bottom>"
of_nat_in_Nats: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero n \<in> semiring_1.Nats one plus zero"
same_cs_2: fixes c :: "('p, 's, 'm) configuration"   and d :: "('p, 's, 'm) configuration"   and i :: "nat"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration" assumes "\<forall>p. (ps c p \<noteq> None) = (ps d p \<noteq> None)"   and "cs c i = cs d i"   and "c \<turnstile> ev \<mapsto> c'"   and "d \<turnstile> ev \<mapsto> d'" shows "cs c' i = cs d' i"
del0_def: fixes k :: "int"   and t :: "'b tree23" shows "del0 k t = (case del (Some k) t of None \<Rightarrow> t | Some (x, xa, t') \<Rightarrow> t')"
plus_finite_3_def: fixes x :: "finite_3"   and y :: "finite_3" shows "x + y = (case (x, y) of (a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>1, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>1, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>3 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>1) \<Rightarrow> a\<^sub>3 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>2)"
bind_gpv_assoc: fixes f :: "('e, 'b, 'c) gpv"   and g :: "'e \<Rightarrow> ('d, 'b, 'c) gpv"   and h :: "'d \<Rightarrow> ('a, 'b, 'c) gpv" shows "f \<bind> g \<bind> h = f \<bind> (\<lambda>x. g x \<bind> h)"
open_imp_locally_connected: fixes S :: "'a set" assumes "open S" shows "locally connected S"
infdist_notempty: fixes A :: "'b set"   and x :: "'b" assumes "A \<noteq> {}" shows "infdist x A = Inf (dist x ` A)"
nat_filtration_singleton: fixes n :: "nat"   and w :: "bool stream" assumes "pseudo_proj_True n w = w" shows "pseudo_proj_True n -` {w} \<in> sets (nat_filtration n)"
I'_def: shows "I' \<equiv> \<lambda>n. of_nat n powr (1 - s) / (1 - s)"
abs_eq: fixes x :: "'f" shows "Pred x = Abs_formula (Pred\<^sub>\<alpha> x)"
fpxs_val_geI: fixes f :: "'b fpxs"   and r' :: "rat" assumes "f \<noteq> 0"   and "\<And>r. r < r' \<Longrightarrow> fpxs_nth f r = (0::'b)" shows "r' \<le> fpxs_val f"
True_True_steps_concI: fixes p :: "bool list"   and q :: "bool list"   and L :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list"   and R :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "(p, q) \<in> steps L w" shows "(True # p, True # q) \<in> steps (RegExp2NA.conc L R) w"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "Left.arr f"   and "Left.cod f = b" shows "Left.comp b f = f"
sup_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> x \<sqinter> y = x"
ex_set_conv: fixes xs :: "'b list" shows "(\<exists>x. x \<in> set xs) = (xs \<noteq> [])"
lower_one_def: fixes k :: "nat"   and A :: "'a mat"   and i :: "nat"   and j :: "nat" shows "local.lower_one k A i j = (j = k \<longrightarrow> A $$ (i, j) = (0::'a) \<or> i = j \<or> A $$ (i, j) = (1::'a) \<and> j = Suc i \<and> A $$ (i, i) = A $$ (j, j))"
00: shows "coeff (Var x\<^sub>j) x = 0"
M4: fixes p :: "'a"   and q :: "'a"   and s :: "'a"   and r :: "'a" assumes "p \<parallel> q"   and "q \<parallel> s"   and "p \<parallel> r"   and "r \<parallel> s" shows "q = r"
Skip: shows "\<turnstile>\<^sub>3\<^sub>a {$ 1} SKIP { $ 0}"
uminus_r_inv_vec: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v + - v = 0\<^sub>v n"
axioms_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data" assumes "span_in_category C S" shows "category C"
x'_def: shows "x' = (if x \<in> T1 then x else - \<i> *\<^sub>C x)"
in_rel: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "('a, 'c, 'e) Vardi.var"   and b :: "('b, 'd, 'e) Vardi.var" shows "rel_var R1 R2 a b = (\<exists>z. z \<in> {x. set1_var x \<subseteq> {(x, y). R1 x y} \<and> set2_var x \<subseteq> {(x, y). R2 x y}} \<and> map_var fst fst z = a \<and> map_var snd snd z = b)"
x: shows "x \<in> TER\<^bsub>quotient_web \<Gamma> f\<^esub> g"
statEqWeakBisim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<approx> Q"   and "\<Psi> \<simeq> \<Psi>'" shows "\<Psi>' \<rhd> P \<approx> Q"
find_closest_pair_eq_val_find_closest_pair_tm: fixes c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and ps :: "(int \<times> int) list" shows "Time_Monad.val (find_closest_pair_tm (c\<^sub>0, c\<^sub>1) ps) = find_closest_pair (c\<^sub>0, c\<^sub>1) ps"
L4_2a: fixes \<phi> :: "'b ltlr"   and qs :: "'b node set"   and \<sigma> :: "nat \<Rightarrow> 'b node"   and \<mu> :: "'b ltlr"   and \<eta> :: "'b ltlr" assumes "create_gba_from_nodes_precond \<phi> qs"   and "ipath (g_E (create_gba_from_nodes \<phi> qs)) \<sigma>"   and "\<mu> U\<^sub>r \<eta> \<in> old (\<sigma> 0)" shows "(\<forall>i. {\<mu>, \<mu> U\<^sub>r \<eta>} \<subseteq> old (\<sigma> i) \<and> \<eta> \<notin> old (\<sigma> i)) \<or> (\<exists>j. (\<forall>i<j. {\<mu>, \<mu> U\<^sub>r \<eta>} \<subseteq> old (\<sigma> i)) \<and> \<eta> \<in> old (\<sigma> j))"
scaleR_minus1_left: fixes x :: "'a" shows "- 1 *\<^sub>R x = - x"
assms: shows "(w', w'') \<in> relations (gen_model M w) a"
that: fixes c :: "'d" assumes "homotopic_with_canon (\<lambda>h. True) S U (g \<circ> f) (\<lambda>x. c)" shows "thesis"
dup_is_natural_transformation: shows "natural_transformation (\<cdot>) (\<cdot>) local.map (local.Prod \<circ> Diag) dup"
rep_eq: fixes x :: "'a covector"   and xa :: "'a covector" shows "(x < xa) = (Rep_covector x < Rep_covector xa)"
equality: fixes r :: "'a DiskBlock_scheme"   and r' :: "'a DiskBlock_scheme" assumes "mbal r = mbal r'"   and "bal r = bal r'"   and "inp r = inp r'"   and "DiskBlock.more r = DiskBlock.more r'" shows "r = r'"
orderE: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" obtains "a = inf_class.inf a b"
simps_2: fixes dims :: "nat list"   and M0 :: "complex mat"   and M1 :: "complex mat"   and WS :: "complex mat \<Rightarrow> complex mat"   and n :: "nat"   and P :: "complex mat" shows "state_sig.wlp_while_n dims M0 M1 WS (Suc n) P = adjoint M0 * P * M0 + adjoint M1 * WS (state_sig.wlp_while_n dims M0 M1 WS n P) * M1"
obtain_fresh: fixes V :: "'var set"   and XS :: "('c, 'd, 'varSort, 'var, 'e) qTerm set set"   and AS :: "('c, 'd, 'varSort, 'var, 'e) qAbs set set"   and Rho :: "('varSort \<Rightarrow> 'var \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm set option) set"   and xs :: "'varSort" assumes "|V| <o |UNIV| \<or> finite V"   and "|XS| <o |UNIV| \<or> finite XS"   and "|AS| <o |UNIV| \<or> finite AS"   and "|Rho| <o |UNIV| \<or> finite Rho"   and "Ball XS good"   and "Ball AS goodAbs"   and "Ball Rho goodEnv" shows "\<exists>z. z \<notin> V \<and> (\<forall>X\<in>XS. fresh xs z X) \<and> (\<forall>A\<in>AS. freshAbs xs z A) \<and> (\<forall>rho\<in>Rho. freshEnv xs z rho)"
Ri_limit_Sup: fixes Ns :: "'a literal multiset set llist" assumes "chain sr_ext.derive Ns" shows "sr_ext_Ri (Liminf_llist Ns) = sr_ext_Ri (Sup_llist Ns)"
dip_Some: fixes ip :: "'a word \<times> nat" assumes "ipcidr_conjunct dip1 dip2 = Some ip" shows "simple_match_ip ip (p_dst p) = (simple_match_ip dip1 (p_dst p) \<and> simple_match_ip dip2 (p_dst p))"
r: shows "\<forall>x\<in>S. (x, L) \<in> r"
Assume: fixes A :: "fmla"   and H :: "fmla set" shows "insert A H \<turnstile> A"
fcomp_vunion_right: fixes r :: "V"   and s :: "V"   and t :: "V" shows "r \<circ>\<^sub>\<bullet> (s \<union>\<^sub>\<circ> t) = r \<circ>\<^sub>\<bullet> s \<union>\<^sub>\<circ> r \<circ>\<^sub>\<bullet> t"
higman: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "wqo_on P A" shows "wqo_on (list_emb P) (lists A)"
f_def: fixes x :: "nat" shows "f x = s_ce n x"
bounded: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "bounded_sesquilinear prod" shows "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
roundup_help1: fixes l :: "nat" assumes "0 < roundup l 160" shows "roundup l 160 - 1 + 1 = roundup l 160"
r_shrink_prim: fixes f :: "recf" assumes "prim_recfn 2 f" shows "prim_recfn 1 (r_shrink f)"
reversepath_reversepath: fixes g :: "real \<Rightarrow> 'a" shows "reversepath (reversepath g) = g"
is_sig_red_regularD_lt: fixes top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_red (\<prec>\<^sub>t) top_tail {f} p" shows "lt f \<prec>\<^sub>t lt p"
prems_2: fixes S' :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> U <: S"   and "TVarB U \<Colon> \<Gamma> \<turnstile> s : S'"   and "TVarB U \<Colon> \<Gamma> \<turnstile> S' <: U'" shows "P"
ord_spmf_bind_reflI: fixes p :: "'d spmf"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e spmf"   and g :: "'d \<Rightarrow> 'e spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> ord_spmf R (f x) (g x)" shows "ord_spmf R (p \<bind> f) (p \<bind> g)"
rebalance_middle_tree_height: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and t :: "'b btree"   and sub :: "'b btree"   and rs :: "('b btree \<times> 'b) list"   and k :: "nat"   and ls :: "('b btree \<times> 'b) list"   and sep :: "'b" assumes "split split"   and "height t = height sub"   and "case rs of [] \<Rightarrow> True | (rsub, rsep) # list \<Rightarrow> height rsub = height t" shows "height (split.rebalance_middle_tree k ls sub sep rs t) = height (Node (ls @ (sub, sep) # rs) t)"
seq_char: fixes g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "seq g f = (arr f \<and> arr g \<and> Dom g = Cod f)"
flowD_finite_OUT: fixes f :: "'v option \<times> 'v option \<Rightarrow> ennreal"   and x :: "'v option" assumes "flow \<Psi> f"   and "x \<noteq> source \<Psi>"   and "x \<noteq> sink \<Psi>" shows "d_OUT f x \<noteq> \<top>"
supp_aezfun_diff: fixes x :: "('b, 'a) aezfun"   and y :: "('b, 'a) aezfun" shows "supp (aezfun (x - y)) \<subseteq> supp (aezfun x) \<union> supp (aezfun y)"
ind_hyp: fixes n :: "nat"   and x :: "'a" shows "(index es n = Some x) = (n < length es \<and> x = es ! n)"
sing_fac_pow: fixes w :: "'b list"   and a :: "'b"   and v :: "'b list" assumes "w \<in> [a]*"   and "v \<le>f w" shows "v \<in> [a]*"
ce_rel_lm_29: fixes A :: "nat set"   and B :: "nat set" assumes "A \<in> ce_sets"   and "B \<in> ce_sets" shows "A \<times> B \<in> ce_rels"
prime_power_inj: fixes a :: "'a"   and m :: "nat"   and n :: "nat" assumes "prime a"   and "a ^ m = a ^ n" shows "m = n"
l_set_val_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_val type_wf set_val set_val_locs \<equiv> ((\<forall>character_data_ptr tag h h'. writes (set_val_locs character_data_ptr) (set_val character_data_ptr tag) h h') \<and> (\<forall>h character_data_ptr tag. type_wf h \<longrightarrow> character_data_ptr |\<in>| character_data_ptr_kinds h \<longrightarrow> h \<turnstile> ok set_val character_data_ptr tag)) \<and> (\<forall>w character_data_ptr h h'. w \<in> set_val_locs character_data_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> object_ptr_kinds h = object_ptr_kinds h') \<and> (\<forall>w character_data_ptr h h'. w \<in> set_val_locs character_data_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> type_wf h = type_wf h')"
set_eq_def: fixes S :: "('a, 'b) eq_object_scheme"   and A :: "'a set"   and B :: "'a set" shows "(A {.=}\<^bsub>S\<^esub> B) = ((\<forall>x\<in>A. x .\<in>\<^bsub>S\<^esub> B) \<and> (\<forall>x\<in>B. x .\<in>\<^bsub>S\<^esub> A))"
redT_ts_Some_inv: fixes ls :: "('l, 't) locks"   and ts :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option"   and m :: "'m"   and ws :: "'t \<Rightarrow> 'w wait_set_status option"   and "is" :: "'t set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and ls' :: "('l, 't) locks"   and ts' :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option"   and m' :: "'m"   and ws' :: "'t \<Rightarrow> 'w wait_set_status option"   and is' :: "'t set"   and t' :: "'t"   and x :: "'x \<times> 'l \<Rightarrow>f nat" assumes "\<langle>ls, (ts, m), ws, is\<rangle> -t\<triangleright>ta\<rightarrow> \<langle>ls', (ts', m'), ws', is'\<rangle>"   and "t \<noteq> t'"   and "ts t' = \<lfloor>x\<rfloor>" shows "ts' t' = \<lfloor>x\<rfloor>"
set_disconnected_nodes_get_parent: fixes ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "\<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_parent_locs. r h h')"
qAbs_pick_Abs: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "qAbs xs x (pick X) $= pick (Abs xs x X)"
i_join_nth: fixes f :: "nat \<Rightarrow> 'b"   and I :: "nat set"   and n :: "nat" shows "(f \<Join>\<^sub> I) n = f (I \<rightarrow> n)"
trans: shows "local.trans (xs !! j) (xs !! Suc j)"
qmodule_is_ag: fixes H :: "'a set" assumes "submodule R M H" shows "aGroup (M /\<^sub>m H)"
senior_senior: fixes x :: "nat"   and n :: "nat" shows "senior (senior x n) n = senior x n"
a: shows "real ar / 2 ^ r = real a / 2 ^ n"
tJ_def: 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 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 fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and Ops :: "('b \<Rightarrow> 'b) set"   and tenc :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b"   and F :: "('b \<Rightarrow> 'b) set"   and encF :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and N :: "'b \<Rightarrow> 'b"   and ssap :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and \<psi> :: "'c" assumes "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv enc Ops tenc F encF N ssap" shows "Jeroslow_Diagonalization.tJ tenc encF ssap \<psi> \<equiv> encF (ssap \<psi>) (tenc (encF (ssap \<psi>)))"
strict_evaluation_functor_axioms: shows "strict_evaluation_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V"
getParts_nonempty_elems: fixes rs :: "'a simple_rule list" shows "\<forall>w\<in>set (getParts rs). \<not> wordinterval_empty w"
assms_3: fixes n :: "int" assumes "n \<in> \<K>'" shows "2 \<le> n"
2: shows "Left.in_hom \<l>[Left.dom \<mu>] (trg \<mu> \<star> local.dom \<mu>) (local.dom \<mu>)"
irreducible_normalize_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "comm_semiring_1.irreducible times one zero (normalize x) = comm_semiring_1.irreducible times one zero x"
prime_card: assumes "class.prime_card TYPE('b)" shows "prime CARD('b)"
unit_char_2: fixes a :: "'a" assumes "C.obj a" shows "D.iso (G.unit a)"
chine_assoc'_props_3: shows "Prj\<^sub>0\<^sub>1 \<cdot> chine_assoc' = Prj\<^sub>1\<^sub>0"
mpoly_map_vars_sum: fixes f :: "nat \<Rightarrow> nat"   and g :: "'b \<Rightarrow> 'a mpoly"   and A :: "'b set" shows "mpoly_map_vars f (sum g A) = (\<Sum>x\<in>A. mpoly_map_vars f (g x))"
le_infE: fixes w\<^sub>1 :: "'action stream"   and w\<^sub>2 :: "'action stream"   and v\<^sub>1 :: "'action list" assumes "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2"   and "v\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>1" obtains "v\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"
lift_gi_closed: fixes f :: "'e \<Rightarrow> 'a"   and gens :: "'e set"   and gi :: "bool \<times> 'e" assumes "f \<in> gens \<rightarrow> carrier G"   and "snd gi \<in> gens" shows "G.lift_gi f gi \<in> carrier G"
simps_3: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "median_le_5 [a, b, c] = median_3 a b c"
poly_subst_one: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_subst f 1 = 1"
sorted_wrt_quicksort_by_rel: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "\<And>x y. R x y \<or> R y x"   and "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z" shows "sorted_wrt R (quicksort_by_rel R [] xs)"
ord_dvd_Carmichael: fixes n :: "nat"   and k :: "nat" assumes "1 < n"   and "coprime n k" shows "ord n k dvd Carmichael n"
intros: fixes rest :: "('p, 's, 'q, 't, 'r, 'u, 'p) rest_scheme"   and \<I>_adv :: "('q, 'r) \<I>"   and I :: "'p \<Rightarrow> bool"   and \<I>_usr :: "('t, 'u) \<I>" assumes "\<And>s x y es s'. \<lbrakk>((y, es), s') \<in> set_spmf (rfunc_adv rest s x); x \<in> outs_\<I> \<I>_adv; I s\<rbrakk> \<Longrightarrow> y \<in> responses_\<I> \<I>_adv x \<and> I s'"   and "\<And>s x y es s'. \<lbrakk>((y, es), s') \<in> set_spmf (rfunc_usr rest s x); x \<in> outs_\<I> \<I>_usr; I s\<rbrakk> \<Longrightarrow> y \<in> responses_\<I> \<I>_usr x \<and> I s'"   and "I (rinit rest)" shows "WT_rest \<I>_adv \<I>_usr I rest"
p_or_not_p: fixes P :: "'b \<Rightarrow> bool" shows "(P \<or> \<not> P) = true"
leadsTo_Union: fixes S :: "'d set set"   and F :: "'d program"   and B :: "'d set" assumes "\<And>A. A \<in> S \<Longrightarrow> F \<in> A \<longmapsto> B" shows "F \<in> \<Union> S \<longmapsto> B"
annotate_named_loop_inv_fix: fixes b :: "'e set"   and name :: "'h"   and V :: "('e \<times> 'e) set"   and c :: "('e, 'f, 'g) com"   and I :: "'i \<Rightarrow> 'e set" shows "whileAnno b (named_loop name) V c = whileAnnoFix b I (\<lambda>_. V) (\<lambda>_. c)"
that: fixes l :: "'s"   and u :: "'c \<Rightarrow> real" assumes "s = (l, u)"   and "u \<in> [u]\<^sub>\<R>"   and "[u]\<^sub>\<R> \<in> \<R>" shows "thesis"
nn_integral_map_spmf: fixes f :: "'e \<Rightarrow> 'd"   and p :: "'e spmf"   and g :: "'d \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_spmf (map_spmf f p)) g = integral\<^sup>N (measure_spmf p) (g \<circ> f)"
hom_one: fixes a :: "'b" shows "poly 1 a = (1::'b)"
simps_9: fixes f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and f3 :: "'g \<Rightarrow> 'f \<Rightarrow> 'e" shows "(case Done of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = f1"
dw_def: shows "dw \<equiv> dropWhile ((\<noteq>) u) tjs"
induct: fixes P :: "'b list \<Rightarrow> bool"   and list :: "'b list" assumes "P []"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (x1 # x2)" shows "P list"
is_interval_existence_ivl: fixes t0 :: "real"   and x0 :: "'a" shows "is_interval (existence_ivl t0 x0)"
contains_predE: fixes A :: "'b set"   and x :: "'b"   and y :: "unit" assumes "pred.eval (Predicate_Compile.contains_pred A x) y" obtains "Predicate_Compile.contains A x"
coeff_in_coeffs: fixes p :: "'a poly"   and n :: "nat" assumes "p \<noteq> 0"   and "n \<le> degree p" shows "coeff p n \<in> set (coeffs p)"
Mapping_lookup_filter_Some_P: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "('b, 'a) mapping"   and k :: "'b"   and v :: "'a" assumes "Mapping.lookup (Mapping.filter P m) k = Some v" shows "P k v"
substs_preserve_well_constrainedness: fixes C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list" assumes "well_constrained C" shows "well_constrained (subst_ecl C \<sigma>)"
eru1: shows "u\<^sub>1 \<equiv>\<^sub>E ru1"
disj_simps_5: fixes P :: "fm" shows "Ferrack.disj P P = P"
assms: fixes x :: "\<nu>" shows "[\<phi> x in dw] = [\<psi> x in dw]"
xsa_def: shows "xsa = take j xs' @ drop (Suc j) xs'"
right: fixes \<sigma>' :: "'state" assumes "\<sigma>' \<in> small_nstep P \<sigma> (Suc (Suc n))" shows "\<exists>\<sigma>1. \<sigma>1 \<in> small P \<sigma> \<and> \<sigma>' \<in> small_nstep P \<sigma>1 (Suc n)"
extensionalityI: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> extensional A"   and "g \<in> extensional A"   and "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "f = g"
put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def: fixes ptr :: "(_) object_ptr"   and obj :: "(_) Object"   and h :: "(_) heap" shows "put ptr obj h = Heap (fmupd ptr obj (the_heap h))"
face_cycle_set_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and x :: "'f" assumes "digraph_isomorphism hom"   and "x \<in> iso_arcs hom ` arcs G" shows "pre_digraph_map.face_cycle_set (map_iso hom) x = iso_arcs hom ` face_cycle_set (iso_arcs (inv_iso hom) x)"
suc_underS: fixes B :: "'b set"   and b :: "'b" assumes "B \<subseteq> Field s"   and "s.AboveS B \<noteq> {}"   and "b \<in> B" shows "b \<in> s.underS (s.suc B)"