hcju/isabelleps huggingface dataset:
inject_2: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat"   and x25 :: "nat"   and y21 :: "nat"   and y22 :: "nat"   and y23 :: "nat"   and y24 :: "nat"   and y25 :: "nat" shows "(Rrep x21 x22 x23 x24 x25 = Rrep y21 y22 y23 y24 y25) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24 \<and> x25 = y25)"
invar_adjust: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and lv :: "nat"   and r :: "('b \<times> nat) tree" assumes "pre_adjust \<langle>l, (a, lv), r\<rangle>" shows "invar (adjust \<langle>l, (a, lv), r\<rangle>)"
span_eq_dim: fixes S :: "'b set"   and T :: "'b set" assumes "local.span S = local.span T" shows "local.dim S = local.dim T"
set_child_nodes_impl: shows "set_child_nodes = a_set_child_nodes"
u_eq_u_stop: fixes x :: "real" assumes "NO_MATCH local.t_stop x"   and "local.t_stop \<le> x" shows "u x = u_max"
subgroup_iso_DirProds_IDirProds: fixes J :: "'a set"   and S :: "'d \<Rightarrow> 'a set"   and I :: "'d set" assumes "subgroup J G"   and "is_idirprod J S I"   and "finite I" shows "(\<lambda>x. finprod G x I) \<in> Group.iso (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) (G\<lparr>carrier := J\<rparr>)"
Gromov_converging_at_boundary_converges: fixes u :: "nat \<Rightarrow> 'a" assumes "Gromov_converging_at_boundary u" shows "\<exists>x\<in>Gromov_boundary. (\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x"
in_set_gb_red_auxI: fixes p :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e"   and q :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e"   and ps :: "((('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list"   and h :: "'t \<Rightarrow>\<^sub>0 'd"   and bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" assumes "(p, q) \<in> set ps"   and "h = trdsp (map fst bs) (p, q)"   and "h \<noteq> 0" shows "h \<in> set (gb_red_aux bs ps)"
plus_pres_lens_indep: fixes X :: "'a \<Longrightarrow> 'b"   and Z :: "'c \<Longrightarrow> 'b"   and Y :: "'d \<Longrightarrow> 'b" assumes "X \<bowtie> Z"   and "Y \<bowtie> Z" shows "X +\<^sub>L Y \<bowtie> Z"
induct: fixes x1 :: "('Var, 'AExp, 'BExp) Stmt"   and x2 :: "Mode \<Rightarrow> 'Var set"   and x3 :: "'Var \<Rightarrow> 'Val"   and lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and P :: "('Var, 'AExp, 'BExp) Stmt \<Rightarrow> (Mode \<Rightarrow> 'Var set) \<Rightarrow> ('Var \<Rightarrow> 'Val) \<Rightarrow> bool" assumes "\<langle>x1, x2, x3\<rangle> \<in> loc_reach lc"   and "P (fst (fst lc)) (snd (fst lc)) (snd lc)"   and "\<And>c' mds' mem' c'' mds'' mem''. \<lbrakk>\<langle>c', mds', mem'\<rangle> \<in> loc_reach lc; P c' mds' mem'; \<langle>c', mds', mem'\<rangle> \<leadsto> \<langle>c'', mds'', mem''\<rangle>\<rbrakk> \<Longrightarrow> P c'' mds'' mem''"   and "\<And>c' mds' mem' mem''. \<lbrakk>\<langle>c', mds', mem'\<rangle> \<in> loc_reach lc; P c' mds' mem'; \<forall>x. var_asm_not_written mds' x \<longrightarrow> mem' x = mem'' x \<and> dma mem' x = dma mem'' x\<rbrakk> \<Longrightarrow> P c' mds' mem''" shows "P x1 x2 x3"
bot_least: fixes bot :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.order_bot bot less_eq less" shows "less_eq bot a"
vc_iterate_foreach: fixes S :: "'v set"   and V :: "'v set"   and v :: "'v"   and it :: "'v set"   and V' :: "'v set"   and v' :: "'v"   and Vr'' :: "'v set \<times> bool" assumes "fe_inv S V v it (V', False)"   and "rpost (insert v S) (V', v') Vr''" shows "fe_inv S V v (it - {v'}) Vr''"
resolvable_design_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and partition :: "'b set multiset multiset" shows "resolvable_design point_set block_collection partition \<equiv> design point_set block_collection \<and> resolvable_design_axioms point_set block_collection partition"
has_parametrized_subprob_density_cexpr_sem_integral: fixes V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and M :: "(nat \<Rightarrow> val) \<Rightarrow> val measure"   and t :: "pdf_type"   and t' :: "pdf_type"   and f :: "cexpr"   and \<rho> :: "nat \<Rightarrow> val" assumes "has_parametrized_subprob_density (state_measure V' \<Gamma>) M (stock_measure t) (\<lambda>\<rho> x. \<integral>\<^sup>+ xa. ennreal (eval_cexpr f (case_nat x \<rho>) xa) \<partial>stock_measure t')"   and "nonneg_cexpr (shift_var_set (shift_var_set V')) (case_nat t' (case_nat t \<Gamma>)) f"   and "case_nat t' (case_nat t \<Gamma>) \<turnstile>\<^sub>c f : REAL"   and "free_vars f \<subseteq> shift_var_set (shift_var_set V')"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "AE x in stock_measure t. \<integral>\<^sup>+ xa. ennreal (eval_cexpr f (case_nat x \<rho>) xa) \<partial>stock_measure t' = ennreal (eval_cexpr (\<integral>\<^sub>c f \<partial>t') \<rho> x)"
affine_hull_convex_hull: fixes S :: "'a set" shows "affine hull (convex hull S) = affine hull S"
spr_simAction_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'f, 'e) GC list"   and envVal :: "('b, 'c, 'd) BEState \<Rightarrow> 'f \<Rightarrow> bool"   and envObs :: "'b \<Rightarrow> ('b, 'c, 'd) BEState \<Rightarrow> 'g \<times> 'd option" shows "SPRViewDet.spr_simAction jkbp envVal envObs \<equiv> \<lambda>a ec. concat (map (\<lambda>gc. if eval envVal envObs ec (guard gc) then [action gc] else []) (jkbp a))"
x2: shows "lincomb_list (lc(j := lc j + lc i * c)) us = (lc j + lc i * c) \<cdot>\<^sub>v us ! j + (lc i \<cdot>\<^sub>v us ! i + M1)"
basisI: fixes x :: "'b list"   and G :: "'b list set" assumes " x \<in>B G " shows "x \<in> \<BB> G"
marker_not_vanishing_means_always_present: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and j :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "i \<le> j"   and "j \<le> length t"   and "Marker \<in> set (msgs (S t i) cid)"   and "Marker \<in> set (msgs (S t j) cid)"   and "channel cid = Some (p, q)" shows "\<forall>k. i \<le> k \<and> k \<le> j \<longrightarrow> Marker \<in> set (msgs (S t k) cid)"
neg_pf_Suc: fixes pf :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and n :: "nat"   and w :: "bool stream"   and m :: "nat" assumes "stock_portfolio Mkt pf"   and "trading_strategy pf"   and "self_financing Mkt pf"   and "1 + r < u"   and "d < 1 + r"   and "cls_val_process Mkt pf n w < 0"   and "n \<le> m" shows "\<exists>y. cls_val_process Mkt pf m y < 0"
l9_19_R1: fixes X :: "'p"   and Y :: "'p"   and P :: "'p"   and A :: "'p"   and B :: "'p" assumes "Col X Y P"   and "Col A B P"   and "X Y OS A B" shows "P Out A B \<and> \<not> Col X Y A"
currentD_OUT_IN: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "current \<Gamma> f"   and "x \<notin> A \<Gamma>" shows "d_OUT f x \<le> d_IN f x"
run_V0: fixes G :: "('e, 'f) graph_rec_scheme"   and r :: "nat \<Rightarrow> 'e" assumes "graph_defs.is_run G r" shows "r 0 \<in> g_V0 G"
prems_2: shows "\<forall>a1 a2. a1 \<in> insert x A \<and> a2 \<in> insert x A \<and> a1 \<noteq> a2 \<longrightarrow> comm_monoid_mult_class.coprime a1 a2"
encode_kind_Cn: fixes f :: "recf" assumes "encode_kind (encode f) = 3" shows "\<exists>n f' gs. f = Cn n f' gs"
cut_greater_absorb: fixes I :: "'a set"   and t :: "'a" shows "I \<down>> t \<down>> t = I \<down>> t"
nat_le_0: fixes z :: "int" assumes "z \<le> 0" shows "nat z = 0"
arity_generalize_fundef: fixes fd2 :: "('dyn, 'a, 'b, 'c, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" shows "arity (Subx.generalize_fundef fd2) = arity fd2"
cont: shows "\<forall>b\<in>basis. \<forall>(k, \<gamma>2)\<in>one_chain. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
Complete_\<pi>_fix: fixes k :: "nat"   and T :: "(symbol \<times> character list) set"   and I :: "item set" shows "Complete k (\<pi> k T I) = \<pi> k T I"
no_stat_ref_comp: shows "\<not> statically_complete_calculus Bot Inf (\<Turnstile>Q) Red_I Red_F"
Ands: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and l :: "('msaux, 'muaux) mformula list"   and l_pos :: "Formula.formula list"   and l_neg :: "Formula.formula list"   and l' :: "Formula.formula list"   and buf :: "event_data option list set list list"   and A_pos :: "nat set list"   and A_neg :: "nat set list" assumes "list_all2 (wf_mformula \<sigma> j P V n R) l (l_pos @ map remove_neg l_neg)"   and "wf_mbufn (Monitor.progress \<sigma> P (formula.Ands l') j) (map (\<lambda>\<psi>. Monitor.progress \<sigma> P \<psi> j) (l_pos @ map remove_neg l_neg)) (map (\<lambda>\<psi> i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<psi>)) (l_pos @ map remove_neg l_neg)) buf"   and "(l_pos, l_neg) = partition safe_formula l'"   and "l_pos \<noteq> []"   and "list_all safe_formula (map remove_neg l_neg)"   and "A_pos = map fv l_pos"   and "A_neg = map fv l_neg"   and "\<Union> (set A_neg) \<subseteq> \<Union> (set A_pos)" shows "wf_mformula \<sigma> j P V n R (MAnds A_pos A_neg l buf) (formula.Ands l')"
wf_terminates: fixes R :: "('a \<times> 'a) set"   and g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)" assumes "wf R"   and "\<And>s. fst g s \<Longrightarrow> (snd (snd g s), s) \<in> R" shows "terminates g"
rel_inject_2: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x2 :: "'d"   and y2 :: "'e" shows "rel_option R (Some x2) (Some y2) = R x2 y2"
to_vec_inverse: fixes x :: "'a sq_mtx" shows "to_mtx (to_vec x) = x"
acomplete_bal_tree: fixes n :: "nat"   and t :: "'a tree" assumes "n \<le> size t" shows "acomplete (bal_tree n t)"
q_def: fixes t :: "real" shows "other.q t = s\<^sub>o + v\<^sub>o * t"
prv_pls_cong: fixes t1 :: "'trm"   and t1' :: "'trm"   and t2 :: "'trm"   and t2' :: "'trm" assumes "t1 \<in> atrm"   and "t1' \<in> atrm"   and "t2 \<in> atrm"   and "t2' \<in> atrm"   and "prv (eql t1 t1')"   and "prv (eql t2 t2')" shows "prv (eql (pls t1 t2) (pls t1' t2'))"
rel2p_inv_1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel2p (p2rel P) = P"
least_le: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and A :: "'c set"   and a :: "'c" assumes "least L x A"   and "a \<in> A" shows "x \<sqsubseteq>\<^bsub>L\<^esub> a"
lens_indep_vwb_iff: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'d \<Longrightarrow> 'c" assumes "vwb_lens x"   and "vwb_lens y" shows "(x \<bowtie> y) = (\<forall>u v \<sigma>. put\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) u = put\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> \<sigma> u) v)"
eq_fold: fixes M :: "'a multiset" shows "prod_mset M = fold_mset (*) (1::'a) M"
filterlim_at_left_to_right: fixes f :: "real \<Rightarrow> 'a"   and F :: "'a filter"   and a :: "real" shows "filterlim f F (at_left a) = (LIM x at_right (- a). f (- x) :> F)"
abs_eq: fixes x :: "'a ltl" shows "\<up>Unf (Abs x) = Abs (Unf x)"
rep_eq: fixes x :: "uint" shows "lsb x = lsb (Rep_uint x)"
set_add_defs_1: fixes G :: "('c, 'd) ring_scheme" shows "(<+>\<^bsub>G\<^esub>) = (<#>\<^bsub>add_monoid G\<^esub>)"
unit_counit_adjunction_axioms: shows "unit_counit_adjunction CC.comp (\<cdot>) Diag local.Prod dup \<pi>.map"
counit_simps_3: shows "D.cod (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)) = trg\<^sub>D (F f)"
const_def: fixes X :: "'d \<Rightarrow> 'e" shows "const X \<equiv> \<forall>\<tau> \<tau>'. X \<tau> = X \<tau>'"
IH: fixes init :: "('a, 'b) fmap" assumes "fmpred P init"   and "\<And>x. x \<in> set xs \<Longrightarrow> fmpred P x" shows "fmpred P (foldl (++\<^sub>f) init xs)"
elims: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and x :: "nat"   and xa :: "nat"   and y :: "'c" assumes "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x xa = y" obtains xb :: "nat" and n :: "nat" where "x = xb" and "xa = n" and "y = run \<delta> q\<^sub>0 (suffix xb w) (n - xb)"
weakenTransitionCases: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and Prop :: "'a action \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"   and "Prop (\<tau>) P"   and "\<And>P''' P''. \<lbrakk>\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'''; \<Psi> \<rhd> P''' \<longmapsto> \<alpha> \<prec> P''; \<Psi> \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'\<rbrakk> \<Longrightarrow> Prop \<alpha> P'" shows "Prop \<alpha> P'"
basic_subtype_x_UnlimitedNatural: fixes \<tau> :: "'a basic_type" assumes "\<tau> \<sqsubset>\<^sub>B UnlimitedNatural" obtains "\<tau> = OclVoid"
GT_def: shows "GT \<equiv> Ordering_abs\<cdot>(sinr\<cdot>(sinr\<cdot>ONE))"
assms_2: fixes G :: "'a list_graph"   and nP :: "'a \<Rightarrow> 'b" assumes "wf_list_graph G" shows "sinvar_spec (list_graph_to_graph G) nP = sinvar_impl G nP"
leD: fixes y :: "'x \<Rightarrow>\<^sub>0 nat"   and x :: "'x \<Rightarrow>\<^sub>0 nat" assumes "y \<preceq> x" shows "\<not> x \<prec> y"
connected_iff_eq_connected_component_set: fixes S :: "'c set" shows "connected S = (\<forall>x\<in>S. connected_component_set S x = S)"
alt_fail1: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "'b" assumes "monad_fail_alt return bind fail alt" shows "alt fail m = m"
list_before_rel_on_elems: fixes l :: "'b list" shows "list_before_rel l \<subseteq> set l \<times> set l"
mset_eq_perm: fixes xs :: "'b list"   and ys :: "'b list" shows "(mset xs = mset ys) = (xs <~~> ys)"
assms_3: shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (unlabel D) \<subseteq> X"
worklist_tree_aux_Some_foldl: fixes succs :: "'c \<Rightarrow> 'c list"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and ws :: "'c list"   and s :: "'d"   and ws' :: "'c list"   and s' :: "'d" assumes "worklist_tree_aux succs f (ws, s) = Some (ws', s')" shows "\<exists>rs. set rs = (Rel succs)\<^sup>* `` set ws \<and> s' = foldl (\<lambda>s x. f x s) s rs"
Pred\<^sub>\<alpha>: fixes \<phi> :: "'b" shows "Formula.hereditarily_fs (Formula.Pred\<^sub>\<alpha> \<phi>)"
simps_2: fixes a :: "('a, 'b) Node"   and b :: "('a, 'b) Node" shows "FingerTreeStruc.digitToList (Two a b) = FingerTreeStruc.nodeToList a @ FingerTreeStruc.nodeToList b"
nrm_dot: fixes x :: "'a"   and y :: "'a" shows "(x - y) \<bullet> rev.rot (x - y) = 0"
Y: fixes y :: "'b" assumes "y \<in> topspace Y"   and "P (prod_topology X Y)" shows "P (subtopology (prod_topology X Y) (topspace X \<times> {y}))"
dvd_triv_left: fixes times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and a :: "'m"   and b :: "'m" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times a (times a b)"
surjective: fixes r :: "'b ibody_scheme" shows "r = \<lparr>access = access r, imethods = imethods r, \<dots> = ibody.more r\<rparr>"
deg_shifts_def: fixes d :: "nat"   and fs :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" shows "deg_shifts d fs = concat (map (\<lambda>f. map (\<lambda>t. local.punit.monom_mult (1::'b) t f) (punit.pps_to_list (deg_le_sect X (d - poly_deg f)))) fs)"
rcs_eq: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G" shows "(b \<cdot> \<rho> a \<in> H) = (H \<bullet> a = H \<bullet> b)"
null_space_is_preserved: fixes P :: "(('b, 'c) vec, 'd) vec"   and A :: "(('b, 'e) vec, 'c) vec" assumes "invertible P" shows "null_space (P ** A) = null_space A"
try_spmf_bind_pmf: fixes p :: "'e pmf"   and f :: "'e \<Rightarrow> 'd spmf"   and q :: "'d spmf" shows "TRY p \<bind> f ELSE q = p \<bind> (\<lambda>x. TRY f x ELSE q)"
subterm_trm_ord: fixes t :: "'a trm"   and p :: "indices list"   and s :: "'a trm" assumes "subterm t p s"   and "p \<noteq> []" shows "(s, t) \<in> trm_ord"
local_ring_at_is_local: fixes R :: "'c set"   and I :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "pr_ideal R I addition multiplication zero unit" shows "local_ring R \<^bsub>I addition multiplication zero\<^esub> (pr_ideal.add_local_ring_at R I addition multiplication zero) (pr_ideal.mult_local_ring_at R I addition multiplication zero) (pr_ideal.zero_local_ring_at R I addition multiplication zero unit) (pr_ideal.one_local_ring_at R I addition multiplication zero unit)"
lim_5: shows "integrable (M1 \<Otimes>\<^sub>M M2) f"
lexs_def: fixes p :: "real \<times> real"   and q :: "real \<times> real" shows "lexs p q = (Counterclockwise_2D_Arbitrary.lex p q \<and> p \<noteq> q)"
redundant_def: fixes g :: "'g" shows "uninst_code.ssa.redundant uses phis g \<equiv> \<exists>v\<in>uninst_code.ssa.allVars uses phis g. uninst_code.ssa.trivial uses phis g v"
split_option_ex: fixes P :: "'c option \<Rightarrow> bool" shows "(\<exists>x. P x) = (P None \<or> (\<exists>x. P (Some x)))"
eventually_at_left: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and y :: "'a"   and x :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "class.linorder_topology less_eq less open"   and "less y x" shows "eventually P (topological_space.at_within open x (ord.lessThan less x)) = (\<exists>b. less b x \<and> (\<forall>y. less b y \<longrightarrow> less y x \<longrightarrow> P y))"
type_definition_IF2: shows "type_definition Rep_IF2 Abs_IF2 (car_init2 undefined)"
conv_radius_ratio_limit_nonzero: fixes c' :: "ereal"   and c :: "real"   and f :: "nat \<Rightarrow> 'a" assumes "c' = ereal c"   and "c \<noteq> 0"   and "(\<lambda>n. norm (f n) / norm (f (Suc n))) \<longlonglongrightarrow> c" shows "conv_radius f = c'"
in_interval_to_interval: fixes a :: "'a" shows "a \<in>\<^sub>i interval_of a"
snd_hd: fixes xs :: "'x" assumes "fst (list_of_oalist xs) \<noteq> []" shows "snd (local.hd xs) = lookup xs (fst (local.hd xs))"
dimX: fixes n :: "nat" shows "X n \<in> carrier_mat m m"
lang_def: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula" shows "lang idx \<phi> = L idx {\<AA>. \<AA> \<Turnstile> \<phi> \<and> #\<^sub>V \<AA> = idx}"
finite_rderivs_same: fixes a :: "'a"   and r :: "'b rexp" shows "finite {\<guillemotleft>rderivs (replicate m a) r\<guillemotright> |m. True}"
dghm_ArrMap_vrange: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) \<subseteq>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
inv_approx: fixes E :: "'b set set"   and S :: "'b set"   and r :: "nat" assumes "Graph_E E"   and "independent_vertices E S"   and "card (Graph_E.vertices E) \<le> card S * r" shows "Graph_E.approximation_miv E r S"
hyp1: shows "matrix_to_iarray (fst (Gauss_Jordan_upt_k_PA A k)) = fst (Gauss_Jordan_upt_k_iarrays_PA (matrix_to_iarray A) k)"
simps: fixes L :: "'a list"   and i :: "nat" shows "closeI L i = close L (L ! i)"
cases: fixes a1 :: "'d list"   and a2 :: "'d list" assumes "cancel1 a1 a2" obtains xs :: "'d list" and gs :: "'d list" and gs' :: "'d list" where "a1 = gs @ xs @ xs @ gs'" and "a2 = gs @ xs @ gs'" and "xs \<noteq> []"
prj_simps_5: shows "C.dom \<mu>\<nu>.prj\<^sub>0 = \<mu>.leg0 \<down>\<down> \<nu>.leg1"
lset_lfilter: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lset (lfilter P xs) = {x \<in> lset xs. P x}"
assms_2: fixes n' :: "nat" assumes "\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'" shows "\<exists>n''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n'\<^esub>. n'' \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n'\<^esub> \<and> eval c t t' n'' \<gamma>"
w_strictly_prefers_def: fixes w :: "W"   and \<mu> :: "(M \<times> W) set" shows "Non_Strict.w_strictly_prefers w \<mu> = {m' \<in> Field (Pw w). \<forall>m\<in>{m. (m, w) \<in> \<mu>}. (m, m') \<in> Pw w \<and> (m', m) \<notin> Pw w}"
l0: shows "limsup (\<lambda>x. ereal (real C / real x)) = 0"
tranclp_unfold: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" shows "r\<^sup>+\<^sup>+ a b \<equiv> (a, b) \<in> {(x, y). r x y}\<^sup>+"
in_set_upd_eq_aux: fixes i :: "nat"   and l :: "'a list"   and x :: "'a"   and y :: "'a" assumes "i < length l" shows "(x \<in> set (l[i := y])) = (x = y \<or> (\<forall>y. x \<in> set (l[i := y])))"
deriv_ln_Gamma_complex: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "deriv ln_Gamma z = Digamma z"
not_Ref_eq: fixes x :: "'b ref" shows "(\<forall>y. x \<noteq> Ref y) = (x = Null)"
fresh_ty_dom: shows "X \<sharp> ty_dom \<Gamma>"
prems_4: shows "card {q. prime q \<and> q < p'} = n"
path_cons_simp: fixes v :: "nat"   and v' :: "nat"   and e :: "(nat \<times> (form, nat) out_port) \<times> nat \<times> (form, nat) antecedent"   and pth :: "((nat \<times> (form, nat) out_port) \<times> nat \<times> (form, nat) antecedent) list" shows "task.path v v' (e # pth) = (fst (fst e) = v \<and> e \<in> task_edges \<and> task.path (fst (snd e)) v' pth)"
t0_1: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
r_not_updated_stays_the_same: fixes r :: "nat"   and U :: "(nat \<times> vname aexp) list"   and c :: "vname \<Rightarrow> value option"   and d :: "nat \<Rightarrow>f value option" assumes "r \<notin> fst ` set U" shows "apply_updates U c d $ r = d $ r"
invKey_eq: fixes K :: "key"   and K' :: "key" shows "(invKey K = invKey K') = (K = K')"
eq_val_eq_ideal: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier (Vr K v)"   and "y \<in> carrier (Vr K v)"   and "v x = v y" shows "Vr K v \<diamondsuit>\<^sub>p x = Vr K v \<diamondsuit>\<^sub>p y"
integrable_null_measure_iff: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" shows "integrable (null_measure M) f = (f \<in> borel_measurable M)"
reachable_induce_ss: fixes G :: "('c, 'd) pre_digraph"   and S :: "'c set"   and u :: "'c"   and v :: "'c"   and T :: "'c set" assumes "wf_digraph G"   and "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v"   and "S \<subseteq> T" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> T\<^esub> v"
nf_invar_axioms_def: fixes C :: "nat set"   and N :: "nat set" shows "nf_invar_axioms C N \<equiv> C = {} \<longrightarrow> N = {}"
suminf_pos_iff: fixes f :: "nat \<Rightarrow> 'b" assumes "summable f"   and "\<And>n. (0::'b) \<le> f n" shows "((0::'b) < suminf f) = (\<exists>i. (0::'b) < f i)"
divide_closed: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "y \<noteq> \<zero>" shows "local.divide x y \<in> carrier Zp"
subtermeq_set_if_vars: fixes x :: "'a"   and M :: "('b, 'a) Term.term set" assumes "x \<in> fv\<^sub>s\<^sub>e\<^sub>t M" shows "Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M"
simps_1: fixes D :: "('d, 'e) ctxt" shows "\<box> \<circ>\<^sub>c D = D"
pair_graphI_complete: fixes n :: "nat"   and G :: "'b pair_pre_digraph" assumes "K\<^bsub>n\<^esub> (with_proj G)" shows "pair_graph G"
AssertionStatImpEqvt: fixes p :: "(name \<times> name) list"   and \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "p \<bullet> (\<Psi> \<hookrightarrow> \<Psi>') = p \<bullet> \<Psi> \<hookrightarrow> p \<bullet> \<Psi>'"
gen_dfs_def: shows "gen_dfs \<equiv> gds_init gds \<bind> WHILE gen_cond gen_step"
lift_imp_trans: fixes A :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool"   and C :: "'b \<Rightarrow> bool" assumes "\<turnstile> A \<longrightarrow> B"   and "\<turnstile> B \<longrightarrow> C" shows "\<turnstile> A \<longrightarrow> C"
times_class: fixes x :: "'b"   and y :: "'b" shows "x \<otimes>\<^bsub>cring_class_ops\<^esub> y = x * y"