hcju/isabelleps huggingface dataset:
inv_of_domain_ss0: fixes U :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "continuous_on U f"   and "inj_on f U"   and "f ` U \<subseteq> S"   and "subspace S"   and "dim S = DIM('b)"   and "openin (top_of_set S) U" shows "openin (top_of_set S) (f ` U)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and hd :: "('c, 'b) hd" shows "P (case hd of Var x \<Rightarrow> f1 x | Sym x \<Rightarrow> f2 x) = ((\<forall>x1. hd = Var x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. hd = Sym x2 \<longrightarrow> P (f2 x2)))"
Melem_subseteq_Union_mset: fixes x :: "'b multiset"   and T :: "'b multiset multiset" assumes "x \<in># T" shows "x \<subseteq># \<Sum>\<^sub># T"
nAct_mono: fixes n :: "enat"   and n' :: "enat"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "cnf llist" assumes "n \<le> n'" shows "dynamic_component.nAct tCMP active c n t \<le> dynamic_component.nAct tCMP active c n' t"
pts1_def: shows "pts1 \<equiv> pts \<inter> s"
trace_callee_eq_def: fixes callee1 :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and callee2 :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf"   and A :: "'b set"   and p :: "'a spmf"   and q :: "'d spmf" shows "trace_callee_eq callee1 callee2 A p q = (\<forall>xs. set xs \<subseteq> A \<times> UNIV \<longrightarrow> (\<forall>x\<in>A. trace_callee callee1 p xs x = trace_callee callee2 q xs x))"
antisymI: fixes r :: "('d \<times> 'd) set" assumes "\<And>x y. \<lbrakk>(x, y) \<in> r; (y, x) \<in> r\<rbrakk> \<Longrightarrow> x = y" shows "antisym r"
continuous_on_of_real: fixes s :: "'b set"   and g :: "'b \<Rightarrow> real" assumes "continuous_on s g" shows "continuous_on s (\<lambda>x. of_real (g x))"
real_mon_conv_le: fixes x :: "nat \<Rightarrow> real"   and y :: "real"   and i :: "nat" assumes "x\<up>y" shows "x i \<le> y"
not_in_path_image_join: fixes x :: "'b"   and g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "x \<notin> path_image g1"   and "x \<notin> path_image g2" shows "x \<notin> path_image (g1 +++ g2)"
map_fun_apply: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'b"   and h :: "'d \<Rightarrow> 'e"   and x :: "'c" shows "(f ---> g) h x = g (h (f x))"
mat_zeror: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "\<delta> \<oplus> f = f"
pdevs_val_unop_linear: fixes f :: "'b \<Rightarrow> 'c"   and e :: "nat \<Rightarrow> real"   and x :: "'b pdevs" assumes "linear f" shows "pdevs_val e (unop_pdevs f x) = f (pdevs_val e x)"
has_domain_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C" shows "(partial_magma.domains (span_vertical_category.vcomp C) \<mu> \<noteq> {}) = arrow_of_spans C \<mu>"
iso_unit: fixes a :: "'a" assumes "B.obj a" shows "B.iso \<i>[a]"
hom_connected_2: fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "(\<nu> \<star> \<mu> \<noteq> null) = (\<nu> \<star> local.dom \<mu> \<noteq> null)"
tail_of_expansion: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R" shows "(1\<^sub>r \<plusminus> x)^\<^bsup>R Suc n\<^esup> = \<Sigma>\<^sub>e R (\<lambda>i.  \<^bsub>Suc n\<^esub>C\<^bsub>Suc i\<^esub> \<times>\<^bsub>R\<^esub> x^\<^bsup>R Suc i\<^esup>) n \<plusminus> 1\<^sub>r"
arrE: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VV.arr f" shows "arr (fst f) \<and> arr (snd f) \<and> src (fst f) = trg (snd f)"
powr_log_cancel: fixes a :: "real"   and x :: "real" assumes "0 < a"   and "a \<noteq> 1"   and "0 < x" shows "a powr Transcendental.log a x = x"
dim_col_mat_2: fixes nr :: "nat"   and nc :: "nat"   and g :: "nat \<Rightarrow> 'a Matrix.vec" shows "dim_col (mat\<^sub>r nr nc g) = nc"
bind_is_OK_pure_I: fixes f :: "('x, 'y, 'z) prog"   and h :: "'x"   and g :: "'z \<Rightarrow> ('x, 'y, 'aa) prog" assumes "pure f h"   and "h \<turnstile> ok f"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> h \<turnstile> ok g x" shows "h \<turnstile> ok Heap_Error_Monad.bind f g"
exhaust: fixes y :: "'d \<times> 'e" obtains a :: "'d" and b :: "'e" where "y = (a, b)"
map_map: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "map f (map g xs) = map (f \<circ> g) xs"
d_IN_split_Vertex: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "antiparallel_edges \<Delta>" shows "d_IN (antiparallel_edges.split f) (Vertex x) = d_IN f x"
simple_path2_aux: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" obtains ns' :: "'node list" where "g \<turnstile> n-ns'\<rightarrow>m" and "distinct ns'" and "set ns' \<subseteq> set ns" and "length ns' \<le> length ns"
monomial_rel_order_map: fixes x :: "String.literal list \<times> int"   and a :: "char list list"   and b :: "int"   and y :: "String.literal list \<times> int"   and aa :: "char list list"   and bb :: "int" assumes "(x, a, b) \<in> monomial_rel"   and "(y, aa, bb) \<in> monomial_rel" shows "(fst x \<le> fst y) = (a \<le> aa)"
add_le_cancel_left: fixes c :: "'d multiset"   and a :: "'d multiset"   and b :: "'d multiset" shows "(c + a \<subseteq># c + b) = (a \<subseteq># b)"
is_zero_row_def': fixes i :: "'d"   and A :: "(('e, 'f) vec, 'd) vec" shows "is_zero_row i A = (\<forall>j. A $ i $ j = (0::'e))"
join_commute: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlat le" shows "dlat.join le x y = dlat.join le y x"
at_exists_fresh': fixes x :: "'b" assumes "at TYPE('a)"   and "finite (supp x)" shows "\<exists>c. c \<sharp> x"
n_split: shows "[0..<n] = [0..<i] @ i # [Suc i..<n]"
UNIV_tau_i_is_empty: fixes \<tau> :: "'a set \<Rightarrow> 'a set" assumes "finite UNIV"   and "mono \<tau>"   and "\<forall>i. (\<tau> ^^ (i + 1)) UNIV \<subset> (\<tau> ^^ i) UNIV" shows "(\<tau> ^^ card UNIV) UNIV = {}"
dim_subset_UNIV: fixes S :: "'b set" shows "dim S \<le> dimension"
gen_length_code_2: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "List.gen_length n (x # xs) = List.gen_length (Suc n) xs"
alluopairs_ex: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "\<forall>x y. P x y = P y x" shows "(\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y) = (\<exists>(x, y)\<in>set (alluopairs xs). P x y)"
Pow_iff: fixes A :: "'g set"   and B :: "'g set" shows "(A \<in> Pow B) = (A \<subseteq> B)"
fmdom_fmmerge: fixes g :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and xm :: "'d \<rightharpoonup>\<^sub>f 'f"   and ym :: "'d \<rightharpoonup>\<^sub>f 'g" shows "fmdom (fmmerge g xm ym) = fmdom xm |\<inter>| fmdom ym"
concat_append: fixes xs :: "'d list list"   and ys :: "'d list list" shows "concat (xs @ ys) = concat xs @ concat ys"
P2: shows "Bet C B C' \<and> Cong B C' B C"
better_Cut_substn: fixes a :: "coname"   and c :: "coname"   and P :: "trm"   and x :: "name"   and y :: "name"   and M :: "trm"   and N :: "trm" assumes "a \<sharp> [c].P"   and "x \<sharp> (y, P)" shows "Cut <a>.M x.N{y:=<c>.P} = (if M = Ax y a then Cut <c>.P x.N{y:=<c>.P} else Cut <a>.M{y:=<c>.P} x.N{y:=<c>.P})"
bij_coord: fixes X :: "(('a, 'b) vec, 'b) vec" assumes "is_basis (set_of_vector X)" shows "bij (coord X)"
any_moves_remain_winning_Even: fixes p :: "'a list"   and \<sigma> :: "'a list \<Rightarrow> 'a"   and m :: "'a" assumes "odd (length p)"   and "strategy_winning_by_Even \<sigma> p" shows "strategy_winning_by_Even \<sigma> (p @ [m])"
assms_2: shows "case_sum f1 g1 \<noteq> case_sum f2 g2"
F\<eta>_inverse_3: fixes b :: "'d" assumes "D.ide b" shows "C.inverse_arrows (F (\<eta> b)) (\<epsilon>' (F b))"
l_set_shadow_root_get_child_nodes_axioms_def: fixes set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_shadow_root_get_child_nodes_axioms set_shadow_root_locs get_child_nodes_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_shadow_root_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
fresh_vsubst_ident: fixes Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and x1 :: "'var" assumes "good Y"   and "fresh xs x Y" shows "Y #[x1 // x]_xs = Y"
abs_eq: shows "top = Filters.filter.Abs_filter UNIV"
geq2: shows "r2 \<le> r'\<^sub>2"
backward_simulation_pow: fixes step :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and final :: "'d \<Rightarrow> bool"   and order :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and match :: "'e \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool" assumes "backward_simulation step step final final order match" shows "backward_simulation step step final final (lexp order\<^sup>+\<^sup>+) (rel_comp_pow match)"
pos_prod_lt: fixes c :: "'a"   and x :: "'a" assumes "(0::'a) < c" shows "c * x < (0::'a) \<equiv> x < (0::'a)"
convex_closure_rel_interior: fixes S :: "'a set" assumes "convex S" shows "closure (rel_interior S) = closure S"
b0: shows "local.prefixes t\<^sub>0 j \<down>= b\<^sub>0"
higher_deriv_power: fixes j :: "nat"   and z :: "'b"   and n :: "nat"   and w :: "'b" shows "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)"
prems: shows "(u @ [a]) @ v\<^sub>1 =\<^sub>F (u @ [a]) @ v\<^sub>2"
is_heap_partial2_def: fixes xs :: "('b \<times> 'c) list"   and k :: "nat" shows "is_heap_partial2 xs k = (\<forall>i j. eq_pred i j \<longrightarrow> j < length xs \<longrightarrow> j \<noteq> k \<longrightarrow> snd (xs ! i) \<le> snd (xs ! j))"
valid_sops_axioms: shows "valid_sops ts\<^sub>s\<^sub>b"
ego_other_conv: fixes t :: "real" assumes "bounded t" shows "(ego.p t = other.p t) = (ego_other.p t = 0)"
E_Abs: fixes v\<^sub>2 :: "trm"   and T\<^sub>1\<^sub>1 :: "POPLmark.type"   and t\<^sub>1\<^sub>2 :: "trm" assumes "v\<^sub>2 \<in> value" shows "(\<lambda>:T\<^sub>1\<^sub>1. t\<^sub>1\<^sub>2) \<bullet> v\<^sub>2 \<longmapsto> t\<^sub>1\<^sub>2[0 \<mapsto> v\<^sub>2]"
proc_ord_proc_eq_spec: fixes P :: "'b process"   and Q :: "'b process" assumes "P \<sqsubseteq> Q"   and "D P \<subseteq> D Q" shows "P = Q"
has_finite_limits: shows "category_with_finite_limits (\<cdot>)"
strict_border_def: fixes xs :: "'a list"   and ys :: "'a list" shows "strict_border xs ys = (border xs ys \<and> length xs < length ys)"
nn_integral_mult_bounded_inf: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and c :: "ennreal"   and g :: "'c \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "integral\<^sup>N M f < \<infinity>"   and "c \<noteq> \<infinity>"   and "AE x in M. g x \<le> c * f x" shows "integral\<^sup>N M g < \<infinity>"
unique_identity1: fixes g :: "'a" assumes "g \<in> G"   and "\<forall>x\<in>G. binop g x = x" shows "g = e"
absorb1: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "inf a b = a"
pull_up_configuration_senior: fixes x :: "nat"   and q :: "'b"   and n :: "nat"   and y :: "nat" assumes "x \<in> configuration q n"   and "y \<in> configuration q n" shows "senior x n = senior y n"
piecewise_smooth_path_imp_reverse: fixes g :: "real \<Rightarrow> 'a" assumes "g piecewise_C1_differentiable_on {0..1}" shows "reversepath g piecewise_C1_differentiable_on {0..1}"
cases: fixes a :: "'a tree" assumes "tfinite a" obtains t :: "'a tree" where "a = t" and "\<And>t'. t' |\<in>| cont t \<Longrightarrow> tfinite t'"
pmdl_Macaulay_list: fixes ps :: "(('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" shows "MPoly_Type_Class.punit.pmdl (set (pp_pm.extended_ord.punit.Macaulay_list ps)) = MPoly_Type_Class.punit.pmdl (set ps)"
enab_alpha2: shows "\<turnstile> $pc2 = #a \<longrightarrow> #0 < $sem \<longrightarrow> Enabled alpha2"
v: shows "v = - (e ^ 3 / (27::'a))"
assign_elim: fixes x :: "var\<^sub>C"   and e :: "aexp\<^sub>C"   and mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat"   and c' :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds' :: "Mode \<Rightarrow> var\<^sub>C set"   and mem' :: "var\<^sub>C \<Rightarrow> nat" assumes "eval_abv\<^sub>C \<langle>x \<leftarrow> e, mds, mem\<rangle>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C" shows "c' = Stop \<and> mds = mds' \<and> mem' = mem(x := ev\<^sub>A\<^sub>C mem e)"
AnnoTry: fixes is_lub :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and P :: "'b J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and e1 :: "'b expr"   and e1' :: "'b expr"   and V :: "String.literal"   and C :: "String.literal"   and e2 :: "'b expr"   and e2' :: "'b expr" assumes "is_lub,P,E \<turnstile> e1 \<leadsto> e1'"   and "is_lub,P,E(V \<mapsto> Class C) \<turnstile> e2 \<leadsto> e2'" shows "is_lub,P,E \<turnstile> try e1 catch(C V) e2 \<leadsto> try e1' catch(C V) e2'"
top_edge_smooth: shows "(\<lambda>x. twoC (1, x)) piecewise_C1_differentiable_on {0..1}"
pow_cong: fixes x :: "int"   and n :: "nat" shows "(x mod int p) [^]\<^bsub>R\<^esub> n = x ^ n mod int p"
simps_3: fixes n :: "nat" shows "harry_sum (Suc (Suc n)) = 2 * harry_sum (Suc n) + (2 * n + 4) * 2 ^ n"
finished_imp_succ_discovered: fixes v :: "'v"   and w :: "'v" assumes "v \<in> dom (finished s)"   and "w \<in> E `` {v}" shows "w \<in> dom (discovered s)"
premc: fixes \<omega> :: "variable \<Rightarrow> real"   and I :: "interp" shows "\<omega> \<in> fml_sem I \<phi>"
fst_Gauss_Jordan_column_k: fixes i :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" assumes "i \<le> nrows A" shows "fst (Gauss_Jordan_column_k (i, A) k) \<le> nrows A"
jacobian_def: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and net :: "('a, 'b) vec filter" shows "jacobian f net = matrix (frechet_derivative f net)"
distinct_7: fixes list :: "char list"   and form_list :: "form_list"   and list' :: "char list"   and var' :: "NominalSequents.var"   and form' :: "form" shows "Cpd0 list form_list \<noteq> list' \<nabla> [var'].form'"
hyps_4: fixes s :: "dB" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "\<exists>t'a. s => t'a \<and> t'a \<rightarrow>\<^sub>\<eta>\<^sup>* t'"
one_is_id: fixes m :: "'c" assumes "m \<in> M" shows "\<phi> \<one> m = m"
functors_preserve_arrows: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme"   and F :: "('d, 'e, 'g, 'h) functor"   and f :: "'e" assumes "Functor F : AA \<longrightarrow> BB"   and "f \<in> Ar\<^bsub>AA\<^esub>" shows "F \<^bsub>\<a>\<^esub> f \<in> Ar\<^bsub>BB\<^esub>"
orthogonal_matrix_orthonormal_rows: fixes A :: "((real, 'b) vec, 'b) vec" shows "orthogonal_matrix A = ((\<forall>i. norm (row i A) = 1) \<and> (\<forall>i j. i \<noteq> j \<longrightarrow> orthogonal (row i A) (row j A)))"
choose_step_def: fixes ode_ops :: "ode_ops"   and X0 :: "'c set"   and h :: "real" shows "choose_step ode_ops X0 h = method_spec \<bind> (\<lambda>mid. if mid = 2 then rk2_step ode_ops X0 h else euler_step ode_ops X0 h)"
simps_7: fixes i :: "nat"   and \<phi> :: "atom fm" shows "push_forall (ExN i \<phi>) = ExN i (push_forall \<phi>)"
sub1: shows "(\<Union>i\<in>{i |i. Inl i \<in> SIGO ODE1}. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a) \<subseteq> (\<Union>i\<in>{i |i. Inl i \<in> SIGO (OProd ODE1 ODE2)}. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a)"
strong_barbed_bisimulation_impl_weak_respection_of_barbs: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "(strong_reduction_simulation Rel (Calculus CWB) \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>Calculus CWB Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Calculus CWB P' \<and> (P', Q') \<in> Rel))) \<and> rel_respects_binary_pred Rel (\<lambda>a. hasBarb a CWB)" shows "rel_weakly_respects_barbs Rel CWB"
assms: fixes S :: "'a set"   and T :: "'a set" assumes "P S"   and "openin X T"   and "x \<in> T"   and "T \<subseteq> S" shows "P T"
eq_onp_True: shows "eq_onp (\<lambda>_. True) = (=)"
validNF_def: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> ('d \<times> 'c) set \<times> bool"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>! \<equiv> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace> \<and> no_fail P f"
unit_simps_5: shows "trg\<^sub>D (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) = src\<^sub>D (F f)"
finite_surj: fixes A :: "'a set"   and B :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "B \<subseteq> f ` A" shows "finite B"
chamber_vertex_outside_facet_image: fixes v :: "'a"   and z :: "'a set" assumes "v \<notin> z"   and "folding_g.chamber (insert v z)" shows "g v \<notin> g ` z"
filter_mset_empty_if_finite_and_filter_set_empty: fixes X :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "{x \<in> X. P x} = {}"   and "finite X" shows "filter_mset P (mset_set X) = {#}"
trancl_trans: fixes x :: "'d"   and y :: "'d"   and r :: "('d \<times> 'd) set"   and z :: "'d" assumes "(x, y) \<in> r\<^sup>+"   and "(y, z) \<in> r\<^sup>+" shows "(x, z) \<in> r\<^sup>+"
commute: fixes a :: "'b"   and b :: "'b" shows "inf a b = inf b a"
rref_implies_rref_upt: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" assumes "reduced_row_echelon_form A" shows "reduced_row_echelon_form_upt_k A k"
linear_span_image: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec"   and S :: "('b, 'c) vec set" assumes "Vector_Spaces.linear (*s) (*s) f" shows "vec.span (f ` S) = f ` vec.span S"
conc_empty: fixes xs :: "'b seq" shows "conc xs Empty = xs"
elims_3: fixes x :: "'a val" assumes "\<not> is_Intg x" obtains "x = Unit"   | "x = Null"   | v :: "bool" where "x = Bool v__"   | v :: "'a" where "x = Addr v"
admits_diagonal_reduction_eq_exists_algorithm_is_SNF: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat" assumes "A \<in> carrier_mat m n" shows "admits_diagonal_reduction A = (\<exists>algorithm. is_SNF A (algorithm A))"
dist_stereographic': fixes A :: "complex_homo"   and B :: "complex_homo" shows "dist_fs A B = dist_riemann_sphere' (inv_stereographic A) (inv_stereographic B)"
landau_product_preprocess_5: fixes n :: "num" shows "num.One + num.Bit0 n = num.Bit1 n"
gens_span_free_group: fixes gens :: "'c set" shows "\<langle>\<iota> ` gens\<rangle>\<^bsub>\<F>\<^bsub>gens\<^esub>\<^esub> = carrier \<F>\<^bsub>gens\<^esub>"
remove_sorted: fixes t :: "tree"   and x :: "nat" assumes "BinaryTree_TacticStyle.sorted t" shows "BinaryTree_TacticStyle.sorted (remove x t)"
monad_catch_state_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and catch :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" shows "monad_catch_state return bind fail catch get put \<equiv> monad_catch return bind fail catch \<and> monad_state return bind get put \<and> monad_catch_state_axioms catch get put"
simps_2: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "real" shows "aEvalUni (LessUni (a, b, c)) x = (a * x\<^sup>2 + b * x + c < 0)"
Limit_plus_omega_vsubset_Limit: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Limit \<alpha>"   and "Limit \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<alpha> + \<omega> \<subseteq>\<^sub>\<circ> \<beta>"
unit_def: fixes p :: "fm" shows "unit p \<equiv> let p' = zlfm p; l = \<zeta> p'; q = And (Dvd l (CN 0 1 (C 0))) (a_\<beta> p' l); d = \<delta> q; B = remdups (map simpnum (\<beta> q)); a = remdups (map simpnum (\<alpha> q)) in if length B \<le> length a then (q, B, d) else (mirror q, a, d)"
rel_inject: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'c list"   and x2 :: "nat"   and y1 :: "'d list"   and y2 :: "nat" shows "rel_fundef R (Fundef x1 x2) (Fundef y1 y2) = (list_all2 R x1 y1 \<and> x2 = y2)"
delta_scalars_lincomb_eq_nth: fixes ms :: "'v list"   and n :: "nat" assumes "set ms \<subseteq> V"   and "n < length ms" shows "(replicate (length ms) (0::'f))[n := 1::'f] \<bullet>\<cdot> ms = ms ! n"
ChiPlus_subset_States: fixes a :: "('a, 'b, 'c) hierauto"   and t :: "'a" shows "ChiPlus a `` {t} \<subseteq> \<Union> (States ` SAs a)"
dvdnumcoeff_aux: fixes t :: "MIR.num"   and m :: "int" assumes "ismaxcoeff t m"   and "0 \<le> m"   and "1 < numgcdh t m" shows "dvdnumcoeff t (numgcdh t m)"
preserves_comp: fixes g :: "'d"   and f :: "'d" assumes "D.seq g f" shows "\<I>\<^sub>D \<otimes>\<^sub>D g \<cdot>\<^sub>D f = (\<I>\<^sub>D \<otimes>\<^sub>D g) \<cdot>\<^sub>D (\<I>\<^sub>D \<otimes>\<^sub>D f)"
normalize_ex_bound: fixes cs :: "(String.literal \<times> String.literal) set"   and s :: "String.literal set"   and c :: "String.literal" assumes "wf_subclass_loc cs"   and "finite s"   and "c \<in> s"   and "c \<notin> normalize_sort cs s"   and "sort_ex cs s" obtains c' :: "String.literal" where "c' \<in> normalize_sort cs s" and "class_les cs c' c"
from_vec_to_vec: fixes M :: "('c, 'd) sq_matrix" shows "from_vec (to_vec M) = M"
hd_concat_tl: fixes ws :: "'b list list" assumes "ws \<noteq> \<epsilon>" shows "hd ws \<cdot> concat (tl ws) = concat ws"
P_sup_def: fixes s :: "'s \<times> ('c \<Rightarrow> real)"   and P :: "('s \<times> ('c \<Rightarrow> real)) stream \<Rightarrow> bool" shows "MDP.P_sup s P = (\<Squnion>cfg\<in>MDP.cfg_on s. emeasure (MDP.T cfg) {x \<in> space MDP.St. P x})"
hm_fold_custom_empty_sz_1: fixes sz :: "nat" shows "op_map_empty = op_hm_empty_sz sz"
ceiling_correct: fixes x :: "'a" shows "of_int \<lceil>x\<rceil> - (1::'a) < x \<and> x \<le> of_int \<lceil>x\<rceil>"
T_A_def: fixes n :: "nat" shows "T_A n = sum t_A {..<n}"
left_unit: fixes a :: "('a set \<times> 'b) set" assumes "a \<in> stx.carrier_direct_lim" shows "stx.add_rel (stx.class_of X (zero_str\<^sub>X X)) a = a"
2: shows "x\<^sup>T ; - (x\<^sup>T \<cdot> 1') \<le> - (x\<^sup>T \<cdot> 1')"
common_part_in_product: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "P z x \<and> P z y" shows "P z (x \<otimes> y)"
vs: fixes j :: "nat" assumes "j < dim_row B'" shows "R (vs (integer_of_nat j)) (vs' j)"
matrix_vector_mult_zero_eq: fixes P :: "(('b, 'c) vec, 'd) vec"   and A :: "(('b, 'e) vec, 'c) vec"   and x :: "('b, 'e) vec" assumes "invertible P" shows "(P ** A *v x = 0) = (A *v x = 0)"
prime_cong_eq_0_iff: fixes p :: "int"   and a :: "int" assumes "prime p" shows "[a = 0] (mod p) = (\<not> coprime a p)"
preserves_dom: fixes f :: "'c" assumes "C.arr f" shows "C.dom (src\<^sub>C f) = src\<^sub>C (C.dom f)"
ctor1_diff: fixes x :: "('d, 'd IF1, 'd IF2) F1"   and y :: "('d, 'd IF1, 'd IF2) F1" shows "(ctor1 x = ctor1 y) = (x = y)"
spectrum_imp_cxt_quotient_ring: fixes \<pp> :: "'a set" assumes "\<pp> \<in> Spec" shows "Comm_Ring.quotient_ring (R\<setminus>\<pp>) R (+) (\<cdot>) \<zero> \<one>"
zero_less_double_add_iff_zero_less_single_add: fixes a :: "'c" shows "((0::'c) < a + a) = ((0::'c) < a)"
map2fun_def: fixes v :: "(nat, 'a) mapping" shows "\<langle>v\<rangle> \<equiv> \<lambda>x. case Mapping.lookup v x of None \<Rightarrow> 0::'a | Some y \<Rightarrow> y"
C: shows "\<forall>\<^sub>F n in sequentially. (1 / 2) ^ n < epsilon"
is_singletonE: fixes A :: "'a set" assumes "is_singleton A" obtains x :: "'a" where "A = {x}"
topology_c2ka_axioms: shows "topology_c2ka (\<circ>) \<lambda> Orb Orb\<^sub>S Stab fixed (\<lless>\<^sub>\<K>) (\<lless>\<^sub>\<S>) (\<lhd>) (\<sim>\<^sub>\<K>)"
heap_del_min: fixes q :: "'a tree" assumes "heap q" shows "heap (del_min q)"