hcju/isabelleps huggingface dataset:
not_reach_a: shows "\<not> tail G a \<rightarrow>\<^sup>*\<^bsub>H\<^esub> head G a"
that: fixes T :: "real" assumes "0 \<le> T"   and "T \<le> t"   and "flow0 x ` {0..T} \<subseteq> M"   and "\<And>s'. \<lbrakk>0 \<le> s'; s' \<le> t; flow0 x ` {0..s'} \<subseteq> M\<rbrakk> \<Longrightarrow> s' \<le> T" shows "thesis"
maybe_or_idempotent: fixes a :: "trilean" shows "a \<or>? a = a"
x: fixes z :: "'a" assumes "z \<in> set xs"   and "P z" shows "False"
basic_superposition_axioms: shows "basic_superposition trm_ord sel pos_ord vars filter_trms"
simps_1: fixes M :: "nat" shows "input_sizes (Input M) = [M]"
leq_antisym: fixes x :: "'h spmf"   and y :: "'h spmf" assumes "ord_spmf (=) x y"   and "ord_spmf (=) y x" shows "x = y"
integral_FTC_Icc: fixes a :: "real"   and b :: "real"   and F :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (F has_vector_derivative f x) (at x within {a..b})"   and "continuous_on {a..b} f" shows "LBINT x. indicat_real {a..b} x *\<^sub>R f x = F b - F a"
b_e_type_comp: fixes \<C> :: "t_context"   and es :: "b_e list"   and e :: "b_e"   and t1s :: "t list"   and t4s :: "t list" assumes "\<C> \<turnstile> es @ [e] : t1s _> t4s" shows "\<exists>ts'. \<C> \<turnstile> es : t1s _> ts' \<and> \<C> \<turnstile> [e] : ts' _> t4s"
simple_rule_conj_def: shows "simple_rule_conj \<equiv> (uncurry SimpleRule \<circ>\<circ> apsnd) (uncurry simple_action_conj)"
retract_inverse: fixes x :: "'a" assumes "x \<in> carrier \<X>\<^bsub>G\<^esub>" shows "\<pi>\<^sub>*\<^bsub>G\<^esub> (\<pi>\<^sup>*\<^bsub>G\<^esub> x) = x"
simps: fixes \<Delta> :: "('a, 'b) network_scheme"   and S :: "'a set" shows "cut \<Delta> S = (source \<Delta> \<in> S \<and> sink \<Delta> \<notin> S)"
trms\<^sub>s\<^sub>s\<^sub>t_unlabel_dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_eq: fixes A :: "('g strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
ltln_weak_strong_1: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> W\<^sub>n \<psi> = w \<Turnstile>\<^sub>n G\<^sub>n \<phi> or\<^sub>n (\<phi> U\<^sub>n \<psi>)"
App_def: fixes f :: "'c \<Rightarrow> 'd" shows "App f \<equiv> f"
MakeFaceFinal_facesAt_eq: fixes f :: "face"   and g :: "graph" assumes "f \<in> \<F> g"   and "minGraphProps g" shows "facesAt_eq (makeFaceFinal f g)"
simps_1: fixes div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and H :: "'a mat"   and P :: "'a mat"   and K :: "'a mat" shows "local.reduce_column_aux div_op [] H (P, K) = (P, K)"
inf_absorb2: fixes y :: "'b"   and x :: "'b" assumes "y \<le> x" shows "x \<sqinter> y = y"
IH1_5: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M1" shows "interpret_slp slp1 xs ! slp_index_lookup slp1 M1 f = interpret_floatarith f xs"
is_inj_def: fixes x :: "'a" shows "is_inj x \<equiv> x ; x\<^sup>\<smile> \<le> 1'"
img_bound: shows "range_vars \<delta> \<subseteq> fv\<^sub>s\<^sub>t (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')"
INF_empty: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and f :: "'c::type \<Rightarrow> 'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(Inf::'b::type set \<Rightarrow> 'b::type) ((f::'c::type \<Rightarrow> 'b::type) ` {}) = (top::'b::type)"
sincreasing_grD: fixes s :: "'b stream"   and i :: "nat"   and x :: "'b" assumes "sincreasing s" shows "\<exists>j>i. x < s !! j"
inv_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<otimes> y = \<one>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "y \<otimes> x = \<one>"
Abs_sat'_nat_of: fixes n :: "'b sat" shows "Abs_sat' (nat_of n) = n"
l_sort_inv_2_def: fixes x :: "'b"   and xs :: "'b list"   and X :: "'b list \<times> 'b list \<times> 'b list" shows "l_sort_inv_2 x xs X \<equiv> (fst X = [] \<longrightarrow> fst (snd X) = []) \<and> l_count x (fst X) + l_count x (fst (snd X)) + l_count x (snd (snd X)) = l_count x xs"
sint_eq_uint: fixes x :: "'b word" assumes "\<not> msb x" shows "sint x = uint x"
bal_split_min: fixes t :: "'c tree234"   and x :: "'c"   and t' :: "'c up\<^sub>d" assumes "split_min t = (x, t')"   and "bal t"   and "0 < height t" shows "bal (tree\<^sub>d t')"
is_empty_empty: fixes q :: "('c, 'd) pq" shows "is_empty q = (q = PQ.empty)"
aligned_add_offset_no_wrap: fixes x :: "'b word"   and sz :: "nat"   and off :: "'b word" assumes "is_aligned x sz"   and "off < 2 ^ sz" shows "unat x + unat off < 2 ^ LENGTH('b)"
ocondition_def: fixes c :: "'a \<Rightarrow> bool"   and L :: "'a \<Rightarrow> 'b"   and R :: "'a \<Rightarrow> 'b" shows "ocondition c L R \<equiv> \<lambda>s. if c s then L s else R s"
cdi_is_cd_icdi: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and k :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and j :: "nat"   and i :: "nat" assumes "IFC prob"   and "IFC_def.is_icdi prob k \<pi> j" shows "IFC_def.is_cdi prob k \<pi> i = (IFC_def.is_cdi prob j \<pi> i \<or> i = j)"
prems_1: shows "\<forall>a\<in>set (A # As). (0::'a) \<in> a"
division_ofD_1: fixes s :: "'a set set"   and i :: "'a set" assumes "s division_of i" shows "finite s"
dd_def: shows "dd = of_int n * g z powr of_int (n - 1) * gd"
elims_1: fixes x :: "('d, 'e) fmap"   and xa :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list"   and y :: "bool" assumes "sat_precond_as x xa = y" obtains s :: "('d, 'e) fmap" where "x = s__" and "xa = []" and "y"   | s :: "('d, 'e) fmap" and a :: "('d, 'e) fmap \<times> ('d, 'e) fmap" and as :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list" where "x = s" and "xa = a # as" and "y = (fst a \<subseteq>\<^sub>f s \<and> sat_precond_as (state_succ s a) as)"
smult_diff_right: fixes a :: "'d"   and p :: "'d poly"   and q :: "'d poly" shows "smult a (p - q) = smult a p - smult a q"
fps_lr_inverse_minus_2: fixes f :: "'a fps"   and x :: "'a" shows "fps_right_inverse (- f) (- x) = - fps_right_inverse f x"
get_heap_is_OK: fixes h :: "'a"   and f :: "'a \<Rightarrow> ('a, 'b, 'c) prog" shows "h \<turnstile> ok Heap_Error_Monad.bind get_heap f = h \<turnstile> ok f h"
prems_2: shows "gc' \<rightarrow>\<^bsub>bs\<^esub> gc''"
v: shows "a \<in> \<Z> (charslength p) v"
update_get: fixes f :: "('k \<Rightarrow> 'a option) \<Rightarrow> 'k \<Rightarrow> 'a option"   and g :: "('k \<Rightarrow> 'a option) \<Rightarrow> 'm" shows "update f (get g) = get (update f \<circ> g \<circ> f)"
simps_83: fixes x101 :: "stmt"   and x102 :: "stmt"   and x41 :: "stmt"   and x42 :: "stmt" shows "x101 Finally x102 \<noteq> x41;; x42"
normal_vector_3: fixes W :: "'a vec set"   and w :: "'a vec" assumes "finite W"   and "Suc (card W) = n"   and "lin_indpt W"   and "W \<subseteq> carrier_vec n"   and "w \<in> W" shows "w \<bullet> normal_vector W = (0::'a)"
compose_def: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c" shows "Cring_Poly.compose R f g = eval R (UP R) (to_polynomial R) g f"
intros_39: fixes n :: "nat"   and i :: "nat"   and vs :: "v list" shows "\<lparr>[Local n i vs [Trap]]\<rparr> \<leadsto> \<lparr>[Trap]\<rparr>"
cases: fixes a1 :: "('rule \<times> 'form) NatRule"   and a2 :: "'form fset \<times> 'form"   and a3 :: "('form fset \<times> 'form) fset" assumes "eff a1 a2 a3" obtains con :: "'form" and \<Gamma> :: "'form fset" where "a1 = Axiom" and "a2 = (\<Gamma>__ \<turnstile> con)" and "a3 = {||}" and "con |\<in>| \<Gamma>__"   | rule :: "'rule \<times> 'form" and c :: "'form" and ants :: "('form, 'var) antecedent fset" and \<Gamma> :: "'form fset" and a :: "nat" and s :: "'subst" where "a1 = NatRule rule" and "a2 = (\<Gamma>'__ \<turnstile> subst s (freshen a c))" and "a3 = (\<lambda>ant. (\<lambda>p. subst s (freshen a p)) |`| a_hyps ant |\<union>| \<Gamma>'__ \<turnstile> subst s (freshen a (a_conc ant))) |`| ants" and "natEff_Inst rule c ants" and "\<And>ant f. \<lbrakk>ant |\<in>| ants; f |\<in>| \<Gamma>'__\<rbrakk> \<Longrightarrow> freshenLC a ` a_fresh ant \<inter> lconsts f = {}" and "\<And>ant. ant |\<in>| ants \<Longrightarrow> freshenLC a ` a_fresh ant \<inter> subst_lconsts s = {}"   | \<Gamma> :: "'form fset" and c' :: "'form" where "a1 = Cut" and "a2 = (\<Gamma> \<turnstile> c')" and "a3 = {|\<Gamma> \<turnstile> c'|}"
b: shows "length (bvxor (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))) (MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))))) @ sha1 (generate_M' (sha1 m) salt) @ BC) = length (bvxor (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))) (MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt)))))))) + 168"
arr_iff_in_hom: fixes f :: "('o, 'a) arr" shows "arr f = in_hom f (local.dom f) (cod f)"
nonempty_graph_def: fixes G :: "nat set \<times> nat set set" shows "nonempty_graph G \<equiv> uverts G \<noteq> {} \<and> uedges G \<noteq> {}"
f_inv_on_f: fixes y :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "y \<in> f ` A" shows "f (inv_on f A y) = y"
vdomain_atD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "\<exists>b\<in>elts (\<R>\<^sub>\<circ> r). r\<lparr>a\<rparr> = b"
H_eq_2: fixes s :: "'s" shows "(\<not> recurrent s) = (H s s = 0)"
order_refl: fixes less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and x :: "'a::type" assumes "class.preorder (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (x::'a::type) x"
cut_geE: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<in> I \<down>\<ge> t" obtains "x \<in> I" and "t \<le> x"
primes_M_bigo: shows "\<MM> \<in> O(ln)"
nth_fls_subdegree_nonzero: fixes f :: "'b fls" assumes "f \<noteq> 0" shows "fls_nth f (fls_subdegree f) \<noteq> (0::'b)"
psubst_cong: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "good X"   and "goodEnv rho"   and "goodEnv rho'"   and "\<And>ys y. fresh ys y X \<or> rho ys y = rho' ys y" shows "X #[rho] = X #[rho']"
Root_def: fixes F :: "('a, 'b, 'c) seqauto set"   and G :: "'a \<Rightarrow> ('a, 'b, 'c) seqauto set option" shows "Root F G = (SOME A. A \<in> F \<and> A \<notin> \<Union> (ran G))"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> char list option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_attribute get_attribute_locs"   and "l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_attribute set_attribute_locs" shows "l_set_attribute_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_attribute get_attribute_locs set_attribute set_attribute_locs"
sub_q11_array: fixes i :: "nat" assumes "i \<in> {..<11}" shows "q11_array !! i = (i \<in> q11)"
1: shows "v * top * e = v * v\<^sup>T * e"
obsnecs: shows "obsp \<sigma> i \<noteq> obsp \<sigma>' i' \<or> cs\<^bsup>\<pi>\<^esup> i \<noteq> cs\<^bsup>\<pi>'\<^esup> i'"
lborel_f_measurable_restrict: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "(\<lambda>f. restrict f {..<m}) \<in> lborel_f n \<rightarrow>\<^sub>M lborel_f m"
one_Arity_def: shows "1 \<equiv> to_Arity 1"
eq_elem_in: fixes a :: "'e"   and A :: "'e set"   and b :: "'e" assumes "a \<in> A"   and "a = b" shows "b \<in> A"
path_connected_component_set: fixes S :: "'b set" shows "path_connected S = (\<forall>x\<in>S. path_component_set S x = S)"
left_right_simps_2: fixes l :: "'a"   and r :: "'a" shows "right (app l r) = r"
swap_def: fixes p :: "'c \<times> 'b" shows "prod.swap p = (time p, hamlet p)"
polysub_def: fixes p :: "poly"   and q :: "poly" shows "p -\<^sub>p q = p +\<^sub>p ~\<^sub>p q"
gauss_cnj_0: shows "gauss_cnj 0 = 0"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 r f"
atMost_shift: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "prod g {..n} = g 0 * (\<Prod>i<n. g (Suc i))"
sorted_del_list: fixes xs :: "'b list"   and x :: "'b" assumes "Sorted_Less.sorted xs" shows "Sorted_Less.sorted (del_list x xs)"
subset_\<J>: fixes k' :: "nat"   and k :: "nat"   and u' :: "nat"   and u :: "nat" assumes "k' < k \<or> k' = k \<and> u' \<le> u" shows "\<J> k' u' \<subseteq> \<J> k u"
coinduct: fixes X :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'b"   and x2 :: "'b"   and R :: "('b \<times> 'b) set" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>x y z. x1 = x \<and> x2 = z \<and> (x, y) \<in> R \<and> (X y z \<or> (y, z) \<in> Omega_Algebra_Models.omega R)" shows "(x1, x2) \<in> Omega_Algebra_Models.omega R"
open_cover_of_subset_def: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and U :: "'a set"   and I :: "real set"   and C :: "real \<Rightarrow> 'a set" shows "open_cover_of_subset X is_open U I C \<equiv> topological_space X is_open \<and> cover_of_subset X U I C \<and> open_cover_of_subset_axioms is_open I C"
ceiling_le: fixes x :: "'a"   and a :: "int" assumes "x \<le> of_int a" shows "\<lceil>x\<rceil> \<le> a"
assms_1: shows "\<forall>a. a + 1 < min (dim_row A) (dim_col A) \<longrightarrow> A $$ (a, a) dvd A $$ (a + 1, a + 1)"
not_has_disc_NegPos_map: fixes disc :: "'c \<Rightarrow> bool"   and C :: "'b \<Rightarrow> 'c"   and ls :: "'b negation_type list" assumes "\<forall>a. \<not> disc (C a)" shows "\<forall>a\<in>set (NegPos_map C ls). \<not> has_disc disc (negation_type_to_match_expr a)"
numeral_monom: fixes n :: "num" shows "numeral n = Polynomial.monom (numeral n) 0"
zero_cap_simp: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<notin> E" shows "c (u, v) = (0::'capacity)"
dd_closed: fixes n' :: "'a LDCFG_node"   and V :: "'d"   and n'' :: "'a LDCFG_node"   and S :: "'a LDCFG_node set" assumes "CFGExit_wf.data_dependence n' V n''"   and "n'' \<in> CFGExit_wf.wod_backward_slice S" shows "n' \<in> CFGExit_wf.wod_backward_slice S"
enc_atom_dec: fixes n :: "nat"   and w :: "('a \<times> bool list) list"   and FO :: "nat set"   and p :: "nat" assumes "wf_word n w"   and "\<forall>i\<in>FO. i < n \<longrightarrow> (\<exists>!p. p < length w \<and> snd (w ! p) ! i)"   and "p < length w" shows "enc_atom (dec_interp n FO w) p (fst (w ! p)) = w ! p"
case_1: fixes f1 :: "'c btree \<Rightarrow> 'b"   and f2 :: "'c btree \<Rightarrow> 'c \<Rightarrow> 'c btree \<Rightarrow> 'b"   and x1 :: "'c btree" shows "(case T\<^sub>i x1 of T\<^sub>i x \<Rightarrow> f1 x | Up\<^sub>i x xa xb \<Rightarrow> f2 x xa xb) = f1 x1"
inj_on_list_of_vec: fixes n :: "nat" shows "inj_on list_of_vec (carrier_vec n)"
connected_component_def: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" shows "connected_component S x y \<equiv> \<exists>T. connected T \<and> T \<subseteq> S \<and> x \<in> T \<and> y \<in> T"
subgroup_inverse_equality: fixes u :: "'a" assumes "u \<in> G" shows "inverse u = sub.inverse u"
exec_impl_steps_Normal_Abrupt: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "'b"   and t :: "'b" assumes "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> Abrupt t" shows "\<Gamma>\<turnstile> (c, Normal s) \<rightarrow>\<^sup>* (Throw, Normal t)"
bind_advantage: fixes \<A> :: "'g \<times> 'a \<Rightarrow> (('j \<times> 'd) \<times> 'e \<times> (('e \<times> 'k) \<times> 'e \<times> 'f) \<times> 'e \<times> ('e \<times> 'k) \<times> 'e \<times> 'f) spmf" shows "OR_\<Sigma>_commit.abstract_com.bind_advantage \<A> \<le> OR_\<Sigma>_commit.rel_advantage (OR_\<Sigma>_commit.adversary \<A>)"
set_mset_vars_term_ms: fixes t :: "('b, 'a) Term.term" shows "set_mset (vars_term_ms t) = vars_term t"
rep_inverse: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and equal :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and rep_nat :: "'b \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'b"   and x :: "'b" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "abs_nat (rep_nat x) = x"
CARD: shows "Card_order r \<and> Card_order r' \<and> Card_order p"
simps_4: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "ty"   and va :: "ty list" shows "(P,E,h \<turnstile> [] [:]\<^bsub>NT\<^esub> v # va) = False"
IH: fixes b :: "V" assumes "vfinite b" shows "vfinite (a' \<times>\<^sub>\<bullet> b)"
of_nat_0_eq_iff: 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_char_0 one times plus zero" shows "(zero = semiring_1.of_nat one plus zero n) = (0 = n)"
insert_DiffM: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "add_mset x (M \<ominus> x) = M"
convergence_G_less_1: fixes z :: "'a"   and x :: "'s"   and y :: "'s" assumes "norm z < 1" shows "convergence_G x y z"
size_median_of_medians_aux: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "R \<in> {(<), \<lambda>x y. y < x}" shows "size {#y \<in># mset xs. R y M#} \<le> nat \<lceil>7 / 10 * real (length xs) + 3\<rceil>"
step1: shows "Bell (n + m) = card {P. partition_on A P}"
lstrict_prefix_code_1: shows "lstrict_prefix LNil LNil = False"
ub1': fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "('d \<times> 'c) list"   and y :: "'c"   and a :: "'d"   and b :: "'c" assumes "Semilat A r f"   and "\<forall>(p, s)\<in>set S. s \<in> A"   and "y \<in> A"   and "(a, b) \<in> set S" shows "b \<sqsubseteq>\<^bsub>r\<^esub> map snd (filter (\<lambda>(p', t'). p' = a) S) \<Squnion>\<^bsub>f\<^esub> y"
real_of_3_remdups_equal_3: fixes xs :: "real_alg_3 list" shows "real_of_3 ` set (Complex_Algebraic_Numbers.remdups_gen equal_3 xs) = real_of_3 ` set xs"
axioms_1: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah" assumes "inverse_transformation A B F G \<tau>" shows "category A"
in_set_butlastI: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs"   and "x \<noteq> last xs" shows "x \<in> set (butlast xs)"
rv_branching_edges_slice_kinds_False: fixes a :: "'edge"   and ax :: "'edge"   and S :: "'node SDG_node set"   and as :: "'edge list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and asx :: "'edge list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "valid_edge a"   and "valid_edge ax"   and "sourcenode a = sourcenode ax"   and "targetnode a \<noteq> targetnode ax"   and "intra_kind (kind a)"   and "intra_kind (kind ax)"   and "preds (slice_kinds S (a # as)) s"   and "preds (slice_kinds S (ax # asx)) s'"   and "length s = length s'"   and "snd (hd s) = snd (hd s')"   and "\<forall>V\<in>rv S (CFG_node (sourcenode a)). state_val s V = state_val s' V" shows "False"
precedesFilter: fixes a :: "'a"   and b :: "'a"   and list :: "'a list"   and f :: "'a \<Rightarrow> bool" assumes "precedes a b list"   and "f a"   and "f b" shows "precedes a b (filter f list)"
supr_least: fixes r :: "('a \<times> 'a) set"   and B :: "'a set"   and a :: "'a" assumes "wo_rel r"   and "B \<subseteq> Field r"   and "a \<in> Field r"   and "\<And>b. b \<in> B \<Longrightarrow> (b, a) \<in> r" shows "(wo_rel.supr r B, a) \<in> r"
polynomial_in_carrier: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list" assumes "ring R"   and "subring K R"   and "polynomial\<^bsub>R\<^esub> K p" shows "set p \<subseteq> carrier R"
5: shows "(1::'a) \<sqinter> (v \<sqinter> - r) * (v \<sqinter> - r)\<^sup>T \<le> t\<^sup>T * t"
cdf_unique: fixes M1 :: "real measure"   and M2 :: "real measure" assumes "real_distribution M1"   and "real_distribution M2"   and "cdf M1 = cdf M2" shows "M1 = M2"
StepSeq: fixes c\<^sub>1 :: "cmd"   and c\<^sub>2 :: "cmd"   and l :: "nat"   and s :: "char list \<Rightarrow> val option" assumes "labels (c\<^sub>1;; c\<^sub>2) l (Skip;; c\<^sub>2)"   and "labels (c\<^sub>1;; c\<^sub>2) #:c\<^sub>1 c\<^sub>2"   and "l < #:c\<^sub>1" shows "c\<^sub>1;; c\<^sub>2 \<turnstile> \<langle>Skip;; c\<^sub>2,s,l\<rangle> \<leadsto> \<langle>c\<^sub>2,s,#:c\<^sub>1\<rangle>"
\<beta>sdec: shows "sorted_wrt (\<lambda>x y. y < x) \<beta>s"
uncurry_preserves_functors: fixes F :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "functor (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) F" shows "functor (\<cdot>\<^sub>J\<^sub>x\<^sub>A) (\<cdot>\<^sub>B) (Curry.uncurry F)"
fixp_induct_uc: fixes U :: "'c \<Rightarrow> 'a \<Rightarrow> 'b set"   and F :: "'c \<Rightarrow> 'c"   and C :: "('a \<Rightarrow> 'b set) \<Rightarrow> 'c"   and f :: "'c"   and P :: "('a \<Rightarrow> 'b set) \<Rightarrow> bool" assumes "\<And>x. set.mono_body (\<lambda>f. U (F (C f)) x)"   and "f \<equiv> C (set.fixp_fun (\<lambda>f. U (F (C f))))"   and "\<And>f. U (C f) = f"   and "set.admissible P"   and "P (\<lambda>_. \<Union> {})"   and "\<And>f. P (U f) \<Longrightarrow> P (U (F f))" shows "P (U f)"
num_reachable_def: fixes G :: "'b graph"   and v :: "'b" shows "num_reachable G v = card (succ_tran G v)"
termMOD_igSwapIGAbs: shows "igSwapIGAbs termMOD"
DSTEP_star_mono: fixes s :: "('c + 'd) list"   and t :: "('c + 'd) list"   and G :: "('c \<times> ('c, 'd) RHS) list" assumes "s -G\<rightarrow>\<^sup>* t" shows "length s \<le> length t"
fds_right_inverse_unique: fixes f :: "'b fds"   and g :: "'b fds" assumes "f * g = 1" shows "g = inverse f"
compl_sets: fixes a :: "'a set"   and M :: "'a measure" assumes "a \<in> sets M" shows "space M - a \<in> sets M"
strong_postdominate_refl: fixes n :: "'node" assumes "valid_node n"   and "n \<noteq> (_Exit_)" shows "n strongly-postdominates n"
Vd_def: fixes d :: "nat" shows "Vd d \<equiv> {v. local.connected src v \<and> ndist v = d}"
monad_fmap: fixes f :: "'f \<rightarrow> 'd"   and xs :: "'f\<cdot>'e" shows "fmap\<cdot>f\<cdot>xs = xs \<bind> (\<Lambda> x. return\<cdot>(f\<cdot>x))"