hcju/isabelleps huggingface dataset:
comp2_2: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "f \<in> PrimRec2"   and "g \<in> PrimRec2"   and "h \<in> PrimRec2" shows "(\<lambda>x y. f (g x y) (h x y)) \<in> PrimRec2"
Arr: shows "\<AA>\<lparr>Arr\<rparr> = []\<^sub>\<circ>"
distinct_46: fixes name' :: "name"   and pi' :: "pi"   and pi :: "pi" shows "<\<nu>name'>pi' \<noteq> \<tau>.pi"
kronecker_factorization_def: fixes p :: "int poly" shows "kronecker_factorization p = kronecker_factorization_main divisors_int divisors_int_pos (degree p div 2) p"
H_def: shows "H = fps_poly_swap2 hensel_fpxs2"
ringE_1: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "abelian_group R"
nths_reencode_eq3: fixes C :: "nat set"   and B :: "nat set"   and A :: "nat set"   and xs :: "'a list" assumes "C \<subseteq> B"   and "B \<subseteq> A" shows "nths (nths xs (ind_in_set A ` B)) (ind_in_set B ` C) = nths xs (ind_in_set A ` C)"
sgn_def: fixes x :: "'a" shows "Matrix_Equation_Construction.sgn x = (if (0::'a) < x then 1 else if x < (0::'a) then - 1 else 0)"
cut_le_Min_eq: fixes I :: "'a set"   and t :: "'a" assumes "I \<down>\<le> t \<noteq> {}" shows "iMin (I \<down>\<le> t) = iMin I"
axioms_2: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "total_pseudo_ordered_set A less_eq" shows "antisymmetric A less_eq"
finite_PLS: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "finite (PLS s as)"
d_IN_h_le_value: fixes x :: "'v" shows "d_IN h x \<le> value_flow \<Delta> h"
assms_2: fixes exp :: "exp"   and var :: "Vars.var" assumes "y = App exp var" shows "P"
simps_1: fixes f :: "'c \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat" shows "map_idx f [] n = []"
iso_reduced_homology_group_upper_hemisphere: fixes k :: "nat"   and n :: "nat"   and p :: "int" assumes "k \<le> n" shows "hom_induced p (nsphere n) {} (nsphere n) {x. 0 \<le> x k} id \<in> Group.iso (reduced_homology_group p (nsphere n)) (relative_homology_group p (nsphere n) {x. 0 \<le> x k})"
dg_Par_is_arrE: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V"   and S :: "V" assumes "S : A \<mapsto>\<^bsub>dg_Par \<alpha>\<^esub> B" obtains "arr_Par \<alpha> S" and "S\<lparr>ArrDom\<rparr> = A" and "S\<lparr>ArrCod\<rparr> = B"
return_state_def: fixes return :: "'i \<times> 'g \<Rightarrow> 'h"   and x :: "'i" shows "return_state return x = StateT (\<lambda>s. return (x, s))"
order_iff_strict: fixes b :: "'a"   and a :: "'a" shows "lesseq_a b a = (less_a b a \<or> a = b)"
assms_6: fixes e :: "real" assumes "0 < e" shows "\<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm (\<lambda>v. f' x v - f' a v) < e"
complete_uniform: fixes S :: "'a set" shows "complete S = (\<forall>F\<le>principal S. F \<noteq> bot \<longrightarrow> cauchy_filter F \<longrightarrow> (\<exists>x\<in>S. F \<le> nhds x))"
af_decompose_2: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and w :: "'a set list" shows "af (\<phi> or\<^sub>n \<psi>) w = af \<phi> w or\<^sub>n af \<psi> w"
rbt_delete_def: fixes k :: "'a"   and t :: "('a, 'c) rbt" shows "rbt_delete k t = paint B (rbt_del k t)"
fv_Let: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "fv (Terms.Let \<Gamma> e) = fv \<Gamma> \<union> fv e - domA \<Gamma>"
a_host_shadow_root_rel_shadow_root: fixes h :: "(_) heap"   and host :: "(_) element_ptr"   and shadow_root_option :: "(_) shadow_root_ptr option"   and shadow_root :: "(_) shadow_root_ptr" assumes "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r shadow_root_option" shows "(shadow_root_option = Some shadow_root) = ((cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root) \<in> local.a_host_shadow_root_rel h)"
set_2: fixes x2 :: "'a" shows "set_extended \<lbrakk>x2\<rbrakk> = {x2}"
s1_1_is_pr: shows "s1_1 \<in> PrimRec2"
advantage_def: fixes key_gen :: "'a spmf"   and encrypt :: "'a \<Rightarrow> 'b \<Rightarrow> 'c spmf"   and decrypt :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option"   and msg_predicate :: "'b \<Rightarrow> bool"   and \<A> :: "(bool, 'b \<times> 'b + 'c, 'c option + 'b option) gpv" shows "IND_CCA2_sym.ind_cca.advantage key_gen encrypt decrypt msg_predicate \<A> = \<bar>spmf (IND_CCA2_sym.ind_cca.game key_gen encrypt decrypt msg_predicate \<A>) True - 1 / 2\<bar>"
inject: fixes x21 :: "'c tree"   and x22 :: "'c"   and x23 :: "'c tree"   and y21 :: "'c tree"   and y22 :: "'c"   and y23 :: "'c tree" shows "(\<langle>x21, x22, x23\<rangle> = \<langle>y21, y22, y23\<rangle>) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
refine: fixes ai :: "'b list list"   and a :: "'c set"   and sctni :: "real list sctn"   and sctn :: "'c sctn" assumes "(ai, a) \<in> clw_rel appr_rel"   and "(sctni, sctn) \<in> \<langle>lv_rel\<rangle>sctn_rel" shows "(below_sctn_coll_impl ai sctni, below_sctn_coll $ a $ sctn) \<in> \<langle>bool_rel\<rangle>nres_rel"
prob_space_bind: fixes M :: "'b measure"   and N :: "'b \<Rightarrow> 'c measure"   and S :: "'c measure" assumes "prob_space M"   and "AE x in M. prob_space (N x)"   and "N \<in> M \<rightarrow>\<^sub>M subprob_algebra S" shows "prob_space (M \<bind> N)"
f3: shows "pre (oid (i, Store e1 mo2 i)) (Broadcast (i, Store e1 mo2 i)) prefix of oid (i, Store e1 mo2 i)"
Rep_bit0_inverse: fixes x :: "'b bit0" shows "Abs_bit0 (Rep_bit0 x) = x"
inj_on_inv_into: fixes F :: "'a \<Rightarrow> 'b"   and P :: "'a set" assumes "inj_on F P" shows "inv_into P F ` F ` P = P"
reset_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat"   and k :: "nat"   and d :: "'c" shows "reset M n k d = (\<lambda>i j. if i = k \<and> j = 0 then Le d else if i = 0 \<and> j = k then Le (- d) else if i = k \<and> j \<noteq> k then \<infinity> else if i \<noteq> k \<and> j = k then \<infinity> else if i = k \<and> j = k then M k k else min (dbm_add (M i k) (M k j)) (M i j))"
alive_alloc_exhaust: fixes x :: "Value"   and s :: "Store"   and t :: "New" shows "alive x (s\<langle>t\<rangle>) = (alive x s \<or> x = new s t)"
interior_unique: fixes T :: "'a set"   and S :: "'a set" assumes "T \<subseteq> S"   and "open T"   and "\<And>T'. \<lbrakk>T' \<subseteq> S; open T'\<rbrakk> \<Longrightarrow> T' \<subseteq> T" shows "interior S = T"
suminfI: fixes f :: "nat \<Rightarrow> 'a"   and k :: "'a" assumes "f sums k" shows "suminf f = k"
prv_nprv0I: fixes \<phi> :: "'fmla" assumes "prv \<phi>"   and "\<phi> \<in> fmla" shows "nprv {} \<phi>"
has_integral_null_eq: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and i :: "'b" assumes "content (cbox a b) = 0" shows "(f has_integral i) (cbox a b) = (i = (0::'b))"
rep_eq: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b fset" assumes "semilattice_fset f" shows "semilattice_fset.F f x = semilattice_set.F f (fset x)"
blinding_of_on_axioms: shows "blinding_of_on A h bo"
cart_prod_split: fixes x :: "'e"   and y :: "'f"   and A :: "'e set"   and B :: "'f set" shows "((x, y) \<in> A \<times> B) = (x \<in> A \<and> y \<in> B)"
emeasure_density_add: fixes X :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "X \<in> sets M"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "emeasure (density M f) X + emeasure (density M g) X = emeasure (density M (\<lambda>x. f x + g x)) X"
le_diff_conv2: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b" shows "(c \<le> b - a) = (c + a \<le> b)"
inorder_splay_maxD: fixes t :: "('c \<times> 'd) tree"   and l :: "('c \<times> 'd) tree"   and a :: "'c \<times> 'd"   and r :: "('c \<times> 'd) tree" assumes "splay_max t = \<langle>l, a, r\<rangle>"   and "sorted1 (inorder t)" shows "inorder l @ [a] = inorder t \<and> r = \<langle>\<rangle>"
hom_ik: shows "\<forall>l. \<forall>m\<in>M l. homogeneous\<^sub>l\<^sub>s\<^sub>t m \<A> Sec \<and> m \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
distinct_923: fixes x281 :: "t"   and x282 :: "testop"   and x311 :: "t"   and x312 :: "cvtop"   and x313 :: "t"   and x314 :: "sx option" shows "Testop x281 x282 \<noteq> Cvtop x311 x312 x313 x314"
F_apply: fixes a :: "'a"   and b :: "'b"   and a' :: "'a" assumes "a \<in> basisA'"   and "b \<in> basisB"   and "a' \<in> basisA'" shows "F a b *\<^sub>V a' = (if a' = a then b else (0::'b))"
index_minus_mat_3: fixes A :: "'b mat"   and B :: "'b mat" shows "dim_col (A - B) = dim_col B"
assms_3: shows "j < length (func_t (s_inst \<S> ! i))"
IH_1: assumes "\<not> complete l" shows "size1 l < 2 ^ height l"
prems_3: shows "(yys, zzs) \<in> lexord r"
enum_def: fixes v :: "'a" shows "enum v = (if dist v = \<infinity> \<or> dist v = - \<infinity> then \<infinity> else enat (num v))"
simps_2: fixes uv :: "le_rel" shows "Lt_Rel + uv = Lt_Rel"
list_def: fixes n :: "nat"   and A :: "'c set" shows "list n A = {xs. length xs = n \<and> set xs \<subseteq> A}"
terminates_def: fixes g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)" shows "terminates g = (terminates_on g = UNIV)"
piJ: shows "path_image J = path_image J1 \<union> path_image J2"
simps_4: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and P :: "'b"   and \<alpha> :: "'d"   and t :: "('e, 'c, 'd) Tree" assumes "nominal_ts satisfies transition" shows "nominal_ts.valid_Tree satisfies transition P (tAct \<alpha> t) = (\<exists>\<alpha>'' t' P'. tAct \<alpha> t =\<^sub>\<alpha> tAct \<alpha>'' t' \<and> transition P \<langle>\<alpha>'',P'\<rangle> \<and> nominal_ts.valid_Tree satisfies transition P' t')"
os__coplanar: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B OS C D" shows "Coplanar A B C D"
map_diff_def: fixes f :: "'c \<Rightarrow> 'd option"   and ks :: "'c fset" shows "map_diff f ks = f |` (- fset ks)"
rel_cases: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "('d, 'f) optionT"   and b :: "('e, 'g) optionT" assumes "rel_optionT' R1 R2 a b" obtains xa :: "'f" and ya :: "'g" where "a = OptionT xa" and "b = OptionT ya" and "R2 xa ya"
Int_Un_distrib2: fixes B :: "'b set"   and C :: "'b set"   and A :: "'b set" shows "(B \<union> C) \<inter> A = B \<inter> A \<union> C \<inter> A"
interaction_bounded_by_exec_gpv_count: fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('d, 'a, 'b) gpv"   and n :: "enat"   and x :: "'d"   and s' :: "'s"   and s :: "'s"   and count :: "'s \<Rightarrow> enat" assumes "interaction_bounded_by consider gpv n"   and "(x, s') \<in> set_spmf (exec_gpv callee gpv s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); I s; consider x; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> count s' \<le> eSuc (count s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); I s; \<not> consider x; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> count s' \<le> count s"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "count s' \<le> n + count s"
sp_cf_len: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and a :: "'a" assumes "pol_coeff S c"   and "a \<in> carrier S" shows "fst (sp_cf S a c) = fst c"
snth_smap: fixes f :: "'c \<Rightarrow> 'b"   and s :: "'c stream"   and n :: "nat" shows "smap f s !! n = f (s !! n)"
distinct_29: fixes x9 :: "('a, String.literal) phantom" shows "OclVoid \<noteq> Enum x9"
insert_ident: fixes x :: "'k"   and A :: "'k set"   and B :: "'k set" assumes "x \<notin> A"   and "x \<notin> B" shows "(insert x A = insert x B) = (A = B)"
generate_M'_def: fixes mHash :: "bit list"   and salt_new :: "bit list" shows "generate_M' mHash salt_new = bv_prepend 64 \<zero> [] @ mHash @ salt_new"
prems_3: shows "d \<in> set (markedElements (m # M'))"
span_on_def: fixes b :: "'b set" shows "local.span b = {\<Sum>a\<in>t. r a *s a |t r. finite t \<and> t \<subseteq> b}"
weak_ranking_permute: fixes carrier :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c" assumes "finite_total_preorder_on carrier le"   and "f permutes carrier" shows "weak_ranking (map_relation (inv f) le) = map ((`) f) (weak_ranking le)"
assms_1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> set xs"   and "y \<in> set xs" shows "P x y = Q x y"
Min_gr_iff: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}" shows "(x < Min A) = (\<forall>a\<in>A. x < a)"
r9: fixes x :: "name"   and s :: "trm" assumes "x \<sharp> s" shows "s to x in [Var x] \<mapsto> s"
equivD: fixes c :: "com"   and c' :: "com"   and \<pi> :: "char list \<Rightarrow> com option"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "c \<sim> c'" shows "\<pi>: (c, s) \<Rightarrow> t = \<pi>: (c', s) \<Rightarrow> t"
hyps_2: fixes cstate :: "call \<times> (nat \<Rightarrow> contour option) \<times> ((nat \<times> char list) \<times> contour \<Rightarrow> d option) \<times> contour"   and cstate_a :: "call \<times> (nat \<Rightarrow> 'c option) \<times> ((nat \<times> char list) \<times> 'c \<Rightarrow> 'c \<aproc> set) \<times> 'c" assumes "|cstate| \<lessapprox> cstate_a" shows "|evalC\<cdot>(Discr cstate)| \<lessapprox> \<aC>\<cdot>(Discr cstate_a)"
cod_in_codomains: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "partial_magma C"   and "partial_magma.codomains C f \<noteq> {}" shows "partial_magma.cod C f \<in> partial_magma.codomains C f"
polynomial_pow_degree: fixes p :: "'a list"   and n :: "nat" assumes "p \<in> carrier (poly_ring R)" shows "degree (p [^]\<^bsub>poly_ring R\<^esub> n) = n * degree p"
component_le_infnorm_cart: fixes x :: "(real, 'a) vec"   and i :: "'a" shows "\<bar>x $ i\<bar> \<le> infnorm x"
sstt: fixes n :: "nat" assumes "n < length cl" shows "s \<approx> t"
simps: fixes a1 :: "nat"   and a2 :: "Regions.intv" shows "Regions.valid_intv a1 a2 = ((\<exists>d c. a1 = c \<and> a2 = Regions.intv.Const d \<and> 0 \<le> d \<and> d \<le> c) \<or> (\<exists>d c. a1 = c \<and> a2 = Regions.intv.Intv d \<and> 0 \<le> d \<and> d < c) \<or> (\<exists>c. a1 = c \<and> a2 = Regions.intv.Greater c))"
equals_supr: fixes B :: "'a set"   and a :: "'a" assumes "B \<subseteq> Field r"   and "a \<in> Field r"   and "\<And>b. b \<in> B \<Longrightarrow> (b, a) \<in> r"   and "\<And>a'. \<lbrakk>a' \<in> Field r; \<And>b. b \<in> B \<Longrightarrow> (b, a') \<in> r\<rbrakk> \<Longrightarrow> (a, a') \<in> r" shows "a = local.supr B"
vconverse_atI: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = b" shows "\<langle>b, a\<rangle> \<in>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>)\<inverse>\<^sub>\<circ>"
cont2cont_Pair: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "cont f"   and "cont g" shows "cont (\<lambda>x. (f x, g x))"
Grp_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
rel_fun_iff_leq_vimage2p: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and S :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j" shows "(R ===> S) f g = (R \<le> BNF_Def.vimage2p f g S)"
lagrange_interpolation_poly: fixes xs_ys :: "('a \<times> 'a) list"   and p :: "'a poly"   and x :: "'a"   and y :: "'a" assumes "distinct (map fst xs_ys)"   and "p = lagrange_interpolation_poly xs_ys"   and "(x, y) \<in> set xs_ys" shows "poly p x = y"
configuration_token: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "x \<le> n" shows "x \<in> semi_mojmir_def.configuration \<delta> q\<^sub>0 w (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n) n"
unit_abs_eta_conv: fixes f :: "unit \<Rightarrow> 'b" shows "(\<lambda>u. f ()) = f"
mul_not_mask_eq_neg_shiftl: fixes n :: "nat" shows "NOT (mask n) = - 1 << n"
pathfinish_pathstart_partcirclepath_simps_8: fixes z0 :: "complex"   and r :: "real"   and st :: "real" shows "pathfinish (part_circlepath z0 r st 0) = z0 + complex_of_real r"
pred_cong: fixes x :: "'a list"   and ya :: "'a list"   and P :: "'a \<Rightarrow> bool"   and Pa :: "'a \<Rightarrow> bool" assumes "x = ya"   and "\<And>z. z \<in> set ya \<Longrightarrow> P z = Pa z" shows "list_all P x = list_all Pa ya"
return_time0: fixes A :: "'a set" shows "return_time_function A -` {0} = UNIV - recurrent_subset A"
apfst_def: fixes f :: "'h \<Rightarrow> 'j" shows "apfst f = map_prod f id"
d1_def: shows "c.d1 = prod_list c.dims1"
normalize_div: fixes a :: "'a" shows "normalize a div a = (1::'a) div unit_factor a"
aux3: fixes tx :: "real \<times> 'a"   and h :: "real \<times> 'a" shows "blinfun_apply (ode_d2 (snd tx)) (snd h) o\<^sub>L snd_blinfun = blinfun_apply (flip_blinfun (flip_blinfun (ode_d2 (snd tx) o\<^sub>L snd_blinfun) o\<^sub>L snd_blinfun)) h"
l_delete_shadow_root_get_disconnected_nodes_get_disconnected_nodes_locs: shows "l_delete_shadow_root_get_disconnected_nodes get_disconnected_nodes_locs"
narrow: fixes G :: "prog"   and S :: "ty"   and T :: "ty" assumes "G\<turnstile>S\<succ>T" shows "G\<turnstile>S\<preceq>? T"
diff_union_single_conv: fixes a :: "'a"   and J :: "'a multiset"   and I :: "'a multiset" assumes "a \<in># J" shows "I + J - {#a#} = I + (J - {#a#})"
nu_rec_total_pure_rule: fixes R :: "('d \<times> 'd) set"   and B :: "('e, 'f) urel \<Rightarrow> ('e, 'f) urel"   and Pre :: "('e, 'f) urel"   and e :: "('d, 'e) uexpr"   and Post :: "('e, 'f) urel" assumes "wf R"   and "Monotonic B"   and "\<And>f st. \<lbrakk>(Pre \<and> (\<lceil>e\<rceil>\<^sub><, \<guillemotleft>st\<guillemotright>)\<^sub>u \<in>\<^sub>u \<guillemotleft>R\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> f; \<nu> B = f\<rbrakk> \<Longrightarrow> (Pre \<and> \<lceil>e\<rceil>\<^sub>< =\<^sub>u \<guillemotleft>st\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> B f" shows "(Pre \<Rightarrow> Post) \<sqsubseteq> \<nu> B"
feels_safe_def: fixes s :: "event list"   and r :: "room" shows "feels_safe s r = (\<exists>s\<^sub>1 s\<^sub>2 s\<^sub>3 g c c'. s = s\<^sub>3 @ [Enter g r c] @ s\<^sub>2 @ [Check_in g r c'] @ s\<^sub>1 \<and> no_Check_in (s\<^sub>3 @ s\<^sub>2) r \<and> isin (s\<^sub>2 @ [Check_in g r c] @ s\<^sub>1) r = {})"
stutter_extend_build: fixes u :: "nat \<Rightarrow> 'b"   and v :: "nat \<Rightarrow> 'b"   and a :: "'b" assumes "u \<approx> v" shows "a ## u \<approx> a ## v"
powr_float_interval_eqI: fixes p :: "nat"   and X :: "float interval"   and Y :: "float interval"   and IVL :: "float interval"   and x :: "real"   and y :: "real" assumes "powr_float_interval p X Y = Some IVL"   and "x \<in>\<^sub>r X"   and "y \<in>\<^sub>r Y" shows "x powr y \<in>\<^sub>r IVL"
Sym: fixes P :: "pi"   and Q :: "pi" assumes "P \<equiv>\<^sub>s Q" shows "Q \<equiv>\<^sub>s P"
notin: shows "\<nexists>l'. cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>'\<^esup> l'"
fls_subdegree_fls_to_fps_gt0: fixes f :: "'b fps" shows "0 \<le> fls_subdegree (fps_to_fls f)"
cross3_simps_10: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b * c = a * (b * c)"
idiff_enat_0: fixes n :: "enat" shows "enat 0 - n = enat 0"
meet_bot: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<sqinter> \<bottom> = \<bottom>"
p4_def: shows "p4 = p \<circ>\<^sub>p [:Complex (Re lb) (Im ub), Complex 0 (Im lb - Im ub):]"
r_update_def: shows "r_update \<equiv> Cn 3 r_update_aux [Cn 3 r_length [recf.Id 3 0], recf.Id 3 0, recf.Id 3 1, recf.Id 3 2]"
valid_pimp_def: fixes ts :: "((stmt \<times> nat) \<times> instr list \<times> (nat \<Rightarrow> 'a option) \<times> (stmt \<times> nat) memref list \<times> 'b \<times> 'c) list" shows "valid_pimp ts \<equiv> separated_tmps ts"
not_fic_nrename: fixes M :: "trm"   and x :: "name"   and y :: "name"   and c :: "coname" assumes "\<not> fic (M[x\<turnstile>n>y]) c" shows "\<not> fic M c"
fps_expansion_def: fixes f :: "complex \<Rightarrow> complex"   and z0 :: "complex" shows "fps_expansion f z0 = Abs_fps (\<lambda>n. (deriv ^^ n) f z0 / fact n)"
surjective_axioms: shows "surjective (inv (inv (inv (inv hom))))"
sum_if_inner: fixes i :: "'a"   and j :: "'a"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "i \<in> Basis"   and "j \<in> Basis" shows "(\<Sum>k\<in>Basis. if k = i then f i *\<^sub>R i else g k *\<^sub>R k) \<bullet> j = (if j = i then f j else g j)"
residualInject'_1: fixes \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi"   and M :: "'d"   and N :: "'d"   and Q :: "('d, 'e, 'f) psi" shows "(\<alpha> \<prec> P = RIn M N Q) = (P = Q \<and> \<alpha> = M\<lparr>N\<rparr>)"
vectorspace_def: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme" shows "vectorspace K V \<equiv> module K V \<and> field K"
simps: fixes dd :: "nat"   and c :: "int"   and g :: "int poly"   and mm :: "int"   and k :: "nat"   and kk :: "nat" shows "local.mahler_approximation_main dd c g mm k kk = (let mmm = mahler_landau_graeffe_approximation kk dd g; new_mm = if k = 0 then mmm else min mm mmm in if bnd \<le> k then new_mm else local.mahler_approximation_main (dd * dd) c (graeffe_one_step c g) new_mm (Suc k) (2 * kk))"
dpd1: fixes x :: "'a" shows "x \<le> x \<cdot> antidomain_semiringl.ads_d antirange_op x"
Y_Z': fixes n :: "nat" assumes "1 \<le> n" shows "Y n \<subseteq> Z' n"
lunit_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.ide (free_monoidal_category.comp C) a" shows "partial_magma.in_hom (free_monoidal_category.comp C) (free_monoidal_category.lunit\<^sub>F\<^sub>M\<^sub>C C a) (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C (free_monoidal_category.unity\<^sub>F\<^sub>M\<^sub>C C) a) a"
range_ex1_eq: fixes f :: "'d \<Rightarrow> 'e"   and b :: "'e" assumes "inj f" shows "(b \<in> range f) = (\<exists>!x. b = f x)"
ext_cont_def: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c" shows "ext_cont f a b = (\<lambda>x. f (clamp a b x))"
mscf: shows "matrix_seq d (\<lambda>n. complex_of_real c \<cdot>\<^sub>m f n)"
f_def: fixes s :: "(nat \<times> form) list list"   and n :: "nat" shows "f s n = iter (\<lambda>x. flatten (map subs x)) s n"
m_rcancel: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly"   and c :: "'a mod_ring poly" assumes "a \<noteq> \<zero>\<^bsub>R\<^esub>"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "(b \<otimes>\<^bsub>R\<^esub> a = c \<otimes>\<^bsub>R\<^esub> a) = (b = c)"
vcpowerI: fixes f :: "V"   and n :: "V"   and A :: "V" assumes "f \<in>\<^sub>\<circ> (\<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>n. A)" shows "f \<in>\<^sub>\<circ> A ^\<^sub>\<times> n"
rep_eq: fixes x :: "'b"   and xa :: "'c"   and xb :: "('b, 'c) alist" shows "alist.impl_of (DAList.update x xa xb) = AList.update x xa (alist.impl_of xb)"
word_sle_eq: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "(a \<le>s b) = (sint a \<le> sint b)"
pa_eq: shows "proj_arcs_H ` A = proj_arcs_H ` B"
intros_3: fixes A :: "nat \<Rightarrow> 'c set"   and \<Omega> :: "'c set"   and M :: "'c set set" assumes "range A \<in> Pow (smallest_ccdi_sets \<Omega> M)"   and "A 0 = {}"   and "\<And>n. A n \<subseteq> A (Suc n)" shows "\<Union> (range A) \<in> smallest_ccdi_sets \<Omega> M"
trivial_group_def: fixes G :: "('a, 'b) monoid_scheme" shows "trivial_group G \<equiv> Group.group G \<and> carrier G = {\<one>\<^bsub>G\<^esub>}"