hcju/isabelleps huggingface dataset:
basis_reduction_swap_def: fixes i :: "nat"   and state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray" shows "local.basis_reduction_swap i state = (let di = d_state state i; dsi = d_state state (Suc i); dim1 = d_state state (i - 1); fi = fi_state state; fim1 = fim1_state state; dmu_i_im1 = dmu_ij_state state i (i - 1); fi' = fim1; fim1' = fi in case state of (f, dmus, djs) \<Rightarrow> (False, i - 1, dec_i (update_im1 (update_i f fi') fim1'), local.swap_mu dmus i dmu_i_im1 dim1 di dsi, iarray_update djs i ((dsi * dim1 + dmu_i_im1 * dmu_i_im1) div di)))"
R1_def: shows "\<R> \<equiv> {\<lambda>x. eval r [x] |r. recfn 1 r \<and> Partial_Recursive.total r}"
ring_simprules_10: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"
map_add_dom_app_simps_3: fixes m :: "'a"   and l2 :: "'a \<Rightarrow> 'b option"   and l1 :: "'a \<Rightarrow> 'b option" assumes "m \<notin> dom l2" shows "(l2 \<Oplus> l1) m = l1 m"
Stone_Weierstrass_real_polynomial_function: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real"   and e :: "real" assumes "compact S"   and "continuous_on S f"   and "0 < e" obtains g :: "'c \<Rightarrow> real" where "real_polynomial_function g" and "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x - g x\<bar> < e"
simps_14: fixes t :: "Syntax.term"   and f :: "(Syntax.var, Syntax.term) fmap" shows "psubst_term (Unroll t) f = Unroll (psubst_term t f)"
param_rec_nat: fixes R :: "('a \<times> 'b) set" shows "(rec_nat, rec_nat) \<in> R \<rightarrow> (nat_rel \<rightarrow> R \<rightarrow> R) \<rightarrow> nat_rel \<rightarrow> R"
assms_1: shows "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
sumf_pos: fixes x :: "'a" assumes "x \<in> carrier" shows "0 < (\<Sum>i\<in>{j \<in> N. x \<in> W j}. f i x)"
cut_le_Int_conv: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<le> t = I \<inter> {..t}"
contour_integral_reversepath: fixes g :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" assumes "valid_path g" shows "contour_integral (reversepath g) f = - contour_integral g f"
venv_lookup: fixes \<Gamma> :: "(nat \<times> ty) list"   and x :: "nat"   and A :: "ty" assumes "lookup \<Gamma> x = Some A" shows "lookup (Venv \<Gamma>) x = Some (V A)"
R: shows "(\<Pi>\<^sub>E i\<in>I. topspace (X i)) = {} \<or> (\<forall>i\<in>I. Hausdorff_space (X i))"
valid_dbm_pos: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and v :: "'e \<Rightarrow> nat" assumes "valid_dbm M n" shows "[M]\<^bsub>v,n\<^esub> \<subseteq> {u. \<forall>c. v c \<le> n \<longrightarrow> (0::'d) \<le> u c}"
interleave_cons_1: fixes l :: "'b list"   and a :: "'b list"   and b :: "'b list"   and x :: "'b" assumes "l \<in> a \<otimes> b" shows "x # l \<in> x # a \<otimes> b"
le: fixes t :: "real" assumes "A \<le> t" shows "cmod (local.laplace_integrand s t) \<le> M * exp (- (Re s - c) * t)"
\<N>_2: shows "\<Union> \<N> = \<Union> \<M>"
sig_red_def: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "sig_red sing_reg top_tail F p q = (\<exists>f\<in>F. \<exists>t. sig_red_single sing_reg top_tail p q f t)"
g_to_list_def: fixes m :: "('b, 'c) assoc_list" shows "lm_basic.g_to_list m \<equiv> lm_basic.iterate m (#) []"
restrict_extensional: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "restrict f A \<in> extensional A"
y_max: fixes t :: "('a \<times> 'a) set" assumes "t \<in> S"   and "s \<subseteq> t" shows "s = t"
assms_5: shows "\<p>\<^sub>1[f, g] \<cdot> h = \<p>\<^sub>1[f, g] \<cdot> h'"
simps_6: fixes ux :: "'a" shows "(extC.Top \<le> extC.Val ux) = False"
rep_eq: fixes x :: "'b" shows "(*\<^sub>V) (cblinfun_mult_right x) = (*) x"
take_map: fixes n :: "nat"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "take n (map f xs) = map f (take n xs)"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_db\<^sub>l\<^sub>s\<^sub>s\<^sub>t: fixes A :: "('f strand_label \<times> ('e, 'd) stateful_strand_step) list" shows "db'\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = db'\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
acc_relfunD: fixes s :: "'a cfg"   and s' :: "'a cfg" assumes "(s, s') \<in> MC.acc" obtains n :: "nat" where "((\<lambda>a b. b \<in> set_pmf (K_cfg a)) ^^ n) s s'"
psubst_idEnv: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" assumes "good X" shows "X #[idEnv] = X"
is_dghm_axioms': fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<alpha>' :: "V"   and \<AA>' :: "V"   and \<BB>' :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<alpha>' = \<alpha>"   and "\<AA>' = \<AA>"   and "\<BB>' = \<BB>" shows "\<FF> : \<AA>' \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>'\<^esub> \<BB>'"
graph_Entry_def: fixes \<alpha>e :: "'d \<Rightarrow> ('e \<times> 'f \<times> 'e) set"   and \<alpha>n :: "'d \<Rightarrow> 'e list"   and invar :: "'d \<Rightarrow> bool"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and Entry :: "'d \<Rightarrow> 'e" shows "graph_Entry \<alpha>e \<alpha>n invar inEdges' Entry \<equiv> graph_path \<alpha>e \<alpha>n invar inEdges' \<and> graph_Entry_axioms \<alpha>n invar inEdges' Entry"
intro: fixes R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and X :: "'e set"   and is_open :: "'e set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'e set \<Rightarrow> 'f set"   and \<rho> :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'f \<Rightarrow> 'f"   and b :: "'f"   and add_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str :: "'e set \<Rightarrow> 'f"   and one_str :: "'e set \<Rightarrow> 'f" assumes "comm_ring R addition multiplication zero unit"   and "locally_ringed_space X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str"   and "scheme_axioms R addition multiplication zero unit X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str" shows "scheme R addition multiplication zero unit X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str"
cart2_append1_z: fixes p :: "proj2" assumes "z_non_zero p" shows "cart2_append1 p $ 3 = 1"
eq_general: fixes B :: "'a set"   and A :: "'b set"   and h :: "'b \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> 'c"   and \<phi> :: "'b \<Rightarrow> 'c" assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>!x. x \<in> A \<and> h x = y"   and "\<And>x. x \<in> A \<Longrightarrow> h x \<in> B \<and> \<gamma> (h x) = \<phi> x" shows "prod \<phi> A = prod \<gamma> B"
keys_def: fixes t :: "('c, 'd) rbt" shows "RBT_Impl.keys t = list.map fst (RBT_Impl.entries t)"
simps_2: fixes v :: "nat \<Rightarrow> 'a"   and l :: "linear_poly"   and r :: "'a" shows "v \<Turnstile>\<^sub>n\<^sub>s GEQ_ns l r = (r \<le> l \<lbrace> v \<rbrace>)"
p_coprime_left_int: fixes p :: "nat"   and a :: "int" assumes "residues_prime p" shows "coprime (int p) a = (\<not> int p dvd a)"
dpd4: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and domain_op :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.domain_pre_dioid_one plus times one less_eq less domain_op" shows "domain_op (plus x y) = plus (domain_op x) (domain_op y)"
M_par_def: fixes s1 :: "('c, 'd) alpha_rp_scheme"   and x1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "('c, 'd) alpha_rp_scheme"   and x2 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and cs :: "'c set" shows "M_par s1 x1 s2 x2 cs = ((\<lambda>(S, S'). diff_tr S' S \<in> ParMerge (diff_tr s1 S) (diff_tr s2 S) cs \<and> ev_eq (tr_filter (tr s1) cs) (tr_filter (tr s2) cs)) \<and> ((\<lambda>(S, S'). (wait s1 \<or> wait s2) \<and> ref S' \<subseteq> (ref s1 \<union> ref s2) \<inter> cs \<union> (ref s1 \<inter> ref s2 - cs)) \<triangleleft> wait \<circ> snd \<triangleright>  (\<lambda>(S, S'). \<not> wait s1 \<or> \<not> wait s2) \<and> MergeSt (alpha_rp.more s1, x1) (alpha_rp.more s2, x2)))"
matrix_to_iarray_echelon_form_of_column_k: fixes k :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" assumes "k < ncols A"   and "i \<le> nrows A" shows "matrix_to_iarray (fst (echelon_form_of_column_k bezout (A, i) k)) = fst (echelon_form_of_column_k_iarrays (matrix_to_iarray A, i, bezout) k)"
heaps: shows "(\<N>\<lbrace>\<Gamma>\<rbrace>)|\<^sup>\<circ>\<^bsub>r\<^esub> \<sqsubseteq> \<N>\<lbrace>delete x \<Gamma>\<rbrace>"
ComplI: fixes c :: "'c"   and A :: "'c set" assumes "c \<in> A \<Longrightarrow> False" shows "c \<in> - A"
composite_simps_2: shows "src ((f \<star> g) \<star> h) = src h"
ran_clearjunk: fixes al :: "('d \<times> 'c) list" shows "ran (map_of (clearjunk al)) = ran (map_of al)"
polyn_deg_add4: fixes p :: "'a"   and q :: "'a"   and n :: "nat" assumes "p \<in> carrier R"   and "q \<in> carrier R"   and "deg R S X p \<le> an n"   and "deg R S X q \<le> an n" shows "deg R S X (p \<plusminus> q) \<le> an n"
mkArr_restrict_eq': fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "S.arr (S.mkArr A B (restrict F A))" shows "S.mkArr A B (restrict F A) = S.mkArr A B F"
vec_inv_def: fixes v :: "'b vec" shows "vec_inv v = (1::'b) / (v \<bullet>c v) \<cdot>\<^sub>v conjugate v"
sup_ge2: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and y :: "'e"   and x :: "'e" assumes "class.semilattice_sup sup less_eq less" shows "less_eq y (sup x y)"
inducts: assumes "res.wf_residual_network" obtains "\<And>x y. (x, y) \<in> \<^bold>E\<^bsub>residual_network f\<^esub> \<Longrightarrow> (y, x) \<notin> \<^bold>E\<^bsub>residual_network f\<^esub>" and "(source (residual_network f), sink (residual_network f)) \<notin> \<^bold>E\<^bsub>residual_network f\<^esub>"
iwlsFSwSTR_qFreshAll_igFreshAll_qInitAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and y :: "'var"   and qX :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm"   and qA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs" assumes "iwlsFSwSTR MOD" shows "(qFresh ys y qX \<longrightarrow> igFresh MOD ys y (qInit MOD qX)) \<and> (qFreshAbs ys y qA \<longrightarrow> igFreshAbs MOD ys y (qInitAbs MOD qA))"
affine_hull_eq_empty: fixes S :: "'a set" shows "(affine hull S = {}) = (S = {})"
find_None_iff: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "(find P xs = None) = (\<nexists>x. x \<in> set xs \<and> P x)"
to_from_enum_alt: fixes x :: "'b" shows "toEnumAlt (fromEnumAlt x) = x"
surjective: fixes r :: "'a heap_scheme" shows "r = \<lparr>arrays = arrays r, refs = refs r, lim = lim r, \<dots> = more r\<rparr>"
subst_subst4: fixes x :: "name"   and P :: "trm"   and N :: "trm"   and y :: "name"   and M :: "trm"   and a :: "coname"   and c :: "coname" assumes "x \<sharp> (P, N, y)"   and "y \<sharp> (M, N)"   and "a \<sharp> (c, P, M)"   and "c \<sharp> (P, a)"   and "M \<noteq> Ax x c" shows "N{a:=x.M}{c:=y.P} = N{c:=y.P}{a:=x.M{c:=y.P}}"
vh_v'h: shows "v \<Turnstile>\<^sub>n\<^sub>s h = v' \<Turnstile>\<^sub>n\<^sub>s h"
at_le: fixes "open" :: "'a set \<Rightarrow> bool"   and s :: "'a set"   and t :: "'a set"   and x :: "'a" assumes "class.topological_space open"   and "s \<subseteq> t" shows "topological_space.at_within open x s \<le> topological_space.at_within open x t"
has_sgnx_imp_sgnx_able: fixes f :: "real \<Rightarrow> real"   and c :: "real"   and F :: "real filter" assumes "(f has_sgnx c) F" shows "f sgnx_able F"
phiDefs_def: fixes g :: "'g"   and n :: "'node" shows "other.phiDefs g n \<equiv> {v. (n, v) \<in> dom (phis' g)}"
n_inj_polymap_times: fixes p :: "'f poly"   and q :: "'f poly" assumes "\<not> inj_on (polymap (p * q)) V" shows "\<not> inj_on (polymap p) V \<or> \<not> inj_on (polymap q) V"
imult_infinity_right: fixes n :: "enat" assumes "0 < n" shows "n * \<infinity> = \<infinity>"
establish_invarI: fixes I :: "'v tarjan_state \<Rightarrow> bool" assumes "on_init tarjan_params \<le>\<^sub>n SPEC (\<lambda>x. I (empty_state x))"   and "\<And>s s' v0. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s = []; v0 \<in> V0; v0 \<notin> dom (discovered s); s' = new_root v0 s\<rbrakk> \<Longrightarrow> on_new_root tarjan_params v0 s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s \<noteq> []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s\<rbrakk> \<Longrightarrow> on_finish tarjan_params u s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u v. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s); v \<in> dom (discovered s); v \<in> dom (finished s); s' = cross_edge u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<rbrakk> \<Longrightarrow> on_cross_edge tarjan_params u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u v. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s); v \<in> dom (discovered s); v \<notin> dom (finished s); s' = back_edge u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<rbrakk> \<Longrightarrow> on_back_edge tarjan_params u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u v. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s); v \<notin> dom (discovered s); s' = discover u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<rbrakk> \<Longrightarrow> on_discover tarjan_params u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))" shows "is_invar I"
PolysD_1: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'a set" assumes "p \<in> P[X]" shows "keys p \<subseteq> .[X]"
af\<^sub>G_ltl_continuation_suffix: fixes \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set"   and i :: "nat" assumes "\<forall>\<psi>\<in>\<^bold>G \<phi>. w \<Turnstile> \<psi> = suffix i w \<Turnstile> \<psi>" shows "w \<Turnstile> \<phi> = suffix i w \<Turnstile> af\<^sub>G \<phi> (prefix i w)"
drlex_pm_def: shows "drlex_pm \<equiv> dord_pm (\<lambda>s t. lex_pm t s)"
not_acc_down: fixes x :: "'g"   and R :: "('g \<times> 'g) set" assumes "x \<notin> Wellfounded.acc R" obtains z :: "'g" where "(z, x) \<in> R" and "z \<notin> Wellfounded.acc R"
coprime_prod: fixes A :: "'b mod_ring set"   and g :: "'b mod_ring \<Rightarrow> 'b mod_ring poly"   and a :: "'b mod_ring" assumes "\<forall>x\<in>A. comm_monoid_mult_class.coprime (g a) (g x)" shows "comm_monoid_mult_class.coprime (g a) (prod g A)"
select_convs_1: fixes carrier :: "'c set"   and rel :: "('c \<times> 'c) set"   and more :: "'d" shows "rel \<lparr>carrier = carrier, rel = rel, \<dots> = more\<rparr> = rel"
compact_insert: fixes s :: "'b set"   and x :: "'b" assumes "compact s" shows "compact (insert x s)"
scc_root_unique_root: fixes v :: "'v"   and scc :: "'v set"   and v' :: "'v" assumes "scc_root s' v scc"   and "scc_root s' v' scc" shows "v = v'"
\<iota>_in_hom: shows "\<guillemotleft>\<iota>\<^sub>1 : \<I>\<^sub>1 \<otimes>\<^sub>D \<I>\<^sub>1 \<rightarrow>\<^sub>D \<I>\<^sub>1\<guillemotright>"
setcompr_eq_image: fixes f :: "'e \<Rightarrow> 'd"   and P :: "'e \<Rightarrow> bool" shows "{f x |x. P x} = f ` {x. P x}"
worse_in_no_better: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "rational_preference carrier relation"   and "x \<succeq>[relation] y" shows "y \<in> no_better_than y carrier relation"
inf_sup_ord_3: fixes inf :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sup :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and x :: "'k"   and y :: "'k" assumes "class.lattice inf less_eq less sup" shows "less_eq x (sup x y)"
r': shows "Catch r' c\<^sub>2 \<in> redexes c''"
spmf_True_conv_False: fixes p :: "bool spmf" assumes "lossless_spmf p" shows "spmf p True = 1 - spmf p False"
finite_INT: fixes I :: "'e set"   and A :: "'e \<Rightarrow> ('i \<Rightarrow> 'a) set" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> generator" shows "\<Inter> (A ` I) \<in> generator"
conseqPre: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P' :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P' c Q,A"   and "P \<subseteq> P'" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
in_replicate_mset: fixes x :: "'k"   and n :: "nat"   and y :: "'k" shows "(x \<in># replicate_mset n y) = (0 < n \<and> x = y)"
conversep_iff: fixes r :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'f"   and b :: "'e" shows "r\<inverse>\<inverse> a b = r b a"
assms_2: shows "(x, y) \<in> set (cycle_edges p)"
c_tl_eq_tl: fixes ls :: "nat list" shows "c_tl (list_to_nat ls) = list_to_nat (tl ls)"
simps_2: fixes a :: "real"   and as :: "real list"   and b :: "real"   and bs :: "real list" shows "compare_lists (a # as) (b # bs) = (if a < b then LT else if b < a then GT else compare_lists as bs)"
dim_hyperplane: fixes a :: "'c" assumes "a \<noteq> (0::'c)" shows "dim {x. a \<bullet> x = 0} = DIM('c) - 1"
simps_16: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case Public of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f4"
smallest_k_ex: fixes f :: "nat \<Rightarrow> 'a"   and a\<^sub>1 :: "'a"   and a :: "'a"   and a\<^sub>n :: "'a"   and Y :: "'a set"   and b :: "'a" assumes "[f[a\<^sub>1 .. a ..  a\<^sub>n]Y]"   and "b \<notin> Y"   and "[[a\<^sub>1 b a\<^sub>n]]" shows "\<exists>k>0. [[a\<^sub>1 b f k]] \<and> k < card Y \<and> \<not> (\<exists>k'<k. [[a\<^sub>1 b f k']])"
r: fixes b :: "trm" assumes "SNa b" shows "P a b"
assms_5: shows "fadd (0::'c) (0::'c) = (0::'c)"
span_base: fixes a :: "'d"   and S :: "'d set" assumes "a \<in> S" shows "a \<in> span S"
landau_function_family_pair_trans_powr: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> real"   and H :: "('a \<Rightarrow> real) set"   and f :: "'a \<Rightarrow> real" assumes "landau_function_family_pair F (powr_closure g) H (\<lambda>x. g x powr 1)"   and "filterlim f at_top F"   and "\<And>p. (\<lambda>x. g x powr p) \<in> o[F](f)" shows "landau_function_family_pair F (powr_closure f) (powr_closure g * H) (\<lambda>x. f x powr 1)"
ennreal_spmf_lub_spmf: fixes Y :: "'a spmf set"   and x :: "'a" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y"   and "Y \<noteq> {}" shows "ennreal (spmf (lub_spmf Y) x) = (\<Squnion>p\<in>Y. ennreal (spmf p x))"
valid_def: fixes as :: "'b list"   and t :: "'b tree" shows "valid as t = (well_shaped t \<and> well_valued as t)"
walk_comp: fixes xs :: "'a list"   and ys :: "'a list" assumes "H.walk xs"   and "H.walk ys"   and "xs = [] \<or> ys = [] \<or> last xs \<rightarrow>\<^bsub>H\<^esub> hd ys" shows "H.walk (xs @ ys)"
oreachable_init: fixes s :: "'e \<times> 'f"   and A :: "('e \<times> 'f, 'g) automaton"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "s \<in> init A" shows "s \<in> oreachable A S U"
assms: shows "[\<lparr>O!,x\<rparr> in v]"
natural_isomorphism_\<ll>: shows "natural_isomorphism (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.L B.map B.\<ll>"
init_state_def: fixes src :: "nat" shows "Graph.init_state src \<equiv> return (False, [src \<mapsto> src], {src}, {}, 0)"
degree_aform_err_inverse_aform_err: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real"   and Y :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" assumes "inverse_aform_err p X = Some Y"   and "degree_aform_err X \<le> n" shows "degree_aform_err Y \<le> n"
wf_less_nmultiset_depth: fixes i :: "nat" shows "wf {(X, Y). |X| = i \<and> |Y| = i \<and> X < Y}"
sd_distl_aux: fixes Y :: "('c \<times> 'b) set"   and p :: "'b"   and x :: "'a" assumes "Y \<noteq> {}" shows "p \<squnion> \<Squnion>{\<alpha> x (snd y) |y. y \<in> Y} = \<Squnion>{p \<squnion> \<alpha> x (snd y) |y. y \<in> Y}"
continuous_def: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set" shows "continuous f S T \<equiv> func f S T \<and> continuous_axioms f S T"
val_process_def: fixes Mkt :: "('b, 'c) discrete_market"   and p :: "'c \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real" shows "val_process Mkt p = (if \<not> portfolio p then \<lambda>n w. 0 else (\<lambda>n w. \<Sum>x\<in>support_set p. prices Mkt x n w * p x (Suc n) w))"
assms_1: fixes x :: "'a"   and y :: "'a" shows "dist (f x) (f y) = dist x y"
hyperpow_hypnat_of_nat: fixes x :: "'b star"   and n :: "nat" shows "x pow hypnat_of_nat n = x ^ n"
PiE_empty_domain: fixes T :: "'b \<Rightarrow> 'c set" shows "Pi\<^sub>E {} T = {\<lambda>x. undefined}"
mat1_vert_block: fixes b :: "brick list" shows "rat_poly.matrix_mult (blockmat b) (blockmat (make_vert_block (nat (codomain_block b)))) = blockmat b"
many_strongly_connected_iff_9: fixes x :: "'a" assumes "is_inj x" shows "many_strongly_connected x = (x\<^sup>T ; x \<le> x\<^sup>+)"
\<pi>_apply_setmonotone: fixes x :: "item"   and I :: "item set"   and k :: "nat"   and T :: "(symbol \<times> character list) set" assumes "x \<in> I" shows "x \<in> \<pi> k T I"
to_tree_order_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "ptr |\<in>| object_ptr_kinds h"   and "known_ptrs h"   and "type_wf h" shows "h \<turnstile> ok to_tree_order ptr"
callee_lcl_This_simp: fixes C :: "qtname"   and sig :: "sig"   and m :: "methd" shows "callee_lcl C sig m This = (if is_static m then None else Some (Class C))"
fls_as_fps: fixes f :: "'b fls"   and n :: "int" assumes "- fls_subdegree f \<le> n" obtains f' :: "'b fps" where "f = fls_shift n (fps_to_fls f')"
alphaAbs_all_distinct_qAFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" assumes "FixVars TYPE('d) TYPE('e)" shows "alphaAbs_all_distinct_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
zero_order_1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq zero x"