hcju/isabelleps huggingface dataset:
isDERIV_FDERIV_floatarith_linear: fixes i :: "nat"   and fa :: "floatarith"   and xs :: "real list"   and hs :: "real list" assumes "\<And>k. k < DIM('a) \<Longrightarrow> isDERIV i (DERIV_floatarith k fa) xs"   and "max_Var_floatarith fa \<le> DIM('a)"   and "length xs = DIM('a)"   and "length hs = DIM('a)" shows "isDERIV i (FDERIV_floatarith fa [0..<DIM('a)] (map Var [DIM('a)..<2 * DIM('a)])) (xs @ hs)"
\<gamma>: shows "\<gamma> = ladder_\<gamma> (\<alpha> @ \<delta>) D L (Suc index)"
analytically_valid_x: fixes s :: "(real \<times> real) set"   and F :: "real \<times> real \<Rightarrow> 'a" assumes "analytically_valid s F j" shows "(\<lambda>x. integral UNIV (\<lambda>y. partial_vector_derivative F j (x, y) * indicator s (x, y))) \<in> borel_measurable lborel"
q_def: shows "q = t \<^bsub>\<^esub>\<bowtie>\<^bsub>a\<^esub> p"
Cauchy_altdef: fixes f :: "nat \<Rightarrow> 'a" shows "Cauchy f = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e)"
subset_eq: fixes A :: "'b set"   and B :: "'b set" shows "order.greater_eq B A = (\<forall>x\<in>A. x \<in> B)"
arr_inv: fixes f :: "'d" assumes "D.iso f" shows "D.arr (D.inv f)"
inject_4: fixes x61 :: "'b mrexp2"   and x62 :: "bool"   and y61 :: "'b mrexp2"   and y62 :: "bool" shows "(Star2 x61 x62 = Star2 y61 y62) = (x61 = y61 \<and> x62 = y62)"
that: fixes s :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(red F)\<^sup>*\<^sup>* (p + r) s"   and "(red F)\<^sup>*\<^sup>* (q + r) s" shows "thesis"
Pbij_compose_Dom: fixes \<beta> :: "(Location \<times> Location) set"   and \<gamma> :: "(Location \<times> Location) set" shows "Pbij_Dom (Pbij_compose \<beta> \<gamma>) \<subseteq> Pbij_Dom \<beta>"
transpose_four_block_mat: fixes A :: "'b mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'b mat"   and nc2 :: "nat"   and C :: "'b mat"   and nr2 :: "nat"   and D :: "'b mat" assumes "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2" shows "(four_block_mat A B C D)\<^sup>T = four_block_mat A\<^sup>T C\<^sup>T B\<^sup>T D\<^sup>T"
Image_Un: fixes R :: "('c \<times> 'b) set"   and A :: "'c set"   and B :: "'c set" shows "R `` (A \<union> B) = R `` A \<union> R `` B"
simps_4: fixes uu :: "variable \<times> typ"   and uv :: "nat"   and v :: "String.literal"   and va :: "typ" shows "bind_fv2 uu uv (Ct v va) = Ct v va"
dom_char: fixes f :: "'a \<times> 'a" shows "C.VV.dom f = (if C.VV.arr f then C.VxV.dom f else C.VxV.null)"
fact_div_fact_ffact_nat: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "fact n div fact (n - k) = ffact k n"
invar2_init_init: shows "prim_invar2_init initQ init\<pi>"
zero_Arity_def: shows "0 \<equiv> to_Arity 0"
simps_4: fixes P' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and uu :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and uv :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre ({P'/uu/uv} CONSEQ C) Q = P'"
cycle_freeI: fixes g :: "'b ugraph" assumes "\<And>p u. \<lbrakk>path g u p u; p \<noteq> []; simple p\<rbrakk> \<Longrightarrow> False" shows "cycle_free g"
D'_def: shows "D' = codomain.closest_supchamber (f ` F) (f ` C)"
full_polys_assn_def: shows "full_polys_assn = hr_comp (hr_comp polys_assn (\<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel)) polys_rel"
compile_id_correct: fixes n :: "nat"   and m :: "nat"   and xs :: "nat list"   and ap :: "abc_inst list"   and arity :: "nat"   and fp :: "nat"   and r :: "nat"   and anything :: "nat list" assumes "n < m"   and "length xs = m"   and "rec_ci (recf.id m n) = (ap, arity, fp)"   and "rec_exec (recf.id m n) xs = r" shows "{\<lambda>nl. nl = xs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ r # 0 \<up> (fp - Suc arity) @ anything}"
P''Chain: shows "P''[x::=y] \<Longrightarrow>\<^sub>\<tau> P'"
cases: fixes x :: "'a tree \<times> 'a" obtains w\<^sub>b :: "nat" and b :: "'a" and a :: "'a" where "x = (Leaf w\<^sub>b__ b__, a__)"   | w :: "nat" and w\<^sub>b :: "nat" and b :: "'a" and w\<^sub>c :: "nat" and c :: "'a" and a :: "'a" where "x = (Node w__ (Leaf w\<^sub>b b) (Leaf w\<^sub>c c), aa__)"   | w :: "nat" and v :: "nat" and va :: "'a tree" and vb :: "'a tree" and t\<^sub>2 :: "'a tree" and a :: "'a" where "x = (Node wa__ (Node v__ va__ vb__) t\<^sub>2, ab__)"   | w :: "nat" and t\<^sub>1 :: "'a tree" and v :: "nat" and va :: "'a tree" and vb :: "'a tree" and a :: "'a" where "x = (Node w t\<^sub>1 (Node v va vb), a)"
fMax_ge: fixes x :: "'b"   and A :: "'b fset" assumes "x |\<in>| A" shows "x \<le> fMax A"
alpha_set: fixes bs :: "atom set"   and x :: "'b"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom set"   and y :: "'b" shows "(bs, x) \<approx>set R f p (cs, y) = (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs)"
delete_node_def: fixes v :: "'d"   and g :: "('d, 'e, 'f) graph_scheme" shows "delete_node v g \<equiv> \<lparr>nodes = nodes g - {v}, edges = edges g \<inter> (- {v}) \<times> UNIV \<times> - {v}\<rparr>"
widen_trans_lemma: fixes G :: "prog"   and S :: "ty"   and U :: "ty"   and T :: "ty" assumes "G\<turnstile>S\<preceq>U"   and "\<forall>C. is_class G C \<longrightarrow> G\<turnstile>C\<preceq>\<^sub>C Object"   and "G\<turnstile>U\<preceq>T" shows "G\<turnstile>S\<preceq>T"
assms: shows "t \<in> trms\<^sub>s\<^sub>s\<^sub>t S"
scale_zero_right: fixes a :: "complex" shows "a *\<^sub>C (0::'b) = (0::'b)"
hs_is_it_def: fixes s :: "'b set"   and hs :: "('b, unit) hashtable"   and its :: "'b set"   and it :: "nat \<times> ('b \<times> unit) list \<times> ('b, unit) hashtable" shows "hs_is_it s hs its it \<equiv> hm_is_it (map_of_set s) hs (map_of_set its) it"
map_prod_o_convol: fixes h1 :: "'h \<Rightarrow> 'f"   and h2 :: "'i \<Rightarrow> 'g"   and f :: "'e \<Rightarrow> 'h"   and g :: "'e \<Rightarrow> 'i" shows "map_prod h1 h2 \<circ> <f, g> = <h1 \<circ> f, h2 \<circ> g>"
pnderiv_alt: fixes a :: "'a"   and r :: "'a rexp" shows "pnderiv a r = pnorm (deriv a r)"
C2l: fixes x :: "'a"   and y :: "'a" assumes "x = y \<odot> x" shows "x = y\<^sup>\<star> \<odot> x"
sum_list_pair_relator_halflist_flip: fixes s :: "'w"   and t :: "'w" assumes "s \<in> S"   and "t \<in> S" shows "sum_list (pair_relator_halflist s t) = sum_list (pair_relator_halflist t s)"
ad_rel_def: fixes R :: "('a \<times> 'b) set" shows "ad_rel R = {(x, x) |x. \<nexists>y. s2p R (x, y)}"
simps_3: fixes x2 :: "'c" shows "\<lfloor>x2\<rfloor> \<noteq> \<bottom>"
obj_initial_def: fixes \<CC> :: "V" shows "obj_initial \<CC> \<equiv> obj_terminal (op_smc \<CC>)"
S\<^sub>i\<^sub>v: fixes x :: "i"   and y :: "i" shows "(E x \<^bold>\<and> E y \<^bold>\<leftarrow> E (x \<cdot> y)) \<^bold>\<and> (E x \<^bold>\<leftarrow> E (dom x)) \<^bold>\<and> (E y \<^bold>\<leftarrow> E (cod y))"
l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs"
main: fixes t :: "real" shows "\<forall>\<^sub>F n in sequentially. cmod (\<phi> n t - complex_of_real ((1 + - t\<^sup>2 / 2 / real n) ^ n)) \<le> t\<^sup>2 / (6 * \<sigma>\<^sup>2) * \<mu>.expectation (\<lambda>x. min (6 * x\<^sup>2) (\<bar>t / sqrt (\<sigma>\<^sup>2 * real n)\<bar> * \<bar>x\<bar> ^ 3))"
timesdistributesoverpdifference: fixes m :: "nat"   and a :: "nat"   and b :: "nat" shows "m * pdifference a b = pdifference (m * a) (m * b)"
normalize_prod_mset: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a multiset" assumes "class.normalization_semidom_multiplicative divide plus minus zero times one unit_factor normalize" shows "normalize (comm_monoid_mult.prod_mset times one A) = comm_monoid_mult.prod_mset times one (image_mset normalize A)"
even_nat_iff: fixes k :: "int" assumes "0 \<le> k" shows "even (nat k) = even k"
submap_imp_state_succ_submap_b: fixes s1 :: "('c, 'd) fmap"   and s2 :: "('c, 'd) fmap"   and s0 :: "('c, 'd) fmap" assumes "s1 \<subseteq>\<^sub>f s2" shows "s0 ++ s1 \<subseteq>\<^sub>f s0 ++ s2"
Abs_real_int_cases: fixes x :: "real_int" obtains y :: "real \<times> real" where "x = Abs_real_int y" and "y \<in> {r. fst r \<le> snd r}"
assms_2: shows "s0 \<bullet> (1::'a) < s \<bullet> (1::'a)"
is_fair_run_def: fixes s :: "nat \<Rightarrow> 'a" shows "is_fair_run s \<equiv> \<exists>l. is_lrun l s \<and> is_lfair l s"
fermat: fixes p :: "nat"   and m :: "nat" assumes "prime p"   and "m mod p \<noteq> 0" shows "m ^ (p - 1) mod p = 1"
dom_mkArr: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set"   and B :: "'b set"   and F :: "'b \<Rightarrow> 'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> A B F)" shows "partial_magma.dom S (set_category_given_img.mkArr S img \<AA> A B F) = set_category_given_img.mkIde S img \<AA> A"
resT_abs_def: shows "resT_abs \<equiv> PRJ('e\<cdot>'f resT) oo EMB('e\<^sub>\<bottom> \<oplus> ('e\<cdot>'f resT\<cdot>'f)\<^sub>\<bottom>)"
ordered_resolution_is_complete_model_resolution: fixes Sel :: "'at set" shows "Complete (ordered_model_resolvent Sel)"
prob_space_axioms: fixes i :: "'b" shows "prob_space (distr M borel (Y i))"
pointbase_is_mono: fixes f :: "'b set \<Rightarrow> 'c set" shows "mono (pointbase f)"
distinct_35: fixes x4 :: "key"   and x71 :: "key"   and x72 :: "msg" shows "Key x4 \<noteq> Crypt x71 x72"
additive_principal_oSuc: fixes a :: "ordinal" shows "additive_principal (oSuc a) = (a = 0)"
fresh_gen_set: fixes X :: "W.tvar"   and Xs :: "W.tvar list"   and T :: "ty" assumes "X \<in> set Xs" shows "X \<sharp> gen T Xs"
has_field_mono: fixes P :: "'b prog"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal"   and C' :: "String.literal" assumes "P \<turnstile> C has F:T (fm) in D"   and "P \<turnstile> C' \<preceq>\<^sup>* C" shows "P \<turnstile> C' has F:T (fm) in D"
rep_eq: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xa :: "'b set_rbt" shows "fold1 x xa = RBT_Impl_fold1 x (mapping_rbt.impl_of xa)"
sub_refl: fixes s :: "('a, 'b) tm" shows "sub s s"
e0_2: fixes xa :: "'a \<times> 'b" assumes "dist xa (x, y) < e0" shows "f' xa o\<^sub>L embed2_blinfun \<in> invertibles_blinfun"
br'_while_algo: fixes \<delta> :: "('a, 'b) ta_rule set" assumes "finite \<delta>" shows "while_algo (br'_algo \<delta>)"
Ostrowski_elem_not_one: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and x :: "'a" assumes "vals_nonequiv K (Suc n) vv"   and "x \<in> carrier K"   and "Ostrowski_elem K (Suc n) vv x" shows "1\<^sub>r \<plusminus> -\<^sub>a x \<noteq> \<zero>"
zero_least1: fixes x :: "'a" shows "o \<oplus> n x = n x"
fps_X_mult_right_nth: fixes a :: "'a fps"   and n :: "nat" shows "fps_nth (a * fps_X) n = (if n = 0 then 0::'a else fps_nth a (n - 1))"
NE_intT_forget: fixes \<sigma> :: "'tp" shows "\<exists>a. intT \<sigma> a"
effective_matrix_match_condn_2: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zer :: "'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inver :: "'b \<Rightarrow> 'b"   and A1 :: "'b list list"   and A2 :: "'b list list"   and B1 :: "'b list list"   and B2 :: "'b list list" assumes "plus_mult id f zer g inver"   and "plus_mult.matrix_match A1 A2 B1 B2" shows "\<forall>i j. i < mult.row_length A1 * mult.row_length B1 \<and> j < length A2 * length B2 \<longrightarrow> mat_multI zer g f (mult.row_length (mult.Tensor f A1 B1)) (mult.Tensor f A1 B1) (mult.Tensor f A2 B2) ! j ! i = plus_mult.scalar_product f zer g (mult.vec_vec_Tensor f (row A1 (i div mult.row_length B1)) (row B1 (i mod mult.row_length B1))) (mult.vec_vec_Tensor f (col A2 (j div length B2)) (col B2 (j mod length B2)))"
simps: fixes a :: "'c llist" shows "lfinite a = (a = LNil \<or> (\<exists>x xs. a = LCons x xs \<and> lfinite xs))"
l3: shows "(abscissa X - abscissa A)\<^sup>2 + (ordinate X - ordinate A)\<^sup>2 = (abscissa B - abscissa C)\<^sup>2 + (ordinate B - ordinate C)\<^sup>2"
Tag_EphK: fixes X :: "tag"   and X' :: "ephkey" shows "Tag X \<noteq> EphK X'"
Hide_def: fixes P :: "('c, 'd) action"   and cs :: "'c set" shows "P \ cs \<equiv> action_of (R (\<lambda>(S, S'). \<exists>s. diff_tr S' S = tr_filter (s - tr S) cs \<and> action.relation_of P (S, S'\<lparr>tr := s, ref := ref S' \<union> cs\<rparr>)) ;; action.relation_of Skip)"
simps: fixes t :: "'b" shows "inverse (\<tau>\<^sub>c\<^sub>s\<^sub>t t) = \<tau>\<^sub>c\<^sub>s\<^sub>t (inverse t)"
suffix_def: fixes k :: "nat"   and x :: "nat \<Rightarrow> 'c" shows "suffix k x \<equiv> \<lambda>n. x (k + n)"
empty: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "Semantics.matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Empty], Undecided\<rangle> \<Rightarrow> Undecided"
well_valued0_t1: shows "well_valued0 (ys @ as) t1"
functor_axioms: shows "functor VoV.comp (\<cdot>) (\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star> snd \<mu>\<nu>)"
to_fun_plus: fixes g :: "nat \<Rightarrow> nat \<Rightarrow> int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and x :: "nat \<Rightarrow> int" assumes "g \<in> carrier (UP Zp)"   and "f \<in> carrier (UP Zp)"   and "x \<in> carrier Zp" shows "(f \<oplus>\<^bsub>UP Zp\<^esub> g) \<bullet> x = f \<bullet> x \<oplus> g \<bullet> x"
cases: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and S :: "'a set" assumes "orthogonal_current \<Gamma> f S" obtains "\<And>x. \<lbrakk>x \<in> S; x \<notin> A \<Gamma>\<rbrakk> \<Longrightarrow> weight \<Gamma> x \<le> d_IN f x" and "\<And>x. \<lbrakk>x \<in> S; x \<in> A \<Gamma>; x \<notin> B \<Gamma>\<rbrakk> \<Longrightarrow> d_OUT f x = weight \<Gamma> x" and "\<And>u v. \<lbrakk>v \<in> RF\<^bsub>\<Gamma>\<^esub> S; u \<notin> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> S\<rbrakk> \<Longrightarrow> f (u, v) = 0"
i_Exec_Stream_Init_Pre_Post1: fixes c_n :: "'a"   and trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b"   and c :: "'a"   and n :: "nat"   and x_n :: "'b"   and P1 :: "'b \<Rightarrow> bool"   and P2 :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "c_n = i_Exec_Comp_Stream_Init trans_fun input c n"   and "x_n = input n" shows "(P1 x_n \<and> P2 c_n \<longrightarrow> Q (i_Exec_Comp_Stream_Init trans_fun input c (Suc n))) = (P1 x_n \<and> P2 c_n \<longrightarrow> Q (trans_fun x_n c_n))"
d_implies_ruc: fixes P :: "'a process" assumes "deterministic P" shows "ref_union_closed P"
assms: shows "fmdom' (snd (action_proj a vs)) = {}"
trans: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "a \<preceq> b"   and "b \<preceq> c" shows "a \<preceq> c"
him_mult_right: fixes q :: "quat"   and a :: "real" shows "HIm (q * quat_of_real a) = a *\<^sub>R HIm q"
simple_ownership_distinct: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'p"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "unit"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'p"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "unit"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "j < length ts"   and "i \<noteq> j"   and "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "\<O>\<^sub>i \<inter> \<O>\<^sub>j = {}"
measurable_force_enabled: fixes x :: "'s" shows "force_enabled x \<in> stream_space (count_space UNIV) \<rightarrow>\<^sub>M stream_space (count_space UNIV)"
fset_exhaust: fixes S :: "'a fset" obtains "S = {||}"   | x :: "'a" and S' :: "'a fset" where "S = finsert x S'"
k: shows "\<mu> = (SUP \<beta>\<in>elts \<mu>. \<beta>)"
Cons_mem_set_subseqsD: fixes y :: "'c"   and ys :: "'c list"   and xs :: "'c list" assumes "y # ys \<in> set (subseqs xs)" shows "y \<in> set xs"
sgn_scaleR: fixes r :: "real"   and x :: "'b" shows "sgn (r *\<^sub>R x) = sgn r *\<^sub>R sgn x"
sel_3: fixes x1 :: "'b"   and x2 :: "'c"   and x3 :: "'a"   and x4 :: "'b" shows "outOf (Trans x1 x2 x3 x4) = x3"
hyps_1: shows "(a, b) \<in> A\<^sup>*"
noAbrupt_start: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and t :: "('b, 'c) xstate"   and s' :: "'b" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "\<forall>t'. t \<noteq> Abrupt t'" shows "s \<noteq> Abrupt s'"
axioms_1: fixes G :: "('b, 'c) pre_digraph" assumes "loopfree_digraph G" shows "wf_digraph G"
deriv_to_deriv2: fixes C :: "'b sequent"   and R :: "('b sequent list \<times> 'b sequent) set" assumes "C \<in> derivable' R" shows "\<exists>n. (C, n) \<in> derivable R"
assms_2: shows "s\<^sub>2' = s\<^sub>1(r' \<mapsto> ls)"
power_decreasing_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and b :: "'c"   and m :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less zero b"   and "less b one" shows "less_eq (power.power one times b m) (power.power one times b n) = (n \<le> m)"
measure_def: fixes M :: "'d measure"   and A :: "'d set" shows "Sigma_Algebra.measure M A = enn2real (emeasure M A)"
QuoteP_imp_ConstP: fixes x :: "tm"   and y :: "tm" shows "{QuoteP x y} \<turnstile> ConstP y"
LeadsTo_Un_distrib: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "(F \<in> A \<union> B \<longmapsto>w C) = (F \<in> A \<longmapsto>w C \<and> F \<in> B \<longmapsto>w C)"
IH: fixes gpv :: "('d, 'b, 'c) gpv"   and s :: "'a" assumes "\<I> \<turnstile>g gpv \<surd>" shows "expectation_gpv'' gpv \<le> \<integral>\<^sup>+ a. (case a of (x, s) \<Rightarrow> f x) \<partial>measure_spmf (exec_gpv' gpv s)"
simps_8: fixes e\<^sub>1 :: "cnt_expr"   and e\<^sub>2 :: "cnt_expr"   and R :: "nat \<times> nat \<Rightarrow> bool" shows "\<lbrakk> \<lceil>e\<^sub>1, e\<^sub>2\<rceil> \<in> R \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = {\<rho>. R (\<lbrakk> \<rho> \<turnstile> e\<^sub>1 \<rbrakk>\<^sub>c\<^sub>n\<^sub>t\<^sub>e\<^sub>x\<^sub>p\<^sub>r, \<lbrakk> \<rho> \<turnstile> e\<^sub>2 \<rbrakk>\<^sub>c\<^sub>n\<^sub>t\<^sub>e\<^sub>x\<^sub>p\<^sub>r)}"
d1: fixes x :: "'a" shows "x \<le> d x \<sqinter> x"
X: shows "X \<subseteq> Restr (ns\<^sup>* O s O ns\<^sup>*) (((ns\<^sup>* O s O ns\<^sup>*)\<^sup>*)\<inverse> `` A) `` X"
uniformity_dist: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
2: fixes J :: "'a formula" assumes "J \<in> set (subformulae F)" shows "J \<in> snd ` set (tseytin_assmt F)"
Trans_fst_sndI: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and s :: "'a"   and a :: "'b" shows "IO_Automaton.valid step [Trans s a (fst (step s a)) (snd (step s a))]"
distinct_10: fixes x41 :: "'g"   and x42 :: "('e, 'f, 'g) Tree"   and x2 :: "('e, 'f, 'g) Tree" shows "tAct x41 x42 \<noteq> tNot x2"
minus_one_power_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus one) n = (if even n then one else uminus one)"
trivial_homomorphism_image: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and f :: "'a \<Rightarrow> 'c" assumes "Group.group G" shows "trivial_homomorphism G H f = (f \<in> hom G H \<and> f ` carrier G = {\<one>\<^bsub>H\<^esub>})"
v: fixes r :: "rat"   and a :: "'a atom" assumes "(r, a) \<in> set C" shows "v \<Turnstile>\<^sub>l\<^sub>e lec_of_nsc (r *R nsc_of_atom a)"
sub_smult: fixes f :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "q \<in> carrier P"   and "a \<in> carrier R" shows "Cring_Poly.compose R (a \<odot>\<^bsub>P\<^esub> f) q = a \<odot>\<^bsub>P\<^esub> Cring_Poly.compose R f q"
restrictto_def: fixes X :: "(variable \<Rightarrow> real) set"   and V :: "variable set" shows "restrictto X V = {\<nu>. \<exists>\<omega>. \<omega> \<in> X \<and> Vagree \<omega> \<nu> V}"
qOp_alpha_iff: fixes delta :: "'e"   and inp :: "'a \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm option"   and binp :: "'b \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs option"   and X' :: "('a, 'b, 'c, 'd, 'e) qTerm" shows "(qOp delta inp binp #= X') = (\<exists>inp' binp'. X' = qOp delta inp' binp' \<and> sameDom inp inp' \<and> sameDom binp binp' \<and> liftAll2 (#=) inp inp' \<and> liftAll2 ($=) binp binp')"
coeff_scaleRat: fixes k :: "rat"   and lp1 :: "linear_poly"   and var :: "nat" shows "coeff (k *R lp1) var = k * coeff lp1 var"
vimage_vdomain: fixes r :: "V" shows "r `\<^sub>\<circ> \<D>\<^sub>\<circ> r = \<R>\<^sub>\<circ> r"
sstrip_comb_termination: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b"   and ts :: "'b list"   and t :: "'b"   and t' :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "(f, ts) = term.strip_comb unapp t"   and "t' \<in> set ts" shows "size t' < size t"
transfer_FOREACHci_nres: fixes iterate :: "('c dres \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c dres \<Rightarrow> 'c dres) \<Rightarrow> 'c dres \<Rightarrow> 'c dres"   and s :: "'d set"   and fi :: "'d \<Rightarrow> 'c \<Rightarrow> 'c dres"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and c :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c"   and I :: "'d set \<Rightarrow> 'c \<Rightarrow> bool" assumes "set_iterator iterate s"   and "\<And>x \<sigma>. nres_of (fi x \<sigma>) \<le> f x \<sigma>" shows "nres_of (iterate (case_dres False False c) (\<lambda>x s. s \<bind> fi x) (dRETURN \<sigma>)) \<le> FOREACH\<^sub>C\<^bsup>I\<^esup> s c f \<sigma>"
fbx_def: shows "fbx \<equiv> \<lambda>D. cbox (f (zf D) - (B * real DIM('M) * prj1 (vf D - uf D)) *\<^sub>R One) (f (zf D) + (B * real DIM('M) * prj1 (vf D - uf D)) *\<^sub>R One)"
simps_2: fixes x :: "('a, 'b) Combinators"   and y :: "('a, 'b) Combinators"   and xs :: "('a, 'b) Combinators list" shows "singleCombinators (x \<oplus> y # xs) = False"
connected_closed_set: fixes S :: "'b set" assumes "closed S" shows "connected S = (\<nexists>A B. closed A \<and> closed B \<and> A \<noteq> {} \<and> B \<noteq> {} \<and> A \<union> B = S \<and> A \<inter> B = {})"