hcju/isabelleps huggingface dataset:
HomFtorMor_3: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<in> mor\<^bsub>C\<^esub>" shows "cod\<^bsub>SET\<^esub> (Hom\<^bsub>C\<^esub>[X,f]) = Hom\<^bsub>C\<^esub> X cod\<^bsub>C\<^esub> f"
croot_power: fixes n :: "nat"   and x :: "complex" assumes "n \<noteq> 0" shows "croot n x ^ n = x"
exhaust: fixes y :: "'a literal" obtains x11 :: "char list" and x12 :: "'a list" where "y = Pos x11 x12"   | x21 :: "char list" and x22 :: "'a list" where "y = Neg x21 x22"
subd: fixes p :: "nat"   and s :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" assumes "simplicial_chain p (standard_simplex s) c" shows "(\<forall>r g. simplicial_simplex s (standard_simplex r) g \<longrightarrow> chain_map (Suc p) g (subd p c) = subd p (chain_map p g c)) \<and> simplicial_chain (Suc p) (standard_simplex s) (subd p c) \<and> chain_boundary (Suc p) (subd p c) + subd (p - Suc 0) (chain_boundary p c) = simplicial_subdivision p c - c"
groupF_concat_set: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "set (concat (groupF f xs)) = set xs"
cos_expansion_lemma: fixes x :: "real"   and m :: "nat" shows "cos (x + real (Suc m) * pi / 2) = - sin (x + real m * pi / 2)"
cont: fixes Y :: "'a set" assumes "Complete_Partial_Order.chain leq Y"   and "Y \<noteq> {}"   and "\<And>x. x \<in> Y \<Longrightarrow> \<not> leq x bound" shows "f (lub Y) = lub' (f ` Y)"
higher_deriv_iter_top_lemma: fixes s :: "complex set"   and f :: "complex \<Rightarrow> complex"   and z :: "complex"   and n :: "nat"   and m :: "nat" assumes "open s"   and "f holomorphic_on s"   and "f ` s \<subseteq> s"   and "z \<in> s"   and "f z = z"   and "deriv f z = 1"   and "1 < n"   and "\<And>i. \<lbrakk>1 < i; i < n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f z = 0" shows "(deriv ^^ n) (f ^^ m) z = of_nat m * (deriv ^^ n) f z"
gauss_int_norm_1: shows "gauss_int_norm 1 = 1"
simps_2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'a" shows "bisum f [] = f (0::'b) (0::'b)"
s_top_elem: fixes H :: "'a set"   and K :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "G \<guillemotright> K "   and "a \<in> H"   and "b \<in> K" shows "a \<cdot> b \<in> H \<diamondop> K"
word_and_1_shiftls_1: fixes x :: "'b word" shows "x AND 1 = (if x !! 0 then 1 << 0 else 0)"
IH: assumes "set agents' \<subseteq> agents" shows "RSD_pareto_eqclass (set agents') alts R (rsd_winners R alts agents')"
loop_induct: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> (('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)"   and "nearly_healthy (wlp body)"   and "\<And>S. \<lbrakk>\<forall>x\<in>S. P (fst x) (snd x); \<forall>x\<in>S. feasible (fst x); \<forall>x\<in>S. \<forall>Q. unitary Q \<longrightarrow> unitary (snd x Q)\<rbrakk> \<Longrightarrow> P (Sup_trans (fst ` S)) (Inf_utrans (snd ` S))"   and "\<And>t u. \<lbrakk>P t u; feasible t; \<And>Q. unitary Q \<Longrightarrow> unitary (u Q)\<rbrakk> \<Longrightarrow> P (wp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) (wlp (body ;; Embed u \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"   and "\<And>t t' u u'. \<lbrakk>P t u; equiv_trans t t'; equiv_utrans u u'\<rbrakk> \<Longrightarrow> P t' u'" shows "P (wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) (wlp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
prefix_split_first: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" obtains as :: "'a list" where "prefix (as @ [x]) xs" and "x \<notin> set as"
insert_noteq_member: fixes b :: "'a"   and B :: "'a multiset"   and c :: "'a"   and C :: "'a multiset" assumes "add_mset b B = add_mset c C"   and "b \<noteq> c" shows "c \<in># B"
rm_vars_empty_2: fixes s :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "rm_vars (set []) s = s"
less_divide_eq_1_pos: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less one (divide b a) = less a b"
simps_95: fixes x91 :: "'f"   and x92 :: "'d set"   and x93 :: "('d, 'e, 'f) com"   and x61 :: "'d set"   and x62 :: "('d, 'e, 'f) com" shows "Guard x91 x92 x93 \<noteq> While x61 x62"
rep_eq: shows "fromQ 0 = \<lparr>mag = 0::'a, dim = QD('c), unit_sys = unit\<rparr>"
mstep_mverdicts: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and R :: "'a list set"   and st :: "'a mstate"   and tdb :: "(char list \<times> 'a list) set \<times> nat"   and v :: "'a option list"   and i :: "nat" assumes "wf_mstate \<phi> \<pi> R st"   and "last_ts \<pi> \<le> snd tdb"   and "mem_restr R v" shows "((i, v) \<in> fst (mstep tdb st)) = ((i, v) \<in> M (psnoc \<pi> tdb) - M \<pi>)"
intros: fixes F1 :: "'fenv_inca"   and F2 :: "'fenv_ubx"   and st1 :: "('fun, 'dyn) frame list"   and st2 :: "('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame list"   and H :: "'a" assumes "rel_fundefs (Finca_get F1) (Fubx_get F2)"   and "sp_fundefs (Fubx_get F2)"   and "rel_stacktraces (Fubx_get F2) st1 st2 None" shows "State F1 H st1 \<sim> State F2 H st2"
p_intnatpow_prod: fixes n :: "int"   and m :: "nat" assumes "0 \<le> n" shows "[p] \<cdot> \<one> [^] n \<otimes> [p] \<cdot> \<one> [^] m = [p] \<cdot> \<one> [^] (int m + n)"
uminus_closed: fixes A :: "'q set"   and x :: "'q" assumes "ring_closed A"   and "x \<in> A" shows "- x \<in> A"
action_Cfg: fixes s :: "'a"   and d' :: "'a pmf"   and c' :: "'a \<Rightarrow> 'a cfg" shows "action (Cfg s d' c') = d'"
lift_def: fixes h :: "'e \<Rightarrow> 'd"   and inp :: "'c \<Rightarrow> 'e option" shows "lift h inp \<equiv> \<lambda>i. case inp i of None \<Rightarrow> None | Some v \<Rightarrow> Some (h v)"
diff_neutralizes: fixes G :: "('d, 'e) monoid_scheme"   and H :: "'d set"   and R :: "'d set"   and r1 :: "'d"   and r2 :: "'d" assumes "Group.group G"   and "subgroup H G"   and "R \<in> rcosets\<^bsub>G\<^esub> H"   and "r1 \<in> R"   and "r2 \<in> R" shows "r1 \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> r2 \<in> H"
isLimOrd_def: shows "isLimOrd \<equiv> \<not> isSuccOrd"
conj_cancel_right: fixes x :: "'a" shows "x \<^bold>\<sqinter> \<sim> x = \<zero>"
at_within_Icc_at_left: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.order_topology less_eq less open"   and "less a b" shows "topological_space.at_within open b (ord.atLeastAtMost less_eq a b) = topological_space.at_within open b (ord.lessThan less b)"
add_strict_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "class.strict_ordered_ab_semigroup_add plus less_eq less"   and "less a b"   and "less c d" shows "less (plus a c) (plus b d)"
dfk: fixes k :: "nat" assumes "k < n" shows "f k \<in> carrier_mat d d"
ch0_def: shows "ch0 = Cn 4 r_ifeq [Cn 4 rs [Cn 4 r_snoc [recf.Id 4 1, r_constn 3 0]], Cn 4 rs [recf.Id 4 1], ch1, r_constn 3 0]"
\<Phi>o_def: fixes yx :: "'d \<times> 'c" shows "\<Phi>o yx = S.mkArr (HomC.set (F (fst yx), snd yx)) (HomD.set (fst yx, G (snd yx))) (inD \<circ> \<phi> (fst yx) \<circ> HomC.\<psi> (F (fst yx), snd yx))"
bdd2: shows "small ((\<lambda>x. \<Squnion> (f ` B x)) ` A)"
P1: shows "Bet B A A0 \<and> Cong A A0 B A \<and> Bet B C C0 \<and> Cong C C0 B C' \<and> Bet B A A1 \<and> Cong A A1 B A \<and> Bet B C' C1 \<and> Cong C' C1 B C \<and> Cong A0 C0 A1 C1"
delta_pos_neg_least_1: fixes rm :: "('am \<times> 'bm) admS \<times> ('av \<times> 'bv) admS"   and rp :: "('am \<times> 'bm) admS \<times> ('av \<times> 'bv) admS" assumes "rm \<le> F (dual rp, rm)"   and "F (dual rm, rp) \<le> rp" shows "delta_neg \<le> dual rm"
axiom10_H: fixes P :: "'b \<Rightarrow> bool" shows "pred_H P = (\<lambda>x. Ball (set_H x) P)"
order_unique: fixes p :: "'a list"   and a :: "'a"   and n :: "nat" assumes "poly p \<noteq> poly []"   and "[- a, 1::'a] %^ n divides p"   and "\<not> [- a, 1::'a] %^ Suc n divides p" shows "n = order a p"
induct: fixes x1 :: "('p, 'v, 's) configuration"   and x2 :: "'p set"   and x3 :: "('p, 'v, 's) configuration"   and P :: "('p, 'v, 's) configuration \<Rightarrow> 'p set \<Rightarrow> ('p, 'v, 's) configuration \<Rightarrow> bool" assumes "qReachable x1 x2 x3"   and "\<And>cfg1 Q. P cfg1 Q cfg1"   and "\<And>cfg1 Q cfg2 msg cfg3. \<lbrakk>qReachable cfg1 Q cfg2; P cfg1 Q cfg2; cfg2 \<turnstile> msg \<mapsto> cfg3; \<exists>p\<in>Q. isReceiverOf p msg\<rbrakk> \<Longrightarrow> P cfg1 Q cfg3" shows "P x1 x2 x3"
vimageI: fixes f :: "'e \<Rightarrow> 'd"   and a :: "'e"   and b :: "'d"   and B :: "'d set" assumes "f a = b"   and "b \<in> B" shows "a \<in> f -` B"
WT1: shows "\<I>1, \<I>1' \<turnstile>\<^sub>C conv1 \<surd>"
reachable_def: shows "C.reachable \<equiv> C.step\<^sup>*\<^sup>* cs\<^sub>0"
fs_gs_N_N: assumes "m \<noteq> 0" shows "real_of_rat fs.gs.N = real N"
gbm_add_edge_impl: shows "graph_add_edge gbm_\<alpha> gbm_invar gbm_add_edge"
seqI: fixes Q :: "'b \<Rightarrow> bool"   and d :: "'b \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b" assumes "\<lbrace>Q\<rbrace> d \<lbrace>R\<rbrace>"   and "\<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>" shows "\<lbrace>P\<rbrace> c ;; d \<lbrace>R\<rbrace>"
tendsto_fst: fixes f :: "'b \<Rightarrow> 'c \<times> 'd"   and a :: "'c \<times> 'd"   and F :: "'b filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. fst (f x)) \<longlongrightarrow> fst a) F"
finite_pderivs_lang_UNIV: fixes r :: "'a rexp" shows "finite (pderivs_lang UNIV r)"
bind_spmf_to_nat_on: fixes p :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf" shows "map_spmf (to_nat_on (set_spmf p)) p \<bind> (\<lambda>n. f (from_nat_into (set_spmf p) n)) = p \<bind> f"
simps_7: fixes ve :: "('a, 'b) rbt"   and vf :: "'a"   and vg :: "'b"   and vh :: "('a, 'b) rbt"   and w :: "'a"   and x :: "'b"   and b :: "('a, 'b) rbt"   and s :: "'a"   and t :: "'b"   and c :: "('a, 'b) rbt"   and y :: "'a"   and z :: "'b"   and va :: "('a, 'b) rbt"   and vb :: "'a"   and vc :: "'b"   and vd :: "('a, 'b) rbt" shows "balance (Branch R (Branch B ve vf vg vh) w x (Branch R b s t c)) y z (Branch B va vb vc vd) = Branch R (Branch B (Branch B ve vf vg vh) w x b) s t (Branch B c y z (Branch B va vb vc vd))"
treeEquation: fixes subs :: "'a \<Rightarrow> 'a set"   and gamma :: "'a" shows "tree subs gamma = insert (0, gamma) (\<Union>delta\<in>subs gamma. incLevel ` tree subs delta)"
bounded_iff: fixes b :: "'c DBMEntry"   and c :: "'c DBMEntry"   and a :: "'c DBMEntry" shows "(linordered_monoid.max b c \<preceq> a) = (b \<preceq> a \<and> c \<preceq> a)"
trans_from_sterms: fixes \<xi> :: "'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, p), a, \<xi>', q) \<in> seqp_sos \<Gamma>"   and "wellformed \<Gamma>" shows "\<exists>p'\<in>sterms \<Gamma> p. ((\<xi>, p'), a, \<xi>', q) \<in> seqp_sos \<Gamma>"
edges_irrefl: fixes g :: "'b ugraph" shows "irrefl (edges g)"
intro: fixes R :: "('d, 'e) Ring_scheme" assumes "(\<plusminus>\<^bsub>R\<^esub>) \<in> carrier R \<rightarrow> carrier R \<rightarrow> carrier R"   and "\<And>a b c. \<lbrakk>a \<in> carrier R; b \<in> carrier R; c \<in> carrier R\<rbrakk> \<Longrightarrow> a \<plusminus>\<^bsub>R\<^esub> b \<plusminus>\<^bsub>R\<^esub> c = a \<plusminus>\<^bsub>R\<^esub> (b \<plusminus>\<^bsub>R\<^esub> c)"   and "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R\<rbrakk> \<Longrightarrow> a \<plusminus>\<^bsub>R\<^esub> b = b \<plusminus>\<^bsub>R\<^esub> a"   and "mop R \<in> carrier R \<rightarrow> carrier R"   and "\<And>a. a \<in> carrier R \<Longrightarrow> -\<^sub>a\<^bsub>R\<^esub> a \<plusminus>\<^bsub>R\<^esub> a = \<zero>\<^bsub>R\<^esub>"   and "\<zero>\<^bsub>R\<^esub> \<in> carrier R"   and "\<And>a. a \<in> carrier R \<Longrightarrow> \<zero>\<^bsub>R\<^esub> \<plusminus>\<^bsub>R\<^esub> a = a"   and "(\<cdot>\<^sub>r\<^bsub>R\<^esub>) \<in> carrier R \<rightarrow> carrier R \<rightarrow> carrier R"   and "\<And>a b c. \<lbrakk>a \<in> carrier R; b \<in> carrier R; c \<in> carrier R\<rbrakk> \<Longrightarrow> a \<cdot>\<^sub>r\<^bsub>R\<^esub> b \<cdot>\<^sub>r\<^bsub>R\<^esub> c = a \<cdot>\<^sub>r\<^bsub>R\<^esub> (b \<cdot>\<^sub>r\<^bsub>R\<^esub> c)"   and "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R\<rbrakk> \<Longrightarrow> a \<cdot>\<^sub>r\<^bsub>R\<^esub> b = b \<cdot>\<^sub>r\<^bsub>R\<^esub> a"   and "1\<^sub>r\<^bsub>R\<^esub> \<in> carrier R"   and "\<And>a b c. \<lbrakk>a \<in> carrier R; b \<in> carrier R; c \<in> carrier R\<rbrakk> \<Longrightarrow> a \<cdot>\<^sub>r\<^bsub>R\<^esub> (b \<plusminus>\<^bsub>R\<^esub> c) = a \<cdot>\<^sub>r\<^bsub>R\<^esub> b \<plusminus>\<^bsub>R\<^esub> a \<cdot>\<^sub>r\<^bsub>R\<^esub> c"   and "\<And>a. a \<in> carrier R \<Longrightarrow> 1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>r\<^bsub>R\<^esub> a = a" shows "Ring R"
abs_le_D2: fixes abs :: "'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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq (abs a) b" shows "less_eq (uminus a) b"
trimmed_pos_ms_iff: fixes xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" shows "trimmed_pos (MS xs f) = (case xs of MSLNil \<Rightarrow> False | MSLCons x xs \<Rightarrow> trimmed_pos (fst x))"
and_nonnegative_int_iff: fixes k :: "int"   and l :: "int" shows "(0 \<le> k AND l) = (0 \<le> k \<or> 0 \<le> l)"
f: shows "(\<Sum>a\<in>B. f a *\<^sub>R a) = x"
orbit_contains_rep: fixes m :: "'c" assumes "m \<in> M" shows "m \<in> orbit m"
next_def: shows "next = {(a, b). (\<exists>i. link0 i a = b) \<and> a \<noteq> nil \<and> b \<noteq> nil \<and> label0 a = none}"
porder_happens_before: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and P :: "'c prog" shows "porder_on (actions E) (happens_before P E)"
fst_normalize_polynomial_subset: fixes p :: "(char list list \<times> int) list" shows "mononoms (merge_coeffs p) \<subseteq> mononoms p"
TwoRings_def: fixes R :: "('c, 'd) Ring_scheme"   and R' :: "('e, 'f) Ring_scheme" shows "TwoRings R R' \<equiv> Ring R \<and> TwoRings_axioms R'"
intensional_simps_7: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "P \<noteq> Q \<equiv> P = \<not> Q"
hyps: assumes "winning_path p P"   and "enat n < llength P" shows "winning_path p (ldropn n P)"
I_and: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm"   and xs :: "'a list" shows "DLO.I (and \<phi>\<^sub>1 \<phi>\<^sub>2) xs = DLO.I (And \<phi>\<^sub>1 \<phi>\<^sub>2) xs"
add_scale_eq_noteq: fixes 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 r :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.comm_semiring_1_cancel_crossproduct plus minus zero times one"   and "r \<noteq> zero"   and "a = b \<and> c \<noteq> d" shows "plus a (times r c) \<noteq> plus b (times r d)"
continuous_within: fixes x :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous (at x within s) f = (f \<longlongrightarrow> f x) (at x within s)"
add_mset_add_mset_same_iff: fixes a :: "'k"   and A :: "'k multiset"   and B :: "'k multiset" shows "(add_mset a A = add_mset a B) = (A = B)"
pre_split_face_p_between: fixes oldF :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and newVertexList :: "nat list" assumes "pre_split_face oldF ram1 ram2 newVertexList" shows "pre_between (vertices oldF) ram1 ram2"
decr_grading: fixes d :: "'b \<Rightarrow> nat" assumes "hom_grading d" shows "hom_grading_fun d (decr_grading d)"
of_int_1_le_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq one (ring_1.of_int one plus zero minus z) = (1 \<le> z)"
F: fixes i :: "'i" assumes "i \<in> A" shows "F i \<in> sets borel"
subtop: fixes S :: "'b set set"   and T :: "'b set set"   and s :: "'b set" assumes "subtopology S T" shows "(s open\<^bsub>S\<^esub>) = (\<exists>t. t open\<^bsub>T\<^esub> \<and> s = t \<inter> carrier\<^bsub>S\<^esub>)"
target: fixes T :: "'procT" shows "TargetTerm T \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L TargetTerm T"
prj_simps_6: shows "C.dom \<nu>\<pi>\<rho>.Prj\<^sub>0 = HH\<nu>\<pi>\<rho>.chine"
dsj_CI: fixes B :: "fmla"   and H :: "fmla set"   and A :: "fmla" assumes "insert (neg B) H \<turnstile> A" shows "H \<turnstile> A OR B"
content_ff_divides_ff: fixes x :: "'b fract"   and p :: "'b fract poly" assumes "x \<in> set (coeffs p)" shows "divides_ff (content_ff p) x"
seq_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "B.VxV.seq g f = (B.seq (fst g) (fst f) \<and> B.seq (snd g) (snd f))"
notin_Path_update: fixes u :: "ref"   and ps :: "ref list"   and p :: "ref"   and f :: "ref \<Rightarrow> ref"   and v :: "ref"   and q :: "ref" assumes "u \<notin> set ps" shows "Path p (f(u := v)) q ps = Path p f q ps"
order_less_subst2: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c"   and c :: "'c" assumes "a < b"   and "f b < c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "f a < c"
simps_13: fixes \<C> :: "t_context"   and ts :: "checker_type" shows "check_single \<C> Drop ts = type_update ts [TAny] (Type [])"
assms: shows "Fun h S \<in> set \<langle>a --\<guillemotright> b\<rangle>\<langle>Fun f T\<rangle>"
ring_simprules_18: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus> x \<oplus> (x \<oplus> y) = y"
keys_single: fixes k :: "'a"   and v :: "'b" shows "keys (Poly_Mapping.single k v) = (if v = (0::'b) then {} else {k})"
insort_wrt_insort_key: fixes a :: "'c"   and A :: "'c set"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "a \<in> A"   and "set xs \<subseteq> A" shows "insert_wrt (linorder_from_keys A f) a xs = insort_key f a xs"
sfirst_eq_Inf: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "sfirst P \<omega> = \<Sqinter> {enat i |i. P (sdrop i \<omega>)}"
SN_intro: fixes t :: "lam" assumes "\<And>t'.  t \<longrightarrow>\<^sub>\<beta> t' \<Longrightarrow> SN t'" shows "SN t"
unique_inverse: fixes g :: "'a" assumes "g \<in> G" shows "\<exists>!h. h \<in> G \<and> binop g h = e \<and> binop h g = e"
emb_LNil2E: fixes xs :: "'b llist" assumes "emb xs LNil" obtains "xsa__ = LNil"   | xs :: "'b llist" and ys :: "'b llist" and x :: "'b" and zs :: "'b list" where "xsa__ = LCons x xs" and "LNil = prepend zs (LCons x ys)" and "emb xs ys"
l3_inv4_init: shows "init l3 \<subseteq> l3_inv4"
assms: shows "(s, a, s') \<in> node_sos S"
test_one_top: fixes p :: "'a" shows "\<iota> p \<le> (1::'b)"
common_boundary_sudivision_exists_def: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and one_chain2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" shows "common_boundary_sudivision_exists one_chain1 one_chain2 \<equiv> \<exists>subdiv. chain_subdiv_chain one_chain1 subdiv \<and> chain_subdiv_chain one_chain2 subdiv \<and> (\<forall>(k, \<gamma>)\<in>subdiv. valid_path \<gamma>) \<and> boundary_chain subdiv"
snd_foldl_Gauss_Jordan_column_k_eq: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "snd (foldl Gauss_Jordan_column_k_PA (mat (1::'a), 0, A) [0..<k]) = foldl Gauss_Jordan_column_k (0, A) [0..<k]"
extract_splitres_l_def: fixes r :: "('a, 'b) splitres" shows "extract_splitres_l r \<equiv> case Rep_splitres r of (l, a, r) \<Rightarrow> Abs_FingerTree l"
subgroupE_1: fixes G :: "('f, 'g) monoid_scheme"   and H :: "'f set" assumes "Group.group G"   and "subgroup H G" shows "H \<subseteq> carrier G"
assms_2: shows "\<Delta> \<noteq> \<Empt>"
vsv_appI: fixes a :: "V"   and b :: "V"   and A :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<circ> A" shows "(r \<restriction>\<^sup>r\<^sub>\<circ> A)\<lparr>a\<rparr> = b"
tendsto_le: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'a"   and x :: "'a"   and g :: "'c \<Rightarrow> 'a"   and y :: "'a" assumes "F \<noteq> bot"   and "(f \<longlongrightarrow> x) F"   and "(g \<longlongrightarrow> y) F"   and "\<forall>\<^sub>F x in F. g x \<le> f x" shows "y \<le> x"
map_sel_2: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d stream" shows "stl (smap f a) = smap f (stl a)"
fixespointwiseI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = id a" shows "fixespointwise f A"
simps_6: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and p :: "('a, 'b) form"   and q :: "('a, 'b) form" shows "subc c s (Or p q) = Or (subc c s p) (subc c s q)"
has_derivative_inverse: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and f' :: "'b \<Rightarrow> 'a"   and S :: "'b set" assumes "f x \<noteq> (0::'a)"   and "(f has_derivative f') (at x within S)" shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x)))) (at x within S)"
tpOfV_getVars_nth: fixes i :: "nat"   and \<sigma>l :: "'tp list" assumes "i < length \<sigma>l" shows "TE.tpOfV (TE.getVars \<sigma>l ! i) = \<sigma>l ! i"
Min_le: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and x :: "'d" assumes "class.linorder less_eq less"   and "finite A"   and "x \<in> A" shows "less_eq (linorder.Min less_eq A) x"
zs2: shows "remdups_adj (xs @ xs2) = remdups_adj xs @ zs2"
from_hma_to_hma\<^sub>m: fixes A :: "'b mat" assumes "A \<in> carrier_mat CARD('c) CARD('d)" shows "Mod_Type_Connect.from_hma\<^sub>m (Mod_Type_Connect.to_hma\<^sub>m A) = A"
busy_squareI: fixes A :: "state \<times> state \<Rightarrow> bool"   and s :: "state"   and t :: "state"   and v :: "state \<Rightarrow> 'a" assumes "A (s, t)" shows "(s, t) \<Turnstile> [A]_v"
fls_subdegree_fls_to_fps_gt0: fixes f :: "'a fps" shows "0 \<le> fls_subdegree (fps_to_fls f)"
l_2: shows "\<forall>n>0. real n * l \<le> - u n"
const_vector_cart: fixes d :: "real" shows "(\<chi>i. d) = sum ((*\<^sub>R) d) Basis"
S1safe: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S1 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<longrightarrow> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)"