hcju/isabelleps huggingface dataset:
assms_1: shows "(\<not>\<^sup>c \<gamma>) t n"
w_cmpser_gr: fixes G :: "('e, 'f) Group_scheme"   and r :: "nat"   and f :: "nat \<Rightarrow> 'e set"   and i :: "nat" assumes "Group G"   and "0 < r"   and "w_cmpser G r f"   and "i \<le> r" shows "G \<guillemotright> f i "
basis_def: fixes A :: "'a vec set" shows "basis A = (lin_indpt A \<and> local.span A = carrier_vec n \<and> A \<subseteq> carrier_vec n)"
inc\<^sub>1\<sigma>: shows "l\<^sub>1 \<cdot>\<^sub>l \<sigma> \<in> C\<^sub>1 \<cdot>\<^sub>l\<^sub>s \<sigma>"
mat_plus_index: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'c list list"   and m2 :: "'c list list"   and i :: "nat"   and j :: "nat"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "mat nr nc m1"   and "mat nr nc m2"   and "i < nc"   and "j < nr" shows "mat_plusI pl m1 m2 ! i ! j = pl (m1 ! i ! j) (m2 ! i ! j)"
I_def: shows "I = (\<lambda>n. complex_of_real (real n + a) powr (1 - s) / (1 - s))"
plus_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> A"   and "b \<in> A"   and "c \<in> A" shows "a \<squnion>\<^bsub>f\<^esub> (b \<squnion>\<^bsub>f\<^esub> c) = a \<squnion>\<^bsub>f\<^esub> b \<squnion>\<^bsub>f\<^esub> c"
1: shows "start_points x ; (x\<^sup>+ \<cdot> 1') ; 1 \<le> x\<^sup>T ; 1"
cring_simprules_14: fixes x :: "int"   and y :: "int" shows "x * y = y * x"
map_comp: fixes g :: "'f \<Rightarrow> 'e"   and f :: "'g \<Rightarrow> 'f"   and v :: "'d \<Rightarrow> 'g" shows "g \<circ> (f \<circ> v) = g \<circ> f \<circ> v"
comp_wb_lens: fixes x :: "'f \<Longrightarrow> 'g"   and y :: "'g \<Longrightarrow> 'h" assumes "wb_lens x"   and "wb_lens y" shows "wb_lens (x ;\<^sub>L y)"
h_def: shows "h = (\<lambda>t. g (dist x y - t))"
ub: fixes x :: "'a"   and y :: "'a" shows "\<exists>z\<ge>x. y \<le> z"
lir_fps_aux: fixes m :: "nat"   and c :: "nat \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> 'b"   and N :: "nat"   and g :: "nat \<Rightarrow> 'b" assumes "\<And>n. m \<le> n \<Longrightarrow> (\<Sum>k\<le>N. c k * f (n + k)) = g (n + N)"   and "c N \<noteq> (0::'b)" shows "Abs_fps f = (fps_of_poly (Poly (map (\<lambda>k. (\<Sum>i\<le>min N k. c (N - i) * f (k - i)) - g k) [0..<N + m])) + Abs_fps g) / fps_of_poly (Poly (map (\<lambda>k. c (N - k)) [0..<Suc N]))"
unlift_b_e: assumes "\<S>\<bullet>\<C> \<turnstile> $* [e1, e2] : t1s _> t2s" shows "\<C> \<turnstile> [e1, e2] : t1s _> t2s"
ind1: fixes xs :: "nat list" assumes "length xs = n" shows "terminate f xs"
segment_1: fixes a :: "'b"   and b :: "'b" shows "open_segment a b \<equiv> closed_segment a b - {a, b}"
simps: fixes n :: "nat"   and j :: "nat"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.step_2_main n j A = (if n \<le> j then A else let ev = A $$ (j, j) in Let (case ??.Jordan_Normal_Form_Existence.lookup_ev ev j A of None \<Rightarrow> A | Some i \<Rightarrow> ??.Jordan_Normal_Form_Existence.swap_cols_rows_block (Suc i) j A) (??.Jordan_Normal_Form_Existence.step_2_main n (Suc j)))"
domain: shows "Domainp pcr_mod_ring = (\<lambda>x. \<exists>y. x = y \<and> y \<in> {0..<int CARD('b)})"
OUTfromV_VARto_def: shows "OUTfromV_VARto \<equiv> \<forall>x v. (v \<in> OUTfromV x) = (x \<in> VARto v)"
GB_alt_2_finite: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "finite G" shows "is_Groebner_basis G = (\<forall>f\<in>pmdl G. f \<noteq> 0 \<longrightarrow> is_red G f)"
leg1_prj_in_hom: shows "\<guillemotleft>S.leg1 \<cdot> \<p>\<^sub>1[S.leg0, T.leg1] : S.leg0 \<down>\<down> T.leg1 \<rightarrow> S.trg\<guillemotright>"
kD_nsqn: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> kD rt" shows "nsqn rt dip = nsqn\<^sub>r (the (rt dip))"
partial_order_onD_1: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "partial_order_on A r" shows "refl_on A r"
int_sum: fixes f :: "'c \<Rightarrow> nat"   and A :: "'c set" shows "int (sum f A) = (\<Sum>x\<in>A. int (f x))"
compact_closure: fixes S :: "'a set" shows "compact (closure S) = bounded S"
Proj_range: fixes S :: "'a ccsubspace" shows "Proj S *\<^sub>S \<top> = S"
path_start_in_topspace: fixes X :: "'a topology"   and g :: "real \<Rightarrow> 'a" assumes "pathin X g" shows "g 0 \<in> topspace X"
empty_empty: fixes xs :: "('c, 'd) bintree option list" assumes "normalized xs" shows "(xs = []) = PQ.is_empty (pqueue xs)"
sum_decode_def: fixes n :: "nat" shows "sum_decode n = (if even n then Inl (n div 2) else Inr (n div 2))"
assms_3: fixes g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "g \<in> L F h" shows "(\<lambda>x. t x (g x)) \<in> L F (\<lambda>x. t x (h x))"
elims_3: fixes x :: "'vertex \<times> ('form, 'var) antecedent" assumes "\<not> valid_in_port x" obtains v :: "'vertex" and p :: "('form, 'var) antecedent" where "x = (v, p)" and "\<not> (v |\<in>| vertices \<and> p |\<in>| inPorts (nodeOf v))"
bn0: shows "pochhammer (- b) n \<noteq> (0::'a)"
reflp_on_reflclp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "reflp_on P\<^sup>=\<^sup>= A"
agop_gop: fixes A :: "('c, 'd) aGroup_scheme" assumes "aGroup A" shows "(\<cdot>\<^bsub>b_ag A\<^esub>) = (\<plusminus>\<^bsub>A\<^esub>)"
w1Leq': shows "sint w1 \<le> 2 ^ 31 - 1"
subset_imp_less_zmset: fixes M :: "'b zmultiset"   and N :: "'b zmultiset" assumes "M \<subset>#\<^sub>z N" shows "M < N"
col_tranpose: fixes M :: "'a mat"   and n :: "nat"   and i :: "nat" assumes "dim_row M = n"   and "i < n" shows "col M\<^sup>t i = row M i"
correctCompositionDiffLevelsA9: shows "correctCompositionDiffLevels sA9"
Dirichlet_L_conv_eval_fds_weak: fixes s :: "complex" assumes "1 < Re s" shows "Dirichlet_L n \<chi> s = eval_fds (fds \<chi>) s"
add_nat_pow_monom: fixes a :: "'a"   and n :: "nat"   and k :: "nat" assumes "a \<in> carrier R" shows "[n] \<cdot>\<^bsub>P\<^esub> monom P a k = monom P ([n] \<cdot> a) k"
hyp2_rep_abs: fixes v :: "(real, 2) vec" assumes "norm v < 1" shows "hyp2_rep (hyp2_abs v) = v"
privateKey_in_initStateCA: fixes b :: "bool"   and A :: "agent" shows "Key (invKey (publicKey b A)) \<in> initState A"
crit_pair_cs_imp_crit_pair_cbelow_on: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "F \<subseteq> punit.dgrad_p_set d m"   and "p \<in> punit.dgrad_p_set d m"   and "q \<in> punit.dgrad_p_set d m"   and "relation.cs (punit.red F) (fst (punit.crit_pair p q)) (snd (punit.crit_pair p q))" shows "punit.crit_pair_cbelow_on d m F p q"
induct: fixes P :: "'b list \<Rightarrow> bool"   and a0 :: "'b list" assumes "\<And>x x' xs. P (x' # xs) \<Longrightarrow> P (x # x' # xs)"   and "P []"   and "\<And>v. P [v]" shows "P a0"
stream_stream_eqI: fixes xs :: "('b \<times> 'c list) stream"   and ys :: "('b \<times> 'c list) stream" assumes "\<forall>(uu_, x)\<in>sset xs. x \<noteq> []"   and "\<forall>(uu_, x)\<in>sset ys. x \<noteq> []"   and "smap (\<lambda>(uu_, x). hd x) xs = smap (\<lambda>(uu_, x). hd x) ys"   and "smap \<pi> xs = smap \<pi> ys" shows "xs = ys"
wf_mhead_def: fixes G :: "prog"   and P :: "pname" shows "wf_mhead G P = (\<lambda>sig mh. length (parTs sig) = length (pars mh) \<and> (\<forall>T\<in>set (parTs sig). is_acc_type G P T) \<and> is_acc_type G P (resTy mh) \<and> distinct (pars mh))"
givenBy_o_eq_extend_set: fixes v :: "'a \<Rightarrow> 'd" shows "givenBy (v \<circ> f) = extend_set h ` givenBy v"
pred_iffI: fixes A :: "'a pred"   and B :: "'a pred" assumes "\<And>x. eval A x \<Longrightarrow> eval B x"   and "\<And>x. eval B x \<Longrightarrow> eval A x" shows "A = B"
preserves_ide: fixes a :: "'a" assumes "B.ide a" shows "B.ide (B.R a)"
var: fixes t :: "'trm" assumes "set (y # map snd txs) \<subseteq> var"   and "set (s # map fst txs) \<subseteq> trm"   and "set [\<phi>] \<subseteq> fmla"   and "t \<in> set (s # map fst txs)" shows "set (getFrN (y # map snd txs) (s # map fst txs) [\<phi>] (length txs)) \<inter> set (y # map snd txs) = {}"
notn': shows "\<forall>i. (\<pi> @\<^bsup>l\<^esup> \<pi>') i \<noteq> n"
addrow_mat_Unit: fixes k :: "nat"   and n :: "nat"   and l :: "nat"   and a :: "'c"   and b :: "'d" assumes "k < n"   and "l < n"   and "k \<noteq> l" shows "addrow_mat n a k l \<in> Units (ring_mat TYPE('c) n b)"
prefix_is_continuation: fixes k :: "nat"   and u :: "nat"   and p :: "(symbol \<times> character list) list"   and ts :: "(symbol \<times> character list) list"   and ts' :: "(symbol \<times> character list) list" assumes "is_continuation k u p ts"   and "is_prefix ts' ts" shows "is_continuation k u p ts'"
eventually_returns_to_continuousI: fixes P :: "'a set"   and x :: "'a"   and S :: "'a set" assumes "Y.returns_to P x"   and "closed P"   and "continuous (at x within S) (Y.return_time P)" shows "eventually (Y.returns_to P) (at x within S)"
mult_oner: fixes x :: "('b \<times> 'b) set" shows "Id O x = x"
Idempotence_axioms: shows "Idempotence (\<bullet>) \<gamma> \<bottom> id1 id2"
communicating_def: shows "K2.communicating = K2.acc \<inter> K2.acc\<inverse>"
latin_rect_def: fixes s :: "(latin_type \<Rightarrow> nat) set"   and m :: "nat"   and n :: "nat" shows "latin_rect s m n \<equiv> m \<le> n \<and> partial_latin_square s n \<and> bij_betw row_col s ({0..<m} \<times> {0..<n}) \<and> bij_betw num_row s ({0..<n} \<times> {0..<m})"
strong_induct: fixes x1 :: "(name \<times> ty) list"   and x2 :: "lam"   and x3 :: "ty"   and P :: "'a \<Rightarrow> (name \<times> ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<turnstile> x2 : x3"   and "\<And>\<Gamma> x T y. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P y \<Gamma> (VAR x) T"   and "\<And>\<Gamma> t\<^sub>1 T\<^sub>1 T\<^sub>2 t\<^sub>2 y. \<lbrakk>\<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1 \<rightarrow> T\<^sub>2; \<And>z. P z \<Gamma> t\<^sub>1 T\<^sub>1 \<rightarrow> T\<^sub>2; \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>1; \<And>z. P z \<Gamma> t\<^sub>2 T\<^sub>1\<rbrakk> \<Longrightarrow> P y \<Gamma> (APP t\<^sub>1 t\<^sub>2) T\<^sub>2"   and "\<And>x \<Gamma> T\<^sub>1 t T\<^sub>2 y. \<lbrakk>x \<sharp> y; x \<sharp> \<Gamma>; (x, T\<^sub>1) # \<Gamma> \<turnstile> t : T\<^sub>2; \<And>z. P z ((x, T\<^sub>1) # \<Gamma>) t T\<^sub>2\<rbrakk> \<Longrightarrow> P y \<Gamma> LAM [x].t T\<^sub>1 \<rightarrow> T\<^sub>2"   and "\<And>\<Gamma> n y. P y \<Gamma> (NUM n) tINT"   and "\<And>\<Gamma> t\<^sub>1 t\<^sub>2 y. \<lbrakk>\<Gamma> \<turnstile> t\<^sub>1 : tINT; \<And>z. P z \<Gamma> t\<^sub>1 tINT; \<Gamma> \<turnstile> t\<^sub>2 : tINT; \<And>z. P z \<Gamma> t\<^sub>2 tINT\<rbrakk> \<Longrightarrow> P y \<Gamma> t\<^sub>1 -- t\<^sub>2 tINT"   and "\<And>\<Gamma> t\<^sub>1 t\<^sub>2 y. \<lbrakk>\<Gamma> \<turnstile> t\<^sub>1 : tINT; \<And>z. P z \<Gamma> t\<^sub>1 tINT; \<Gamma> \<turnstile> t\<^sub>2 : tINT; \<And>z. P z \<Gamma> t\<^sub>2 tINT\<rbrakk> \<Longrightarrow> P y \<Gamma> t\<^sub>1 ++ t\<^sub>2 tINT"   and "\<And>\<Gamma> y. P y \<Gamma> TRUE tBOOL"   and "\<And>\<Gamma> y. P y \<Gamma> FALSE tBOOL"   and "\<And>\<Gamma> t1 t2 T t3 y. \<lbrakk>\<Gamma> \<turnstile> t1 : tBOOL; \<And>z. P z \<Gamma> t1 tBOOL; \<Gamma> \<turnstile> t2 : T; \<And>z. P z \<Gamma> t2 T; \<Gamma> \<turnstile> t3 : T; \<And>z. P z \<Gamma> t3 T\<rbrakk> \<Longrightarrow> P y \<Gamma> (IF t1 t2 t3) T"   and "\<And>\<Gamma> t y. \<lbrakk>\<Gamma> \<turnstile> t : tINT; \<And>z. P z \<Gamma> t tINT\<rbrakk> \<Longrightarrow> P y \<Gamma> (ZET t) tBOOL"   and "\<And>\<Gamma> t1 t2 y. \<lbrakk>\<Gamma> \<turnstile> t1 : tINT; \<And>z. P z \<Gamma> t1 tINT; \<Gamma> \<turnstile> t2 : tINT; \<And>z. P z \<Gamma> t2 tINT\<rbrakk> \<Longrightarrow> P y \<Gamma> (EQI t1 t2) tBOOL"   and "\<And>x \<Gamma> T t y. \<lbrakk>x \<sharp> y; x \<sharp> \<Gamma>; (x, T) # \<Gamma> \<turnstile> t : T; \<And>z. P z ((x, T) # \<Gamma>) t T\<rbrakk> \<Longrightarrow> P y \<Gamma> FIX [x].t T" shows "P z x1 x2 x3"
convex_polychain_map_mirror: fixes hs :: "((real \<times> real) \<times> real \<times> real) list"   and x :: "real \<times> real" assumes "convex_polychain hs" shows "convex_polychain (map (pairself (mirror_point x)) hs)"
FreeGroup_genby_set_lconjby_set_rconjby_closed: fixes g :: "'b freeword"   and S :: "'b set"   and x :: "'b freeword"   and A :: "'b freeword set" assumes "g \<in> FreeGroup S"   and "x \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` A\<rangle>" shows "rconjby g x \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` A\<rangle>"
prj_simps_10: shows "C.arr fgf.Prj\<^sub>1"
induct: fixes P :: "'b tree list \<Rightarrow> 'b tree list \<Rightarrow> bool"   and a0 :: "'b tree list"   and a1 :: "'b tree list" assumes "\<And>ts\<^sub>1. P ts\<^sub>1 []"   and "\<And>v va. P [] (v # va)"   and "\<And>t\<^sub>1 ts\<^sub>1 t\<^sub>2 ts\<^sub>2. \<lbrakk>rank t\<^sub>1 < rank t\<^sub>2 \<Longrightarrow> P ts\<^sub>1 (t\<^sub>2 # ts\<^sub>2); \<lbrakk>\<not> rank t\<^sub>1 < rank t\<^sub>2; rank t\<^sub>2 < rank t\<^sub>1\<rbrakk> \<Longrightarrow> P (t\<^sub>1 # ts\<^sub>1) ts\<^sub>2; \<lbrakk>\<not> rank t\<^sub>1 < rank t\<^sub>2; \<not> rank t\<^sub>2 < rank t\<^sub>1\<rbrakk> \<Longrightarrow> P ts\<^sub>1 ts\<^sub>2\<rbrakk> \<Longrightarrow> P (t\<^sub>1 # ts\<^sub>1) (t\<^sub>2 # ts\<^sub>2)" shows "P a0 a1"
INT_extend_simps_10: fixes B :: "'d \<Rightarrow> 'b set"   and f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "(\<Inter>a\<in>A. B (f a)) = \<Inter> (B ` f ` A)"
bhv_hn_in: fixes tid :: "'nid"   and n :: "nat"   and t :: "nat \<Rightarrow> cnf" assumes "honest tid"   and "\<exists>n'\<ge>n. \<parallel>tid\<parallel>\<^bsub>t n'\<^esub>"   and "\<exists>n'. latestAct_cond tid t n n'"   and "\<not> (\<exists>b\<in>pin (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>). length (bc (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>)) < length b)" shows "\<not> mining (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<rightarrow> t\<rangle>\<^bsub>n\<^esub>) \<and> bc (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<rightarrow> t\<rangle>\<^bsub>n\<^esub>) = bc (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>) \<or> mining (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<rightarrow> t\<rangle>\<^bsub>n\<^esub>) \<and> (\<exists>b. bc (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<rightarrow> t\<rangle>\<^bsub>n\<^esub>) = bc (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>) @ [b])"
rot_diamond_gen_div_diamond_gen: shows "gen_division (cubeImage diamond_cube_gen) (cubeImage ` {rot_diamond_cube})"
trace_reject_later: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and t :: "(String.literal \<times> value list) list" assumes "\<forall>(s', T)|\<in>|possible_steps e s r l i. \<not> recognises_execution e s' (evaluate_updates T i r) t" shows "\<not> recognises_execution e s r ((l, i) # t)"
FL_bisimilar_def: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and F :: "'d fs_set"   and P :: "'a"   and Q :: "'a" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.FL_bisimilar satisfies transition effect_apply L F P Q \<equiv> \<exists>R. effect_nominal_ts.is_L_bisimulation satisfies transition effect_apply L R \<and> R F P Q"
terminates_maps_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and f :: "'f \<Rightarrow> ('g, 'h) generator \<times> 'h" assumes "terminates g" shows "terminates (maps_raw f g)"
bangInduct: fixes P :: "pi"   and Rs :: "residual"   and F :: "'a \<Rightarrow> pi \<Rightarrow> residual \<Rightarrow> bool"   and C :: "'a" assumes "!P \<longmapsto> Rs"   and "\<And>a x P' C. \<lbrakk>P \<longmapsto> a<\<nu>x> \<prec> P'; x \<sharp> P; x \<sharp> C\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (a<\<nu>x> \<prec> (P' \<parallel> !P))"   and "\<And>\<alpha> P' C. P \<longmapsto> \<alpha> \<prec> P' \<Longrightarrow> F C (P \<parallel> !P) (\<alpha> \<prec> P' \<parallel> !P)"   and "\<And>a x P' C. \<lbrakk>!P \<longmapsto> a<\<nu>x> \<prec> P'; x \<sharp> P; x \<sharp> C; \<And>C. F C (!P) (a<\<nu>x> \<prec> P')\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (a<\<nu>x> \<prec> (P \<parallel> P'))"   and "\<And>\<alpha> P' C. \<lbrakk>!P \<longmapsto> \<alpha> \<prec> P'; \<And>C. F C (!P) (\<alpha> \<prec> P')\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (\<alpha> \<prec> P \<parallel> P')"   and "\<And>a P' b P'' C. \<lbrakk>P \<longmapsto> a<b> \<prec> P'; !P \<longmapsto> a[b] \<prec> P''; \<And>C. F C (!P) (a[b] \<prec> P'')\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (\<tau> \<prec> P' \<parallel> P'')"   and "\<And>a b P' P'' C. \<lbrakk>P \<longmapsto> a[b] \<prec> P'; !P \<longmapsto> a<b> \<prec> P''; \<And>C. F C (!P) (a<b> \<prec> P'')\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (\<tau> \<prec> P' \<parallel> P'')"   and "\<And>a x P' P'' C. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; !P \<longmapsto> a<\<nu>x> \<prec> P''; x \<sharp> P; x \<sharp> C; \<And>C. F C (!P) (a<\<nu>x> \<prec> P'')\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (\<tau> \<prec> <\<nu>x>(P' \<parallel> P''))"   and "\<And>a x P' P'' C. \<lbrakk>P \<longmapsto> a<\<nu>x> \<prec> P'; !P \<longmapsto> a<x> \<prec> P''; x \<sharp> P; x \<sharp> C; \<And>C. F C (!P) (a<x> \<prec> P'')\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (\<tau> \<prec> <\<nu>x>(P' \<parallel> P''))"   and "\<And>Rs C. \<lbrakk>P \<parallel> !P \<longmapsto> Rs; \<And>C. F C (P \<parallel> !P) Rs\<rbrakk> \<Longrightarrow> F C (!P) Rs" shows "F C (!P) Rs"
prems_3: fixes j :: "nat" assumes "j \<le> Suc i" shows "suffix n w \<Turnstile>\<^sub>n af \<psi> (w [j \<rightarrow> n])[X]\<^sub>\<nu>"
Arg_eq_pi_iff: fixes z :: "complex" shows "(Arg z = pi) = (z \<in> \<real> \<and> Re z < 0)"
VPi_def: fixes A :: "V"   and B :: "V \<Rightarrow> V" shows "VPi A B \<equiv> ZFC_in_HOL.set {f \<in> elts (VPow (VSigma A B)). elts A \<subseteq> Domain (pairs f) \<and> single_valued (pairs f)}"
convex_hull_convex_sum: fixes A :: "'b vec set"   and n :: "nat"   and x :: "'b vec"   and y :: "'b vec"   and a :: "'b" assumes "A \<subseteq> carrier_vec n"   and "x \<in> gram_schmidt.convex_hull n A"   and "y \<in> gram_schmidt.convex_hull n A"   and "(0::'b) \<le> a"   and "a \<le> (1::'b)" shows "a \<cdot>\<^sub>v x + ((1::'b) - a) \<cdot>\<^sub>v y \<in> gram_schmidt.convex_hull n A"
sum_group: fixes I :: "'b set"   and G :: "'b \<Rightarrow> ('c, 'd) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)" shows "Group.group (sum_group I G)"
alt_writer_def: fixes alt :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and m :: "('g, 'h, 'i) writerT"   and m' :: "('g, 'h, 'i) writerT" shows "alt_writer alt m m' = WriterT (alt (run_writer m) (run_writer m'))"
frestrictionD_3: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<bullet> A" shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
f_2: shows "(f has_derivative f'') (at x within box a b)"
strict_trans2: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<sqsubset> a"   and "c \<sqsubseteq> b" shows "c \<sqsubset> a"
F_nontrivial: fixes F :: "'a filter"   and H :: "('a \<Rightarrow> real) set" assumes "landau_function_family F H" shows "F \<noteq> bot"
tySub: shows "typ_of (subst_bv x t) = Some propT"
oner: fixes x :: "'a" shows "x \<oplus> (1::'a) = x"
resNilLeft_2: fixes Rel :: "(pi \<times> pi) set"   and x :: "name"   and b :: "name"   and P :: "pi" shows "\<zero> \<leadsto>[Rel] <\<nu>x>x{b}.P"
Tree\<^sub>\<alpha>_rep_abs: fixes t :: "('b, 'c, 'd, 'e) Tree" shows "rep_Tree\<^sub>\<alpha> (abs_Tree\<^sub>\<alpha> t) =\<^sub>\<alpha> t"
nat_not_le_less: fixes n :: "nat"   and m :: "nat" shows "(\<not> n \<le> m) = (m < n)"
Cignorabimus_finite: fixes \<alpha> :: "game"   and V :: "variable set" shows "finite (Cignorabimus \<alpha> V)"
homogeneous_set_IntI: fixes A :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and B :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" assumes "homogeneous_set A"   and "homogeneous_set B" shows "homogeneous_set (A \<inter> B)"
check_induct: fixes s :: "('i, 'a) state"   and P :: "('i, 'a) state \<Rightarrow> ('i, 'a) state \<Rightarrow> bool" assumes "\<nabla> s"   and "\<triangle> (\<T> s)"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<And>s. \<U> s \<Longrightarrow> P s s"   and "\<And>s. \<lbrakk>\<not> \<U> s; min_lvar_not_in_bounds s = None\<rbrakk> \<Longrightarrow> P s s"   and "\<And>s x\<^sub>i dir I. \<lbrakk>dir = Positive \<or> dir = Negative; \<not> \<U> s; min_lvar_not_in_bounds s = Some x\<^sub>i; \<lhd>\<^sub>l\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x\<^sub>i) (LB dir s x\<^sub>i); min_rvar_incdec dir s x\<^sub>i = Inl I\<rbrakk> \<Longrightarrow> P s (set_unsat I s)"   and "\<And>s x\<^sub>i x\<^sub>j l\<^sub>i. \<lbrakk>\<triangle> (\<T> s); \<nabla> s; x\<^sub>i \<in> lvars (\<T> s); x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)\<rbrakk> \<Longrightarrow> P s (pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s)"   and "\<And>si sj sk. \<lbrakk>P si sj; P sj sk\<rbrakk> \<Longrightarrow> P si sk" shows "P s (check s)"
distinct_1: fixes x11 :: "e list"   and x12 :: "e list"   and x21 :: "e list"   and x22 :: "nat"   and x23 :: "e list"   and x24 :: "Lholed"   and x25 :: "e list" shows "LBase x11 x12 \<noteq> LRec x21 x22 x23 x24 x25"
UINF_is_exists: fixes P :: "'e \<Rightarrow> 'd upred" shows "(\<Sqinter> x \<bullet> P x) = ushEx P"
diff_eq_diff_less_eq: fixes plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and zero :: "'r"   and minus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and uminus :: "'r \<Rightarrow> 'r"   and less_eq :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and less :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and a :: "'r"   and b :: "'r"   and c :: "'r"   and d :: "'r" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "minus a b = minus c d" shows "less_eq a b = less_eq c d"
simps_1: fixes a :: "'ent atom" shows "wf_fmla (Atom a) = wf_atom a"
sel_spec_f4_sel: shows "sel_spec punit.f4_sel"
inj_on_apply_chart: fixes f :: "('c, 'd) chart" shows "inj_on (apply_chart f) (domain f)"
empty: fixes g :: "'d \<Rightarrow> 'c" shows "aggregation.sum_0 g {} = zero"
mod_eq_dvd_iff: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a mod c = b mod c) = (c dvd a - b)"
Q_def: shows "Q = (\<Prod>i<n. [:- Root i, 1:])"
llength_eq_infty_conv_lfinite: fixes xs :: "'b llist" shows "(llength xs = \<infinity>) = (\<not> lfinite xs)"
igAbsCongUSTR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igAbsCongUSTR MOD \<equiv> \<forall>xs x x' y X X'. igFresh MOD xs y X \<and> igFresh MOD xs y X' \<and> igSubst MOD xs (igVar MOD xs y) x X = igSubst MOD xs (igVar MOD xs y) x' X' \<longrightarrow> igAbs MOD xs x X = igAbs MOD xs x' X'"
subset_iff: fixes A :: "'h set"   and B :: "'h set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
monoid_list_axioms: shows "monoid_list (\<^bold>*) \<^bold>1"
filter_neq_takeWhile: fixes a :: "'d"   and l :: "'d list" assumes "filter ((\<noteq>) a) l \<noteq> takeWhile ((\<noteq>) a) l" shows "\<exists>i j. i < j \<and> j < length l \<and> l ! i = a \<and> l ! j \<noteq> a"
map_eq_const_coin_spmf: fixes c :: "bool" shows "map_spmf ((=) c) coin_spmf = coin_spmf"
inj_on_map_sum: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'e \<Rightarrow> 'f"   and B :: "'e set" assumes "inj_on f A"   and "inj_on g B" shows "inj_on (map_sum f g) (A <+> B)"