hcju/isabelleps huggingface dataset:
zero_le_divide_1_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq zero (divide one a) = less_eq zero a"
classic_circline: fixes H :: "circline"   and A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex" assumes "H = mk_circline A B C D"   and "hermitean (A, B, C, D) \<and> (A, B, C, D) \<noteq> mat_zero" shows "circline_set H - {\<infinity>\<^sub>h} = of_complex ` circline (Re A) B (Re D)"
choose_two_int: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and index :: "int"   and x :: "int" assumes "symmetric_bibd point_set block_collection u_block_size index"   and "0 \<le> x" shows "int (nat x choose 2) = x * (x - 1) div 2"
tr_x_def: fixes T :: "x \<Rightarrow> x option"   and x :: "x" shows "tr_x T x = (case T x of None \<Rightarrow> x | Some x' \<Rightarrow> x')"
times_enat_def: fixes m :: "enat"   and n :: "enat" shows "m * n = (case m of enat m \<Rightarrow> case n of enat n \<Rightarrow> enat (m * n) | \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity>)"
uncurry_curry0_id: fixes g :: "unit \<Rightarrow> 'a" shows "uncurry0 (curry0 g) = g"
Suc_le': fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and a :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "plus a one \<noteq> zero" shows "less a (plus a one)"
h: shows "h \<in> ring_hom (law_restrict A) (law_restrict B)"
assms: shows "FI [] (SLN * P) (SLN * Q) SLN SLN F"
lincomb_def: fixes a :: "'a Matrix.vec \<Rightarrow> 'a"   and A :: "'a Matrix.vec set" shows "NC.lincomb a A = (\<Oplus>\<^bsub>NC.V\<^esub>v\<in>A. a v \<cdot>\<^sub>v v)"
filtermap_times_pos_at_top: fixes e :: "real" assumes "0 < e" shows "filtermap ((*) e) at_top = at_top"
in_carrier_atlasI: fixes c :: "('a, 'e) chart"   and x :: "'a" assumes "c \<in> src.atlas"   and "x \<in> domain c" shows "x \<in> src.carrier"
le_bound_of: fixes f :: "'b \<Rightarrow> real"   and x :: "'b" assumes "Expectations.bounded f" shows "f x \<le> bound_of f"
lm_imp_m_or_eq: fixes n :: "nat" assumes "m < length (f n)" shows "length (f (Suc n)) \<le> m \<or> length (f n) = length (f (Suc n))"
weakTransitiveWeakCoinduct: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "eqvt X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<lessapprox><X> Q"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<leadsto><{(\<Psi>, P, Q) |\<Psi> P Q. \<exists>P' Q'. \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}> Q"   and "\<And>\<Psi> P Q \<Psi>'. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi>, Q, P) \<in> X" shows "\<Psi> \<rhd> P \<approx> Q"
rep_eq: shows "rbt.impl_of RBT.empty = rbt.Empty"
semiring_norm_172: fixes v :: "num"   and w :: "num"   and y :: "'a" shows "- numeral v * (- numeral w * y) = numeral (v * w) * y"
cases: fixes a1 :: "'node list"   and a2 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a3 :: "'node list"   and a4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and S :: "'node SDG_node set" assumes "((a1, a2), a3, a4) \<in> WS S" obtains ms :: "'node list" and ms' :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and msx :: "'node list" and mx :: "'node" where "a1 = ms" and "a2 = s" and "a3 = ms'" and "a4 = s'" and "Ball (set ms) valid_node" and "Ball (set ms') valid_node" and "length ms = length s" and "length ms' = length s'" and "s \<noteq> []" and "s' \<noteq> []" and "ms = msx @ mx # tl ms'" and "get_proc mx = get_proc (hd ms')" and "\<forall>m\<in>set (tl ms'). \<exists>m'. call_of_return_node m m' \<and> m' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" and "msx \<noteq> [] \<longrightarrow> (\<exists>mx'. call_of_return_node mx mx' \<and> mx' \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>)" and "\<forall>i<length ms'. snd (s ! (length msx + i)) = snd (s' ! i)" and "Ball (set (tl ms)) return_node" and "\<forall>i<length ms'. \<forall>V\<in>rv S (CFG_node ((mx # tl ms') ! i)). fst (s ! (length msx + i)) V = fst (s' ! i) V" and "obs ms \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = obs ms' \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
h_h': fixes z :: "complex" shows "(h has_field_derivative h' z) (at z)"
weakPsiCongE_2: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<doteq> Q" shows "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q"
extended_distance_pos: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c Bonk_Schramm_extension_unfolded"   and b :: "'c Bonk_Schramm_extension_unfolded" assumes "class.metric_space dist uniformity open"   and "a \<in> metric_space.extended_distance_set dist"   and "b \<in> metric_space.extended_distance_set dist" shows "0 \<le> metric_space.extended_distance dist a b"
star_of_diff: fixes x :: "'a"   and y :: "'a" shows "star_of (x - y) = star_of x - star_of y"
assms_2: fixes conv :: "('a, 'b, 'c, 'd) converter"   and q :: "'a"   and r :: "'b"   and conv' :: "('a, 'b, 'c, 'd) converter" assumes "X conv"   and "q \<in> outs_\<I> \<I>" shows "\<I>' \<turnstile>g run_converter conv q \<surd> \<and> ((r, conv') \<in> results_gpv \<I>' (run_converter conv q) \<longrightarrow> r \<in> responses_\<I> \<I> q \<and> (X conv' \<or> \<I>, \<I>' \<turnstile>\<^sub>C conv' \<surd>))"
min_int_poly_complex_def: fixes x :: "complex" shows "min_int_poly_complex x = (if algebraic x then if Im x = 0 then min_int_poly_real (Re x) else the (find (\<lambda>f. x \<in> set (complex_roots_of_int_poly f)) (complex_poly (min_int_poly (Re x)) (min_int_poly (Im x)))) else [:0, 1:])"
eval_wp_Apply: fixes f :: "'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> real" shows "wp (Apply f) P = P \<circ> f"
closure_approachable: fixes x :: "'c"   and S :: "'c set" shows "(x \<in> closure S) = (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
prime_elem_dvd_power: fixes p :: "'a"   and x :: "'a"   and n :: "nat" assumes "prime_elem p"   and "p dvd x ^ n" shows "p dvd x"
is_Prog\<^sub>s\<^sub>b_simps_1: fixes volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "is_Prog\<^sub>s\<^sub>b (Read\<^sub>s\<^sub>b volatile a t v) = False"
bin_rest_numeral_simps_6: fixes w :: "num" shows "numeral (num.Bit1 w) div 2 = numeral w"
cfun_below_iff: fixes f :: "'c \<rightarrow> 'd"   and g :: "'c \<rightarrow> 'd" shows "(f \<sqsubseteq> g) = (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)"
Im2_Hv: fixes v :: "(real, 3) vec" shows "Im2 (Hv v) = v $ 2"
Gromov_product_at_continuous: fixes u :: "'a \<Rightarrow> 'b"   and x :: "'b"   and F :: "'a filter"   and v :: "'a \<Rightarrow> 'b"   and y :: "'b"   and w :: "'a \<Rightarrow> 'b"   and z :: "'b" assumes "(u \<longlongrightarrow> x) F"   and "(v \<longlongrightarrow> y) F"   and "(w \<longlongrightarrow> z) F" shows "((\<lambda>n. Gromov_product_at (u n) (v n) (w n)) \<longlongrightarrow> Gromov_product_at x y z) F"
singleton_Int1_if: fixes a :: "'a"   and A :: "'a set" shows "{a} \<inter> A = (if a \<in> A then {a} else {})"
map_graph_preserves_restricted: fixes G :: "('e, 'f) labeled_graph"   and f :: "('f \<times> 'g) set" assumes "graph G" shows "graph (map_graph f G)"
set_less_eq_total: fixes A :: "'a set"   and B :: "'a set" shows "A \<sqsubseteq> B \<or> B \<sqsubseteq> A"
fps_zero_nth: fixes n :: "nat" shows "fps_nth 0 n = (0::'a)"
order_trans_rules_19: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x < y"   and "y < z" shows "x < z"
decr_grading_term_splus: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat"   and t :: "'a"   and v :: "'t" assumes "hom_grading d" shows "decr_grading_term d n (t \<oplus> v) = decr_grading d n t \<oplus> decr_grading_term d n v"
br_invar_def: fixes \<delta> :: "('d, 'e) ta_rule set" shows "br_invar \<delta> \<equiv> {(Q, W). W \<subseteq> Q \<and> Q \<subseteq> b_accessible \<delta> \<and> bacc_step \<delta> (Q - W) \<subseteq> Q \<and> finite Q}"
knC_init: fixes runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and K :: "key"   and al :: "atom list" assumes "runz Ra = Some (Init, [A, B], aKey K # al)" shows "(K, A) \<in> knC runz"
less: shows "\<forall>i\<in>load_tmps is. i < n"
vec_index_vCons_Suc: fixes a :: "'a"   and v :: "'a Matrix.vec"   and n :: "nat" shows "vCons a v $ Suc n = v $ n"
nMTSort: fixes p :: "((int \<times> int) set set, int) Combinators list"   and l :: "(int \<times> int) set set set list" assumes "none_MT_rules C p" shows "none_MT_rules C (FWNormalisationCore.sort p l)"
mult_map_scale_conv_mult: fixes s :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'b" shows "Poly_Mapping.map ((*) s) p = Poly_Mapping.single (0::'a) s * p"
mult_left_less_imp_less: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_semiring plus minus less_eq less zero times"   and "less (times c a) (times c b)"   and "less_eq zero c" shows "less a b"
lang_flatten_INTERSECT: fixes X :: "'a atom rexp set"   and n :: "nat"   and w :: "('a \<times> bool list) list" assumes "finite X"   and "X \<noteq> {}"   and "\<forall>r\<in>X. local.wf n r" shows "(w \<in> lang n (INTERSECT (sorted_list_of_set X))) = (\<forall>r\<in>X. w \<in> lang n r)"
lepoll_funcset_left: fixes B :: "'a set"   and A :: "'b set"   and A' :: "'c set" assumes "B \<noteq> {}"   and "A \<lesssim> A'" shows "A \<rightarrow>\<^sub>E B \<lesssim> A' \<rightarrow>\<^sub>E B"
simps_1: fixes x2 :: "expr"   and y2 :: "expr" shows "(Expr x2 = Expr y2) = (x2 = y2)"
leg1_simps_5: shows "local.dom r\<^sub>0s\<^sub>1.p\<^sub>1 = r\<^sub>0s\<^sub>1.p\<^sub>1"
gfp_fusion_inf_pres: fixes h :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'c" assumes "Inf_pres h"   and "mono f"   and "mono g"   and "h \<circ> f = g \<circ> h" shows "h (gfp f) = gfp g"
mat_of_cblinfun_minus: fixes M :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c"   and N :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "mat_of_cblinfun (M - N) = mat_of_cblinfun M - mat_of_cblinfun N"
nonzero_less_TC: fixes x :: "V" assumes "x \<noteq> 0" shows "0 \<sqsubset> x"
CHECK_simps_1: fixes e :: "'c" shows "CHECK True e = ERETURN ()"
cleanup_def: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal" shows "cleanup f = (\<lambda>(a, b). if f (b, a) < f (a, b) then f (a, b) - f (b, a) else 0)"
subdomain_axioms_def: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" shows "subdomain_axioms H R \<equiv> \<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub> \<and> (\<forall>h1 h2. h1 \<in> H \<longrightarrow> h2 \<in> H \<longrightarrow> h1 \<otimes>\<^bsub>R\<^esub> h2 = \<zero>\<^bsub>R\<^esub> \<longrightarrow> h1 = \<zero>\<^bsub>R\<^esub> \<or> h2 = \<zero>\<^bsub>R\<^esub>)"
cone_subset_PolysD_1: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and X :: "'x set" assumes "cone (h, U) \<subseteq> P[X]" shows "h \<in> P[X]"
assms: shows "set (injectionsAlg xs Y) = injections (set xs) (set Y)"
var_rename_fv_set_disjoint: fixes M :: "('a, 'b \<times> nat) Term.term set"   and s :: "('a, 'b \<times> nat) Term.term"   and t :: "('c, 'b \<times> nat) Term.term" assumes "finite M"   and "s \<in> M" shows "fv s \<inter> fv (t \<cdot> var_rename (max_var_set (fv\<^sub>s\<^sub>e\<^sub>t M))) = {}"
rel_intros_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "'a"   and y2 :: "'b" assumes "R x2 y2" shows "rel_option R (Some x2) (Some y2)"
induct: fixes G :: "prog"   and x1 :: "qtname"   and x2 :: "qtname"   and P :: "qtname \<Rightarrow> qtname \<Rightarrow> bool" assumes "G\<turnstile>x1\<leadsto>x2"   and "\<And>C J. G\<turnstile>C\<leadsto>1J \<Longrightarrow> P C J"   and "\<And>C I J. \<lbrakk>G\<turnstile>C\<leadsto>1I; G\<turnstile>I\<preceq>I J\<rbrakk> \<Longrightarrow> P C J"   and "\<And>C D J. \<lbrakk>G\<turnstile>C\<prec>\<^sub>C1D; G\<turnstile>D\<leadsto>J; P D J\<rbrakk> \<Longrightarrow> P C J" shows "P x1 x2"
zero_mat_def: fixes nr :: "nat"   and nc :: "nat" shows "0\<^sub>m nr nc \<equiv> Matrix.mat nr nc (\<lambda>ij. 0::'d)"
block_size_gt_t: fixes k :: "int" assumes "k \<in> {\<k>}" shows "2 \<le> k"
flatten_set_group_hom: fixes G :: "('c set, 'd) monoid_scheme"   and rep :: "'c set \<Rightarrow> 'e" assumes "Group.group G"   and "inj_on rep (carrier G)" shows "rep \<in> hom G (flatten G rep)"
relcomp_UNION_distrib: fixes s :: "('a \<times> 'c) set"   and r :: "'d \<Rightarrow> ('c \<times> 'b) set"   and I :: "'d set" shows "s ; \<Union> (r ` I) = (\<Union>i\<in>I. s ; r i)"
Sem_def: fixes s :: "Var \<Rightarrow> Val"   and c :: "IMP"   and t :: "Var \<Rightarrow> Val" shows "s , c \<Down> t  = (\<exists>n.  s , c \<rightarrow>\<^sub>n  t )"
add_less_cancel_left: fixes c :: "'d multiset"   and a :: "'d multiset"   and b :: "'d multiset" shows "(c + a \<subset># c + b) = (a \<subset># b)"
shadow_root_ptr_kinds_eq_h: shows "shadow_root_ptr_kinds h = shadow_root_ptr_kinds h2"
rel_F1_F2: shows "rel_fundefs (Finca_get F1) (Fubx_get F2)"
homogeneous_ideal': fixes F :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "p \<in> ideal F"   and "q \<in> hom_components p" shows "q \<in> ideal F"
proj_llength: fixes c :: "'id"   and t :: "cnf llist" shows "llength (\<pi>\<^bsub>c\<^esub>t) \<le> llength t"
J_2: fixes i :: "nat" assumes "i \<in> J" shows "F' i \<in> sets (M i)"
redT_silent_move_invariant: fixes final :: "'c \<Rightarrow> bool"   and r :: "('f, 'd, 'c, 'e, 'g, 'h) semantics"   and \<tau>move :: "('f, 'd, 'c, 'e, 'g, 'h) \<tau>moves"   and P :: "'i \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> bool"   and convert_RA :: "'f \<Rightarrow>f nat \<Rightarrow> 'h list"   and s :: "('f, 'd) locks \<times> (('d \<Rightarrow> ('c \<times> 'f \<Rightarrow>f nat) option) \<times> 'e) \<times> ('d \<Rightarrow> 'g wait_set_status option) \<times> 'd set"   and s' :: "('f, 'd) locks \<times> (('d \<Rightarrow> ('c \<times> 'f \<Rightarrow>f nat) option) \<times> 'e) \<times> ('d \<Rightarrow> 'g wait_set_status option) \<times> 'd set"   and Is :: "'d \<Rightarrow> 'i option" assumes "\<tau>lifting_inv final r \<tau>move P"   and "\<tau>trsys.silent_move (multithreaded_base.redT final r convert_RA) (\<tau>multithreaded.m\<tau>move \<tau>move) s s'"   and "ts_inv P Is (thr s) (shr s)" shows "ts_inv P Is (thr s') (shr s')"
tameEnumFilter_def: fixes p :: "nat" shows "tameEnumFilter p = enum_filter_finals next_tame\<^bsub>p\<^esub> [Seed\<^bsub>p\<^esub>]"
index_of_c_pair_n_is_pr: shows "index_of_c_pair_n \<in> PrimRec1"
pbij_region_coregion: fixes S :: "<'d, 'e> \<Longleftrightarrow> 'f" assumes "psym_lens S" shows "pbij_lens (\<V>\<^bsub>S\<^esub> +\<^sub>L \<C>\<^bsub>S\<^esub>)"
unique_rootI: fixes plr :: "int poly \<times> rat \<times> rat"   and x :: "real" assumes "\<And>y. root_cond plr y \<Longrightarrow> x = y"   and "root_cond plr x" shows "\<exists>!x. root_cond plr x"
inter_guards_exec_Fault: fixes c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and s :: "('d, 'f) xstate"   and f :: "'f" assumes "(c1 \<inter>\<^sub>g c2) = Some c"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Fault f" shows "\<Gamma>\<turnstile> \<langle>c1,s\<rangle> \<Rightarrow> Fault f \<or> \<Gamma>\<turnstile> \<langle>c2,s\<rangle> \<Rightarrow> Fault f"
rep_eq: fixes x :: "'d cset cset" shows "rcset (cUnion x) = \<Union> (rcset ` rcset x)"
vrrestriction_vconverse: fixes r :: "V"   and A :: "V" shows "r\<inverse>\<^sub>\<circ> \<restriction>\<^sup>r\<^sub>\<circ> A = (r \<restriction>\<^sup>l\<^sub>\<circ> A)\<inverse>\<^sub>\<circ>"
abs_scaleR: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "real"   and b :: "'b" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "abs (scaleR a b) = scaleR \<bar>a\<bar> (abs b)"
irrefl_def: fixes r :: "('g \<times> 'g) set" shows "irrefl r = (\<forall>a. (a, a) \<notin> r)"
induct: fixes P :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list \<Rightarrow> (('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list \<Rightarrow> bool"   and a0 :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and a1 :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "\<And>D. P [] D"   and "\<And>s t F D. (\<And>x. x \<in> set D \<Longrightarrow> P F D) \<Longrightarrow> P ((s, t) # F) D" shows "P a0 a1"
components_iff: fixes S :: "'a set"   and U :: "'a set" shows "(S \<in> components U) = (\<exists>x. x \<in> U \<and> S = connected_component_set U x)"
Tree\<^sub>\<alpha>_free_1: fixes tset\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha> set['a]"   and t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" shows "Conj\<^sub>\<alpha> tset\<^sub>\<alpha> \<noteq> Not\<^sub>\<alpha> t\<^sub>\<alpha>"
Q2_def: shows "Q2 = matrix_sum d (\<lambda>l. tensor_P (proj_psi_l (l + 1)) (proj_k l)) R"
assms_2: fixes y :: "real" shows "{x \<in> space (lborel_f n). x(n := y) \<in> A} \<in> sets (lborel_f n)"
map_simp_ide: fixes a :: "'a \<times> 'a" assumes "VV.ide a" shows "\<Xi>.map a = CMP (fst a) (snd a)"
basis_wf_singleton: fixes b :: "real \<Rightarrow> real" shows "basis_wf [b] = filterlim b at_top at_top"
i_bound_\<alpha>': shows "i + length (snd r) \<le> length \<alpha>'"
diff_in_polynomial_bool_pac_idealI2: fixes p :: "int mpoly"   and A :: "int mpoly set"   and p' :: "int mpoly" assumes "p \<in> A"   and "p - p' \<in> More_Modules.ideal polynomial_bool" shows "p' \<in> pac_ideal A"
mkt_def: fixes x :: "'a"   and l :: "'a tree"   and r :: "'a tree" shows "mkt x l r = MKT x l r (max (ht l) (ht r) + 1)"
simps_5: fixes va :: "Formula.trm"   and vb :: "Formula.trm" shows "Formula.future_bounded (formula.LessEq va vb) = True"
coboundedI: fixes A :: "'d set"   and a :: "'d" assumes "finite A"   and "a \<in> A" shows "a \<le> Max A"
assms_3: assumes "\<pi>l = []" shows "stateOf \<pi> = s0"
Cons_mem_set_subseqs_sorted: fixes xs :: "'c list"   and y :: "'c"   and ys :: "'c list" assumes "sorted xs"   and "y # ys \<in> set (subseqs xs)" shows "y # ys \<in> set (subseqs (filter ((\<le>) y) xs))"
om_unfold_eq: fixes x :: "'a" shows "x \<cdot> x\<^sup>\<omega> = x\<^sup>\<omega>"
map_ap_conv: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b list" shows "map f x = [f] \<diamondop> x"
fresh_eqvt: fixes pi :: "(name \<times> name) list"   and a :: "name"   and x :: "'d" shows "pi \<bullet> a \<sharp> x = pi \<bullet> a \<sharp> pi \<bullet> x"
strict_iff_order: fixes b :: "'a"   and a :: "'a" shows "(b < a) = (b \<le> a \<and> a \<noteq> b)"
commutative_squares_6: shows "C.commutative_square \<mu>.leg0 (\<nu>\<pi>.leg1 \<cdot> \<nu>\<pi>_\<rho>.prj\<^sub>1) \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>1\<^sub>1 \<langle>\<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0\<^sub>1 \<lbrakk>\<nu>\<pi>.leg0, \<rho>.leg1\<rbrakk> \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0\<rangle>"
AGM_rel: fixes C :: "'a set"   and K :: "'a set"   and B :: "'a set" shows "AGM_relation C K B \<equiv> C = K \<and> B = K \<or> (\<exists>\<phi>. K \<turnstile> \<phi> \<and> C \<in> K .\<bottom>. \<phi>) \<and> (\<exists>\<phi>. K \<turnstile> \<phi> \<and> B \<in> K .\<bottom>. \<phi> \<and> K \<div> \<phi> \<subseteq> B) \<and> (\<forall>\<phi>. K \<turnstile> \<phi> \<and> C \<in> K .\<bottom>. \<phi> \<and> B \<in> K .\<bottom>. \<phi> \<and> K \<div> \<phi> \<subseteq> C \<longrightarrow> K \<div> \<phi> \<subseteq> B)"
map_indets_id: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and f :: "'b \<Rightarrow> 'b" assumes "\<And>x. x \<in> indets p \<Longrightarrow> f x = x" shows "map_indets f p = p"
conga_diff45: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'" shows "A' \<noteq> B'"
cil_\<alpha>n_cons1: fixes w :: "('a, 'b) label list list"   and wa :: "('a, 'b) label list list"   and fg :: "('d, 'a, 'e, 'c, 'f) flowgraph_rec_scheme"   and wb :: "('a, 'b) label list list"   and e :: "('a, 'b) label list" assumes "w \<in> wa \<otimes>\<^bsub>\<alpha>n fg\<^esub> wb"   and "fst (\<alpha>n fg e) \<inter> mon_ww fg wb = {}" shows "e # w \<in> e # wa \<otimes>\<^bsub>\<alpha>n fg\<^esub> wb"
Rat_induct_pos: fixes P :: "rat \<Rightarrow> bool"   and q :: "rat" assumes "\<And>a b. 0 < b \<Longrightarrow> P (Rat.Fract a b)" shows "P q"
fls_lr_inverse_const_1: fixes a :: "'a"   and x :: "'a" shows "fls_left_inverse (fls_const a) x = fls_const x"
simps_2: fixes y :: "'a option" shows "(None \<sqsubseteq> y) = True"
neg_def: fixes \<phi> :: "'a fm" shows "neg \<phi> = (if \<phi> = TrueF then FalseF else if \<phi> = FalseF then TrueF else Neg \<phi>)"
simps_2: fixes n :: "'a"   and l :: "'a tree"   and r :: "'a tree"   and h :: "nat" shows "is_ord (MKT n l r h) = ((\<forall>n'\<in>set_of l. n' < n) \<and> (\<forall>n'\<in>set_of r. n < n') \<and> is_ord l \<and> is_ord r)"
length_mtf: fixes x :: "'b"   and xs :: "'b list" shows "length (mtf x xs) = length xs"
ennreal_less_top: fixes x :: "real" shows "ennreal x < top_class.top"