hcju/isabelleps huggingface dataset:
lincomb_list_index: fixes i :: "nat"   and Xs :: "'a vec list"   and c :: "nat \<Rightarrow> 'a" assumes "i < nr"   and "set Xs \<subseteq> carrier_vec nr" shows "nr.lincomb_list c Xs $ i = (\<Sum>j = 0..<length Xs. c j * Xs ! j $ i)"
assign\<^sub>2: fixes x :: "'Var"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and e :: "'AExp"   and t :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and P :: "'BExp set" assumes "x \<in> dom \<Gamma>"   and "\<Gamma> \<turnstile>\<^sub>a e \<in> t"   and "type_stable \<S> t"   and "P' = assign_post P x e |` {v. stable \<S> v}"   and "x \<notin> snd \<S> \<longrightarrow> t \<le>:\<^sub>P' dma_type x" shows "\<turnstile> \<Gamma>,\<S>,P {x \<leftarrow> e} \<Gamma>(x \<mapsto> t),\<S>,P'"
delete_point_index_zero: fixes G :: "'a set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "G \<in> {g - {x} |g. g \<in> \<G> \<and> g \<noteq> {x}}"   and "y \<in> G"   and "z \<in> G"   and "z \<noteq> y" shows "del_point_blocks x index {y, z} = 0"
size_mset_def: fixes M :: "('c \<times> 'd) multiset" shows "size_mset M = fold_mset ((+) \<circ> pair_size) 0 M"
round_robin_Some_SomeD: fixes I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set"   and n0 :: "nat"   and \<sigma> :: "'t list \<times> nat"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and \<sigma>' :: "'t list \<times> nat" assumes "\<alpha>.deterministic I"   and "\<alpha>.round_robin n0 \<sigma> s = \<lfloor>(t, \<lfloor>(ta, x', m')\<rfloor>, \<sigma>')\<rfloor>"   and "s \<in> I" shows "\<exists>x. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> \<alpha>.r_syntax t x (shr s) ta x' m' \<and> \<alpha>.actions_ok s t ta"
hop12: shows "\<bar>spmf (map_spmf fst (sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> game\<^sub>2 x))) True - spmf (map_spmf fst (sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> game\<^sub>1 x))) True\<bar> \<le> spmf (map_spmf snd (sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> game\<^sub>2 x))) True"
fps_eq_iff: fixes p :: "'b fps"   and q :: "'b fps" shows "(p = q) = (\<forall>n. p $ n = q $ n)"
map_in_bicategory_axioms: shows "map_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f"
prod_def: fixes f :: "'a"   and g :: "'a" shows "local.prod f g \<equiv> tuple (f \<cdot> pr1 (local.dom f) (local.dom g)) (g \<cdot> pr0 (local.dom f) (local.dom g))"
delete_idem: fixes k :: "'c"   and al :: "('c \<times> 'd) list" shows "delete k (delete k al) = delete k al"
Rats_diff: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<rat>"   and "b \<in> \<rat>" shows "a - b \<in> \<rat>"
strict_trans1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "b < c" shows "a < c"
insert_def: fixes x :: "'g"   and xs :: "'g list" shows "List.insert x xs = (if x \<in> set xs then xs else x # xs)"
kruskal_ref_spec: shows "(kruskal getEdges_impl (\<lambda>(u, w, v). return (u, v)), uncurry0 (SPEC (\<lambda>r. distinct r \<and> set r \<subseteq> csuper_E \<and> s.MSF (\<alpha> ` set r)))) \<in> unit_assn\<^sup>k \<rightarrow>\<^sub>a id_assn"
sign_simps_2: fixes af :: "'d"   and bf :: "'d" shows "af + bf = bf + af"
substNext_def: fixes g :: "'g" shows "substNext g \<equiv> \<lambda>v. if v = chooseNext g then substitution g else v"
swapInp_def2: fixes xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var"   and inp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option" shows "inp %[x1 \<and> x2]_xs = asInp (qSwapInp xs x1 x2 (pickInp inp))"
handle2: fixes env :: "v sem_env"   and s1 :: "'c state"   and e :: "exp0"   and s2 :: "'c state"   and v1 :: "v"   and pes :: "(pat \<times> exp0) list"   and e' :: "exp0"   and env' :: "(char list \<times> v) list"   and bv :: "'c state \<times> (v, v) result" assumes "Big_Step_Unclocked.evaluate env s1 e (s2, Rerr (Rraise v1))"   and "match_result env s2 v1 pes v1 = Rval (e', env')"   and "Big_Step_Unclocked.evaluate (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) s2 e' bv" shows "Big_Step_Unclocked.evaluate env s1 (Handle e pes) bv"
eventually_nhds_in_open: fixes "open" :: "'a set \<Rightarrow> bool"   and s :: "'a set"   and x :: "'a" assumes "class.topological_space open"   and "open s"   and "x \<in> s" shows "\<forall>\<^sub>F y in topological_space.nhds open x. y \<in> s"
std_Red_F_eq: shows "std.Red_F = Red_F_\<G>_empty"
Liminf_const: fixes F :: "'a filter"   and c :: "'b" assumes "F \<noteq> \<bottom>" shows "Liminf F (\<lambda>x. c) = c"
converse_relcomp: fixes r :: "('f \<times> 'g) set"   and s :: "('g \<times> 'e) set" shows "(r O s)\<inverse> = s\<inverse> O r\<inverse>"
Collect_mem_eq: fixes A :: "'c set" shows "{x. x \<in> A} = A"
simps_1: fixes uu :: "'b list" shows "remdups_rev_code uu [] = []"
rep_eq: fixes x :: "nat \<Rightarrow> 'b"   and xa :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "insertion_aux x xa = insertion_fun x (lookup xa)"
case_1: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and x1 :: "'c" shows "(case Inl x1 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f1 x1"
env_restr_belowD: fixes S :: "'d set"   and m\<^sub>1 :: "'d \<Rightarrow> 'e"   and m\<^sub>2 :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "m\<^sub>1 f|` S \<sqsubseteq> m\<^sub>2 f|` S"   and "x \<in> S" shows "m\<^sub>1 x \<sqsubseteq> m\<^sub>2 x"
br: shows "class.bezout_ring (*) (1::'a) (+) (0::'a) (-) uminus"
consistent_drop_single: fixes A :: "'b set"   and p :: "('c, 'b) fm"   and ps :: "('c, 'b) fm list"   and a :: "'b"   and S :: "(('c, 'b) fm list \<times> 'b) set" assumes "infinite UNIV"   and "finite A"   and "consistent A ({(p # ps, a)} \<union> S)" shows "consistent A ({(ps, a)} \<union> S)"
matrix_inv_unique: fixes A :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'b) vec" assumes "A ** B = mat (1::'a)"   and "B ** A = mat (1::'a)" shows "matrix_inv A = B"
ivl_disj_int_one_3: fixes l :: "'a"   and u :: "'a" shows "{..l} \<inter> {l<..u} = {}"
stateful_strand_step_subst_inv_cases_5: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term"   and S :: "('a, 'b) stateful_strand_step list"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "delete\<langle>t,s\<rangle> \<in> set (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<sigma>)" shows "\<exists>t' s'. t = t' \<cdot> \<sigma> \<and> s = s' \<cdot> \<sigma> \<and> delete\<langle>t',s'\<rangle> \<in> set S"
ennexp_le1: fixes x :: "ereal" shows "(ennexp x \<le> 1) = (x \<le> 0)"
Gcd_mult: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and c :: "'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Gcd (times c ` A) = normalize (times c (Gcd A))"
Fun: fixes f :: "('b, 'c, 'd) prot_fun"   and T :: "(('b, 'c, 'd) prot_fun, 'e) Term.term list"   and S :: "(('b, 'c, 'd) prot_fun, 'e) Term.term list"   and A :: "'d set set" assumes "\<not> is_Abs f"   and "length T = length S"   and "\<And>i. i < length T \<Longrightarrow> timpls_transformable_to_pred A (T ! i) (S ! i)" shows "timpls_transformable_to_pred A (Fun f T) (Fun f S)"
lborelD: fixes A :: "'b set" assumes "A \<in> sets borel" shows "A \<in> sets lborel"
IN_g: fixes x :: "'v vertex" shows "d_IN g x \<le> value_flow \<Psi> g"
bounded1: fixes fa :: "'b \<Rightarrow> real"   and M' :: "'b measure" assumes "fa \<in> borel_measurable N"   and "\<And>x. x \<in> space N \<Longrightarrow> fa x \<le> B"   and "M' \<in> space M"   and "emeasure M' (space M') \<le> ennreal B'" shows "\<integral>\<^sup>+ x. ennreal (fa x) \<partial>M' \<le> ennreal (B * \<bar>B'\<bar>)"
assoc_naturality: fixes \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" assumes "arr \<mu>"   and "arr \<nu>"   and "arr \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<a>[cod \<mu>, cod \<nu>, cod \<tau>] \<cdot> ((\<mu> \<star> \<nu>) \<star> \<tau>) = (\<mu> \<star> \<nu> \<star> \<tau>) \<cdot> \<a>[local.dom \<mu>, local.dom \<nu>, local.dom \<tau>]"
removen_nth: fixes n :: "nat"   and xs :: "'b list"   and m :: "nat" shows "removen n xs ! m = (if length xs \<le> n then xs ! m else if m < n then xs ! m else if m \<le> length xs then xs ! Suc m else [] ! (m - (length xs - 1)))"
continuous_on_Borsuk_map: fixes a :: "'a"   and s :: "'a set" assumes "a \<notin> s" shows "continuous_on s (\<lambda>x. (x - a) /\<^sub>R norm (x - a))"
ldrop_enat_inf_llist: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "ldrop (enat n) (inf_llist f) = inf_llist (\<lambda>m. f (m + n))"
preferred_alts_altdef: fixes carrier :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "Order_Predicates.preorder_on carrier le" shows "preferred_alts le x = {y \<in> carrier. le x y}"
split_indicator_asm: fixes P :: "'d \<Rightarrow> bool"   and S :: "'e set"   and x :: "'e" shows "P (indicator S x) = (\<not> (x \<in> S \<and> \<not> P (1::'d) \<or> x \<notin> S \<and> \<not> P (0::'d)))"
m_div_def: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" shows "x \<oslash>\<^bsub>G\<^esub> y = (if y = \<zero>\<^bsub>G\<^esub> then \<zero>\<^bsub>G\<^esub> else x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> y)"
inject_5: fixes x51 :: "term"   and x52 :: "term"   and y51 :: "term"   and y52 :: "term" shows "(x51 $ x52 = y51 $ y52) = (x51 = y51 \<and> x52 = y52)"
m_ivl_height: fixes iv :: "ivl" shows "m_ivl iv \<le> 3"
lookup_default: fixes x :: "'c"   and d :: "'b"   and m :: "('c, 'b) mapping" shows "lookup (Mapping.default x d m) x = Some (lookup_default d m x)"
proj_image: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "Equiv_Relations.proj r ` A = A // r"
edges_ins_edge': fixes u :: "'a"   and v :: "'a"   and g :: "'a ugraph" assumes "u \<noteq> v" shows "edges (ins_edge (u, v) g) = {(u, v), (v, u)} \<union> edges g"
it_paths_ConsE: fixes i :: "nat"   and "is" :: "nat list"   and t :: "'a rose_tree" assumes "i # is \<in> it_paths t" obtains "i < length (children t)" and "is \<in> it_paths (children t ! i)"
straighten_downtop_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "straighten_downtop w1 w2" shows "kauff_mat w1 = kauff_mat w2"
eq_iff_6: fixes "term" :: "Syntax.term"   and terma :: "Syntax.term" shows "(Inj2 term = Inj2 terma) = (term = terma)"
ide_char: fixes f :: "'a \<times> 'a" shows "B.VxV.ide f = (B.ide (fst f) \<and> B.ide (snd f))"
x: shows "x = insertion e (map_poly real_of_float p1) + xe"
intros: fixes x :: "'x2"   and m :: "'m2"   and ta :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and x' :: "'x2"   and m' :: "'m2" assumes "\<tau>move2 (x, m) ta (x', m')" shows "r2.init_fin_\<tau>move ((Running, x), m) (convert_TA_initial (convert_obs_initial ta)) ((Running, x'), m')"
convergent_powser_cos_series_stream: shows "convergent_powser (msllist_of_msstream cos_series_stream)"
alpha_trans: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X'' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X \<or> qGood X' \<or> qGood X''"   and "X #= X'"   and "X' #= X''" shows "X #= X''"
inverse_arrowsE: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "category C"   and "category.inverse_arrows C f g" obtains "partial_magma.ide C (C g f)" and "partial_magma.ide C (C f g)"
\<Sigma>_protocols_to_commitments_axioms_def: fixes init :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'e) spmf"   and response :: "'d \<Rightarrow> 'c \<Rightarrow> 'f \<Rightarrow> 'g spmf"   and check :: "'b \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> bool"   and Rel :: "('b \<times> 'c) set"   and S_raw :: "'b \<Rightarrow> 'f \<Rightarrow> ('e \<times> 'g) spmf"   and \<A>ss :: "'b \<Rightarrow> 'e \<times> 'f \<times> 'g \<Rightarrow> 'e \<times> 'f \<times> 'g \<Rightarrow> 'c spmf"   and challenge_space :: "'f set"   and valid_pub :: "'b set"   and G :: "('b \<times> 'c) spmf" shows "\<Sigma>_protocols_to_commitments_axioms init response check Rel S_raw \<A>ss challenge_space valid_pub G \<equiv> (\<Sigma>_protocols_base.\<Sigma>_protocol init response check Rel S_raw \<A>ss challenge_space valid_pub \<and> (\<forall>h w. (h, w) \<in> set_spmf G \<longrightarrow> (h, w) \<in> Rel)) \<and> lossless_spmf G \<and> (\<forall>h w. lossless_spmf (init h w)) \<and> (\<forall>r w e. lossless_spmf (response r w e))"
unlift_b_e: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and b_es :: "b_e list"   and tf :: "tf" assumes "\<S>\<bullet>\<C> \<turnstile> $* b_es : tf" shows "\<C> \<turnstile> b_es : tf"
fv_snd_rcv_strand_subset_2: fixes S :: "('d, 'c) strand_step list" shows "\<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v S)) \<subseteq> fv\<^sub>s\<^sub>t S"
run_stake: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a stream"   and p :: "'b"   and k :: "nat" assumes "transition_system.run execute enabled r p" shows "transition_system.path execute enabled (stake k r) p"
1: shows "(s \<star> w.the_\<theta>) \<cdot> \<a>[s, s\<^sub>0, w] \<cdot> (\<sigma> \<star> w) = ((s \<star> w'.the_\<theta>) \<cdot> \<a>[s, s\<^sub>0, w'] \<cdot> (\<sigma> \<star> w')) \<cdot> w'.the_\<nu> \<cdot> local.inv w.the_\<nu>"
sup_id_star1: fixes x :: "'a ppath set" assumes "pp_one \<subseteq> x" shows "pp_prod x (pp_star x) = pp_star x"
poly_diff: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" shows "poly (p - q) x = poly p x - poly q x"
non_empty_cycle_free: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and v :: "'b \<Rightarrow> nat"   and n :: "nat" assumes "[M]\<^bsub>v,n\<^esub> \<noteq> {}"   and "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)" shows "cycle_free M n"
respection: shows "rel_respects_binary_pred (indRelLSTPO SRel TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
assert_def: fixes p :: "'a"   and q :: "'a" shows "{. p .} q \<equiv> p \<sqinter> q"
mono_step': fixes S :: "ivl Abs_State.st option"   and S' :: "ivl Abs_State.st option"   and c :: "ivl Abs_State.st option acom"   and c' :: "ivl Abs_State.st option acom" assumes "S \<sqsubseteq> S'"   and "c \<sqsubseteq> c'" shows "step_ivl S c \<sqsubseteq> step_ivl S' c'"
du: shows "(f (real_of_float l) - f (real_of_float u) + real_of_float (a * (u - l))) / 2 + be \<in>\<^sub>r divl"
image_cong: fixes M :: "'d set"   and N :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "M = N"   and "\<And>x. x \<in> N \<Longrightarrow> f x = g x" shows "f ` M = g ` N"
cont2contlubE: fixes f :: "'a \<Rightarrow> 'b"   and Y :: "nat \<Rightarrow> 'a" assumes "cont f"   and "chain Y" shows "f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i))"
productive_imp_not_interp: fixes N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "production N C \<noteq> {}" shows "\<not> interp N C \<Turnstile> C"
signed_ucast_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and w :: "'d word" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_1.signed one plus zero minus (ucast w) = ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not (LENGTH('c) - Suc 0) (semiring_1.unsigned one plus zero w)"
component_in_hom: fixes j :: "'b" assumes "J0.arr j" shows "\<guillemotleft>\<pi>0 j : a0 \<rightarrow> D0 (J0.cod j)\<guillemotright>"
c_def: shows "c = integral {1..} (\<lambda>x. - P x) + g 1"
eval_finalsId: fixes es :: "char list exp list"   and P :: "(char list list \<times> char list exp) prog"   and s :: "heap \<times> sfields \<times> sheap" assumes "finals es" shows "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es,s\<rangle>"
mor_cong: fixes f' :: "'d \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e"   and g' :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'g"   and B1 :: "'d set"   and B2 :: "'f set"   and s1 :: "'d \<Rightarrow> ('h, 'd, 'f) F1"   and s2 :: "'f \<Rightarrow> ('h, 'd, 'f) F2"   and B1' :: "'e set"   and B2' :: "'g set"   and s1' :: "'e \<Rightarrow> ('h, 'e, 'g) F1"   and s2' :: "'g \<Rightarrow> ('h, 'e, 'g) F2" assumes "f' = f"   and "g' = g"   and "mor B1 B2 s1 s2 B1' B2' s1' s2' f g" shows "mor B1 B2 s1 s2 B1' B2' s1' s2' f' g'"
section_def: fixes f :: "'a" shows "S.section f \<equiv> \<exists>g. S.ide (g \<cdot>\<^sub>S f)"
strict_monoidal_functor_axioms: shows "strict_monoidal_functor (\<cdot>) \<F>C.T\<^sub>F\<^sub>M\<^sub>C \<F>C.\<alpha>\<^sub>F\<^sub>M\<^sub>C \<F>C.\<iota>\<^sub>F\<^sub>M\<^sub>C (\<cdot>) T\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<alpha>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<iota>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<F>C.D"
assms_1: shows "\<Z>_sgrp \<alpha> \<SS>"
apply_cltn2_abs_in_S: fixes v :: "(real, 3) vec"   and C :: "cltn2" assumes "v \<noteq> 0" shows "(apply_cltn2 (proj2_abs v) C \<in> S) = (v \<bullet> (cltn2_rep C ** M ** Finite_Cartesian_Product.transpose (cltn2_rep C) *v v) = 0)"
witness_tensor: fixes "is" :: "nat list" assumes "is \<lhd> dims Aw" shows "lookup Aw is = (if nths is (Collect even) = nths is {n. odd n} \<and> (\<forall>i\<in>set is. i < last (butlast rs)) then 1 else 0)"
has_integral_set_lebesgue: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "f absolutely_integrable_on S" shows "(f has_integral set_lebesgue_integral lebesgue S f) S"
Equiv_def: fixes P :: "fml"   and Q :: "fml" shows "(P \<leftrightarrow> Q) = (P && Q || ! P && ! Q)"
NotDmd: fixes F :: "'a \<Rightarrow> bool" shows "\<turnstile> (\<not> \<diamond>F) = (\<box>\<not> F)"
substr_simp: fixes s :: "'a list"   and l :: "nat"   and u :: "nat" shows "substr s l u = nths s {k. l < Suc k \<and> Suc k < u}"
rep_eq: shows "comp_fun_idem_apply inf_cfi = inf"
simps_5: fixes i :: "String.literal"   and p :: "fol fm"   and q :: "fol fm" shows "free_in i (Iff p q) = (free_in i p \<or> free_in i q)"
intro: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FinGroupRepresentation G smult V"   and "IrrFinGroupRepresentation_axioms G smult V" shows "IrrFinGroupRepresentation G smult V"
prems: fixes x :: "nat" assumes "x < count_weights True (remove_weights (Pool m1 m2))" shows "f x = extract_weights True (Pool m1 m2) x"
use_exp_def: shows "use_exp \<equiv> (''G_par_1''[] ::= ''n'';; exp_count_up;; ArrayCpy_retv ''n'' ''G_ret_1'');; ''G_par_1''[] ::= ''n'';; exp_count_up;; ArrayCpy_retv ''n'' ''G_ret_1''"
sams2_suma2__conga123: 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 D :: "'a"   and E :: "'a"   and F :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SAMS Bet Cong A B C D E F"   and "Tarski_neutral_dimensionless.SAMS Bet Cong A' B' C' D E F"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I"   and "Tarski_neutral_dimensionless.SumA Bet Cong A' B' C' D E F G H I" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"
divide_strict_right_mono: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a b"   and "less zero c" shows "less (divide a c) (divide b c)"
ac: shows "cbox a c = cbox a b \<union> cbox b c"
Map_comp: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and g :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "arr f"   and "arr g"   and "Dom g = Cod f" shows "Map (g \<cdot> f) = vertical_composite.map (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Map f) (Map g)"
lower_split_float_interval2: fixes X :: "float interval" shows "lower (snd (split_float_interval X)) = mid X"
prime_card_elements_with_ord_eq_totient: fixes n :: "nat"   and d :: "nat" assumes "1 < n"   and "prime n" shows "card {x \<in> totatives n. ord n x = d} = (if d dvd n - 1 then totient d else 0)"
induct: fixes x1 :: "db"   and x2 :: "db"   and P :: "db \<Rightarrow> db \<Rightarrow> bool" assumes "x1 \<Down> x2"   and "\<And>Pa M Q V. \<lbrakk>Pa \<Down> DBAbsN M; P Pa (DBAbsN M); M<Q/0> \<Down> V; P (M<Q/0>) V\<rbrakk> \<Longrightarrow> P (DBApp Pa Q) V"   and "\<And>Pa M Q q V. \<lbrakk>Pa \<Down> DBAbsV M; P Pa (DBAbsV M); Q \<Down> q; P Q q; M<q/0> \<Down> V; P (M<q/0>) V\<rbrakk> \<Longrightarrow> P (DBApp Pa Q) V"   and "\<And>e. val (DBAbsN e) \<Longrightarrow> P (DBAbsN e) (DBAbsN e)"   and "\<And>e. val (DBAbsV e) \<Longrightarrow> P (DBAbsV e) (DBAbsV e)"   and "\<And>Pa V. \<lbrakk>Pa<DBFix Pa/0> \<Down> V; P (Pa<DBFix Pa/0>) V\<rbrakk> \<Longrightarrow> P (DBFix Pa) V"   and "P DBtt DBtt"   and "P DBff DBff"   and "\<And>C T V E. \<lbrakk>C \<Down> DBtt; P C DBtt; T \<Down> V; P T V\<rbrakk> \<Longrightarrow> P (DBCond C T E) V"   and "\<And>C E V T. \<lbrakk>C \<Down> DBff; P C DBff; E \<Down> V; P E V\<rbrakk> \<Longrightarrow> P (DBCond C T E) V"   and "\<And>n. P (DBNum n) (DBNum n)"   and "\<And>Pa n. \<lbrakk>Pa \<Down> DBNum n; P Pa (DBNum n)\<rbrakk> \<Longrightarrow> P (DBSucc Pa) (DBNum (Suc n))"   and "\<And>Pa n. \<lbrakk>Pa \<Down> DBNum (Suc n); P Pa (DBNum (Suc n))\<rbrakk> \<Longrightarrow> P (DBPred Pa) (DBNum n)"   and "\<And>E. \<lbrakk>E \<Down> DBNum 0; P E (DBNum 0)\<rbrakk> \<Longrightarrow> P (DBIsZero E) DBtt"   and "\<And>E n. \<lbrakk>E \<Down> DBNum n; P E (DBNum n); 0 < n\<rbrakk> \<Longrightarrow> P (DBIsZero E) DBff" shows "P x1 x2"
object_ptr_kinds_eq_h: shows "object_ptr_kinds h2 = object_ptr_kinds h |\<union>| {|cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr|}"