hcju/isabelleps huggingface dataset:
lunit_in_vhom: fixes f :: "'a"   and b :: "'a" assumes "ide f"   and "trg f = b" shows "\<guillemotleft>\<l>[f] : b \<star> f \<Rightarrow> f\<guillemotright>"
abd_acd_abcdacbd: fixes a :: "'a"   and b :: "'a"   and d :: "'a"   and c :: "'a" assumes "[[a b d]]"   and "[[a c d]]"   and "b \<noteq> c" shows "[[a b c d]] \<or> [[a c b d]]"
iprev_nth_closed: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "I \<noteq> {}" shows "I \<leftarrow> n \<in> I"
measurable_dens: shows "\<delta> \<in> borel_measurable (state_measure (V \<union> V') \<Gamma>)"
minus_set_def: fixes A :: "'b set"   and B :: "'b set" shows "A - B = Collect ((\<lambda>x. x \<in> A) - (\<lambda>x. x \<in> B))"
degree_max_aform: fixes x :: "(real \<times> real pdevs) \<times> real"   and d :: "nat"   and y :: "(real \<times> real pdevs) \<times> real"   and p :: "nat" assumes "degree_aform_err x \<le> d"   and "degree_aform_err y \<le> d" shows "degree_aform_err (max_aform_err p x y) \<le> d"
poly_all_0_iff_0: fixes p :: "'a poly" shows "(\<forall>x. poly p x = (0::'a)) = (p = 0)"
atLeast1_atMost_eq: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "sum g {Suc 0..n} = (\<Sum>k<n. g (Suc k))"
cmg: shows "continuous_map (subtopology (Euclidean_space n) T) (subtopology (Euclidean_space m) S) g"
itrm_brackets_opaque: fixes extra_rule :: "nat itrm \<Rightarrow> nat itrm \<Rightarrow> bool"   and itrm_bracket :: "nat \<Rightarrow> nat itrm \<Rightarrow> nat itrm option"   and vs :: "nat list"   and x :: "nat itrm"   and x' :: "nat itrm" assumes "itrm_abstraction extra_rule itrm_bracket"   and "itrm_abstraction.itrm_brackets itrm_bracket vs x = Some x'" shows "set (opaque x') = set (opaque x) - set vs"
intros_1: fixes G :: "prog"   and T :: "ty" shows "G\<turnstile>T\<preceq>T"
assms_3: fixes T' :: "'a set" assumes "T' \<subseteq> S"   and "open T'" shows "T' \<subseteq> T"
path: shows "is_path (\<pi> @\<^bsup>n\<^esup> \<pi>')"
union_diff2: fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "finite A"   and "finite B" shows "sum g (A \<union> B) = sum g (A - B) + sum g (B - A) + sum g (A \<inter> B)"
closure_least: fixes s :: "'a set" assumes "s closed" shows "closure s \<subseteq> s"
point_indices_elem_in: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and ps :: "'b set"   and t :: "nat" assumes "incidence_system point_set block_collection"   and "ps \<subseteq> point_set"   and "card ps = t" shows "int (block_collection index ps) \<in> incidence_system.point_indices point_set block_collection (int t)"
A: shows "Z \<in> ipurge_fail_aux_t_set Y"
nn_integral_fst: fixes f :: "'b \<times> (nat \<Rightarrow> 'a) \<Rightarrow> ennreal"   and M1 :: "'b measure" assumes "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M Pi\<^sub>M UNIV (\<lambda>i. M))" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>Pi\<^sub>M UNIV (\<lambda>i. M) \<partial>M1 = integral\<^sup>N (M1 \<Otimes>\<^sub>M Pi\<^sub>M UNIV (\<lambda>i. M)) f"
Res_ok_def: shows "Res_ok = (\<lambda>Z s. even Z = (s<Res> = 0))"
natural_extension_map_def: shows "J.natural_extension_map = int_left_shift"
gmnl_gmft: fixes nl :: "('a, 'b) Node list" assumes "\<forall>x\<in>set nl. FingerTreeStruc.is_measured_node x" shows "FingerTreeStruc.gmft (FingerTreeStruc.nlistToTree nl) = FingerTreeStruc.gmnl nl"
hext_def: fixes h :: "heap"   and h' :: "heap" shows "h \<unlhd> h' \<equiv> \<forall>a C fs. h a = \<lfloor>(C, fs)\<rfloor> \<longrightarrow> (\<exists>fs'. h' a = \<lfloor>(C, fs')\<rfloor>)"
r_2: fixes n :: "nat" shows "\<forall>m\<ge>r n. \<forall>p\<ge>r n. Norm N (v m - v p) < c n"
r: shows "C\<cdot>r = demand (\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<N>\<lbrace>\<Gamma>\<rbrace>\<^esub>)"
sup_commute: fixes sup :: "'l \<Rightarrow> 'l \<Rightarrow> 'l"   and less_eq :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and less :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "'l"   and y :: "'l" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
these_insert_Some: fixes x :: "'d"   and A :: "\<langle>'d\<rangle>\<^sub>\<bottom> set" shows "Option.these (insert \<lfloor>x\<rfloor> A) = insert x (Option.these A)"
summable_LIMSEQ_zero: assumes "summable f" shows "f \<longlonglongrightarrow> (0::'a)"
wp3tl: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "wellformed_policy3Pr p" shows "wellformed_policy3Pr (tl p)"
ide_is_iso: fixes a :: "'d" assumes "Dop.ide a" shows "Dop.iso a"
set_spmf_I_e_d_2: fixes N :: "nat"   and e :: "nat"   and d :: "nat" assumes "((N, e), d) \<in> set_spmf I" shows "1 < d"
join_sym: fixes s :: "'b"   and t :: "'b"   and A :: "('b \<times> 'b) set" assumes "(s, t) \<in> A\<^sup>\<down>" shows "(t, s) \<in> A\<^sup>\<down>"
invar_update: fixes m :: "'t"   and a :: "'a"   and b :: "'b" assumes "invar m" shows "invar (update a b m)"
embed_compat: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" assumes "embed r r' f" shows "compat r r' f"
rm_vars_img_subset: fixes V :: "'d set"   and s :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "subst_range (rm_vars V s) \<subseteq> subst_range s"
rel_fun_iff_leq_vimage2p: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "rel_fun R S f g = (R \<le> BNF_Def.vimage2p f g S)"
homeomorphic_imp_open_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" assumes "homeomorphic_map X Y f" shows "open_map X Y f"
homotopic_with_eq: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and g' :: "'b \<Rightarrow> 'c" assumes "homotopic_with P X Y f g"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f' x = f x"   and "\<And>x. x \<in> topspace X \<Longrightarrow> g' x = g x"   and "\<And>h k. (\<And>x. x \<in> topspace X \<Longrightarrow> h x = k x) \<Longrightarrow> P h = P k" shows "homotopic_with P X Y f' g'"
wf_rderivs: fixes \<Sigma> :: "nat \<Rightarrow> 'c set"   and project :: "'c \<Rightarrow> 'c"   and embed :: "'c \<Rightarrow> 'c list"   and wf_atom :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and n :: "nat"   and r :: "'d rexp"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and ws :: "'c list" assumes "Pi_Derivatives.embed \<Sigma> project embed"   and "alphabet.wf wf_atom n r" shows "alphabet.wf wf_atom n (embed.rderivs lookup embed ws r)"
shadow_root_ptr_kinds_eq: shows "shadow_root_ptr_kinds h' = shadow_root_ptr_kinds h"
pairs_uminus_galois: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c"   and y :: "'d Filters.filter"   and z :: "'c"   and w :: "'d Filters.filter" assumes "triple phi"   and "(x, y) \<in> triple.pairs phi"   and "(z, w) \<in> triple.pairs phi" shows "(triple.pairs_inf (x, y) (z, w) = (bot, Filters.filter.Abs_filter UNIV)) = triple.pairs_less_eq (x, y) (triple.pairs_uminus phi (z, w))"
prems_9: shows "InvariantConflictFlagCharacterization (getConflictFlag state') (getF state') (getM state')"
least_pr_weak'_trans: fixes w :: "'b \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and pr_weak :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and least :: "'b \<Rightarrow> bool"   and scf :: "'b \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'b"   and g :: "'b \<times> nat" assumes "admissible_kbo w w0 pr_strict pr_weak least scf"   and "least f"   and "admissible_kbo.pr_weak' pr_weak least (f, 0) g" shows "least (fst g) \<and> snd g = 0"
var_comp_1: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'c) Term.term" shows "\<sigma> \<circ>\<^sub>s Var = \<sigma>"
step: fixes p :: "('b, 'c, 'a, 'd) seqp"   and pn :: "'a"   and l :: "'d"   and \<xi> :: "'b"   and a :: "'c seq_action"   and \<xi>' :: "'b"   and q :: "('b, 'c, 'a, 'd) seqp"   and l' :: "'d"   and pp :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>"   and "p \<in> ctermsl (\<Gamma> pn)"   and "not_call p"   and "l \<in> labels \<Gamma> p"   and "P (\<xi>, l)"   and "((\<xi>, p), a, \<xi>', q) \<in> seqp_sos \<Gamma>"   and "((\<xi>, p), a, \<xi>', q) \<in> automaton.trans A"   and "l' \<in> labels \<Gamma> q"   and "(\<xi>, pp) \<in> reachable A I"   and "p \<in> sterms \<Gamma> pp"   and "(\<xi>', q) \<in> reachable A I"   and "I a" shows "P (\<xi>', l')"
neutral: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "\<forall>x\<in>A. g x = (1::'e)" shows "prod g A = (1::'e)"
tokens_nth_in_\<Z>: fixes p :: "(symbol \<times> character list) list" assumes "p \<in> \<PP>" shows "\<forall>i<length p. \<exists>u. p ! i \<in> \<Z> (charslength (take i p)) u"
divide_inf_left: fixes z :: "complex_homo" assumes "z \<noteq> \<infinity>\<^sub>h" shows "\<infinity>\<^sub>h :\<^sub>h z = \<infinity>\<^sub>h"
map_add_SomeD: fixes m :: "'d \<Rightarrow> 'c option"   and n :: "'d \<Rightarrow> 'c option"   and k :: "'d"   and x :: "'c" assumes "(m ++ n) k = Some x" shows "n k = Some x \<or> n k = None \<and> m k = Some x"
vifunion_vsingleton_is: fixes i :: "V"   and f :: "V \<Rightarrow> V" shows "\<Union>\<^sub>\<circ> (VLambda (ZFC_in_HOL.set {i}) f `\<^sub>\<circ> ZFC_in_HOL.set {i}) = f i"
index_of_const_inj: fixes n1 :: "nat"   and n2 :: "nat" assumes "index_of_const n1 = index_of_const n2" shows "n1 = n2"
assms_4: shows "x' \<in> set_pmf \<mu>"
loc_reach\<^sub>C: shows "\<langle>c\<^sub>C', mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C \<in> conc.loc_reach \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C"
permutations_of_multiset_def: fixes A :: "'b multiset" shows "permutations_of_multiset A = {xs. mset xs = A}"
succs_of_clause_split: fixes P :: "(lit set \<Rightarrow> lit set) \<Rightarrow> bool"   and l :: "lit"   and x :: "lit uprod" shows "P (succs_of_clause l x) = (\<forall>l1 l2. x = Upair l1 l2 \<longrightarrow> P (succs_of_clause l (Upair l1 l2)))"
Act_eq_iff_perm: fixes f1 :: "'d"   and \<alpha>1 :: "'c"   and x1 :: "('a, 'b, 'c, 'd) FL_Formula.formula"   and f2 :: "'d"   and \<alpha>2 :: "'c"   and x2 :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "(FL_Formula.Act f1 \<alpha>1 x1 = FL_Formula.Act f2 \<alpha>2 x2) = (f1 = f2 \<and> (\<exists>p. supp x1 - bn \<alpha>1 = supp x2 - bn \<alpha>2 \<and> (supp x1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> x1 = x2 \<and> supp \<alpha>1 - bn \<alpha>1 = supp \<alpha>2 - bn \<alpha>2 \<and> (supp \<alpha>1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> \<alpha>1 = \<alpha>2))"
p'_def: fixes t :: "real" shows "ego.p' t = v\<^sub>e + a\<^sub>e * t"
exp_times_self_eqD_2: fixes w :: "real"   and x :: "real" assumes "w * exp w = x" shows "w = Lambert_W x \<or> x < 0 \<and> w = Lambert_W' x"
ncols_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "ncols A = CARD('e)"
steps_z_beta_sound: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>l', Z'\<rangle>"   and "\<forall>c\<in>clk_set A. v c \<le> n"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "Z \<in> V'"   and "Z' \<noteq> {}" shows "\<exists>Z''. A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l', Z''\<rangle> \<and> Z'' \<noteq> {}"
assms_4: fixes l :: "complex" shows "\<not> f \<midarrow>z\<rightarrow> l"
assms_3: shows "cmod (z - z0) \<noteq> r"
foldl_conv_fold: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and s :: "'b"   and xs :: "'c list" shows "foldl f s xs = fold (\<lambda>x s. f s x) xs s"
elims_2: fixes x :: "nat"   and xa :: "'b btree" assumes "invar_inorder x xa" obtains k :: "nat" and t :: "'b btree" where "x = k" and "xa = t" and "bal t \<and> root_order k t"
sums: shows "summable (\<lambda>i. LINT x|M. norm (f i x))"
ex_un_lam_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b"   and g :: "'b" assumes "cartesian_closed_category C"   and "partial_magma.ide C b"   and "partial_magma.ide C c"   and "partial_magma.ide C a"   and "partial_magma.in_hom C g (elementary_category_with_binary_products.prod C (category_with_binary_products.pr0 C) (category_with_binary_products.pr1 C) a b) c" shows "\<exists>!f. partial_magma.in_hom C f a (cartesian_closed_category.exp C b c) \<and> g = C (cartesian_closed_category.eval C b c) (elementary_category_with_binary_products.prod C (category_with_binary_products.pr0 C) (category_with_binary_products.pr1 C) f b)"
less_eq_list_drop: fixes xs :: "'b list"   and ys :: "'b list"   and y :: "'b" assumes "xs \<le> ys" shows "xs \<le> y # ys"
continuous_map_openin_preimage_eq: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "continuous_map X Y f = (f ` topspace X \<subseteq> topspace Y \<and> (\<forall>U. openin Y U \<longrightarrow> openin X (topspace X \<inter> f -` U)))"
confc\<^sub>0: shows "conf_clinit P sh ((stk, loc, C, M, pc, Throwing (C' # Cs) a) # frs)"
leq_antisym: fixes leq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.partial_order leq"   and "leq x y"   and "leq y x" shows "x = y"
inject: fixes x1 :: "nat"   and y1 :: "nat" shows "(Label x1 = Label y1) = (x1 = y1)"
restrict_clm: fixes v :: "view"   and ts :: "traffic"   and c :: "cars" shows "restrict v (clm ts) c \<le> clm ts c"
face_of_disjoint_rel_interior: fixes T :: "'a set"   and S :: "'a set" assumes "T face_of S"   and "T \<noteq> S" shows "T \<inter> rel_interior S = {}"
cil_ileq: fixes w :: "'d list"   and w1 :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and w2 :: "'d list" assumes "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2" shows "w1 \<preceq> w \<and> w2 \<preceq> w"
shorten_position: fixes t :: "('c, 'd) tm"   and p :: "dir list"   and q :: "dir list" assumes "position_of t (p @ q)" shows "position_of t p"
induct: fixes P :: "int \<times> (int \<times> int) \<times> int \<times> int \<Rightarrow> (int \<times> int) list \<Rightarrow> bool"   and a0 :: "int \<times> (int \<times> int) \<times> int \<times> int"   and a1 :: "(int \<times> int) list" assumes "\<And>\<delta> c\<^sub>0 c\<^sub>1. P (\<delta>, c\<^sub>0, c\<^sub>1) []"   and "\<And>\<delta> c\<^sub>0 c\<^sub>1 p. P (\<delta>, c\<^sub>0, c\<^sub>1) [p]"   and "\<And>\<delta> c\<^sub>0 c\<^sub>1 p\<^sub>0 v va. \<lbrakk>\<And>x xa y. \<lbrakk>x = find_closest_bf_code p\<^sub>0 (take 7 (v # va)); (xa, y) = x; \<delta> \<le> xa\<rbrakk> \<Longrightarrow> P (\<delta>, c\<^sub>0, c\<^sub>1) (v # va); \<And>x xa y. \<lbrakk>x = find_closest_bf_code p\<^sub>0 (take 7 (v # va)); (xa, y) = x; \<not> \<delta> \<le> xa\<rbrakk> \<Longrightarrow> P (xa, p\<^sub>0, y) (v # va)\<rbrakk> \<Longrightarrow> P (\<delta>, c\<^sub>0, c\<^sub>1) (p\<^sub>0 # v # va)" shows "P a0 a1"
ts_inv_okD: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and I :: "'f \<Rightarrow> 'i option"   and t :: "'f" assumes "ts_inv_ok ts I" shows "(ts t = None) = (I t = None)"
from_list_Nil: shows "Sinca.Fenv.from_list [] = Finca_empty"
Star_string: fixes s :: "'b list"   and A :: "'b list set" assumes "s \<in> star A" shows "\<exists>ss. concat ss = s \<and> (\<forall>s\<in>set ss. s \<in> A)"
assms_2: shows "\<forall>\<Sigma>. sp_instr F (xs ! n) \<Sigma> = sp_instr F x \<Sigma>"
wf_sub_rel: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" assumes "wf_sub_rel_of g subs" shows "wf_sub_rel subs"
trapped_sol_right: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_forward x K" shows "{0..} \<subseteq> existence_ivl0 x"
\<ll>_ide_simp: fixes a :: "'a" assumes "ide a" shows "\<ll> a = \<l>[a]"
assms: shows "[\<lparr>F,x,y\<rparr> in v]"
IH_2: assumes "v \<notin> ifex_var_set e" shows "val_ifex e (a(v := any)) = val_ifex e a"
pow_eq_id: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and n :: "nat" assumes "Group.group G"   and "x \<in> carrier G" shows "(x [^]\<^bsub>G\<^esub> n = \<one>\<^bsub>G\<^esub>) = (group.ord G x dvd n)"
effect_of_listI: fixes a :: "'a array"   and h' :: "heap"   and xs :: "'a list"   and h :: "heap" assumes "(a, h') = Array.alloc xs h" shows "effect (Array.of_list xs) h h' a"
right_total_UNIV_transfer: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "right_total A" shows "rel_set A (Collect (Domainp A)) UNIV"
mult_isol: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z ; x \<le> z ; y"
exec_WHILE_case: fixes b :: "exp"   and c :: "com"   and s :: "loc \<Rightarrow> nat"   and t :: "loc \<Rightarrow> nat"   and eval :: "((exp \<times> (loc \<Rightarrow> nat)) \<times> nat \<times> (loc \<Rightarrow> nat)) set" assumes "(WHILE b DO c, s) -[eval]-> t" obtains "((b, s), Suc 0, t) \<in> eval"   | s1 :: "loc \<Rightarrow> nat" and s2 :: "loc \<Rightarrow> nat" where "((b, s), 0, s1) \<in> eval" and "(c, s1) -[eval]-> s2" and "(WHILE b DO c, s2) -[eval]-> t"
pairs_phi_less_eq: fixes x :: "'c regular"   and y :: "'c dense Filters.filter" assumes "(x, y) \<in> stone_phi.pairs" shows "stone_phi (- x) \<le> y"
superclass_name_f_def: fixes cld :: "cld" shows "superclass_name_f cld = (case cld of cld_def dcl cl fds mds \<Rightarrow> cl)"
subgroup_is_group: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "subgroup H G"   and "Group.group G" shows "Group.group (G\<lparr>carrier := H\<rparr>)"
vars_finite: fixes g :: "'g" shows "finite (braun_ssa.vars g)"
fun_dual4: fixes f :: "'b \<Rightarrow> 'c set"   and g :: "'b \<Rightarrow> 'c set" shows "(f = (`) \<partial> \<circ> g) = ((`) \<partial> \<circ> f = g)"
deterministicD: fixes R :: "('a \<times> 'b) set"   and x :: "'a"   and y :: "'b"   and z :: "'b" assumes "deterministic R"   and "(x, y) \<in> R"   and "(x, z) \<in> R" shows "y = z"
splitAt_simpE: fixes ram :: "'a"   and b :: "'a list" shows "snd (splitAt ram (ram # b)) = b"
h2: shows "MPoly_Type.degree p var = 1 \<or> MPoly_Type.degree p var = 2"
homotopic_with_sym: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "homotopic_with P X Y f g = homotopic_with P X Y g f"
strict_border_imp_nonempty: fixes xs :: "'a list"   and ys :: "'a list" assumes "strict_border xs ys" shows "ys \<noteq> []"
IH: fixes s :: "'a \<Rightarrow> 'b option" assumes "is_valid_problem_sas_plus \<Psi>"   and "\<forall>op'\<in>set ops'. op' \<in> set ((\<phi> \<Psi> )\<^sub>\<O>)" shows "execute_parallel_operator (\<phi>\<^sub>S \<Psi> s) ops' = \<phi>\<^sub>S \<Psi> execute_parallel_operator_sas_plus s (map (strips_op_to_sasp \<Psi>) ops')"
fmlookup_drop_fset: fixes A :: "'b fset"   and m :: "('b, 'a) fmap"   and x :: "'b" shows "fmlookup (fmdrop_fset A m) x = (if x |\<notin>| A then fmlookup m x else None)"
distinct_57: fixes x41 :: "('d, 'e, 'f) com"   and x42 :: "('d, 'e, 'f) com"   and x61 :: "'d set"   and x62 :: "('d, 'e, 'f) com" shows "Seq x41 x42 \<noteq> While x61 x62"
that: fixes op :: "('a \<times> 'b) strips_operator" assumes "op \<in> set \<pi>"   and "map (strips_op_to_sasp \<Pi>) ops' = [\<phi>\<^sub>O\<inverse> \<Pi> op]" shows "thesis"
M_top: shows "A @\<^sub>c mat_of_col (- b) \<in> carrier_mat nr (n + 1)"
cases: fixes a1 :: "'a"   and a2 :: "'a multiset"   and a3 :: "'b el_step"   and a4 :: "'a"   and a5 :: "'a multiset"   and S :: "(('a \<times> 'a multiset) \<times> 'b \<times> 'a \<times> 'a multiset) set" assumes "((a1, a2), a3, a4, a5) \<in> gtrp S" obtains s :: "'a" and c :: "'a multiset" and e :: "'b" and s' :: "'a" and c' :: "'a multiset" where "a1 = sa__" and "a2 = ca__" and "a3 = LOC e__" and "a4 = s'a__" and "a5 = c'__" and "((sa__, ca__), e__, s'a__, c'__) \<in> S"   | s :: "'a" and sl :: "'a" and c :: "'a multiset" and e :: "'b" and s' :: "'a" and c' :: "'a multiset" where "a1 = sl" and "a2 = add_mset s c" and "a3 = ENV e" and "a4 = sl" and "a5 = add_mset s' c'" and "((s, add_mset sl c), e, s', add_mset sl c') \<in> S"
unique_factorization_m_zero: fixes m :: "int"   and f :: "int poly"   and c :: "int"   and fs :: "int poly multiset" assumes "poly_mod_2 m"   and "poly_mod.unique_factorization_m m f (c, fs)" shows "poly_mod.M m c \<noteq> 0"
ax7: fixes x :: "'a" shows "1\<^sub>\<sigma> \<sqinter> x \<cdot> U + x \<cdot> (0::'a) \<parallel> 1\<^sub>\<sigma> = d x"
post_meas1_def: fixes n :: "nat"   and v :: "complex Matrix.mat"   and i :: "nat" shows "post_meas1 n v i \<equiv> complex_of_real (1 / sqrt (prob1 n v i)) \<cdot>\<^sub>m |Matrix.vec (2 ^ n) (\<lambda>j. if select_index n i j then v $$ (j, 0) else 0)\<rangle>"
set_disconnected_nodes_get_shadow_root_is_l_set_disconnected_nodes_get_shadow_root: shows "l_set_disconnected_nodes_get_shadow_root set_disconnected_nodes_locs get_shadow_root_locs"
drop_Cons: fixes n :: "nat"   and x :: "'c"   and xs :: "'c list" shows "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
ffb_g_odei: fixes P :: "'a set"   and I :: "'a \<Rightarrow> bool"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> bool"   and U :: "'a \<Rightarrow> real set"   and S :: "'a set"   and t\<^sub>0 :: "real"   and Q :: "'a set" assumes "P \<subseteq> {s. I s}"   and "{s. I s} \<subseteq> fb\<^sub>\<F> x\<acute>=f & G on U S @ t\<^sub>0 {s. I s}"   and "{s. I s \<and> G s} \<subseteq> Q" shows "P \<subseteq> fb\<^sub>\<F> x\<acute>=f & G on U S @ t\<^sub>0 DINV I  Q"
assms_1: fixes M :: "'a"   and N :: "'a" assumes "\<alpha> = M\<lparr>N\<rparr>" shows "Prop"
rep_eq: fixes x :: "\<Pi>\<^sub>1" shows "Semantics.d\<^sub>1 x = Some (eval\<Pi>\<^sub>1 x)"
i_expand_nth_mod_eq_0: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" assumes "0 < k"   and "n mod k = 0" shows "(f \<odot> k) n = f (n div k)"
exhaust: fixes y :: "('b, 'a) nondetT" obtains x :: "'a" where "y = NondetT x"
index_uminus_mat_3: fixes A :: "'b mat" shows "dim_col (- A) = dim_col A"
a_Rec_inv_P: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and x :: "Syntax.var"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> e, Rec x eP, eV : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"   and "atom x \<sharp> \<Gamma>" obtains y :: "Syntax.var" and e' :: "Syntax.term" and eP' :: "Syntax.term" and eV' :: "Syntax.term" where "e = Rec x (Lam y e')" and "eP = Lam y eP'" and "eV = Rec x (Lam y eV')" and "atom y \<sharp> (\<Gamma>, x)" and "\<Gamma>(x $$:= Fun \<tau>\<^sub>1 \<tau>\<^sub>2) \<turnstile> Lam y e', Lam y eP', Lam y eV' : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"
d_def: shows "d \<equiv> Suc (Max (poly_deg ` fst ` set (snd ss)))"
plus_perm_def: fixes p :: "'a perm"   and q :: "'a perm" shows "p + q = Abs_perm (Rep_perm p \<circ> Rep_perm q)"
add_assoc: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "semigroup_add_ow S pls"   and "a \<in> S"   and "b \<in> S"   and "c \<in> S" shows "pls (pls a b) c = pls a (pls b c)"
map_poly_hom_monom: fixes aa :: "'c"   and a :: "'c poly"   and i :: "nat" shows "map_poly (\<lambda>p. poly p aa) (monom a i) = monom (poly a aa) i"
generalise_output_preserves_arity: fixes t :: "transition"   and r :: "nat"   and p :: "nat" shows "Arity (generalise_output t r p) = Arity t"
elims: fixes x :: "const"   and xa :: "const"   and y :: "const" assumes "x \<squnion> xa = y" obtains m :: "int" and n :: "int" where "x = Const m" and "xa = Const n" and "y = (if n = m then Const m else Any)"   | uv_ :: "const" where "x = Any" and "xa = uv_" and "y = Any"   | uu_ :: "const" where "x = uu_" and "xa = Any" and "y = Any"
f_smult: fixes r :: "'a"   and m :: "'c" assumes "r \<in> carrier R"   and "m \<in> carrier M" shows "f (r \<odot>\<^bsub>M\<^esub> m) = r \<odot>\<^bsub>N\<^esub> f m"