hcju/isabelleps huggingface dataset:
parPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "(P, Q) \<in> Rel"   and "\<And>P Q R. (P, Q) \<in> Rel \<Longrightarrow> (P \<parallel> R, Q \<parallel> R) \<in> Rel'"   and "\<And>P Q a. (P, Q) \<in> Rel' \<Longrightarrow> (<\<nu>a>P, <\<nu>a>Q) \<in> Rel'"   and "eqvt Rel"   and "eqvt Rel'" shows "P \<parallel> R \<leadsto>\<^sup>^<Rel'> Q \<parallel> R"
183: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> - - (y \<sqinter> z) \<sqinter> y) \<sqinter> z = (x \<squnion> y) \<sqinter> z"
Linear_def: fixes b :: "'g"   and r :: "('f, 'g, 'h, 'i) resumption" shows "Linear b r = Abs_resumption (LINEAR b (Rep_resumption r))"
subgroup_subgroup_generated_iff: fixes H :: "'a set"   and B :: "'a set" shows "subgroup H (subgroup_generated G B) = (subgroup H G \<and> H \<subseteq> carrier (subgroup_generated G B))"
to_int_mod_ring_of_int_mod_ring: fixes x :: "int" assumes "0 \<le> x"   and "x < int CARD('b)" shows "to_int_mod_ring (of_int_mod_ring x) = x"
lookup_update: fixes k :: "'b"   and v :: "'c"   and t :: "('b, 'c) trie" assumes "ID cbl \<noteq> None" shows "lookup (update k v t) = lookup t(k \<mapsto> v)"
sup_absorb2: fixes x :: "'b multiset"   and y :: "'b multiset" assumes "x \<subseteq># y" shows "x \<union># y = y"
pmf_le_1: fixes p :: "'a pmf"   and x :: "'a" shows "pmf p x \<le> 1"
CFG_path_no_Def_equal: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and V :: "'var"   and s :: "'state" assumes "n -as\<rightarrow>* n'"   and "\<forall>n\<in>set (sourcenodes as). V \<notin> Def n"   and "preds (kinds as) s" shows "state_val (transfers (kinds as) s) V = state_val s V"
monom_mult_is_smult: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "p \<in> carrier P" shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
assms: shows "set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (snd a)) \<inter> subst_domain \<theta> = {}"
mkSPRSingleAuto_implements: shows "Robot.SPR.implements (mkSPRSingleAuto kbp envInit envAction envTrans envVal (\<lambda>_. envObs))"
usubstappt_plus_conv: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<theta> :: "trm"   and \<eta> :: "trm" assumes "usubstappt \<sigma> U (Plus \<theta> \<eta>) \<noteq> undeft" shows "usubstappt \<sigma> U \<theta> \<noteq> undeft \<and> usubstappt \<sigma> U \<eta> \<noteq> undeft"
assms: shows "run_one_step d i (s, vs, ves, $Set_local x15) = (s', vs', res)"
image_mset_cong: fixes M :: "'b multiset"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in># M \<Longrightarrow> f x = g x" shows "image_mset f M = image_mset g M"
bal_update: fixes t :: "('c \<times> 'd) tree234"   and x :: "'c"   and y :: "'d" assumes "bal t" shows "bal (update x y t)"
simps_2: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and l :: "'c btree"   and a :: "'c"   and r :: "'c btree" assumes "split split" shows "inorder_up\<^sub>i (Up\<^sub>i l a r) = inorder l @ [a] @ inorder r"
map_minus_plus_commute: fixes g :: "'a \<Rightarrow> 'b option"   and h :: "'a \<Rightarrow> 'b option"   and f :: "'a \<Rightarrow> 'b option" assumes "dom g \<inter> dom h = {}" shows "f -- g ++ h = f ++ h -- g"
sum_nonneg_leq_bound: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and B :: "'a"   and i :: "'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite s"   and "\<And>i. i \<in> s \<Longrightarrow> less_eq zero (f i)"   and "comm_monoid_add.sum plus zero f s = B"   and "i \<in> s" shows "less_eq (f i) B"
case_1: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Ins of Ins \<Rightarrow> f1 | Del \<Rightarrow> f2) = f1"
pp_isotone: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" assumes "x \<preceq> y" shows "\<ominus> \<ominus> x \<preceq> \<ominus> \<ominus> y"
expand_inv_def: fixes \<phi> :: "'d ltlr"   and n_ns :: "('d, 'e) node_scheme \<times> ('d, 'f) node_scheme set" shows "expand_inv \<phi> n_ns \<equiv> case n_ns of (n, ns) \<Rightarrow> expand_inv_node \<phi> n \<and> expand_inv_result \<phi> ns"
v: shows "(v = x \<or> v = y) \<and> O w v"
inner_prod_with_row_bra_vec: fixes u :: "complex vec"   and v :: "complex vec" assumes "dim_vec u = dim_vec v" shows "\<langle>u|v\<rangle> = row (bra_vec u) 0 \<bullet> v"
map_FGcontra_def: fixes l1 :: "'n \<Rightarrow> 'aa"   and co1 :: "'o \<Rightarrow> 'ab"   and co2 :: "'w \<Rightarrow> 'aj"   and co3 :: "'p \<Rightarrow> 'ac"   and co4 :: "'q \<Rightarrow> 'ad"   and co5 :: "'r \<Rightarrow> 'ae"   and contra1 :: "'af \<Rightarrow> 's"   and contra2 :: "'ag \<Rightarrow> 't"   and contra3 :: "'ah \<Rightarrow> 'u"   and contra4 :: "'ai \<Rightarrow> 'v"   and contra5 :: "'ak \<Rightarrow> 'y" shows "map_FGcontra l1 co1 co2 co3 co4 co5 contra1 contra2 contra3 contra4 contra5 = map_F l1 co1 co3 co1 co4 co5 (map_G contra1 contra2 contra3 contra4 co1 co2) contra1 contra5"
k_words_rel: fixes k :: "nat" shows "{w. length w = Suc k \<and> set w \<subseteq> L} = {w. hd w \<in> L \<and> tl w \<in> {w. length w = k \<and> set w \<subseteq> L} \<and> w \<noteq> []}"
implmt_is_class: fixes G :: "prog"   and C :: "qtname"   and I :: "qtname" assumes "G\<turnstile>C\<leadsto>I" shows "is_class G C"
assms_3: shows "w ; v\<^sup>T \<le> x"
not_sub: fixes A :: "'c set"   and B :: "'c set" assumes "\<not> A \<subseteq> B" shows "\<exists>a. a \<in> A \<and> a \<notin> B"
Gamma_integral_complex: fixes z :: "complex" assumes "0 < Re z" shows "((\<lambda>t. complex_of_real t powr (z - 1) / complex_of_real (exp t)) has_integral Gamma z) {0..}"
subset_closed_segment: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "(closed_segment a b \<subseteq> closed_segment c d) = (a \<in> closed_segment c d \<and> b \<in> closed_segment c d)"
subprob_algebra_def: fixes K :: "'b measure" shows "subprob_algebra K = (\<Squnion>A\<in>sets K. vimage_algebra {M. subprob_space M \<and> sets M = sets K} (\<lambda>M. emeasure M A) borel)"
ord_le_eq_trans: fixes a :: "'t"   and b :: "'t"   and c :: "'t" assumes "a \<preceq>\<^sub>t b"   and "b = c" shows "a \<preceq>\<^sub>t c"
assms_2: shows "snd (cs (S t i) cid) = NotStarted"
in_cfl: shows "in_cfl G (\<beta> @ \<alpha>) w"
wf_rexp_of': fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "local.wf n (rexp_of' n \<phi>)"
IH: fixes xa :: "'v \<times> 'v" shows "h_minus i xa \<le> h_plus i xa"
arr_dom: fixes f :: "'a" assumes "C.arr f" shows "C.arr (C.dom f)"
iter_properties: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and n :: "nat"   and w0 :: "nat set"   and ss0 :: "'b list"   and ss' :: "'b list"   and w' :: "nat set" assumes "Semilat A r f"   and "acc r"   and "pres_type step n A"   and "SemilatAlg.mono r step n A"   and "bounded step n"   and "\<And>p. p \<in> w0 \<Longrightarrow> p < n"   and "ss0 \<in> list n A"   and "\<And>p. \<lbrakk>p < n; p \<notin> w0\<rbrakk> \<Longrightarrow> stable r step ss0 p"   and "iter f step ss0 w0 = (ss', w')" shows "ss' \<in> list n A \<and> stables r step ss' \<and> ss0 [\<sqsubseteq>\<^bsub>r\<^esub>] ss' \<and> (\<forall>ts\<in>list n A. ss0 [\<sqsubseteq>\<^bsub>r\<^esub>] ts \<and> stables r step ts \<longrightarrow> ss' [\<sqsubseteq>\<^bsub>r\<^esub>] ts)"
mult_scalar_monomial: fixes t :: "'a"   and c :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'b" shows "punit.mult_scalar (monomial c t) p = punit.monom_mult c t p"
h2: shows "\<forall>i\<in>{m..n}. f i = 0"
OPT2x: fixes x :: "'a"   and \<sigma>' :: "'a list"   and y :: "'a" shows "OPT2 (x # \<sigma>') [x, y] = (0, []) # OPT2 \<sigma>' [x, y]"
imethds_wf_mhead: fixes m :: "qtname \<times> mhead"   and G :: "prog"   and I :: "qtname"   and sig :: "sig" assumes "m \<in> DeclConcepts.imethds G I sig"   and "wf_prog G"   and "is_iface G I" shows "wf_mhead G (pid (decliface m)) sig (mthd m) \<and> \<not> is_static m \<and> accmodi m = Public"
conj_com: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<and>. q) = (A \<turnstile> q .\<and>. p)"
add_data_cache_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and address :: "word32"   and w :: "word32"   and bm :: "word4"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = add_data_cache s1 address w bm \<and> t2 = add_data_cache s2 address w bm" shows "low_equal t1 t2"
eqI: fixes \<tau>' :: "'c \<Rightarrow> 's" assumes "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) (Y a') \<tau>'"   and "\<tau>' a = \<T> e a" shows "\<tau>' = \<T> e"
assms_2: shows "xs \<noteq> ys \<or> x \<notin> {y, y1}"
zero_le_sgn_iff: fixes x :: "real" shows "(0 \<le> sgn x) = (0 \<le> x)"
admissibleI: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and lub :: "'b set \<Rightarrow> 'b" assumes "\<And>A. \<lbrakk>Complete_Partial_Order.chain ord A; A \<noteq> {}; \<forall>x\<in>A. P x\<rbrakk> \<Longrightarrow> P (lub A)" shows "ccpo.admissible lub ord P"
spray_eq: shows "spray x = {y. \<exists>R\<in>SPRAY x. y \<in> R}"
getChoices_def: fixes g :: "gState\<^sub>I"   and p :: "pState" shows "getChoices g p = foldl (\<lambda>E e. if evalHandshake (cond e) (handshake g) g p \<and> evalCond (cond e) g p then (e, p) # E else E) []"
in_MI: fixes y :: "'a"   and x :: "'a" assumes "y \<in> M"   and "x \<in> A"   and "le y x" shows "x \<in> M"
mset_append: fixes xs :: "'k list"   and ys :: "'k list" shows "mset (xs @ ys) = mset xs + mset ys"
False: shows "\<not> (a = 0 \<or> x = 0)"
finite_csucc: fixes a :: "V" assumes "a \<in> elts \<omega>" shows "csucc a = ZFC_in_HOL.succ a"
map_minus_def: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" shows "f -- g = (\<lambda>x. if f x = g x then None else f x)"
add_additive_subgroups: fixes G :: "('a, 'b) ring_scheme"   and H :: "'a set"   and K :: "'a set" assumes "abelian_group G"   and "additive_subgroup H G"   and "additive_subgroup K G" shows "additive_subgroup (H <+>\<^bsub>G\<^esub> K) G"
intros: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "is_App t"   and "fun t \<ge>\<^sub>t s \<or> arg t \<ge>\<^sub>t s" shows "gt_sub t s"
inverse_less_1_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (inverse x) one = (less_eq x zero \<or> less one x)"
contract_closure: fixes Cn :: "'b set \<Rightarrow> 'b set"   and contraction :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "AGM_Contraction Cn contraction"   and "K = Cn A" shows "contraction K \<phi> = Cn (contraction K \<phi>)"
MIN'_2: shows "\<forall>v. enat d \<le> Q_\<alpha> Qi v"
induct: fixes x :: "('fun, 'var) Term.term"   and P :: "('fun, 'var) Term.term \<Rightarrow> bool" assumes "public_ground_wf_term x"   and "\<And>f T. \<lbrakk>public f; arity f = length T; \<And>t. t \<in> set T \<Longrightarrow> public_ground_wf_term t; \<And>t. t \<in> set T \<Longrightarrow> P t\<rbrakk> \<Longrightarrow> P (Fun f T)" shows "P x"
list_rev_it_correct: fixes m :: "'s" assumes "invar m" shows "map_iterator_rev_linord (list_rev_it m) (\<alpha> m)"
fps_const_divide: fixes x :: "'a"   and y :: "'a" shows "fps_const x / fps_const y = fps_const (x / y)"
\<omega>consistentStd1_implies_consistent: assumes "\<omega>consistentStd1" shows "consistent"
far_north_def: shows "far_north = proj2_abs (vector [0, 1, 0])"
sym_c: shows "symmetric (G \<restriction> {v. u \<rightarrow>\<^sup>* v})"
rotate_Suc2: fixes n :: "nat"   and xs :: "'c list" shows "rotate n (rotate1 xs) = rotate (Suc n) xs"
inj1_def: fixes M1 :: "('g, 'e) module"   and M2 :: "('g, 'f) module" shows "inj1 M1 M2 = (\<lambda>v. (v, \<zero>\<^bsub>M2\<^esub>))"
section_char: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category S \<AA>" shows "category.section S f = (partial_magma.arr S f \<and> (set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f) = {} \<longrightarrow> set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.cod S f) = {}) \<and> inj_on (set_category_given_img.Fun S (set_category.some_img S \<AA>) f) (set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f)))"
cases: fixes x :: "(Vars.var \<times> exp) list" obtains "x = []"   | v :: "Vars.var" and e :: "exp" and "binds" :: "(Vars.var \<times> exp) list" where "x = (v, e) # binds"
minorM_def: fixes M :: "(('c, 'd) vec, 'e) vec"   and i :: "'e"   and j :: "'d" shows "minorM M i j = (\<chi>k l. if k = i \<and> l = j then 1::'c else if k = i \<or> l = j then 0::'c else M $ k $ l)"
subintervals_2: shows "snd (split_float_interval ivl) \<le> ivl"
eq_set_perm_remdups: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs = set ys" shows "remdups xs <~~> remdups ys"
oquality_increases: fixes i :: "nat" shows "opaodv i \<Turnstile>\<^sub>A (otherwith quality_increases {i} (orecvmsg (\<lambda>_. rreq_rrep_sn)), other quality_increases {i} \<rightarrow>) onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>((\<sigma>, uu_), uu_, \<sigma>', uu_). \<forall>j. quality_increases (\<sigma> j) (\<sigma>' j))"
R\<^sub>1_def: shows "R\<^sub>1 = \<lbrakk>\<lbrakk>tab\<^sub>1 r\<rbrakk>\<rbrakk>"
Collect_cong: fixes P :: "'b::type \<Rightarrow> bool"   and Q :: "'b::type \<Rightarrow> bool" assumes "\<And>x::'b::type. (P::'b::type \<Rightarrow> bool) x = (Q::'b::type \<Rightarrow> bool) x" shows "{x::'b::type. (P::'b::type \<Rightarrow> bool) x} = {x::'b::type. (Q::'b::type \<Rightarrow> bool) x}"
a_2: shows "a \<notin> \<E>\<^bsub>\<Gamma>\<lparr>weight := weight \<Gamma> - u\<rparr>\<^esub> (TER\<^bsub>\<Gamma>\<lparr>weight := weight \<Gamma> - u\<rparr>\<^esub> f)"
splitAt_in_fst: fixes v :: "'a"   and ram :: "'a"   and vs :: "'a list" assumes "v \<in> set (fst (splitAt ram vs))" shows "v \<in> set vs"
prodag_def: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" shows "a\<Pi>\<^bsub>I\<^esub> A = \<lparr>carrier = carr_prodag I A, pop = prod_pOp I A, mop = prod_mOp I A, zero = prod_zero I A\<rparr>"
ocondition_K_false: fixes T :: "'d \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e" shows "ocondition (\<lambda>_. False) T F = F"
coeffs_map_poly': fixes f :: "'a \<Rightarrow> 'b"   and p :: "'a poly" assumes "\<And>x. x \<noteq> (0::'a) \<Longrightarrow> f x \<noteq> (0::'b)" shows "coeffs (map_poly f p) = map f (coeffs p)"
inputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "a<b>.P \<longmapsto> Rs" obtains x :: "name" and a :: "name" and P :: "pi" where "aa__ = a \<and> [b].Pa__ = [x].P" and "Rs = a<x> \<prec> P" and "x \<noteq> a"
less_rexp_def: fixes r :: "'a rexp"   and s :: "'a rexp" shows "r < s \<equiv> le_rexp r s \<and> r \<noteq> s"
carrier_Fexp: fixes e :: "exp"   and a :: "Arity" shows "carrier (Texp e\<cdot>a) = edom (\<A>\<^bsub>a\<^esub> e)"
ParId: fixes P :: "pi" shows "P \<parallel> \<zero> \<equiv>\<^sub>s P"
OAlist_tc_map2_val_neutr_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "('c, 'd) oalist_tc"   and ys :: "('c, 'd) oalist_tc" shows "OAlist_tc_map2_val_neutr f xs ys = OAlist_tc (map2_val_pair_tc f id id (list_of_oalist_tc xs) (list_of_oalist_tc ys))"
matrix_vector_mul_linear_gen: fixes A :: "(('a, 'b) vec, 'c) vec" shows "Vector_Spaces.linear (*s) (*s) ((*v) A)"
types_stable_def: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set" shows "types_stable \<Gamma> \<S> \<equiv> \<forall>x\<in>dom \<Gamma>. type_stable \<S> (the (\<Gamma> x))"
knows2know_s: fixes A :: "specID"   and m :: "Secrets" assumes "knows A [sE m]" shows "know A (sKS m)"
res_residual_iso: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f"   and "x \<le> y" shows "residual f x \<le> residual f y"
transpose_matrix_diff: fixes A :: "'b matrix"   and B :: "'b matrix" shows "transpose_matrix (A - B) = transpose_matrix A - transpose_matrix B"
inj_Some: fixes A :: "'k set" shows "inj_on Some A"
1: shows "int (card X - Suc j) = int (card X) - int j - 1"
simps_4: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b" shows "(case Err of Err \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = f1"
psubstBinp_preserves_good: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option"   and binp :: "'e \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qAbs set option" assumes "goodEnv rho"   and "goodBinp binp" shows "goodBinp binp %%[rho]"
map2_val_compat_def: fixes f :: "('a \<times> 'e) list \<Rightarrow> ('a \<times> 'f) list" shows "map2_val_compat f = (\<forall>zs. oalist_inv_raw zs \<longrightarrow> oalist_inv_raw (f zs) \<and> fst ` set (f zs) \<subseteq> fst ` set zs)"
ex_ex1I: fixes P :: "'c \<Rightarrow> bool" assumes "\<exists>x. P x"   and "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> x = y" shows "\<exists>!x. P x"
Pool_1: fixes w :: "nat \<Rightarrow> real" assumes "balanced_net m1" shows "shared_weight_net (insert_weights True m1 w)"
wb_prism_def: fixes x :: "'a \<Longrightarrow>\<^sub>\<triangle> 'b" shows "wb_prism x \<equiv> (\<forall>v. match\<^bsub>x\<^esub> (build\<^bsub>x\<^esub> v) = Some v) \<and> (\<forall>s v. match\<^bsub>x\<^esub> s = Some v \<longrightarrow> s = build\<^bsub>x\<^esub> v)"
UP_smult_zero: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "\<zero> \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
in_Use_valid_SDG_node: fixes V :: "'var"   and n :: "'node SDG_node" assumes "V \<in> Use\<^bsub>SDG\<^esub> n" shows "valid_SDG_node n"
iso_iff_section_and_retraction: fixes f :: "'a" shows "S.iso f = (S.section f \<and> S.retraction f)"
nn_integral_dens_ctxt_measure_restrict: fixes \<rho> :: "nat \<Rightarrow> val"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "0 \<le> f \<rho>"   and "f \<in> borel_measurable (state_measure V' \<Gamma>)" shows "\<integral>\<^sup>+ x. f (restrict x V') \<partial>dens_ctxt_measure \<Y> \<rho> = branch_prob \<Y> \<rho> * f \<rho>"
thfg: fixes x :: "'a" assumes "norm x = 1" shows "(if x = (0::'a) then 0::'a else norm x *\<^sub>R f (x /\<^sub>R norm x)) = f x"
simps_1: fixes t :: "t"   and t' :: "t" shows "ct_eq (TSome t) (TSome t') = (t = t')"
c: shows "c = fst (common_denom (coeffs (of_int_poly p)))"
valid_eqvt: fixes P :: "'state"   and x :: "('a, 'pred, 'act) formula"   and p :: "perm" assumes "P \<Turnstile> x" shows "p \<bullet> P \<Turnstile> p \<bullet> x"
diff_add_mset_swap: fixes b :: "'b"   and A :: "'b multiset"   and M :: "'b multiset" assumes "b \<notin># A" shows "(b, M) - A = b, M - A"
Upd_eq_restr_stackD: fixes x :: "Vars.var"   and S :: "stack_elem list"   and V :: "Vars.var set"   and S' :: "stack_elem list" assumes "Upd x # S = restr_stack V S'" shows "x \<in> V"
DSTEP_D: fixes s :: "('c + 'd) list"   and t :: "('c + 'd) list"   and G :: "('c \<times> ('c, 'd) RHS) list" assumes "s -G\<rightarrow> t" shows "\<exists>L N R rhs. s = L \<cdot> [Inl N] \<cdot> R \<and> (N, rhs) \<in> set G \<and> (\<forall>A B. rhs = Branch A B \<longrightarrow> t = L \<cdot> [Inl A, Inl B] \<cdot> R) \<and> (\<forall>x. rhs = Leaf x \<longrightarrow> t = L \<cdot> [Inr x] \<cdot> R)"
sigma_finite_measure_count_space_countable: fixes A :: "'c set" assumes "countable A" shows "sigma_finite_measure (count_space A)"
bind_returns_result_E2: fixes h :: "'m"   and f :: "('m, 'n, 'p) prog"   and g :: "'p \<Rightarrow> ('m, 'n, 'o) prog"   and y :: "'o" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"   and "pure f h" obtains x :: "'p" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>r y"
rel_restrict_def: fixes R :: "('e \<times> 'e) set"   and A :: "'e set" shows "rel_restrict R A \<equiv> {(v, w). (v, w) \<in> R \<and> v \<notin> A \<and> w \<notin> A}"
r_inv: fixes x :: "'b" assumes "x \<in> UNIV" shows "x + - x = (0::'b)"
init_tableau_id: fixes t :: "(nat \<times> linear_poly) list" shows "\<T> (init t) = t"
down: fixes d :: "nat"   and dm :: "nat"   and p :: "(nat \<times> int) list"   and lm :: "nat"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" assumes "d < dm"   and "p \<in> sparsegrid dm lm" shows "down dm lm d \<alpha> p = (\<Sum>p'\<in>parents d (base {d} p) p. \<alpha> p' * l2_\<phi> (p ! d) (p' ! d))"
nonempty_single: fixes x :: "'b multiset" shows "{#x#} \<noteq> {#}"
simps_2: fixes f :: "'c \<Rightarrow> 'a + 'b"   and x :: "'c"   and xs :: "'c list" shows "mapM f (x # xs) = f x \<bind> (\<lambda>y. mapM f xs \<bind> (\<lambda>ys. Inr (y # ys)))"
h': shows "h3 \<turnstile> local.a_insert_node ptr node reference_child \<rightarrow>\<^sub>h h'"
preserves_dom: fixes f :: "'a1" assumes "A1.arr f" shows "A2_B.dom (\<tau> f) = F (A1.dom f)"
hn_rel_def: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and m :: "'b nres" shows "hn_rel P m \<equiv> \<lambda>r. \<exists>\<^sub>Ax. P x r * \<up> (RETURN x \<le> m)"
simps_2: fixes uv :: "nat" shows "catomic (cNumber uv) = True"
erH_map: fixes A :: "('e, 'f) Ring_scheme"   and B :: "('e, 'g) Ring_scheme"   and Y :: "'e"   and h :: "'a \<Rightarrow> 'e"   and n :: "nat"   and c :: "nat \<Rightarrow> 'a" assumes "Idomain S"   and "PolynRg A B Y"   and "h \<in> rHom S B"   and "pol_coeff S (n, c)" shows "erH R S X A B Y h (polyn_expr R X n (n, c)) = polyn_expr A Y n (n, cmp h c)"
alphaEnv_trans: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and rho'' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option" assumes "qGoodEnv rho"   and "rho &= rho'"   and "rho' &= rho''" shows "rho &= rho''"
Ball_Collect: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "Ball (A::'b set) (P::'b \<Rightarrow> bool) = (A \<subseteq> Collect P)"
genideal_minimal: fixes I :: "'a set"   and S :: "'a set" assumes "ideal I R"   and "S \<subseteq> I" shows "Idl S \<subseteq> I"
XObj: shows "X \<in> obj\<^bsub>C\<^esub>"
orthogonal_basis_exists: fixes V :: "'a set" shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> card B = dim V \<and> pairwise orthogonal B"
Guard_def: fixes g :: "'d \<Rightarrow> bool"   and P :: "('c, 'd) action" shows "g `&` P \<equiv> action_of (R (g \<circ> alpha_rp.more \<circ> fst \<longrightarrow> \<not> action.relation_of P\<^sup>f\<^sub>f \<turnstile> g \<circ> alpha_rp.more \<circ> fst \<and> action.relation_of P\<^sup>t\<^sub>f \<or> \<not> g \<circ> alpha_rp.more \<circ> fst \<and> (\<lambda>(A, A'). tr A' = tr A \<and> wait A')))"