hcju/isabelleps huggingface dataset:
ccSp_Append_cdep: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "n cc-ns\<rightarrow>\<^sub>d* n''"   and "n'' \<longrightarrow>\<^bsub>cd\<^esub> n'" shows "n cc-ns @ [n'']\<rightarrow>\<^sub>d* n'"
direct_sum_def: fixes M1 :: "('i, 'g, 'j) module_scheme"   and M2 :: "('i, 'h, 'k) module_scheme" shows "direct_sum M1 M2 = \<lparr>carrier = carrier M1 \<times> carrier M2, mult = \<lambda>v w. (\<zero>\<^bsub>M1\<^esub>, \<zero>\<^bsub>M2\<^esub>), one = (\<zero>\<^bsub>M1\<^esub>, \<zero>\<^bsub>M2\<^esub>), zero = (\<zero>\<^bsub>M1\<^esub>, \<zero>\<^bsub>M2\<^esub>), add = \<lambda>v w. (fst v \<oplus>\<^bsub>M1\<^esub> fst w, snd v \<oplus>\<^bsub>M2\<^esub> snd w), smult = \<lambda>c v. (c \<odot>\<^bsub>M1\<^esub> fst v, c \<odot>\<^bsub>M2\<^esub> snd v)\<rparr>"
Pi_pmf_cong: fixes A :: "'a set"   and A' :: "'a set"   and dflt :: "'b"   and dflt' :: "'b"   and f :: "'a \<Rightarrow> 'b pmf"   and f' :: "'a \<Rightarrow> 'b pmf" assumes "A = A'"   and "dflt = dflt'"   and "\<And>x. x \<in> A \<Longrightarrow> f x = f' x" shows "Pi_pmf A dflt f = Pi_pmf A' dflt' f'"
dependent_alt: fixes B :: "'b set" assumes "B \<subseteq> U\<^sub>M" shows "local.dependent B = (\<exists>f. finite {v \<in> U\<^sub>M. f v \<noteq> (0::'a)} \<and> {v \<in> U\<^sub>M. f v \<noteq> (0::'a)} \<subseteq> B \<and> (\<exists>v\<in>U\<^sub>M. f v \<noteq> (0::'a)) \<and> (\<Sum>x | x \<in> U\<^sub>M \<and> f x \<noteq> (0::'a). f x *s x) = (0::'b))"
sub2: fixes B :: "'d"   and k :: "'d"   and b :: "'d"   and a :: "'d" assumes "B \<equiv> k + b" shows "a - B \<equiv> - k + (a - b)"
oghoare_While: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P (While b c) Q, A" shows "\<exists>r i P' A' Q'. P = AnnWhile r i P' \<and> \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P' c i, A' \<and> r \<subseteq> i \<and> i \<inter> b \<subseteq> pre P' \<and> i \<inter> - b \<subseteq> Q' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A"
openin_trans_full: fixes X :: "'b topology"   and U :: "'b set"   and S :: "'b set" assumes "openin (subtopology X U) S"   and "openin X U" shows "openin X S"
\<phi>_\<psi>: fixes x :: "'b"   and g :: "'c"   and y :: "'c" assumes "B.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>C G' x\<guillemotright>" shows "\<phi>' y (\<psi>' x g) = g"
lessThan_iff: fixes i :: "'b"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b" shows "(i \<in> ord.lessThan less k) = less i k"
assms_2: shows "(\<sigma>, p) \<in> oreachable A S U"
exec_step_ind_CmpEq: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step_ind (StepI CmpEq) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (Bool (hd (tl stk) = hd stk) # tl (tl stk), loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
fpxs_compose_power_add: fixes f :: "'b fpxs"   and g :: "'b fpxs"   and r :: "rat" shows "fpxs_compose_power (f + g) r = fpxs_compose_power f r + fpxs_compose_power g r"
lower_subst_yields_lower_trms: fixes \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list"   and t :: "'a trm"   and x :: "'a" assumes "lower_on \<sigma> \<eta> (vars_of t)"   and "(Var x \<lhd> \<sigma>, Var x \<lhd> \<eta>) \<in> trm_ord"   and "x \<in> vars_of t" shows "(t \<lhd> \<sigma>, t \<lhd> \<eta>) \<in> trm_ord"
raw_ports_compress_src_CannotMatch: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and pss :: "ipt_l4_ports list"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>"   and "l4_ports_compress pss = CannotMatch" shows "\<not> matches (\<beta>, \<alpha>) (alist_and (map (Pos \<circ> Src_Ports) pss)) a p"
Gets_certificate_valid: fixes A :: "agent"   and X :: "msg"   and i :: "nat"   and EKi :: "nat"   and SKi :: "nat"   and evs :: "event list" assumes "Gets A \<lbrace>X, cert (CA i) EKi onlyEnc (priSK RCA), cert (CA i) SKi onlySig (priSK RCA)\<rbrace> \<in> set evs"   and "evs \<in> set_mr" shows "EKi = pubEK (CA i) \<and> SKi = pubSK (CA i)"
prod_encode_pdec': fixes e :: "nat" shows "prod_encode (pdec1' e, pdec2' e) = e"
prems_1: shows "\<forall>i\<le>Suc x. \<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}"
simps_4: fixes v :: "('b, 'c) Combinators"   and va :: "('b, 'c) Combinators"   and l :: "'b set list" shows "smaller (v \<oplus> va) DenyAll l = False"
iffs: fixes lens_get :: "'d \<Rightarrow> 'c"   and lens_put :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and more :: "'e"   and lens_get' :: "'d \<Rightarrow> 'c"   and lens_put' :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and more' :: "'e" shows "(\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr> = \<lparr>lens_get = lens_get', lens_put = lens_put', \<dots> = more'\<rparr>) = (lens_get = lens_get' \<and> lens_put = lens_put' \<and> more = more')"
a_set_mode_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_mode shadow_root_ptr = put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr mode_update"
simps_2: fixes k :: "bool"   and ks :: "bool list"   and t :: "trie" shows "prefix_trie (k # ks) t = (let t' = prefix_trie ks t in Nd False (if k then (Lf, t') else (t', Lf)))"
numeral_eq_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "num"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "(numeral.numeral one plus m = numeral.numeral one plus n) = (m = n)"
lessThan_iff: fixes i :: "'g"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and k :: "'g" shows "(i \<in> ord.lessThan less k) = less i k"
simps: fixes x :: "'c regular"   and y :: "'c dense Filters.filter"   and z :: "'c regular"   and w :: "'c dense Filters.filter" shows "stone_phi.pairs_inf (x, y) (z, w) = (x \<sqinter> z, y \<squnion> w)"
arcosh_def: fixes x :: "'a" shows "arcosh x = ln (x + (x\<^sup>2 - (1::'a)) powr of_real (1 / 2))"
assms: fixes i :: "nat" assumes "i < length Rs" shows "ring (Rs ! i)"
minus_insert_notin_fset: fixes ys :: "'b fset"   and x :: "'b"   and xs :: "'b fset" assumes "x |\<notin>| ys" shows "insert_fset x xs - ys = insert_fset x (xs - ys)"
lstrict_prefix_code_3: fixes x :: "'b"   and xs :: "'b llist" shows "lstrict_prefix (LCons x xs) LNil = False"
Suc_le_length_iff: fixes n :: "nat"   and xs :: "'c list" shows "(Suc n \<le> length xs) = (\<exists>x ys. xs = x # ys \<and> n \<le> length ys)"
weak_greatest_unique: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and A :: "'c set"   and y :: "'c" assumes "weak_partial_order L"   and "greatest L x A"   and "greatest L y A" shows "x .=\<^bsub>L\<^esub> y"
diff_union_swap_zmset: fixes b :: "'b"   and M :: "'b zmultiset"   and a :: "'b" shows "add_zmset b (M - {#a#}\<^sub>z) = add_zmset b M - {#a#}\<^sub>z"
perm_2: fixes pi :: "(name \<times> name) list"   and x1 :: "name" shows "pi \<bullet> BoundOutputS x1 = BoundOutputS (pi \<bullet> x1)"
Stop_Sim: fixes S :: "'e \<Rightarrow> 'd" shows "Stop \<preceq>S Stop"
InitDmd_gen: fixes F :: "'a \<Rightarrow> bool" shows "\<turnstile> Init F \<longrightarrow> \<diamond>F"
gtr_preserve_s: fixes c :: "'f multiset"   and e :: "'g"   and c' :: "'f multiset"   and T :: "(('f \<times> 'f multiset) \<times> 'g \<times> 'f \<times> 'f multiset) set"   and P :: "'f multiset \<Rightarrow> bool" assumes "(c, e, c') \<in> gtr T"   and "P c"   and "\<And>s c s' c' e. \<lbrakk>P (add_mset s c); ((s, c), e, s', c') \<in> T\<rbrakk> \<Longrightarrow> P (add_mset s' c')" shows "P c'"
converse_rtranclE': fixes u :: "'a"   and v :: "'a"   and R :: "('a \<times> 'a) set" assumes "(u, v) \<in> R\<^sup>*" obtains "u = v"   | vh :: "'a" where "u \<noteq> vh" and "(u, vh) \<in> R" and "(vh, v) \<in> R\<^sup>*"
x1_in_type: shows "(left x, s) \<in> {r. fst r \<le> snd r}"
jordan_matrix_upper_triangular: fixes i :: "nat"   and n_as :: "(nat \<times> 'b) list"   and j :: "nat" assumes "i < sum_list (map fst n_as)"   and "j < i" shows "jordan_matrix n_as $$ (i, j) = (0::'b)"
assert_tableau_normalized: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "\<triangle> (\<T> (assert a s))"
assms: shows "eval r_result [t, i, x] \<down>= Suc v"
intros_1: fixes ts :: "dB list"   and x :: "nat" assumes "listall NF ts" shows "NF (dB.Var x \<degree>\<degree> ts)"
sel_3: fixes x31 :: "nat"   and x32 :: "'a"   and x33 :: "'a"   and x34 :: "'b"   and x35 :: "'b"   and x36 :: "'c" shows "occurs_on (Recv x31 x32 x33 x34 x35 x36) = x32"
elims: fixes x :: "('c, 'd) fmap"   and xa :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and y :: "('c, 'd) fmap list" assumes "statelist' x xa = y" obtains s :: "('c, 'd) fmap" where "x = s__" and "xa = []" and "y = [s__]"   | s :: "('c, 'd) fmap" and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap" and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" where "x = s" and "xa = a # as" and "y = s # statelist' (state_succ s a) as"
commute: fixes a :: "'d"   and b :: "'d" shows "inf a b = inf b a"
hyps_for_fimage: fixes r :: "'rule"   and x :: "('form, 'var) antecedent" shows "hyps_for (Rule r) x = (if x |\<in>| f_antecedent r then (\<lambda>f. Hyp f x) |`| a_hyps x else {||})"
assms_3: shows "rk {A', C', P, Q} = 4"
eq: shows "rb_aux (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), z) = ((bs, ss, []), z')"
abs_hIm_le_hcmod: fixes x :: "complex star" shows "\<bar>hIm x\<bar> \<le> hcmod x"
eq_\<I>_gpv_eq_OO2: fixes \<I> :: "('b, 'c) \<I>"   and gpv'' :: "('a, 'b, 'c) gpv"   and gpv' :: "('a, 'b, 'c) gpv"   and A :: "'d \<Rightarrow> 'a \<Rightarrow> bool"   and gpv :: "('d, 'b, 'c) gpv" assumes "eq_\<I>_gpv (=) \<I> gpv'' gpv'"   and "eq_\<I>_gpv A \<I> gpv gpv'" shows "eq_\<I>_gpv A \<I> gpv gpv''"
split_face_edges_f_vs1: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list"   and vs2 :: "nat list" assumes "pre_split_face f ram1 ram2 vs"   and "between (vertices f) ram1 ram2 = []"   and "vs2 = between (vertices f) ram2 ram1"   and "vs2 \<noteq> []" shows "\<E> f = {(last vs2, ram1), (ram1, ram2), (ram2, hd vs2)} \<union> Edges vs2"
assms_3: shows "subterms\<^sub>s\<^sub>e\<^sub>t (subst_range \<theta>) \<inter> (subterms s \<union> subterms t) = {}"
simps: fixes l :: "'b sort_key extended"   and u :: "'b sort_key extended" shows "derive_right (l, u) = (derive_pos (\<gamma> u sort_dir.Left) l, derive_pos (\<gamma> u sort_dir.Left) u)"
run_effectE: fixes c :: "'d Heap"   and \<sigma> :: "heap option"   and \<sigma>' :: "heap option"   and r :: "'d" assumes "run c \<sigma> \<sigma>' r"   and "\<not> is_exn \<sigma>'" obtains h :: "heap" and h' :: "heap" where "\<sigma> = Some h" and "\<sigma>' = Some h'" and "effect c h h' r"
list_rel_induct: fixes l :: "'c list"   and l' :: "'d list"   and R :: "('c \<times> 'd) set"   and P :: "'c list \<Rightarrow> 'd list \<Rightarrow> bool" assumes "(l, l') \<in> \<langle>R\<rangle>list_rel"   and "P [] []"   and "\<And>x x' l l'. \<lbrakk>(x, x') \<in> R; (l, l') \<in> \<langle>R\<rangle>list_rel; P l l'\<rbrakk> \<Longrightarrow> P (x # l) (x' # l')" shows "P l l'"
configuration_Max_in: fixes q :: "'b"   and n :: "nat" assumes "configuration q n \<noteq> {}" shows "Max (configuration q n) \<in> configuration q n"
analz_Un_implInsecSet_aux_aux: fixes G :: "msg set"   and H :: "msg set" assumes "Enc_keys_clean (G \<union> H)" shows "analz (implInsecSet_aux G \<union> H) \<subseteq> implInsecSet_aux G \<union> Tags \<union> synth (analz (G \<union> H))"
y_swap: shows "term_brackets vs (unlift_vars n y) = Some (unlift_vars n y')"
Gromov_completion_inside_limit: fixes x :: "'a Gromov_completion"   and u :: "'b \<Rightarrow> 'a Gromov_completion"   and F :: "'b filter" assumes "x \<notin> Gromov_boundary" shows "(u \<longlongrightarrow> x) F = ((\<lambda>n. extended_Gromov_distance (u n) x) \<longlongrightarrow> 0) F"
strong_induct: fixes x1 :: "ccs"   and x2 :: "act"   and x3 :: "ccs"   and P :: "'a \<Rightarrow> ccs \<Rightarrow> act \<Rightarrow> ccs \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<longmapsto>x2 \<prec> x3"   and "\<And>\<alpha> Pa y. P y (\<alpha>.Pa) \<alpha> Pa"   and "\<And>Pa \<alpha> P' Q y. \<lbrakk>Pa \<longmapsto>\<alpha> \<prec> P'; \<And>z. P z Pa \<alpha> P'\<rbrakk> \<Longrightarrow> P y (Pa \<oplus> Q) \<alpha> P'"   and "\<And>Q \<alpha> Q' Pa y. \<lbrakk>Q \<longmapsto>\<alpha> \<prec> Q'; \<And>z. P z Q \<alpha> Q'\<rbrakk> \<Longrightarrow> P y (Pa \<oplus> Q) \<alpha> Q'"   and "\<And>Pa \<alpha> P' Q y. \<lbrakk>Pa \<longmapsto>\<alpha> \<prec> P'; \<And>z. P z Pa \<alpha> P'\<rbrakk> \<Longrightarrow> P y (Pa \<parallel> Q) \<alpha> (P' \<parallel> Q)"   and "\<And>Q \<alpha> Q' Pa y. \<lbrakk>Q \<longmapsto>\<alpha> \<prec> Q'; \<And>z. P z Q \<alpha> Q'\<rbrakk> \<Longrightarrow> P y (Pa \<parallel> Q) \<alpha> (Pa \<parallel> Q')"   and "\<And>Pa a P' Q Q' y. \<lbrakk>Pa \<longmapsto>a \<prec> P'; \<And>z. P z Pa a P'; Q \<longmapsto>coAction a \<prec> Q'; \<And>z. P z Q (coAction a) Q'; a \<noteq> \<tau>\<rbrakk> \<Longrightarrow> P y (Pa \<parallel> Q) (\<tau>) (P' \<parallel> Q')"   and "\<And>Pa \<alpha> P' x y. \<lbrakk>x \<sharp> y; Pa \<longmapsto>\<alpha> \<prec> P'; \<And>z. P z Pa \<alpha> P'; x \<sharp> \<alpha>\<rbrakk> \<Longrightarrow> P y (\<lparr>\<nu>x\<rparr>Pa) \<alpha> (\<lparr>\<nu>x\<rparr>P')"   and "\<And>Pa \<alpha> P' y. \<lbrakk>Pa \<parallel> !Pa \<longmapsto>\<alpha> \<prec> P'; \<And>z. P z (Pa \<parallel> !Pa) \<alpha> P'\<rbrakk> \<Longrightarrow> P y !Pa \<alpha> P'" shows "P z x1 x2 x3"
WODBackwardSliceCorrect: shows "BackwardSlice src trg knd lve NewEntry lDef lUse state_val CFGExit_wf.wod_backward_slice"
map_swap_sum_plus_oracle: fixes oracle1 :: "'h \<Rightarrow> 'j \<Rightarrow> ('l \<times> 'h) spmf"   and oracle2 :: "'h \<Rightarrow> 'i \<Rightarrow> ('k \<times> 'h) spmf" shows "map_fun id (map_fun swap_sum (map_spmf (map_prod swap_sum id))) (oracle1 \<oplus>\<^sub>O oracle2) = oracle2 \<oplus>\<^sub>O oracle1"
finite_Fvars: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "finite (Fvars \<phi>)"
synth_analz_incr: fixes H :: "msg set" shows "H \<subseteq> synth (analz H)"
of_rat_neg_one: shows "of_rat (- 1) = - (1::'a)"
distinct_1: fixes x11 :: "exp"   and x12 :: "exp"   and x2 :: "Vars.var" shows "Alts x11 x12 \<noteq> Arg x2"
returns_to_implicit_function: fixes S :: "'a set"   and s :: "'a \<Rightarrow> real"   and x :: "'a"   and Ds :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" assumes "returns_to {x \<in> S. s x = 0} x"   and "closed S"   and "\<And>x. (s has_derivative blinfun_apply (Ds x)) (at x)"   and "isCont Ds (poincare_map {x \<in> S. s x = 0} x)"   and "blinfun_apply (Ds (poincare_map {x \<in> S. s x = 0} x)) (f (poincare_map {x \<in> S. s x = 0} x)) \<noteq> 0" obtains u :: "'a \<Rightarrow> real" and e :: "real" where "s (flow0 x (u x)) = 0" and "u x = return_time {x \<in> S. s x = 0} x" and "\<And>y. y \<in> cball x e \<Longrightarrow> s (flow0 y (u y)) = 0" and "continuous_on (cball x e) u" and "(\<lambda>t. (t, u t)) ` cball x e \<subseteq> Sigma X existence_ivl0" and "0 < e" and "(u has_derivative blinfun_apply (- blinfun_scaleR_left (inverse (blinfun_apply (Ds (poincare_map {x \<in> S. s x = 0} x)) (f (poincare_map {x \<in> S. s x = 0} x)))) o\<^sub>L (Ds (poincare_map {x \<in> S. s x = 0} x) o\<^sub>L flowderiv x (return_time {x \<in> S. s x = 0} x)) o\<^sub>L embed1_blinfun)) (at x)"
Lim_ident_at: fixes x :: "'b"   and s :: "'b set" assumes "at x within s \<noteq> bot" shows "Lim (at x within s) (\<lambda>x. x) = x"
ssubst_insert2: fixes p :: "perm"   and Vs :: "name set"   and V :: "name set"   and i :: "name"   and t :: "tm" assumes "quote_perm p Vs"   and "finite V"   and "i \<notin> V" shows "ssubst t (insert i V) (make_F Vs p) = ssubst (subst i (make_F Vs p i) t) V (make_F Vs p)"
assms_2: fixes R :: "('b \<times> 'b) set" assumes "trans R" shows "trans (f R)"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Arr (t \<^bold>\<otimes> u) = (Arr t \<and> Arr u)"
Areflassump2: shows "Arefl R0 \<inter> Arefl (R1 \<union> R2) \<subseteq> {[]}"
vbrelation_vinE: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> s" obtains a :: "V" and b :: "V" where "x = \<langle>a, b\<rangle>" and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s" and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> s"
cases: fixes x :: "'b literal" obtains xa :: "'b equation" where "x = Pos xa__"   | xa :: "'b equation" where "x = Neg xa"
order_iff: fixes b :: "'c multiset"   and a :: "'c multiset" shows "(b \<subseteq># a) = (a = a \<union># b)"
one_or_eq: fixes a :: "'a" shows "(1::'a) OR a = a + of_bool (even a)"
Par_ZObis_discrL1: fixes indis :: "('a \<times> 'a) set"   and c1 :: "('b, 'c) com"   and d :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis"   and "(c1, d) \<in> PL_Indis.ZObis tval aval indis"   and "PL_Indis.discr tval aval indis c2" shows "(c1 | c2, d) \<in> PL_Indis.ZObis tval aval indis"
span_span: fixes A :: "(('c \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'c" assumes "A \<subseteq> dest.tangent_space p" shows "span (span A) = span A"
take_zip: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'c list" shows "take n (xs || ys) = take n xs || take n ys"
induct: fixes P1 :: "freeExp \<Rightarrow> bool"   and P2 :: "freeExp list \<Rightarrow> bool"   and freeExp :: "freeExp" assumes "\<And>x. P1 (VAR x)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (PLUS x1 x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (FNCALL x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 freeExp"
intm: shows "integrable M (\<lambda>x. fm (T x) * g x)"
h_def: shows "h \<equiv> \<lambda>(T, x). let y = max (x k) (- T) in (\<lambda>i. if i = k then y else sqrt (1 - y\<^sup>2) / sqrt (1 - (x k)\<^sup>2) * x i)"
bij_drop_map: fixes i :: "nat" shows "bij (drop_map i)"
symrun_interp_commute: fixes \<Gamma>\<^sub>1 :: "'b constr list"   and \<Gamma>\<^sub>2 :: "'b constr list" shows "\<lbrakk>\<lbrakk> \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<Gamma>\<^sub>2 @ \<Gamma>\<^sub>1 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
thin: fixes H :: "fmla set"   and A :: "fmla"   and H' :: "fmla set" assumes "H \<turnstile> A"   and "H \<subseteq> H'" shows "H' \<turnstile> A"
sccs_verts_impl_correct: fixes G :: "'b list \<times> ('b \<times> 'b) list" assumes "pair_pseudo_graph (list_digraph G)" shows "pre_digraph.sccs_verts (with_proj (list_digraph G)) = sccs_verts_impl G"
bogus_def: shows "local.bogus = [Mkdir user\<^sub>1 perms\<^sub>1 [user\<^sub>1, name\<^sub>1], Mkdir user\<^sub>2 perms\<^sub>2 [user\<^sub>1, name\<^sub>1, name\<^sub>2], Creat user\<^sub>2 perms\<^sub>2 [user\<^sub>1, name\<^sub>1, name\<^sub>2, name\<^sub>3]]"
vcg_post_imp: fixes Q :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and R :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and P :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and c :: "('answer, 'location, 'question, 'state) com" assumes "\<And>s. Q s \<Longrightarrow> R s"   and "\<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>" shows "\<lbrace>P\<rbrace> c \<lbrace>R\<rbrace>"
alt_group_def: fixes n :: "nat" shows "alt_group n = sym_group n\<lparr>carrier := {p. p permutes {1..n} \<and> evenperm p}\<rparr>"
that: fixes k :: "'a \<rightarrow> ('b \<times> 'a)\<cdot>'c" assumes "y = StateT\<cdot>k" shows "thesis"
resume_ABORT: fixes r :: "'g option" shows "Resumption.resumption.resume (Resumption.resumption.Done r) = (\<lambda>inp. ABORT)"
distinct_var_order_Id_var_order: fixes a :: "char list list" assumes "distinct a"   and "sorted_wrt (rel2p (Id \<union> var_order_rel)) a" shows "sorted_wrt var_order a"
proper_def: fixes ct :: "'s \<Rightarrow> 's pmf" shows "proper ct = (ct \<in> Pi\<^sub>E S K \<and> (\<forall>s\<in>S\<^sub>r. 0 < v (simple ct s)))"
matrix_mul_assoc: fixes A :: "(('d, 'g) vec, 'f) vec"   and B :: "(('d, 'h) vec, 'g) vec"   and C :: "(('d, 'e) vec, 'h) vec" shows "A ** (B ** C) = A ** B ** C"
inj_eq: fixes f :: "'g \<Rightarrow> 'h"   and x :: "'g"   and y :: "'g" assumes "inj f" shows "(f x = f y) = (x = y)"
nth_prime_lower_bound_gen: fixes c :: "real"   and n :: "nat" assumes "0 < c"   and "0 < n"   and "\<And>n. 2 \<le> n \<Longrightarrow> \<pi> (real n) < 1 / c * (real n / ln (real n))" shows "c * (real n * ln (real n)) \<le> real (nth_prime (n - 1))"
intros_3: fixes \<pi> :: "nat \<Rightarrow> 'n"   and m :: "nat"   and n :: "nat"   and v :: "'var" assumes "(\<pi>, m) \<in> scp"   and "n dd\<^bsup>\<pi>,v\<^esup>\<rightarrow> m" shows "(\<pi>, n) \<in> scp"
abs_eq: fixes xb :: "'c"   and xa :: "'d"   and x :: "'c \<Rightarrow> 'd option" shows "Mapping.update xb xa (Mapping x) = Mapping (x(xb \<mapsto> xa))"
zip_arg_subterm_split: fixes x :: "'a"   and y :: "'b"   and xs :: "'a list"   and ys :: "'b list" assumes "(x, y) \<in> set (zip xs ys)" obtains xs' :: "'a list" and xs'' :: "'a list" and ys' :: "'b list" and ys'' :: "'b list" where "xs = xs' @ x # xs''" and "ys = ys' @ y # ys''" and "length xs' = length ys'"
diff_ge_0_iff_ge: fixes a :: "'a"   and b :: "'a" shows "((0::'a) \<le> a - b) = (b \<le> a)"
add_diff_assoc2: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "a \<subseteq># b" shows "b - a + c = b + c - a"
Diff_insert: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "A - insert a B = A - B - {a}"
dp: shows "\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0"
i: shows "i < dim_row (mat_of_cols (dim_row A) [col A 0] @\<^sub>c mat_of_cols (dim_row A) (map (col A) [1..<dim_col A]))"
A_ub3: fixes \<alpha> :: "real"   and \<beta> :: "real"   and t :: "'b tree"   and a :: "'b" assumes "Splay_Analysis \<alpha> \<beta>"   and "bst t" shows "Splay_Analysis.A \<alpha> \<beta> a t \<le> log \<alpha> (real (size1 t)) + 1"
solves_Vagree: fixes I :: "interp"   and F :: "real \<Rightarrow> variable \<Rightarrow> real"   and x :: "char"   and \<theta> :: "trm"   and \<zeta> :: "real" assumes "solves_ODE I F x \<theta>" shows "Vagree (F \<zeta>) (F 0) (- {RVar x, DVar x})"
0: shows "prv (exu out (instInpP \<sigma> n \<tau>))"
closed_csubspace_def: fixes A :: "'b set" shows "closed_csubspace A \<equiv> csubspace A \<and> closed A"
simps_1: fixes A :: "nat \<Rightarrow> 'b set" shows "mkdisjoint A 0 = A 0"
left_idem: fixes a :: "'c"   and b :: "'c" shows "min a (min a b) = min a b"
q_Eats_iff: fixes x :: "hf"   and y :: "hf"   and x' :: "hf"   and y' :: "hf" shows "(q_Eats x y = q_Eats x' y') = (x = x' \<and> y = y')"
right_idem: fixes a :: "'a upred"   and b :: "'a upred" shows "((a \<and> b) \<and> b) = (a \<and> b)"
less_irrefl: fixes x :: "'a Word.word" shows "\<not> x <s x"
SeqTermP_Eats: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and s2 :: "tm"   and k1 :: "tm"   and k2 :: "tm"   and t1 :: "tm"   and t2 :: "tm" assumes "atom s \<sharp> (k, s1, s2, k1, k2, t1, t2)"   and "atom k \<sharp> (s1, s2, k1, k2, t1, t2)" shows "{SeqTermP s1 k1 t1, SeqTermP s2 k2 t2} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqTermP (Var s) (Var k) (Q_Eats t1 t2)))"
that: shows "A \<inter> set_spmf p = B \<inter> set_spmf p"
assms_2: shows "H \<turnstile> LstSeqP f1 k1 y1"