hcju/isabelleps huggingface dataset:
opp_chamberD1: fixes C :: "'a set" assumes "C \<in> f \<turnstile> \<C>" shows "(THE D. D \<in> \<C> - f \<turnstile> \<C> \<and> f ` D = C) \<in> \<C> - f \<turnstile> \<C>"
0: shows "Group.group (units_of (residue_ring (p ^ m)))"
add_le_same_cancel2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq (plus a b) b = less_eq a zero"
seq: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and t :: "state"   and rs\<^sub>2 :: "'c rule list"   and t' :: "state" assumes "\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> t"   and "\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow>\<^sub>\<alpha> t'" shows "\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ rs\<^sub>2, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> t'"
quadratic_sub_fm_helper: fixes Dv :: "real"   and Cv :: "real"   and var :: "nat"   and c :: "real mpoly"   and z :: "nat"   and L :: "real list"   and a :: "real mpoly"   and Av :: "real"   and b :: "real mpoly"   and Bv :: "real"   and d :: "real mpoly"   and F :: "atom fm" assumes "Dv \<noteq> 0"   and "0 \<le> Cv"   and "var \<notin> vars c"   and "var + z < length L"   and "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) a = Av"   and "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) b = Bv"   and "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) c = Cv"   and "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) d = Dv" shows "eval F (L[var + z := (Av + Bv * sqrt Cv) / Dv]) = eval (quadratic_sub_fm_helper var a b c d F z) (L[var + z := sqrt Cv])"
assms_4: shows "topspace X \<inter> S \<noteq> {}"
pickE_def: fixes xl :: "Preliminaries.var list"   and al :: "'univ list" shows "pickE xl al \<equiv> SOME \<xi>. wtE \<xi> \<and> (\<forall>i<length xl. \<xi> (xl ! i) = al ! i)"
wary_inf_ary: fixes s :: "('s, 'v) tm" assumes "\<And>\<zeta>. arity_hd \<zeta> = \<infinity>" shows "wary s"
doesnt_modify_def: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and x :: "'Var" shows "doesnt_modify c x = (\<forall>mds mem c' mds' mem'. \<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle> \<longrightarrow> mem x = mem' x)"
3: fixes y :: "nat" assumes "y < x" shows "lookup_pp (rep_nat_pp (PP_oalist xs)) y = lookup_pp (rep_nat_pp (PP_oalist ys)) y"
oseq_no_change_on_send: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "('d, 'e, 'f, 'g) seqp"   and a :: "'e seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "('d, 'e, 'f, 'g) seqp"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'e, 'f, 'g) seqp"   and i :: "nat" assumes "((\<sigma>, s), a, \<sigma>', s') \<in> oseqp_sos \<Gamma> i" shows "case a of receive ma \<Rightarrow> True | \<tau>\<^sub>s \<Rightarrow> True | _ \<Rightarrow> \<sigma>' i = \<sigma> i"
weaken_post: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<turnstile> {P} c {Q}"   and "\<forall>s. Q s \<longrightarrow> Q' s" shows "\<turnstile> {P} c {Q'}"
FactGroup_inj_on: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "group_hom G H h" shows "inj_on (\<lambda>X. the_elem (h ` X)) (carrier (G Mod kernel G H h))"
dom_eq_empty_conv: fixes f :: "'b \<Rightarrow> 'c option" shows "(dom f = {}) = (f = Map.empty)"
distinct_37: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Stmt.If x51 x52 x53"
typ_of_def: fixes t :: "Core.term" shows "typ_of t = (if \<exists>T. \<turnstile>\<^sub>\<tau> t : T then Some (THE T. \<turnstile>\<^sub>\<tau> t : T) else None)"
cod_simp: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.cod f = cod f"
emeasure_real: fixes A :: "'a set" shows "\<exists>r\<ge>0. emeasure (return M x) A = ennreal r"
integrable_bernoulli_pmf: fixes p :: "real"   and f :: "bool \<Rightarrow> 'a" shows "integrable (measure_pmf (bernoulli_pmf p)) f"
smooth_on_scaleR: fixes k :: "enat"   and S :: "'c set"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> 'd" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. f x *\<^sub>R g x)"
Abs_freeword_freelist_uminus_add_in_Q: fixes xs :: "('w \<times> bool) list" assumes "proper_signed_list xs"   and "order.greater_eq S (fst ` set xs)" shows "- Abs_freelistfst xs + Abs_freeword xs \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
wf_inst_formula_atom: fixes a :: "PDDL_STRIPS_Semantics.term atom formula" assumes "wf_fmla_atom (ty_term Q constT) a" shows "wf_fmla_atom objT ((map_formula \<circ> map_atom \<circ> subst_term) f a)"
insert: fixes F :: "'a set"   and x :: "'a" assumes "finite F"   and "F \<noteq> {}"   and "F \<subseteq> S"   and "x \<in> S"   and "x \<notin> F"   and "P F" shows "P (insert x F)"
at_decompose: fixes c :: "('g, 'h, 'i, 'j) com"   and ictxt :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com"   and fctxt :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com list"   and c0 :: "('g, 'h, 'i, 'j) com" assumes "(c, ictxt, fctxt) \<in> decompose_com c0" shows "atC c \<subseteq> atC c0"
IH_3: assumes "\<langle>c: t \<doteq> 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> \<langle>c: t' \<doteq> s'\<rangle> \<in> set S"
sgn_zhmultiset_def: fixes M :: "zhmultiset" shows "sgn M = (if M = 0 then 0 else if 0 < M then 1 else - 1)"
step': fixes Ea :: "('a \<times> ('b, 'a) obs_event action) list"   and obsa :: "('a \<times> ('b, 'a) obs_event action) llist"   and aa :: "'c" assumes "X Ea obsa aa" shows "obsa = LNil \<or> (\<exists>obs' obs''. obsa = lappend obs' obs'' \<and> obs' \<noteq> LNil \<and> ta_hb_consistent P Ea obs' \<and> (lfinite obs' \<longrightarrow> (\<exists>a. X (Ea @ list_of obs') obs'' a) \<or> ta_hb_consistent P (Ea @ list_of obs') obs''))"
del_edge_undirected_degree_plus: fixes g :: "('a, 'b) graph"   and v :: "'a"   and e :: "'b"   and v' :: "'a" assumes "finite (edges g)"   and "(v, e, v') \<in> edges g"   and "(v', e, v) \<in> edges g" shows "degree v (del_unEdge v e v' g) + 1 = degree v g"
counit_simps_5: shows "trg\<^sub>D \<epsilon>' = trg\<^sub>D (F f)"
simps_12: fixes ve :: "nat" shows "(extN.Top \<le> extN.N ve) = False"
lexord_asymmetric: fixes R :: "('g \<times> 'g) set"   and a :: "'g list"   and b :: "'g list" assumes "asym R"   and "(a, b) \<in> lexord R" shows "(b, a) \<notin> lexord R"
1: shows "arrow_of_spans_of_maps (\<cdot>) (\<star>) \<a> \<i> src trg s\<^sub>0 s\<^sub>1 s\<^sub>0 s\<^sub>1 (src s\<^sub>0)"
A_def: shows "A = {x \<in> space M. f (T x) \<le> f x}"
arr_dom: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "C.VV.arr (C.VV.dom f)"
alift_of_remove_min_if: fixes q :: "('c, 'd) pq" shows "alist_of (remove_min q) = (if is_empty q then [] else tl (alist_of q))"
1: shows "prv (imp (cnj \<phi>1 (cnj \<phi> (lcnj (\<phi>1s @ \<phi>2s)))) (cnj \<phi>1 (lcnj (\<phi>1s @ \<phi> # \<phi>2s))))"
pdj: shows "ipd (\<pi> 0) pd\<rightarrow> \<pi> j"
last_state_of_append: fixes e' :: "'b \<times> ('c \<times> 'b) list"   and e :: "'b \<times> ('c \<times> 'b) list" assumes "fst e' = last_state e" shows "last_state (append_exec e e') = last_state e'"
axioms_1: fixes inPorts :: "'a \<Rightarrow> 'b fset"   and outPorts :: "'a \<Rightarrow> 'c fset"   and nodeOf :: "'d \<Rightarrow> 'a"   and hyps :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option"   and vertices :: "'d fset"   and labelsIn :: "'a \<Rightarrow> 'e \<Rightarrow> 'f"   and labelsOut :: "'a \<Rightarrow> 'g \<Rightarrow> 'f"   and freshenLC :: "nat \<Rightarrow> 'h \<Rightarrow> 'h"   and renameLCs :: "('h \<Rightarrow> 'h) \<Rightarrow> 'f \<Rightarrow> 'f"   and lconsts :: "'f \<Rightarrow> 'h set"   and closed :: "'f \<Rightarrow> bool"   and subst :: "'i \<Rightarrow> 'f \<Rightarrow> 'f"   and subst_lconsts :: "'i \<Rightarrow> 'h set"   and subst_renameLCs :: "('h \<Rightarrow> 'h) \<Rightarrow> 'i \<Rightarrow> 'i"   and anyP :: "'f"   and vidx :: "'d \<Rightarrow> nat"   and inst :: "'d \<Rightarrow> 'i"   and edges :: "(('d \<times> 'g) \<times> 'd \<times> 'e) set" assumes "Solution inPorts outPorts nodeOf hyps vertices labelsIn labelsOut freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP vidx inst edges" shows "Instantiation inPorts outPorts hyps vertices freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP vidx"
entails_pure: fixes P :: "assn"   and b :: "bool"   and Q :: "assn" shows "(\<not> (P * \<up> b \<Longrightarrow>\<^sub>A Q)) = (\<not> (P \<Longrightarrow>\<^sub>A Q) \<and> b)"
vsv_vsingleton: fixes a :: "V"   and b :: "V" shows "vsv (set {\<langle>a, b\<rangle>})"
is_constant_seqI: fixes s :: "nat \<Rightarrow> 'a"   and R :: "('a, 'b) ring_scheme"   and a :: "'a" assumes "s \<in> carrier R\<^bsup>\<omega>\<^esup>"   and "\<And>k. s k = a" shows "is_constant_seq R s"
has_integral_inverse_power_to_inf: fixes n :: "nat"   and a :: "real" assumes "1 < n"   and "0 < a" shows "((\<lambda>x. 1 / x ^ n) has_integral 1 / (real (n - 1) * a ^ (n - 1))) {a..}"
fresh_def: fixes "next" :: "'a \<Rightarrow> 'a" shows "fresh next \<equiv> \<forall>x. x < next x"
iso_def: fixes f :: "'c local.term set" shows "local.iso f \<equiv> \<exists>g. inverse_arrows f g"
u_tendsto: shows "u \<midarrow>x\<rightarrow> u x"
R26_bc_2: shows "pmf (sds R26) c = 0"
smallD: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and c :: "real" assumes "f \<in> o[F](g)"   and "0 < c" shows "\<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)"
inject_5: fixes x61 :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)"   and x62 :: "bexp"   and x63 :: "acom"   and y61 :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)"   and y62 :: "bexp"   and y63 :: "acom" shows "({x61} WHILE x62 DO x63 = {y61} WHILE y62 DO y63) = (x61 = y61 \<and> x62 = y62 \<and> x63 = y63)"
assms_1: shows "Q (\<sigma> |\<^sub>s j)"
scale_spmf_return_None: fixes r :: "real" shows "scale_spmf r (return_pmf None) = return_pmf None"
is_unit_hom_iff: fixes hom :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "unit_preserving_hom hom" shows "(hom x dvd (1::'b)) = (x dvd (1::'a))"
arr_Set_ArrCod_in_Vset: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Set \<alpha> T" shows "T\<lparr>ArrCod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
a: shows "\<forall>u. a u \<in> \<rat> \<and> a u < x \<bullet> u \<and> x \<bullet> u - a u < e'"
prems_1: shows "add_mset x M \<in> multisets A"
\<Delta>_simps_1: shows "seq ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<rho>\<sigma>.tab"
summableI: fixes f :: "nat \<Rightarrow> 'b" shows "summable f"
interior_of_subtopology_subset: fixes U :: "'c set"   and X :: "'c topology"   and S :: "'c set" shows "U \<inter> X interior_of S \<subseteq> subtopology X U interior_of S"
40: assumes "\<And>f'. f' \<in> F - {f} \<Longrightarrow> homogeneous f'" shows "hom_decomp ps'"
simps_10: fixes f1 :: "int \<Rightarrow> int list \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> 'a"   and x11 :: "int"   and x12 :: "int list" shows "(case Le x11 x12 of Le x xa \<Rightarrow> f1 x xa | Dvd x xa xb \<Rightarrow> f2 x xa xb | NDvd x xa xb \<Rightarrow> f3 x xa xb) = f1 x11 x12"
mult1_def: fixes r :: "('b \<times> 'b) set" shows "mult1 r = {(N, M). \<exists>a M0 K. M = add_mset a M0 \<and> N = M0 + K \<and> (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)}"
assms: fixes op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "op \<in> set (ast_problem.ast\<delta> prob)" shows "consistent_pres_op op"
wls_nchotomy: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "wls s X" shows "(\<exists>xs x. asSort xs = s \<and> X = Var xs x) \<or> (\<exists>delta inp binp. stOf delta = s \<and> wlsInp delta inp \<and> wlsBinp delta binp \<and> X = Op delta inp binp)"
8: shows "set rs'' \<subseteq> set rs' \<union> set rs"
not_eq_order_implies_strict: fixes a :: "'a list"   and b :: "'a list" assumes "a \<noteq> b"   and "border b a" shows "strict_border b a"
inject: fixes x2 :: "'b"   and y2 :: "'b" shows "(Inner x2 = Inner y2) = (x2 = y2)"
deg_smult: fixes a :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "f \<in> carrier (UP R)"   and "a \<otimes> f (deg R f) \<noteq> \<zero>" shows "deg R (a \<odot>\<^bsub>UP R\<^esub> f) = deg R f"
stream_rel_snth: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and u :: "'b stream"   and v :: "'c stream" shows "stream_all2 P u v = (\<forall>i. P (u !! i) (v !! i))"
awlast_in_verts: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list" assumes "wf_digraph G"   and "u \<in> verts G"   and "set p \<subseteq> arcs G" shows "last (pre_digraph.awalk_verts G u p) \<in> verts G"
simps: fixes P :: "'a state \<Rightarrow> exp0 \<Rightarrow> 'a state \<times> (v, v) result \<Rightarrow> bool"   and a1 :: "'a state"   and a2 :: "exp0 list"   and a3 :: "'a state \<times> (v list, v) result" shows "Big_Step_Unclocked_Single.evaluate_list P a1 a2 a3 = ((\<exists>s. a1 = s \<and> a2 = [] \<and> a3 = (s, Rval [])) \<or> (\<exists>s1 e s2 v es s3 vs. a1 = s1 \<and> a2 = e # es \<and> a3 = (s3, Rval (v # vs)) \<and> P s1 e (s2, Rval v) \<and> Big_Step_Unclocked_Single.evaluate_list P s2 es (s3, Rval vs)) \<or> (\<exists>s1 e s2 err es. a1 = s1 \<and> a2 = e # es \<and> a3 = (s2, Rerr err) \<and> P s1 e (s2, Rerr err)) \<or> (\<exists>s1 e s2 v es s3 err. a1 = s1 \<and> a2 = e # es \<and> a3 = (s3, Rerr err) \<and> P s1 e (s2, Rval v) \<and> Big_Step_Unclocked_Single.evaluate_list P s2 es (s3, Rerr err)))"
K_def: fixes i :: "'a" shows "K i = cone hull {1} \<times> S i"
WTrt_binop_fun: fixes P :: "'c prog"   and T1 :: "ty"   and bop :: "bop"   and T2 :: "ty"   and T :: "ty"   and T' :: "ty" assumes "P \<turnstile> T1\<guillemotleft>bop\<guillemotright>T2 : T"   and "P \<turnstile> T1\<guillemotleft>bop\<guillemotright>T2 : T'" shows "T = T'"
zero_compare_simps_6: fixes a :: "'a"   and b :: "'a" shows "((0::'a) < a \<cdot> b) = ((0::'a) < a \<and> (0::'a) < b \<or> a < (0::'a) \<and> b < (0::'a))"
sparsegrid_length: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and lm :: "nat" assumes "p \<in> sparsegrid dm lm" shows "length p = dm"
perm_sym: fixes xs :: "'d list"   and ys :: "'d list" assumes "xs <~~> ys" shows "ys <~~> xs"
vimage_Compl: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'e set" shows "f -` (- A) = - f -` A"
prems_3: shows "\<sigma>' \<in> small_nstep P \<sigma> n'"
subcringE_2: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subcring H R" shows "\<zero>\<^bsub>R\<^esub> \<in> H"
assms_3: shows "f \<subseteq> edges G \<and> (\<forall>(e1, e2)\<in>f. \<not> p e1 \<le> p e2)"
simpfm_qf: fixes p :: "fm" assumes "qfree p" shows "qfree (simpfm p)"
size_Diff_subset_Int: fixes M :: "'b multiset"   and M' :: "'b multiset" shows "size (M - M') = size M - size (M \<inter># M')"
min_range_mono_def: fixes p :: "nat"   and F :: "floatarith \<Rightarrow> floatarith"   and DF :: "floatarith \<Rightarrow> floatarith"   and l :: "float"   and u :: "float"   and X :: "(real \<times> real pdevs) \<times> real" shows "min_range_mono p F DF l u X = (let L = Num l; U = Num u in approx p (Min (DF L) (DF U)) [] \<bind> (\<lambda>aivl. let a = lower aivl; A = Num a in approx p (Half (F L + F U - A * (L + U))) [] \<bind> (\<lambda>bivl. let (b, be) = mid_err bivl; (B, Be) = (Num (float_of b), Num (float_of be)) in approx p (Half (F U - F L - A * (U - L)) + Be) [] \<bind> (\<lambda>divl. Some (affine_unop p (real_of_float a) b (real_of_float (upper divl)) X)))))"
m0: shows "x' - star_of x \<in> Infinitesimal - {0}"
1: shows "{(f xK, snd xK) |xK. xK \<in> p} tagged_division_of {a..b}"
bigtheta_trans1: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> o[F](g)"   and "g \<in> \<Theta>[F](h)" shows "f \<in> o[F](h)"
simps_4: fixes a :: "nat"   and b :: "nat"   and A :: "atom fm"   and B :: "atom fm" shows "swap_fm a b (And A B) = And (swap_fm a b A) (swap_fm a b B)"
err_semilat_upto_esl: fixes L :: "'b set \<times> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'b \<Rightarrow> 'b err)"   and m :: "nat" assumes "err_semilat L" shows "err_semilat (upto_esl m L)"
cases: fixes x :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" obtains "x = \<langle>\<rangle>"   | l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" and xa :: "'c \<times> 'd" and c :: "tcolor" and uu_ :: "'c \<times> 'd" and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" where "x = \<langle>l, (xa, c, uu_), r\<rangle>"
orthogonal_Gram_Schmidt_matrix2: fixes A :: "(('a, 'c) vec, 'b) vec" shows "pairwise orthogonal (columns (Gram_Schmidt_matrix A))"
1: fixes zs :: "'c list" shows "list_all (\<lambda>(xy, z). case xy of (x, y) \<Rightarrow> P x y z) ((xs || ys) || zs) = list_all (\<lambda>(x, y, z). P x y z) (xs || ys || zs)"
has_vector_derivative_mult_left: fixes f :: "real \<Rightarrow> 'a"   and x :: "'a"   and F :: "real filter"   and a :: "'a" assumes "(f has_vector_derivative x) F" shows "((\<lambda>x. f x * a) has_vector_derivative x * a) F"
simplex_like_no_pseudominimal_in_below_in_imp_singleton: fixes x :: "'a"   and P :: "'a set" assumes "x \<in> P"   and "simplex_like (P .\<^bold>\<le> x)"   and "\<And>z. \<not> pseudominimal_in (P .\<^bold>\<le> x) z" shows "P .\<^bold>\<le> x = {x}"
drinks2_vend_r2_String: fixes r :: "nat \<Rightarrow>f value option"   and x2 :: "String.literal" assumes "r $ 2 = Some (value.Str x2)" shows "possible_steps drinks2 2 r STR ''vend'' [] = {||}"
p'_def: shows "p' \<equiv> \<lambda>(x, y). (x, \<lambda>i. y i - f' x i)"
cons_interleave_cases: fixes a :: "'b"   and l :: "'b list"   and l1 :: "'b list"   and l2 :: "'b list" assumes "a # l \<in> l1 \<otimes> l2" obtains lh :: "'b list" where "l1 = a # lh__" and "l \<in> lh__ \<otimes> l2"   | lh :: "'b list" where "l2 = a # lh" and "l \<in> l1 \<otimes> lh"
psi_convex2: fixes a :: "real \<times> real"   and c :: "real \<times> real"   and b :: "real \<times> real"   and d :: "real \<times> real"   and u :: "real"   and v :: "real" assumes "psi a c b"   and "psi a d b"   and "0 \<le> u"   and "0 \<le> v"   and "u + v = 1" shows "psi a (u *\<^sub>R c + v *\<^sub>R d) b"
a_get_component_def: fixes get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and ptr :: "(_) object_ptr" shows "l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_component get_root_node to_tree_order ptr = Heap_Error_Monad.bind (get_root_node ptr) to_tree_order"
buffon_set'_def: fixes l :: "real"   and d :: "real" shows "buffon_set' l d = {(x, \<phi>). (x, \<phi>) \<in> {0..d / 2} \<times> {0..pi} \<and> d / 2 - sin \<phi> * l / 2 \<le> x}"
case1: assumes "x \<le> z" shows "l_sort_aux (xs, [], ys @ x # z # zs) \<in> l_sort_set (xs, [], ys @ x # z # zs)"
comp_of_ords: fixes le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lt :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.linorder le lt" shows "comparator (comp_of_ords le lt)"
assms_3: shows "\<phi>\<^sub>S \<Psi> G \<subseteq>\<^sub>m execute_parallel_plan (execute_parallel_operator (\<phi>\<^sub>S \<Psi> I) ops') \<pi>"
assms: shows "\<not> fic (M[x\<turnstile>n>y]) c"
Majority_exQ: fixes P :: "Disk \<Rightarrow> bool" assumes "\<exists>D\<in>MajoritySet. \<forall>d\<in>D. P d" shows "\<forall>D\<in>MajoritySet. \<exists>d\<in>D. P d"
p1: shows "\<S>\<^sub>1 = \<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1"
simps: fixes I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and t :: "'b trm"   and s :: "'b trm" shows "validate_ground_eq I (Eq t s) = I t s"
mat_leqb_eqc_split_simplex_correct1: fixes b :: "rat vec"   and A :: "rat mat"   and c :: "rat vec"   and X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec"   and l :: "nat" assumes "dim_vec b = dim_row A"   and "simplex (mat_leqb_eqc A b c) = Inr X"   and "(x, y) = split_i_j_x (dim_col A) l X" shows "[A*\<^sub>vx]\<le>b"
tt_in_keys: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "p \<noteq> 0" shows "ordered_term.tt term_of_pair ord_term p \<in> keys p"
Pi_mem2: fixes f :: "'e \<Rightarrow> 'f"   and S :: "'e set"   and T :: "'f set"   and x :: "'e" assumes "f \<in> S \<rightarrow> T"   and "x \<in> S" shows "f x \<in> T"
assms_19: shows "vars F0 \<subseteq> Vbl"
strand_sem_split'_4: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and S' :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>d' \<theta>" shows "\<lbrakk>M \<union> ik\<^sub>s\<^sub>t S; S'\<rbrakk>\<^sub>d' \<theta>"
length_insert_nth: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "length (insert_nth n x xs) = Suc (length xs)"
INT_Int_distrib: fixes A :: "'b \<Rightarrow> 'a set"   and B :: "'b \<Rightarrow> 'a set"   and I :: "'b set" shows "(\<Inter>i\<in>I. A i \<inter> B i) = \<Inter> (A ` I) \<inter> \<Inter> (B ` I)"
a_2: shows "a \<notin> f ` topspace (nsphere p)"
assms_2: fixes i :: "nat" assumes "i < dim_vec y" shows "(0::'a) \<le> y $ i"
less_card_cut_less_imp_inext_nth_less: fixes n :: "nat"   and I :: "nat set"   and t :: "nat" assumes "n < card (I \<down>< t)" shows "I \<rightarrow> n < t"
Smith_normal_form_of_def: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "Smith_normal_form_of A bezout = (let (P'', D, Q'') = local.diagonalize A bezout; (P', S, Q') = diagonal_to_Smith_PQ D bezout in (P' ** P'', S, Q'' ** Q'))"
subterms_Binop: fixes a :: "floatarith"   and b :: "floatarith" shows "subterms (Binop a b) = insert (Binop a b) (subterms a \<union> subterms b)"
fin_make_untimed_def: fixes x :: "'a list list" shows "fin_make_untimed x \<equiv> concat x"
head_polybound0: fixes p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "polybound0 (head p)"
assoc_agreement: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "C.assoc a b c = assoc a b c"
size_single: fixes b :: "'a" shows "size (\<LM> b  \<RM>) = 1"
hull_eq: fixes S :: "'b set \<Rightarrow> bool"   and s :: "'b set" assumes "\<And>T. Ball T S \<Longrightarrow> S (\<Inter> T)" shows "(S hull s = s) = S s"