hcju/isabelleps huggingface dataset:
choose_v__in_uverts: fixes G :: "nat set \<times> nat set set"   and k :: "nat" assumes "short_cycles G k \<noteq> {}" shows "choose_v G k \<in> uverts G"
WhileFReds: fixes b :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and c :: "expr" assumes "P,E \<turnstile> \<langle>b,s\<rangle> \<rightarrow>* \<langle>false,s'\<rangle>" shows "P,E \<turnstile> \<langle>while (b) c,s\<rangle> \<rightarrow>* \<langle>unit,s'\<rangle>"
2: shows "a \<cdot> (dom a) \<cong> a"
composites_are_identities_2: shows "ide ((f \<star> g) \<star> h)"
scale_one: fixes x :: "'d" shows "(1::'d) * x = x"
characteristic_formula_is_characteristic: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and Q :: "'b"   and P :: "'b" assumes "indexed_nominal_ts TYPE('a) satisfies transition" shows "nominal_ts.valid satisfies transition Q (indexed_nominal_ts.characteristic_formula satisfies transition P) = nominal_ts.bisimilar satisfies transition P Q"
add_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> V"   and "y \<in> V" shows "x + y \<in> V"
mapping_single_eq_iff: fixes a :: "'b"   and b :: "'c"   and c :: "'b"   and d :: "'c" shows "(Poly_Mapping.single a b = Poly_Mapping.single c d) = (b = (0::'c) \<and> d = (0::'c) \<or> a = c \<and> b = d)"
Int_partGen: fixes I :: "'b set"   and J :: "'b set"   and P :: "'b set set" assumes "{I, J} \<subseteq> partGen P"   and "I \<inter> J \<noteq> {}" shows "I = J"
Basic: fixes f :: "'a \<Rightarrow> 'a"   and q :: "'a set" shows "\<parallel>- {s. f s \<in> q} Basic f q"
NullT_leaf: fixes C :: "CTypeId" shows "NullT \<le> CClassT C"
unhalt: shows "{\<lambda>tp. tp = ([], <args>)} p \<up>"
ag_pOp_assoc: fixes A :: "('c, 'd) aGroup_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "z \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> y \<plusminus>\<^bsub>A\<^esub> z = x \<plusminus>\<^bsub>A\<^esub> (y \<plusminus>\<^bsub>A\<^esub> z)"
assms_3: shows "0 \<in>\<^sub>\<circ> \<beta>"
box_conjE: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "(\<box>F) s"   and "(\<box>G) s" obtains "(\<box>(F \<and> G)) s"
lc_not_0: 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.lc term_of_pair ord_term p \<noteq> (0::'f)"
integral_cong: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "M = N"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "integral\<^sup>L M f = integral\<^sup>L N g"
nrename_id: fixes M :: "trm"   and x :: "name" shows "M[x\<turnstile>n>x] = M"
Int: fixes S :: "'a set"   and B :: "'a set set"   and U :: "'a set"   and V :: "'a set" assumes "generated_topology S B U"   and "generated_topology S B V" shows "generated_topology S B (U \<inter> V)"
Inf_from_def: fixes N :: "'g set" shows "no_labels.ground.Inf_from N = {\<iota> \<in> Inf_G. set (prems_of \<iota>) \<subseteq> N}"
set_image_le_iff: fixes A :: "'a set"   and f :: "'a \<Rightarrow> V"   and B :: "V" assumes "small A" shows "(set (f ` A) \<subseteq>\<^sub>\<circ> B) = (\<forall>x\<in>A. f x \<in>\<^sub>\<circ> B)"
dp_consistency_iterator_def: fixes P :: "'a \<Rightarrow> bool"   and lookup :: "'b \<Rightarrow> ('a, 'c option) state"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> ('a, unit) state"   and cnt :: "'b \<Rightarrow> bool"   and nxt :: "'b \<Rightarrow> 'b"   and sizef :: "'b \<Rightarrow> nat" shows "dp_consistency_iterator P lookup update cnt nxt sizef \<equiv> dp_consistency lookup update P \<and> iterator cnt nxt sizef"
f_def: shows "f \<equiv> \<lambda>x. \<Sum>n. P x n / (real (Suc n))\<^sup>2"
obj_sumE: fixes s :: "'e + 'f"   and P :: "bool" assumes "\<forall>x. s = pure x \<longrightarrow> P"   and "\<forall>x. s = Inr x \<longrightarrow> P" shows "P"
assms_2: shows "w\<^sub>2 =\<^sub>I w\<^sub>3"
insert_Diff_subset2: fixes a :: "'b"   and B :: "'b set"   and c :: "'b"   and D :: "'b set" shows "insert a (B - {c}) - D \<subseteq> insert a (B - insert c D)"
ACI_norm_lderivs: fixes ws :: "'a list"   and r :: "'b rexp" shows "\<guillemotleft>lderivs ws \<guillemotleft>r\<guillemotright>\<guillemotright> = \<guillemotleft>lderivs ws r\<guillemotright>"
fls_deriv_sub: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_deriv (f - g) = fls_deriv f - fls_deriv g"
comp_res: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<smile> ; - (x ; y) \<le> - y"
2: shows "\<forall>q'. (q', q) \<in> Ord - Id \<longrightarrow> q' \<notin> Q"
Sum1: fixes P :: "pi"   and Rs :: "Late_Semantics.residual"   and Q :: "pi" assumes "P \<longmapsto>\<^sub>l Rs" shows "P \<oplus> Q \<longmapsto>\<^sub>l Rs"
Cd_Above: fixes d :: "'b"   and X :: "'a set" shows "Bossy'.Cd d X = Above (BPd' d) (X \<inter> Field (BPd' d)) \<inter> X"
v_in_Cs: shows "v \<in> vars_clause_list Cs"
statEqvt: fixes p :: "(name \<times> name) list"   and \<Psi> :: "'b"   and \<Phi> :: "'c" shows "p \<bullet> (\<Psi> \<turnstile> \<Phi>) = p \<bullet> \<Psi> \<turnstile> p \<bullet> \<Phi>"
nres_correctD_1: fixes S :: "'a dres"   and \<Phi> :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "nres_of S \<le> SPEC \<Phi>"   and "S = dRETURN x" shows "\<Phi> x"
AddSA_def: fixes HA :: "('d, 'e, 'f) hierauto"   and SSA :: "'d \<times> ('d, 'e, 'f) seqauto" shows "(HA [++] SSA) = (let (S, SA) = SSA; DNew = HAInitValue HA; FNew = insert SA (SAs HA); ENew = HAEvents HA \<union> SAEvents SA; GNew = CompFun HA [f+] (S, SA) in Abs_hierauto (DNew, FNew, ENew, GNew))"
zero_hom_0_axioms: shows "zero_hom_0 (\<lambda>a. [:a:])"
m_concat_closed: fixes G :: "('d, 'e) monoid_scheme"   and l :: "'d list" assumes "Group.monoid G"   and "set l \<subseteq> carrier G" shows "foldr (\<otimes>\<^bsub>G\<^esub>) l \<one>\<^bsub>G\<^esub> \<in> carrier G"
bmod_assoc_aux1: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'c"   and h :: "'e \<Rightarrow> 'c"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d" shows "\<Squnion>{\<Squnion>{f u \<cdot> g v \<cdot> h w |u v. R y u v} |y w. R x y w} = \<Squnion>{uu_. \<exists>u v y w. uu_ = f u \<cdot> g v \<cdot> h w \<and> R y u v \<and> R x y w}"
h_def: shows "h \<equiv> \<lambda>n z. f (z / of_nat (Suc n))"
lam_pr1: shows "\<guillemotleft>\<Lambda>_pr1 : local.prod a b \<rightarrow> exp b c\<guillemotright>"
C: shows "Crypt (sesK (c * f)) \<lbrace>pubAK (c * (s + a * b)), Auth_Data g b, pubAK g, Crypt (priSK CA) (Hash (pubAK g))\<rbrace> \<in> synth (analz (A \<union> spies evsFR4))"
Suc_1: fixes XSS :: "(real \<times> real pdevs) list list" assumes "prove_nonneg prnt i p (slp_of_fas [fa]) XSS" shows "\<forall>XS\<in>set XSS. \<forall>xs\<in>Joints XS. 0 \<le> interpret_floatarith fa xs"
unifiers_occur_left_not_Var: fixes x :: "'a"   and t :: "('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" assumes "x \<in> vars_term t"   and "t \<noteq> Var x" shows "unifiers (insert (Var x, t) E) = {}"
eq: shows "map (\<lambda>i. v \<bullet> col (polys_to_mat ts ps) i) [0..<length ts] = map (\<lambda>s. v \<bullet> vec_of_list (map (\<lambda>p. lookup p s) ps)) ts"
pdevs_apply_pdev_upd: fixes X :: "'d pdevs"   and n :: "nat"   and x :: "'d" shows "pdevs_apply (pdev_upd X n x) = (pdevs_apply X)(n := x)"
prv_eql_subst_trm_id: fixes y :: "'var"   and \<phi> :: "'fmla"   and n :: "'trm" assumes "y \<in> var"   and "\<phi> \<in> fmla"   and "n \<in> num" shows "prv (imp (eql (Var y) n) (imp \<phi> (subst \<phi> n y)))"
subst_var_inv_wt: fixes \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and X :: "'var set" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>" shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t (subst_var_inv \<delta> X)"
heaps_similar: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "\<lbrace>\<Gamma>\<rbrace> \<triangleleft>\<triangleright>\<^sup>* \<N>\<lbrace>\<Gamma>\<rbrace>"
vwalkI_subgraph: fixes p :: "'a list"   and H :: "('a, 'b) pre_digraph"   and G :: "('a, 'b) pre_digraph" assumes "vwalk p H"   and "subgraph H G" shows "vwalk p G"
Afix_unroll: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ArityAnalysis.Afix Aexp \<Gamma>\<cdot>ae = ArityAnalysis.ABinds Aexp \<Gamma>\<cdot>(ArityAnalysis.Afix Aexp \<Gamma>\<cdot>ae) \<squnion> ae"
assms_5: shows "r2 = ((cmod BC)\<^sup>2 - A * D) / A\<^sup>2"
independent_split_2: fixes Us :: "'a list"   and Vs :: "'a list" assumes "local.independent K (Us @ Vs)" shows "local.independent K Us"
connT: fixes y :: "'b" assumes "y \<in> T" shows "connected (S \<inter> f -` {y})"
is_functor: shows "functor (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) local.map"
weak_lens_def: fixes x :: "'c \<Longrightarrow> 'd" shows "weak_lens x \<equiv> \<forall>\<sigma> v. get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v"
f1: shows "(r |\<^bsub>rel\<^esub> s) \<noteq> {}"
ab_app: assumes "1\<^sub>\<nat> \<in>\<^sub>\<circ> 2\<^sub>\<nat>" shows "ab\<lparr>1\<^sub>\<nat>\<rparr> \<in>\<^sub>\<circ> (if 1\<^sub>\<nat> = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>)\<lparr>Obj\<rparr>"
wf_ord0_leq: shows "wf (ord0_leq O ord0_prec\<^sup>+)"
assms: shows "evaluate_match ck env s v0 pes err_v (s', bv)"
weakening_5: fixes lholed :: "Lholed"   and i :: "nat" shows "\<not> Lfilled 0 lholed [$Br i] (cs @ ($* es))"
scale_eq_0_iff: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and x :: "'c" assumes "vector_space scale" shows "(scale a x = (0::'c)) = (a = (0::'b) \<or> x = (0::'c))"
that: fixes u :: "'a list"   and a :: "'a"   and v :: "'a stream" assumes "w = u @- a ## v"   and "fold (count cs) u k = l"   and "(cs ! l) a" shows "thesis"
Baire: fixes S :: "'b set"   and \<G> :: "'b set set" assumes "closed S"   and "countable \<G>"   and "\<And>T. T \<in> \<G> \<Longrightarrow> openin (top_of_set S) T \<and> S \<subseteq> closure T" shows "S \<subseteq> closure (\<Inter> \<G>)"
smult_cancel: fixes c :: "'d"   and f :: "'d poly"   and g :: "'d poly" shows "(smult c f = smult c g) = (c = (0::'d) \<or> f = g)"
assms_6: fixes s1 :: "'s1"   and x :: "'a" assumes "bad1 s1"   and "x \<in> outs_\<I> \<I>" shows "lossless_spmf (oracle1 s1 x)"
selectlike_def: fixes X :: "(variable \<Rightarrow> real) set"   and \<nu> :: "variable \<Rightarrow> real"   and V :: "variable set" shows "selectlike X \<nu> V = {\<omega> \<in> X. Vagree \<omega> \<nu> V}"
measurable_join1: fixes f :: "'c \<Rightarrow> 'd"   and N :: "'c measure"   and K :: "'d measure"   and M :: "'c measure measure" assumes "f \<in> N \<rightarrow>\<^sub>M K"   and "sets M = sets (subprob_algebra N)" shows "f \<in> join M \<rightarrow>\<^sub>M K"
finfun_Ex_def: fixes P :: "('a, bool) finfun" shows "finfun_Ex P = (\<not> finfun_All (finfun_comp Not P))"
all_total_totalize: fixes F :: "'c program" shows "all_total (totalize F)"
list_all2_lengthD: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list" assumes "list_all2 P xs ys" shows "length xs = length ys"
shiftr_p_def: fixes a :: "'b array"   and key :: "'b"   and i :: "nat ref" shows "shiftr_p a key i = !i \<bind> (\<lambda>i'. if 0 < i' then (a.(i' - 1)) \<bind> (\<lambda>x. return (key < x)) else return False)"
f_approx_bounded_above: obtains c :: "real" where "\<And>x. \<lbrakk>x\<^sub>0 \<le> x; x \<le> x\<^sub>1\<rbrakk> \<Longrightarrow> f_approx x \<le> c" and "0 < c"
count_perm: fixes A :: "'b list"   and B :: "'b list" assumes "\<forall>x. PermutationLemmas.count x A = PermutationLemmas.count x B" shows "A <~~> B"
lcl_conv: fixes h :: "'b"   and l :: "'a" shows "lcl (h, l) = l"
nextl_app: fixes q :: "hf"   and xs :: "'a list"   and ys :: "'a list" shows "MN.nextl q (xs @ ys) = MN.nextl (MN.nextl q xs) ys"
linear_inj_image_is_basis: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and W :: "('b, 'f, 'g) module_scheme"   and T :: "'d \<Rightarrow> 'f"   and B :: "'d set" assumes "linear_map K V W T"   and "inj_on T (carrier V)"   and "T ` carrier V = carrier W"   and "vectorspace.basis K V B"   and "vectorspace.fin_dim K V" shows "vectorspace.basis K W (T ` B)"
finite_vertex_set: shows "finite V\<^bsub>T\<^esub>"
sel_5: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x4 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x5 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x6 :: "'b \<Rightarrow> 'b"   and x7 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x8 :: "'b \<Rightarrow> 'b"   and x9 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x10 :: "'b \<Rightarrow> 'b"   and x11 :: "'b \<Rightarrow> 'b"   and x12 :: "int \<Rightarrow> 'b"   and x13 :: "'b \<Rightarrow> int"   and x14 :: "'b \<Rightarrow> bool" shows "arith_ops_record.minus (Arith_Ops_Record x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14) = x5"
countable_iff_bij: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "countable A = countable B"
ffold_finsert: fixes x :: "val \<times> val"   and A :: "(val \<times> val) fset"   and z :: "nat" assumes "x |\<notin>| A" shows "ffold (vadd \<circ> vprod_size) z (finsert x A) = (vadd \<circ> vprod_size) x (ffold (vadd \<circ> vprod_size) z A)"
UNIV_typedef_to_equivp: fixes Rep :: "'c \<Rightarrow> 'd"   and Abs :: "'d \<Rightarrow> 'c" assumes "type_definition Rep Abs UNIV" shows "equivp (=)"
card_sing_prop: fixes n :: "'b"   and P :: "'b \<Rightarrow> bool" shows "card {i. i = n \<and> P i} = (if P n then 1 else 0)"
le_def: fixes x :: "'a"   and y :: "'a" shows "le x y = (x \<le> y \<and> \<not> y \<le> x)"
bot_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e_def: shows "UML_Types.bot_class.bot \<equiv> Abs_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<bottom>"
vars_iff_subtermeq: fixes x :: "'c"   and t :: "('d, 'c) Term.term" shows "(x \<in> fv t) = (Var x \<sqsubseteq> t)"
no_writes_unchanged: fixes k :: "nat"   and l :: "nat"   and v :: "'var"   and \<sigma> :: "'var \<Rightarrow> 'val" assumes "k \<le> l"   and "\<forall>j\<in>{k..<l}. v \<notin> local.writes (path \<sigma> j)" shows "(\<sigma>\<^bsup>l\<^esup>) v = (\<sigma>\<^bsup>k\<^esup>) v"
N_def: shows "N = \<lceil>(1::'a) / d\<rceil>"
sstls2_def: shows "sstls2 = tl1_to_tl2 s2 sstls1"
list_of_LCons: fixes xs :: "'a llist"   and x :: "'a" assumes "lfinite xs" shows "list_of (LCons x xs) = x # list_of xs"
map_iterator_genord_rule_insert_P: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'e \<Rightarrow> 'f option"   and R :: "'e \<times> 'f \<Rightarrow> 'e \<times> 'f \<Rightarrow> bool"   and I :: "'e set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool" assumes "map_iterator_genord it m R"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> dom m - it; m k = Some v; it \<subseteq> dom m; I it \<sigma>; \<forall>k' v'. k' \<in> dom m - it - {k} \<and> m k' = Some v' \<longrightarrow> R (k, v) (k', v'); \<forall>k' v'. k' \<in> it \<and> m k' = Some v' \<longrightarrow> R (k', v') (k, v)\<rbrakk> \<Longrightarrow> I (insert k it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (dom m) \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> dom m; \<not> c \<sigma>; I it \<sigma>; \<forall>k v k' v'. k \<in> it \<and> m k = Some v \<and> k' \<notin> it \<and> m k' = Some v' \<longrightarrow> R (k, v) (k', v')\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
assms_5: shows "a \<in> \<R>\<^sub>+ \<Psi> v"
coprime_hom: fixes x :: "'a"   and y' :: "'b" shows "coprime (hom x) y' = coprime x (inv hom y')"
sd_distl: fixes act :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and Y :: "('c \<times> 'd) set"   and x :: "'c \<times> 'd" assumes "unital_quantale_module act"   and "Y \<noteq> {}" shows "unital_quantale_module.sd_prod act x (Sup_prod Y) = Sup_prod {unital_quantale_module.sd_prod act x y |y. y \<in> Y}"
extend_state_oracle_parametric: fixes S :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and S' :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_fun (rel_fun S (rel_fun C (rel_spmf (rel_prod R S)))) (rel_fun (rel_prod S' S) (rel_fun C (rel_spmf (rel_prod R (rel_prod S' S))))) extend_state_oracle extend_state_oracle"
M_poly_def: fixes m :: "int"   and p :: "int poly"   and x :: "int" shows "poly_mod.M_poly m p x \<equiv> poly_mod.M m (poly p x)"
antisym: fixes a :: "nat"   and b :: "nat" assumes "(a::nat) dvd (b::nat)"   and "(b::nat) dvd (a::nat)" shows "(a::nat) = (b::nat)"
strict_implies_not_eq: fixes a :: "'b word"   and b :: "'b word" assumes "a <s b" shows "a \<noteq> b"
dX_def: fixes X :: "'a set"   and d :: "'b" shows "PdXXX.dX X d = {x \<in> X. Xd x = d}"
to_fun_X_pow: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R" shows "to_fun (X_poly R [^]\<^bsub>P\<^esub> n) x = x [^] n"
sum_list_lec: fixes ls :: "'b le_constraint list" shows "sum_list ls = Le_Constraint (sum_list (map lec_rel ls)) (sum_list (map lec_poly ls)) (sum_list (map lec_const ls))"
function_mult_def: fixes S :: "'a set"   and R :: "('b, 'c) ring_scheme"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "function_mult S R f g = (\<lambda>x\<in>S. f x \<otimes>\<^bsub>R\<^esub> g x)"
Axiom_of_Powers: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> Vset \<beta>" shows "VPow a \<in>\<^sub>\<circ> Vset \<beta>"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "C.arr (C.L f)"
length_Suc_conv: fixes xs :: "'f list"   and n :: "nat" shows "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
prv_subst_imp_softSubst: fixes x :: "'var"   and t :: "'trm"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "t \<in> trm"   and "\<phi> \<in> fmla"   and "x \<notin> FvarsT t" shows "bprv (imp (subst \<phi> t x) (softSubst \<phi> t x))"
pell_mul_trivial_left: fixes D :: "nat"   and z :: "'a \<times> 'a" assumes "pell D" shows "pell.pell_mul D (1::'a, 0::'a) z = z"
t_bound3: shows "max (te p i) (te q i) \<le> t"
Iff_sym: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A IFF B" shows "H \<turnstile> B IFF A"
joinE: fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set" assumes "(a, b) \<in> A\<^sup>\<down>" obtains c :: "'b" where "(a, c) \<in> A\<^sup>*" and "(b, c) \<in> A\<^sup>*"
simps_88: fixes f1 :: "'a"   and f2 :: "'b basic_type \<Rightarrow> 'a"   and f3 :: "'b basic_type \<Rightarrow> 'a"   and f4 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f5 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f6 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f7 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f8 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f9 :: "String.literal \<rightharpoonup>\<^sub>f 'b OCL_Types.type \<Rightarrow> 'a"   and x8 :: "'b OCL_Types.type" shows "(case Sequence x8 of OclSuper \<Rightarrow> f1 | x[1] \<Rightarrow> f2 x | x[?] \<Rightarrow> f3 x | Collection x \<Rightarrow> f4 x | Set x \<Rightarrow> f5 x | OrderedSet x \<Rightarrow> f6 x | Bag x \<Rightarrow> f7 x | Sequence x \<Rightarrow> f8 x | Tuple x \<Rightarrow> f9 x) = f8 x8"
interval: shows "T = {t0 - e1'..t0 + e1'}"
set_2: fixes x21 :: "'c"   and x22 :: "'c list" shows "set (x21 # x22) = ??.Set.insert x21 (set x22)"
sums_def': fixes f :: "nat \<Rightarrow> 'a"   and s :: "'a" shows "f sums s = (\<lambda>n. sum f {0..n}) \<longlonglongrightarrow> s"
cInf_singleton: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup" shows "Inf {x} = x"
impl_of_inverse: fixes x :: "('a, 'b) RBT.rbt" shows "RBT (impl_of x) = x"
finite_fv: shows "finite (fv\<^sub>s\<^sub>t (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S'))"
Ln_divide_of_real: fixes r :: "real"   and z :: "complex" assumes "0 < r"   and "z \<noteq> 0" shows "Ln (z / complex_of_real r) = Ln z - complex_of_real (ln r)"
last_in_tl: fixes xs :: "'b list" assumes "1 < length xs" shows "last xs \<in> set (tl xs)"
assoc'_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a"   and "partial_magma.ide C b"   and "partial_magma.ide C c" shows "partial_magma.in_hom C (category.inv C (\<alpha> (a, b, c))) (T (a, T (b, c))) (T (T (a, b), c))"
inductive_on_induct: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and Q :: "'b \<Rightarrow> bool" assumes "inductive_on P A"   and "x \<in> A"   and "\<And>y. \<lbrakk>y \<in> A; \<And>x. \<lbrakk>x \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x\<rbrakk> \<Longrightarrow> Q y" shows "Q x"
cut_le_card: fixes I :: "'a set"   and t :: "'a" assumes "finite I" shows "card (I \<down>\<le> t) \<le> card I"
invar_update: fixes m :: "'b"   and a :: "'v"   and b :: "nat" assumes "Q_invar m" shows "Q_invar (Q_update a b m)"
sinh_zero_iff: fixes x :: "'a" shows "(sinh x = (0::'a)) = (exp x \<in> {1::'a, - (1::'a)})"
finsert_ident: fixes x :: "'c"   and A :: "'c fset"   and B :: "'c fset" assumes "x |\<notin>| A"   and "x |\<notin>| B" shows "(finsert x A = finsert x B) = (A = B)"
subspace_span: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set" assumes "module scale" shows "module.subspace scale (module.span scale S)"
div_if: fixes m :: "nat"   and n :: "nat" shows "m div n = (if m < n \<or> n = 0 then 0 else Suc ((m - n) div n))"
adjoint_pair_antipar_4: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.adjoint_pair f g" shows "Span.src g = Span.trg f"
Mapping_Mapping_lookup: fixes m :: "('c, 'd) mapping" shows "Mapping (Mapping.lookup m) = m"
measurable_split_conv: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and A :: "('a \<times> 'b) measure"   and B :: "'c measure" shows "((\<lambda>(x, y). f x y) \<in> A \<rightarrow>\<^sub>M B) = ((\<lambda>x. f (fst x) (snd x)) \<in> A \<rightarrow>\<^sub>M B)"