hcju/isabelleps huggingface dataset:
stateful_strand_step_subst_comp: fixes \<delta> :: "'d \<Rightarrow> ('e, 'd) Term.term"   and x :: "('e, 'd) stateful_strand_step"   and \<theta> :: "'d \<Rightarrow> ('e, 'd) Term.term" assumes "range_vars \<delta> \<inter> set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p x) = {}" shows "x \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta> \<circ>\<^sub>s \<theta> = (x \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>"
IH_2: fixes T1 :: "tree"   and T2 :: "tree" assumes "T2' = Branching T1 T2" shows "\<exists>B. branch (B @ [True]) T2' \<and> branch (B @ [False]) T2'"
length_PartNum_PData: fixes P :: "'b pdata" shows "length (PDataValue P) = PartNum (PDataSpace P)"
arr_Rel_components_2: fixes f :: "V"   and A :: "V"   and B :: "V" shows "[f, A, B]\<^sub>\<circ>\<lparr>ArrDom\<rparr> = A"
enat_of_option_simps_1: fixes x :: "nat option"   and n :: "nat" shows "(Common.enat_of_option x = enat n) = (x = Some n)"
ideal_quotient_def: fixes R :: "('d, 'e) Ring_scheme"   and A :: "'d set"   and B :: "'d set" shows "A \<dagger>\<^sub>R B = {x |x. x \<in> carrier R \<and> (\<forall>b\<in>B. x \<cdot>\<^sub>r\<^bsub>R\<^esub> b \<in> A)}"
eq0_iff: fixes x :: "V" shows "(x = 0) = (\<forall>y. y \<notin> elts x)"
abs_arctan_le: fixes x :: "real" shows "\<bar>arctan x\<bar> \<le> \<bar>x\<bar>"
get_ancestors_pure: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "pure (get_ancestors ptr) h"
all_security_requirements_fulfilled_def: fixes M :: "'a SecurityInvariant_configured list"   and G :: "'a graph" shows "all_security_requirements_fulfilled M G \<equiv> \<forall>m\<in>set M. c_sinvar m G"
irreflexive_inf_arc_asymmetric: fixes x :: "'a" assumes "irreflexive_inf x"   and "arc x" shows "asymmetric x"
prems: shows "atom x \<sharp> (\<Gamma>, Let \<Delta> body)"
pseq_def: fixes fs :: "formula list" shows "pseq fs = ([], map (Pair 0) fs)"
clos: fixes x :: "'a" assumes "x \<in> S" shows "closure (G x) \<subseteq> F x"
SourceOfDSource: fixes Z :: "CSet"   and i :: "AbstrLevelsID"   and S :: "CSet"   and C :: "CSet" assumes "Z \<in> Sources i S"   and "S \<in> DSources i C" shows "Z \<in> Sources i C"
map_poly_cong: fixes p :: "'b poly"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> set (coeffs p) \<Longrightarrow> f x = g x" shows "map_poly f p = map_poly g p"
length_tl: fixes xs :: "'b list" shows "length (tl xs) = length xs - 1"
ipurge_tr_aux_cons: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and x :: "'b"   and xs :: "'b list" shows "ipurge_tr_aux I D U (x # xs) = (if \<exists>u\<in>U. (u, D x) \<in> I then ipurge_tr_aux I D (insert (D x) U) xs else x # ipurge_tr_aux I D U xs)"
rspR_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "rspR_on A f = {(x, y). {x, y} \<subseteq> A \<and> (y, x) \<notin> rsp_on A f}"
trivial_homology_group_empty: fixes X :: "'a topology"   and p :: "int" assumes "topspace X = {}" shows "trivial_group (homology_group p X)"
square_free_factorization_order_root: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list"   and x :: "'a"   and i :: "nat" assumes "square_free_factorization p (c, bs)"   and "p \<noteq> 0" shows "(order x p = i) = (i = 0 \<and> (\<forall>a j. (a, j) \<in> set bs \<longrightarrow> poly a x \<noteq> (0::'a)) \<or> (\<exists>a j. (a, j) \<in> set bs \<and> poly a x = (0::'a) \<and> i = Suc j))"
isoD_1: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2"   and "category.iso (product_category.comp C1 C2) f" shows "category.iso C1 (fst f)"
assms_3: shows "\<Psi> \<rhd> Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P"
that: fixes T :: "'a set" assumes "closed T"   and "T \<subseteq> S"   and "S - T \<in> lmeasurable"   and "emeasure lebesgue (S - T) < ennreal e" shows "thesis"
Auth_Snd: fixes x :: "rat"   and y :: "rat"   and H :: "msg set" assumes "Auth_Data x y \<in> analz H"   and "Pri_AgrK x \<in> analz H" shows "Pri_AgrK y \<in> analz H"
ide_compE: fixes g :: "'s"   and f :: "'s" assumes "S.ide (g \<cdot>\<^sub>S f)" obtains "S.seq g f" and "S.seq f g" and "g \<cdot>\<^sub>S f = S.dom f" and "g \<cdot>\<^sub>S f = S.cod g"
nstut_actrans: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" assumes "NSTUTINV P" shows "NSTUTINV _actrans P v"
gcd_mult_distrib': fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "normalize c * gcd a b = gcd (c * a) (c * b)"
integrable_scaleR_right: fixes c :: "real"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "c \<noteq> 0 \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. c *\<^sub>R f x)"
IH_1: fixes i :: "int"   and f :: "bool"   and n :: "int" assumes "0 \<le> i" shows "succs (bcomp b1 f i) n \<subseteq> {n..n + size (bcomp b1 f i)} \<union> {n + i + size (bcomp b1 f i)}"
\<psi>_in_hom_1: fixes \<mu>' :: "'a" assumes "\<guillemotleft>\<mu>' : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>" shows "\<guillemotleft>\<psi> \<mu>' : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>"
secure_implies_failure_consistency: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list"   and X :: "'a set"   and zs :: "'a list" assumes "secure P I D"   and "(xs, ys) \<in> rel_ipurge_aux P I D (D ` (X \<union> set zs))"   and "(xs @ zs, X) \<in> failures P" shows "(ys @ zs, X) \<in> failures P"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G"
zip_eq: fixes u :: "'c list"   and v :: "'d list"   and r :: "'c list"   and s :: "'d list" assumes "length u = length v"   and "length r = length s" shows "(u || v = r || s) = (u = r \<and> v = s)"
arr_inv: fixes f :: "'a" assumes "A.iso f" shows "A.arr (A.inv f)"
zet_def: shows "zet = {uu_. \<exists>A f z. uu_ = A \<and> inj_on f A \<and> f ` A \<subseteq> explode z}"
min_satisfying_def: fixes P :: "'d \<Rightarrow> bool"   and l :: "'d list" shows "min_satisfying P l \<equiv> let xs = filter P l in if xs = [] then None else Some (list_min xs)"
to_gbg_ext_def: fixes m :: "'b" shows "to_gbg_ext m \<equiv> \<lparr>g_V = V, g_E = E, g_V0 = V0, gbg_F = if F = UNIV then {} else {F}, \<dots> = m\<rparr>"
filternew_flows_state_alt: fixes \<T> :: "'b stateful_policy" shows "filternew_flows_state \<T> = flows_state \<T> - backflows (flows_fix \<T>)"
fixp_same: fixes W1 :: "'a set"   and W2 :: "'a set"   and t :: "'a"   and b :: "'a"   and g :: "'a \<Rightarrow> 'a" assumes "W1 \<subseteq> carrier D"   and "W2 \<subseteq> carrier D"   and "t \<in> W1"   and "b \<in> carrier D"   and "ord_isom (Iod D W1) (Iod (Iod D W2) (segment (Iod D W2) b)) g"   and "\<forall>u\<in>segment (Iod D W1) t. fixp g u" shows "segment (Iod D W1) t = segment (Iod D W2) (g t)"
\<rho>: shows "\<forall>v. (\<lambda>i. \<rho> i\<cdot>v) \<in> R"
power_eq_if: fixes p :: "'a"   and m :: "nat" shows "power.power 1' (;) p m = (if m = zero_class.zero then 1' else p ; power.power 1' (;) p (m - one_class.one))"
UNIV_char_of_nat: shows "UNIV = char_of ` {0..<256}"
lnths_split: fixes xs :: "'c llist"   and A :: "nat set"   and n :: "nat" shows "lnths xs A = lappend (lnths (ltake (enat n) xs) A) (lnths (ldropn n xs) {m. n + m \<in> A})"
distinct_disjoint_shuffles: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "distinct xs"   and "distinct ys"   and "set xs \<inter> set ys = {}"   and "zs \<in> shuffles xs ys" shows "distinct zs"
measure_def: fixes M :: "'c measure"   and A :: "'c set" shows "Sigma_Algebra.measure M A = enn2real (emeasure M A)"
Lambert_W_strict_mono: fixes x :: "real"   and y :: "real" assumes "- exp (- 1) \<le> x"   and "x < y" shows "Lambert_W x < Lambert_W y"
Suc_1: obtains j :: "nat" where "i < length (seq j)"
simps_2: fixes uu :: "'a" shows "changes [uu] = 0"
d_generic_unwinding: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set"   and I :: "('b \<times> 'b) set" assumes "deterministic P"   and "view_partition P D R"   and "d_weakly_future_consistent P I D R"   and "weakly_step_consistent P D R"   and "locally_respects P I D R" shows "secure P I D"
prems_8: shows "history_consistent \<theta> p ((r # sb) @ sb')"
isCont_If_ge: fixes a :: "'a"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "continuous (at_left a) g"   and "(f \<longlongrightarrow> g a) (at_right a)" shows "isCont (\<lambda>x. if x \<le> a then g x else f x) a"
csl\<rho>: shows "cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<rho>\<^esup> l"
assms_2: shows "natural_isomorphism (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G G' \<mu>"
elims: fixes x :: "lit uprod"   and y :: "(lit, lit set) mapping \<Rightarrow> (lit, lit set) mapping" assumes "insert_clause x = y" obtains l1 :: "lit" and l2 :: "lit" where "x = Upair l1 l2" and "y = insert_mapping (negate l1, l2) \<circ> insert_mapping (negate l2, l1)"
update_def: fixes r :: "'a ref"   and v :: "'a" shows "r := v = Heap_Monad.heap (\<lambda>h. ((), Ref.set r v h))"
res: shows "inf_concat n (k + m) = (i, j + m)"
padic_one_in_padic_set: fixes p :: "int" assumes "prime p" shows "padic_one p \<in> padic_set p"
summable_add: assumes "summable f"   and "summable g" shows "summable (\<lambda>n. f n + g n)"
cInf_lower: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and X :: "'b set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> X"   and "preorder.bdd_below less_eq X" shows "less_eq (Inf X) x"
take_all: fixes xs :: "'a list"   and n :: "nat" assumes "length xs \<le> n" shows "take n xs = xs"
single: shows "subst_bv1 (bind_fv2 (idn, ty) lev B) lev t = subst_term [((idn, ty), t)] B"
weakStatImp_def: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<lessapprox><Rel> Q \<equiv> \<forall>\<Psi>'. \<exists>Q' Q''. \<Psi> \<rhd> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q' \<and> insertAssertion (extractFrame P) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q') \<Psi> \<and> \<Psi> \<otimes> \<Psi>' \<rhd> Q' \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q'' \<and> (\<Psi> \<otimes> \<Psi>', P, Q'') \<in> Rel"
Pd_range: shows "\<forall>d. Field (Pd' d) \<subseteq> {x. Xd x = d}"
transp_relcompp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "transp r = (r OO r \<le> r)"
set_bset: fixes x :: "'d bset" shows "set_bset x \<in> {A. |A| \<le>o ctwo}"
upper: fixes x :: "'a"   and y :: "'a" shows "turn_into_distance f x y \<le> f x y"
assms_4: shows "c_msg (exchange_config c) p q \<noteq> []"
ag_eq_diffzero: fixes A :: "('c, 'd) aGroup_scheme"   and a :: "'c"   and b :: "'c" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A" shows "(a = b) = (a \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> b = \<zero>\<^bsub>A\<^esub>)"
pre_arith_simps_3: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(z \<le> max x y) = (z \<le> x \<or> z \<le> y)"
finite_Collect_not: fixes P :: "'a \<Rightarrow> bool" assumes "finite {x. P x}" shows "finite {x. \<not> P x} = finite UNIV"
limproc_is_thelub: fixes S :: "nat \<Rightarrow> 'b process" assumes "chain S" shows "Lub S = lim_proc (range S)"
eval_simps_3: fixes f :: "'a E.term" assumes "E.Arr f" shows "EVAL (E.Trg f) = trg\<^sub>B (EVAL f)"
simps_1: fixes e :: "exp" shows "eta_expand 0 e = e"
sortLemmaR: fixes h :: "'b \<Rightarrow> int"   and t1 :: "'b Tree"   and x :: "'b"   and t2 :: "'b Tree" assumes "sortedTree h (T t1 x t2)" shows "sortedTree h t2"
atom_var_def: fixes a :: "Atoms.var" shows "atom a = Rep_var a"
crdt_ops_reorder_spec: fixes xs :: "('a \<times> 'b) list"   and x :: "'a \<times> 'b"   and deps :: "'b \<Rightarrow> 'a set"   and ys :: "('a \<times> 'b) list"   and zs :: "('a \<times> 'b) list" assumes "spec_ops (xs @ [x]) deps"   and "crdt_ops (ys @ [x] @ zs) deps"   and "set (xs @ [x]) = set (ys @ [x] @ zs)" shows "crdt_ops (ys @ zs @ [x]) deps"
Un_absorb1: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "A \<union> B = B"
k2_2: fixes k' :: "'a" assumes "k' < k2" shows "g k' = h k'"
rc: fixes j :: "nat" assumes "j < nc" shows "row m i ! j = col m j ! i"
simps_1: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "sorted_wrt P [] = True"
b: shows "basis_completion v = v # map (unit_vec n) [0..<k] @ map (unit_vec n) [Suc k..<n]"
Proposition_2: fixes g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'b list" assumes "\<And>n. candidate (@) (map wrap [0..<n + 1]) = ups n"   and "associative g"   and "xs \<noteq> []" shows "candidate g xs = scanl1 g xs"
no_step: shows "\<nexists>e2. CT \<turnstile> e1 \<rightarrow> e2"
g_def: shows "g = (\<lambda>x. c * x + b)"
cardA: shows "card ((\<lambda>x. x\<^sup>2 mod p) ` {0..n}) = card {0..n}"
OclAnd_false1: fixes X :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(false and X) = false"
h2a: shows "(Av + Bv * sqrt Cv = 0) = (Av = - Bv * sqrt Cv)"
discr_Seq: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "discr c1"   and "discr c2" shows "discr (c1 ;; c2)"
Rep_process: fixes x :: "'c process" shows "Rep_process x \<in> {p. is_process p}"
measurable_lift_IntVal: fixes f :: "int \<Rightarrow> int" shows "lift_IntVal f \<in> range IntVal \<rightarrow> range IntVal"
const: fixes \<C> :: "t_context"   and v :: "v" shows "\<C> \<turnstile> [C v] : [] _> [typeof v]"
distinct_filter: fixes xs :: "'e list"   and P :: "'e \<Rightarrow> bool" assumes "distinct xs" shows "distinct (filter P xs)"
SourcesA1_L0: shows "Sources level0 sA1 = {}"
z_def: fixes index_list :: "nat list"   and sign_asg :: "rat list" shows "z index_list sign_asg \<equiv> prod_list (map ((!) sign_asg) index_list)"
disc_2: fixes x2 :: "'d" shows "\<not> isl (Inr x2)"
commute_Un: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and T :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "commute R T"   and "commute S T" shows "commute (sup R S) T"
fup_ccExp_restr_subst': fixes S :: "Vars.var set"   and e :: "exp"   and x :: "Vars.var"   and y :: "Vars.var"   and a :: "Arity\<^sub>\<bottom>" assumes "\<And>a. ccExp_syn a e[x::=y] G|` S = ccExp_syn a e G|` S" shows "ccExp_bot_syn a e[x::=y] G|` S = ccExp_bot_syn a e G|` S"
closure_of_interior_of: fixes X :: "'a topology"   and S :: "'a set" shows "X closure_of S = topspace X - X interior_of (topspace X - S)"
t_sort_sws: fixes s :: "'b list"   and r :: "'b"   and mf :: "nat"   and k :: "'b \<Rightarrow> 'c" shows "t s r (mf, sort_sws k s) \<le> (length s)\<^sup>2 + length s + 1"
mono_onD: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and r :: "'c"   and s :: "'c" assumes "Fun.mono_on f A"   and "r \<in> A"   and "s \<in> A"   and "r \<le> s" shows "f r \<le> f s"
iso_map\<^sub>1_ide: fixes f :: "'c" assumes "C.ide f" shows "C.iso (\<epsilon>\<^sub>1 f)"
anno_type: fixes \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and upd :: "'Var ModeUpd"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and P :: "'BExp set"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>'' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>'' :: "'Var set \<times> 'Var set"   and P'' :: "'BExp set" assumes "\<Gamma>' = \<Gamma> \<oplus>\<^sub>\<S> upd"   and "\<S>' = add_anno_stable \<S> upd"   and "P' = P |` {v. stable \<S>' v}"   and "\<turnstile> \<Gamma>',\<S>',P' {c} \<Gamma>'',\<S>'',P''"   and "c \<noteq> Stop"   and "\<And>x. to_total \<Gamma> x \<le>:\<^sub>P' to_total \<Gamma>' x"   and "anno_type_stable \<Gamma> \<S> upd"   and "anno_type_sec \<Gamma> \<S> P upd" shows "\<turnstile> \<Gamma>,\<S>,P {c@[upd]} \<Gamma>'',\<S>'',P''"
insert_subsetI: fixes x :: "'b"   and A :: "'b set"   and X :: "'b set" assumes "x \<in> A"   and "X \<subseteq> A" shows "insert x X \<subseteq> A"
mfinalE: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x1 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and x :: "'x1"   and ln :: "'a \<Rightarrow>f nat" assumes "r1.mfinal s"   and "thr s t = \<lfloor>(x, ln)\<rfloor>" obtains "final1 x" and "ln = no_wait_locks" and "wset s t = None"
refine: fixes n :: "nat"   and Xi :: "((real list \<times> real list) \<times> 'e) list"   and X :: "'d set"   and a :: "'f \<Rightarrow> 'd set"   and I :: "('e \<times> 'f) set"   and sctnsi :: "real list sctn list"   and sctns :: "'d sctn set" assumes "DIM_precond TYPE('d) n"   and "(Xi, X) \<in> clw_rel (\<langle>I, lvivl_rel\<rangle>invar_rel a)"   and "(sctnsi, sctns) \<in> sctns_rel" shows "(nres_of (restrict_to_halfspaces_invar_clw_impl n sctnsi Xi), restrict_to_halfspaces_invar_clw sctns X) \<in> \<langle>clw_rel (\<langle>I, lvivl_rel\<rangle>invar_rel a)\<rangle>nres_rel"
set_empty2: fixes xs :: "'a list" shows "({} = set xs) = (xs = [])"
finite_dom_map_of: fixes l :: "('b \<times> 'c) list" shows "finite (dom (map_of l))"
relcomp_mono: fixes r' :: "('b \<times> 'c) set"   and r :: "('b \<times> 'c) set"   and s' :: "('c \<times> 'd) set"   and s :: "('c \<times> 'd) set" assumes "r' \<subseteq> r"   and "s' \<subseteq> s" shows "r' O s' \<subseteq> r O s"
noteval_F: shows "\<forall>(e1, e2)\<in>F'. \<not> sinvar \<lparr>nodes = V, edges = E' - F' \<union> {(e1, e2)}\<rparr> nP"
Gp_smult_nsg: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and N :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<triangleright> N" shows "\<natural>\<^bsub>G\<^esub>H \<diamondop>\<^bsub>G\<^esub> N \<triangleright> N"
zdiv_mono1_neg: fixes a :: "int"   and a' :: "int"   and b :: "int" assumes "a \<le> a'"   and "b < 0" shows "a' div b \<le> a div b"
op_dghm_def: fixes \<FF> :: "V" shows "op_dghm \<FF> = [\<FF>\<lparr>ObjMap\<rparr>, \<FF>\<lparr>ArrMap\<rparr>, op_dg (\<FF>\<lparr>HomDom\<rparr>), op_dg (\<FF>\<lparr>HomCod\<rparr>)]\<^sub>\<circ>"
shadows_consts_frees: fixes C_info :: "(name, dt_def) fmap"   and heads :: "name fset"   and t :: "'a" assumes "\<not> pre_constants.shadows_consts C_info heads t" shows "fdisjnt (pre_constants.all_consts C_info heads) (frees t)"
inf_chain_origin_unique: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set"   and g :: "nat \<Rightarrow> 'a" assumes "[f[f 0 ..]X]"   and "[g[g 0 ..]X]" shows "f 0 = g 0"
fresher_refl: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" shows "r \<sqsubseteq> r"
drop_encode: shows "drop k (digit_encode (replicate (Suc k) 2) i) = digit_encode [2] (i div 2 ^ k)"
simps_2: fixes n :: "nat"   and ss :: "('a \<times> 'b) list list"   and ds :: "('a \<times> 'b) list list" shows "ls_copy (Suc n) ss ds = ls_copy n ss (ls_insls (ss ! n) ds)"
integrable_norm_iff: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "f \<in> borel_measurable M" shows "integrable M (\<lambda>x. norm (f x)) = integrable M f"
sg_mult_closedl: fixes H :: "'a set"   and x :: "'a"   and h :: "'a" assumes "G \<guillemotright> H "   and "x \<in> carrier G"   and "h \<in> H" shows "h \<cdot> x \<in> carrier G"
Perp_perm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Perp C D" shows "A B Perp C D \<and> B A Perp C D \<and> A B Perp D C \<and> B A Perp D C \<and> C D Perp A B \<and> C D Perp B A \<and> D C Perp A B \<and> D C Perp B A"
prob_space_erlang_density: fixes l :: "real"   and k :: "nat" assumes "0 < l" shows "prob_space (density lborel (\<lambda>x. ennreal (erlang_density k l x)))"
range_coeffs_to_fract: fixes p :: "'b fract poly"   and i :: "nat" assumes "set (coeffs p) \<subseteq> range to_fract" shows "\<exists>m. coeff p i = to_fract m"
measure_return_pmf: fixes x :: "'c"   and A :: "'c set" shows "measure_pmf.prob (return_pmf x) A = indicat_real A x"
left_add_twice: fixes a :: "'a"   and b :: "'a" shows "a + (a + b) = (2::'a) * a + b"
diff_fun_space_eq: shows "dest.diff_fun_space = {f. diff_fun k charts2 f} \<inter> {f. extensional0 dest.carrier f}"
cong: fixes G :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "right_translations_of_group.translation G composition \<equiv> right_translations_of_group.translation G composition"
compl_inf: fixes x :: "'b"   and y :: "'b" shows "- inf x y = sup (- x) (- y)"
lincomb_snoc0: fixes ms :: "'m list"   and as :: "'r list" assumes "set ms \<subseteq> M" shows "(as @ [0::'r]) \<bullet>\<cdot> ms = as \<bullet>\<cdot> ms"
image_g: shows "(-) (degree f) ` {0..<Suc (degree f)} = {0..<Suc (degree f)}"
norm_Pair1: fixes x :: "'c" shows "norm (0::'b, x) = norm x"
homeomorphic_spheres: fixes d :: "real"   and e :: "real"   and a :: "'a"   and b :: "'a" assumes "0 < d"   and "0 < e" shows "sphere a d homeomorphic sphere b e"
simps_2: fixes t :: "'a tree"   and ts :: "'a tree list" shows "alphabet\<^sub>F (t # ts) = alphabet t \<union> alphabet\<^sub>F ts"
inject: fixes x1 :: "atom_sort"   and x2 :: "nat"   and y1 :: "atom_sort"   and y2 :: "nat" shows "(Atom x1 x2 = Atom y1 y2) = (x1 = y1 \<and> x2 = y2)"
assms_2: shows "fds_abs_converges (fds_norm g) s"
step: shows "\<Gamma>\<turnstile> cfg\<^sub>1 \<rightarrow> cfg''"
dom_distrib: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "d x + d y \<cdot> d z = (d x + d y) \<cdot> (d x + d z)"
finiteE: fixes G :: "'b graph" assumes "wf_graph G" shows "finite (edges G)"
spr_knowledge_def: fixes envObs :: "'b \<Rightarrow> ('b, 'c, 'd) BEState \<Rightarrow> 'e \<times> 'd option"   and cec :: "(('b, 'c, 'd) BEState \<times> ('b, 'c, 'd) BEState) odlist" shows "spr_knowledge envObs cec \<equiv> \<lambda>a s. fromList (concat (map (\<lambda>s'. if (s, s') \<in> SPRViewDet.spr_repRels envObs a then [s'] else []) (toList cec)))"
rQuots_append: fixes v :: "'c list"   and w :: "'c list"   and A :: "'c list set" shows "(v \<in> rQuots w A) = (v @ rev w \<in> A)"
rel_spmf_bindI: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'d spmf"   and q :: "'e spmf"   and P :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f spmf"   and g :: "'e \<Rightarrow> 'g spmf" assumes "rel_spmf R p q"   and "\<And>x y. R x y \<Longrightarrow> rel_spmf P (f x) (g y)" shows "rel_spmf P (p \<bind> f) (q \<bind> g)"
set_take_subset: fixes n :: "nat"   and xs :: "'c list" shows "set (take n xs) \<subseteq> set xs"
all_in_def: fixes x :: "real list"   and I :: "float interval list" shows "(x all_in I) = (length x = length I \<and> (\<forall>i<length I. x ! i \<in>\<^sub>r I ! i))"
swapidseq_ext_finite_expansion: fixes B :: "'a set"   and A :: "'a set"   and n :: "nat"   and p :: "'a \<Rightarrow> 'a" assumes "finite B"   and "swapidseq_ext A n p" shows "swapidseq_ext (A \<union> B) n p"
new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def: fixes h :: "(_) heap" shows "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (let new_document_ptr = document_ptr.Ref (Suc (fMax (finsert 0 (document_ptr.the_ref |`| document_ptrs h)))) in (new_document_ptr, put new_document_ptr (create_document_obj [] None []) h))"
lookup5: fixes a :: "coname"   and \<Delta> :: "(coname \<times> ty) list"   and y :: "name"   and x :: "name"   and c :: "coname"   and P :: "trm" assumes "a \<sharp> idc \<Delta> y" shows "lookupb x a (idc \<Delta> y) c P = Cut <c>.P x.Ax x a"
successive_stepI: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b" assumes "successive P xs"   and "\<not> P x" shows "successive P (x # xs)"
i_def: shows "i \<equiv> \<lambda>x. ((b - x) ^ (p - 1) / fact (p - 1)) *\<^sub>R Df p x"