hcju/isabelleps huggingface dataset:
planning_dimacs_complete_code': fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and \<pi>s :: "char list list"   and h :: "nat" assumes "ast_problem.well_formed prob"   and "\<And>op. op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> consistent_pres_op op"   and "\<And>op. op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> is_standard_operator op"   and "ast_problem.valid_plan prob \<pi>s"   and "length \<pi>s \<le> h" shows "let cnf_formula = SASP_to_DIMACS' h prob in \<exists>dimacs_M. dimacs_model dimacs_M cnf_formula"
integral_real_density: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure"   and g :: "'c \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "g \<in> borel_measurable M" shows "integral\<^sup>L (density M (\<lambda>x. ennreal (f x))) g = LINT x|M. f x * g x"
powser_ms_aux_MSLCons: fixes c :: "real"   and cs :: "real msllist"   and f :: "('a \<times> real) msllist" shows "powser_ms_aux (MSLCons c cs) f = MSLCons (const_expansion c, 0) (times_ms_aux f (powser_ms_aux cs f))"
d_less_e: shows "d_aux a b c \<le> e_aux a b c"
seq': fixes rs :: "'c rule list"   and rs\<^sub>1 :: "'c rule list"   and rs\<^sub>2 :: "'c rule list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and s :: "state"   and t :: "state"   and t' :: "state" assumes "rs = rs\<^sub>1 @ rs\<^sub>2"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t'"
icd_is_the_icd: fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k" shows "k = (THE k. i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k)"
measure_pmf_parametric: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(rel_pmf A ===> rel_pred A ===> (=)) measure_pmf.prob measure_pmf.prob"
bounded_iff: fixes A :: "'b word set"   and x :: "'b word" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le>s signed.Min A) = (\<forall>a\<in>A. x \<le>s a)"
abel_semigroup_axioms: shows "abel_semigroup (\<sqinter>)"
hyps: shows "rel v r = \<Union> (word_rel v ` lang r)"
HoVH_def: fixes \<mu> :: "'a \<times> 'a \<times> 'a" shows "B.HoVH \<mu> \<equiv> if B.VVV.arr \<mu> then (snd (snd \<mu>) \<star> fst (snd \<mu>)) \<star> fst \<mu> else B.null"
secdef: shows "\<forall>nP e1 e2. \<not> P (nP e1) (nP e2) \<longrightarrow> \<not> P \<bottom> (nP e2)"
Local: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and f :: "state \<Rightarrow> state"   and c :: "com"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool"   and g :: "state \<Rightarrow> state \<Rightarrow> state" assumes "\<forall>s'. C \<turnstile> {\<lambda>z s. P z s' \<and> s = f s'} c {\<lambda>z t. Q z (g s' t)}" shows "C \<turnstile> {P} LOCAL f; c; g {Q}"
has_codomain_iff_arr: fixes f :: "'c \<times> 'c" shows "(CopxC.codomains f \<noteq> {}) = CopxC.arr f"
laplace_integrand_integrable_on_Ici_iff: fixes M :: "real"   and c :: "real"   and s :: "complex"   and a :: "real" assumes "exponential_order M c f"   and "c < Re s" shows "(local.laplace_integrand s integrable_on {a..}) = (\<forall>k>a. local.laplace_integrand s integrable_on {a..k})"
valid: shows "\<Turnstile>\<^sub>2\<^sub>' {\<lambda>s. \<up> (P s) + enat n} c {\<lambda>s. \<up> (Q s)}"
map_us_map_IInf_inv: shows "map_us \<circ> map_IInf = id"
OutOfMemory_def: shows "OutOfMemory = STR ''java/lang/OutOfMemoryError''"
max_spec2appl_meths: fixes x :: "(ref_ty \<times> mhead) \<times> ty list"   and G :: "prog"   and S :: "qtname"   and T :: "ref_ty"   and sig :: "sig" assumes "x \<in> max_spec G S T sig" shows "x \<in> appl_methds G S T sig"
intros_8: fixes P :: "pi"   and Q :: "pi" shows "P \<oplus> Q \<equiv>\<^sub>e Q \<oplus> P"
inj_on_image_eqpoll_self: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "f ` A \<approx> A"
abs_eq: fixes xa :: "int extended \<times> int extended"   and x :: "int extended \<times> int extended" shows "abs_ivl xa \<squnion> abs_ivl x = abs_ivl (sup_rep xa x)"
A6: shows "\<forall>SQ TR. SQ \<in>S Q \<and> TR \<in>T R \<longrightarrow> (\<lbrakk>SQ\<rbrakk>, TR) \<in> TRel"
same_append_eq: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" shows "(xs @ ys = xs @ zs) = (ys = zs)"
step_invariant: shows "step_invariant (ws, ps, N) = (ws', ps', N')"
assms_4: shows "\<not> (\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
bal_tree\<^sub>d_del: fixes t :: "('c \<times> 'd) tree234"   and x :: "'c" assumes "bal t" shows "bal (tree\<^sub>d (Tree234_Map.del x t))"
converged_cs_single: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and \<pi>' :: "nat \<Rightarrow> 'a"   and l :: "nat"   and m :: "nat"   and m' :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "IFC_def.is_path prob \<pi>'"   and "l < m"   and "IFC_def.cs prob \<pi> m = IFC_def.cs prob \<pi>' m'"   and "\<pi> 0 = \<pi>' 0"   and "IFC_def.cs prob \<pi> l = [\<pi> l]" shows "\<exists>l'. IFC_def.cs prob \<pi> l = IFC_def.cs prob \<pi>' l'"
prems: shows "read_only \<S> \<subseteq> read_only \<S>'"
hyps_2: shows "P' \<longmapsto> \<tau> \<prec> P''"
3: fixes v :: "'b"   and r :: "'b stream" assumes "v \<in> initial A"   and "run A (w ||| r) v" shows "fins (accepting A) r"
learn_cp_wrt_goedel: fixes \<chi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "goedel_numbering \<chi>"   and "learn_cp \<psi> U s" shows "\<exists>t. learn_cp \<chi> U t"
axioms_1: fixes R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d" assumes "comm_ring R addition multiplication zero unit" shows "ring R addition multiplication zero unit"
children_eq2_h3: fixes ptr' :: "(_) object_ptr" assumes "ptr \<noteq> ptr'" shows "|h3 \<turnstile> get_child_nodes ptr'|\<^sub>r = |h' \<turnstile> get_child_nodes ptr'|\<^sub>r"
chine_assoc'_props_2: shows "fgh.Prj\<^sub>1\<^sub>1 \<cdot> fgh.chine_assoc' = fgh.Prj\<^sub>1"
cod_char: fixes f :: "(bool, bool list) concrete_category.arr" shows "cod f = (if arr f then if f = Zero then Zero else One else null)"
ex_map_conv: fixes ys :: "'b list"   and f :: "'a \<Rightarrow> 'b" shows "(\<exists>xs. ys = map f xs) = (\<forall>y\<in>set ys. \<exists>x. y = f x)"
preimages_conull_2: fixes A :: "'a set" assumes "A \<in> sets M"   and "0 < emeasure M A" shows "space M \<Delta> (\<Union>n. (T ^^ n) --` A) \<in> null_sets M"
us: fixes i :: "nat" assumes "i \<le> j" shows "fs ! i \<in> carrier_vec n"
callee_invariant_on_key_oracle: fixes \<eta> :: "nat" shows "callee_invariant_on (CIPHER.KEY.key_oracle \<eta> \<oplus>\<^sub>O CIPHER.KEY.key_oracle \<eta>) (\<lambda>x. case x of None \<Rightarrow> True | Some x' \<Rightarrow> length x' = \<eta>) (\<I>_uniform UNIV (nlists UNIV \<eta>) \<oplus>\<^sub>\<I> \<I>_full)"
sep_impl_sep_true: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_impl plus sep_disj P (\<lambda>s. True) = (\<lambda>s. True)"
Inf_insert: fixes a :: "'b"   and A :: "'b set" shows "\<Squnion> insert a A = a \<squnion> \<Squnion> A"
lcnj: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and \<phi>s :: "'c list" assumes "Syntax_with_Connectives_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls"   and "set \<phi>s \<subseteq> fmla" shows "Syntax_with_Connectives_False.lcnj cnj imp fls \<phi>s \<in> fmla"
l'_l: shows "\<forall>i. i + 1 < length l' \<longrightarrow> (\<exists>j. j + 1 < length l \<and> l' ! i = l ! j \<and> l' ! (i + 1) = l ! (j + 1))"
special_coset_subset_ordering_iso: fixes w :: "'w" shows "OrderingSetIso dual_order.greater_eq dual_order.greater dual_order.greater_eq dual_order.greater (Pow S) ((+o) w \<circ> genby)"
mrConcEnd: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and b :: "((int \<times> int) set set, int) Combinators"   and p :: "((int \<times> int) set set, int) Combinators list"   and a :: "((int \<times> int) set set, int) Combinators" assumes "applied_rule_rev C x (b # p) = \<lfloor>a\<rfloor>"   and "a \<noteq> b" shows "applied_rule_rev C x p = \<lfloor>a\<rfloor>"
H: shows "\<Gamma> \<turnstile> (\<lambda>:S. s) : U \<rightarrow> U'"
mset_tree_simp_alt: fixes r :: "nat"   and a :: "'b"   and ts :: "'b tree list" shows "mset_tree (Node r a ts) = {#a#} + mset_heap ts"
atLeastAtMost_iff: fixes i :: "'a \<times> 'b"   and l :: "'a \<times> 'b"   and u :: "'a \<times> 'b" shows "s2p (prod.atLeastAtMost l u) i = (less_eq_prod l i \<and> less_eq_prod i u)"
n: shows "n \<equiv> LEAST n. A $ i $ n \<noteq> (0::'b)"
borel_measurable_min: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
remove_all_cycles_distinct: fixes ys :: "'b list"   and xs :: "'b list" assumes "set ys \<subseteq> set xs" shows "distinct (remove_all_cycles xs ys)"
PO_m1_refines_trans_a0n: shows "{R0n0i} TS.trans a0n, TS.trans a0i {> R0n0i}"
false_def: shows "false \<equiv> \<lambda>\<tau>. \<lfloor>\<lfloor>False\<rfloor>\<rfloor>"
NextRecvUpdD_1: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and q :: "'b" assumes "NextRecvUpd' c0 c1 p q" shows "init c0 q"
affine_hull_affine_Int_nonempty_interior: fixes S :: "'a set"   and T :: "'a set" assumes "affine S"   and "S \<inter> interior T \<noteq> {}" shows "affine hull (S \<inter> T) = affine hull S"
lexord_asymmetric: fixes R :: "('a \<times> 'a) set"   and a :: "'a list"   and b :: "'a list" assumes "asym R"   and "(a, b) \<in> lexord R" shows "(b, a) \<notin> lexord R"
exec_stop2: fixes \<sigma> :: "'a control_state_scheme" assumes "return_status \<sigma>" shows "exec_stop \<sigma>"
f: shows "\<forall>i>d. Abstract_Linear_Poly.coeff x i = 0"
of_int_in_nonpos_Ints_iff: fixes n :: "int" shows "(of_int n \<in> \<int>\<^sub>\<le>\<^sub>0) = (n \<le> 0)"
induct: fixes x1 :: "(Location \<times> Location) set"   and x2 :: "Val"   and x3 :: "Val"   and P :: "(Location \<times> Location) set \<Rightarrow> Val \<Rightarrow> Val \<Rightarrow> bool" assumes "(x1, x2, x3) \<in> twiddleVal"   and "\<And>\<beta>. P \<beta> (RVal Nullref) (RVal Nullref)"   and "\<And>l1 l2 \<beta>. (l1, l2) \<in> \<beta> \<Longrightarrow> P \<beta> (RVal (Loc l1)) (RVal (Loc l2))"   and "\<And>i1 i2 \<beta>. i1 = i2 \<Longrightarrow> P \<beta> (IVal i1) (IVal i2)" shows "P x1 x2 x3"
abs_eq: fixes x :: "'a poly fract" assumes "eq_onp (\<lambda>x. constant_term (denominator x) \<noteq> (0::'a)) x x" shows "local.quot_of_ratfps (Abs_ratfps x) = quot_of_fract x"
isFault_simps_2: fixes s :: "'d" shows "isFault (Abrupt s) = False"
fds_eqI: fixes f :: "'b fds"   and g :: "'b fds" assumes "\<And>n. 0 < n \<Longrightarrow> f $ n = g $ n" shows "f = g"
sum_eq: fixes f :: "'f \<Rightarrow> ('d, 'e) vec"   and S :: "'f set" shows "sum f S = (\<chi>i. \<Sum>x\<in>S. f x $ i)"
tndFalse: fixes A :: "('a set set \<times> 'a set set) set"   and B :: "('a set set \<times> 'a set set) set"   and D :: "'a"   and F :: "'a" assumes "\<forall>a b c d. (a, b) \<in> A \<and> (c, d) \<in> B \<longrightarrow> twoNetsDistinct a b c d"   and "\<exists>(a, b)\<in>A. a \<in> subnetsOfAdr D \<and> b \<in> subnetsOfAdr F"   and "\<exists>(a, b)\<in>B. a \<in> subnetsOfAdr D \<and> b \<in> subnetsOfAdr F" shows "False"
remove_cycles_cnt_id: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "x \<noteq> y" shows "cnt y (remove_cycles xs x ys) \<le> cnt y ys + cnt y xs"
\<Phi>_Eps: fixes NL :: "nat set list" assumes "infinite (hd NL)" shows "\<Phi> NL (Eps (\<Phi> NL))"
intros: fixes a :: "'item"   and b :: "'item"   and u :: "'item list"   and v :: "'item list" assumes "ind a b" shows "u @ [a] @ [b] @ v =\<^sub>S u @ [b] @ [a] @ v"
sum_inner_Basis_scaleR: fixes b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "b \<in> Basis" shows "(\<Sum>i\<in>Basis. (i \<bullet> b) *\<^sub>R f i) = f b"
basis_inD_aux: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.basis_in \<E> X" shows "indep_system.basis \<E> (\<E>.indep_in \<E>) X"
validt_restrict_to_validt: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and M :: "'d set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>|\<^bsub>M\<^esub>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
HVZK_def: shows "Schnorr_\<Sigma>.HVZK \<equiv> \<forall>e\<in>challenge_space. (\<forall>(h, w)\<in>R_DL. Schnorr_\<Sigma>.R h w e = Schnorr_\<Sigma>.S h e) \<and> (\<forall>h\<in>valid_pub. \<forall>(a, z)\<in>set_spmf (S2 h e). check h a e z)"
i_set_defs_6: shows "iMOD2_set \<equiv> {[ r, mod m ] |r m. 2 \<le> m}"
chop: shows "v'=v1'--v2' \<and> \<phi> ts v1' \<and> \<psi> ts v2'"
simple_split_1: fixes S :: "('c, 'd) strand_step list"   and S' :: "('c, 'd) strand_step list" assumes "simple (S @ S')" shows "simple S"
vector_derivative_chain_at: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and g :: "real \<Rightarrow> 'b" assumes "f differentiable at x"   and "g differentiable at (f x)" shows "vector_derivative (g \<circ> f) (at x) = vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"
VVp: fixes v :: "'a" assumes "v \<in> S"   and "\<not> deadend v"   and "v \<in> VV p" shows "\<sigma> v \<in> S \<union> T"
valid_path_ltl: fixes P :: "'a llist" assumes "valid_path P" shows "valid_path (ltl P)"
vfsequence_vrange_vcpower: fixes A :: "V" assumes "\<R>\<^sub>\<circ> [a, a]\<^sub>\<circ> \<subseteq>\<^sub>\<circ> A" shows "[a, a]\<^sub>\<circ> \<in>\<^sub>\<circ> A ^\<^sub>\<times> vcard [a, a]\<^sub>\<circ>"
image_mset_subseteq_mono: fixes A :: "'a multiset"   and B :: "'a multiset"   and f :: "'a \<Rightarrow> 'b" assumes "A \<subseteq># B" shows "image_mset f A \<subseteq># image_mset f B"
trans_joinable: fixes R :: "('a \<times> 'a) set" assumes "trans R"   and "Joinable.confluent R" shows "trans (joinable R)"
starfun_mult: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d star"   and g :: "'d \<Rightarrow> 'c" shows "(*f* f) x * (*f* g) x = (*f* (\<lambda>x. f x * g x)) x"
quasi_isometry_betweenD_1: fixes lambda :: "real"   and C :: "real"   and X :: "'c set"   and Y :: "'d set"   and f :: "'c \<Rightarrow> 'd" assumes "lambda C -quasi_isometry_between X Y f" shows "lambda C -quasi_isometry_on X f"
simps: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "('d, 'c, 'a) cmd" assumes "PL_Indis cval indis" shows "PL_Indis.discr aval tval cval indis a = (\<exists>c. a = c \<and> (\<forall>x xa. xa < PL.brn c \<longrightarrow> (x, PL.eff aval tval cval c x xa) \<in> indis \<and> PL_Indis.discr aval tval cval indis (PL.cont aval tval cval c x xa)))"
axioms_2: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> 'e) list"   and \<S> :: "nat \<Rightarrow> bool option"   and valid :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'f list \<times> 'd \<times> nat set \<times> ('g \<Rightarrow> 'h option)) list \<Rightarrow> bool" assumes "initial\<^sub>v ts \<S> valid" shows "read_only_unowned \<S> ts"
SubstForm_def: fixes v :: "hf"   and u :: "hf"   and x :: "hf"   and x' :: "hf" shows "SubstForm v u x x' \<equiv> is_Var v \<and> Term u \<and> (\<exists>s k. SeqSubstForm v u x x' s k)"
HC_1: shows "H \<union> C \<subseteq> carrier_vec n \<inter> \<int>\<^sub>v"
JHS_1: fixes G :: "('e, 'f) Group_scheme"   and E :: "('g, 'h) Group_scheme"   and r :: "nat"   and f :: "nat \<Rightarrow> 'e set"   and s :: "nat"   and g :: "nat \<Rightarrow> 'e set" assumes "Group G"   and "Ugp E"   and "compseries G r f"   and "compseries G s g"   and "0 < r"   and "0 < s" shows "r = r * s - card {i. i \<le> r * s - Suc 0 \<and> Qw_cmpser G (cmp_rfn G r f s g) i \<cong> E}"
simps_6: fixes y :: "name"   and xvec :: "name list"   and Tvec :: "'a list"   and P :: "('a, 'b, 'c) psi" assumes "y \<sharp> xvec"   and "y \<sharp> Tvec" shows "\<lparr>\<nu>y\<rparr>P[xvec::=Tvec] = \<lparr>\<nu>y\<rparr>(P[xvec::=Tvec])"
assms_3: shows "A B C D E F SumA A' B' C'"
tautology_imply_superset: fixes ps :: "'b fm list"   and qs :: "'b fm list"   and r :: "'b fm" assumes "set ps \<subseteq> set qs" shows "tautology (imply ps r \<^bold>\<longrightarrow> imply qs r)"
assms: shows "l \<in> lists (UNIV \<times> gens)"
insert_after_ref: fixes xs :: "'b list"   and ref :: "'b"   and ys :: "'b list"   and oid :: "'b" assumes "distinct (xs @ ref # ys)" shows "insert_spec (xs @ ref # ys) (oid, Some ref) = xs @ ref # oid # ys"
indep_vars_compose: fixes M' :: "'d \<Rightarrow> 'e measure"   and X :: "'d \<Rightarrow> 'a \<Rightarrow> 'e"   and I :: "'d set"   and Y :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and N :: "'d \<Rightarrow> 'f measure" assumes "indep_vars M' X I"   and "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> M' i \<rightarrow>\<^sub>M N i" shows "indep_vars N (\<lambda>i. Y i \<circ> X i) I"
distinct_13: fixes x1 :: "'a" shows "broadcast x1 \<noteq> \<tau>\<^sub>s"
borel_measurable_uminus: fixes g :: "'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "g \<in> borel_measurable M" shows "(\<lambda>x. - g x) \<in> borel_measurable M"
take_eq_Nil: fixes n :: "nat"   and xs :: "'d list" shows "(xs \<down> n = []) = (n = 0 \<or> xs = [])"
Cons_2: shows "InvariantWatchesEl (getF stateA) (getWatch1 stateA) (getWatch2 stateA)"
cdep_SDG_path_realizable: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n cc-ns\<rightarrow>\<^sub>d* n'" shows "realizable n ns n'"
sum_rel_def: fixes Rl :: "('a \<times> 'c) set"   and Rr :: "('b \<times> 'd) set" shows "\<langle>Rl, Rr\<rangle>sum_rel \<equiv> {(Inl a, Inl a') |a a'. (a, a') \<in> Rl} \<union> {(Inr a, Inr a') |a a'. (a, a') \<in> Rr}"
denote_dim: fixes S :: "com"   and \<rho> :: "complex mat" assumes "well_com S"   and "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "denote S \<rho> \<in> carrier_mat d d"
lookup_not_in_vars_eq_zero: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and p :: "'b mpoly"   and i :: "nat" assumes "x \<in> monomials p"   and "i \<notin> vars p" shows "lookup x i = 0"
perform_emb_above_vars0: fixes \<rho> :: "'d \<Rightarrow> ('c, 'd) tm"   and s :: "('c, 'd) tm"   and u :: "('c, 'd) tm" assumes "subst \<rho> s \<unrhd>\<^sub>e\<^sub>m\<^sub>b u" obtains w :: "('c, 'd) tm" where "s \<unrhd>\<^sub>e\<^sub>m\<^sub>b w" and "subst \<rho> w \<unrhd>\<^sub>e\<^sub>m\<^sub>b u" and "\<forall>w'. w \<rightarrow>\<^sub>e\<^sub>m\<^sub>b w' \<longrightarrow> \<not> subst \<rho> w' \<unrhd>\<^sub>e\<^sub>m\<^sub>b u"
pr_weak_refl: fixes fn :: "'f \<times> nat" shows "pr_weak fn fn"
valid_foldl_1: shows "valid_mmuaux' args cur nt (foldl (\<lambda>aux _. eval_step_mmuaux aux) aux tss_list) auxlist"
marked_insertions_def: shows "marked_insertions = (\<lambda>s. \<forall>x. x \<in> set (sys_mem_store_buffers (mutator m) s) \<longrightarrow> marked_insertion x s)"
minus_mat_def: fixes A :: "'d mat"   and B :: "'d mat" shows "A - B \<equiv> Matrix.mat (dim_row B) (dim_col B) (\<lambda>ij. A $$ ij - B $$ ij)"
snd_pseudo_mod_smult_left: fixes x :: "'b"   and p :: "'b poly"   and q :: "'b poly" shows "snd (pseudo_divmod (Polynomial.smult x p) q) = Polynomial.smult x (snd (pseudo_divmod p q))"
setmonotone_cases: fixes f :: "'b set \<Rightarrow> 'b set"   and X :: "'b set" assumes "setmonotone f" shows "f X = X \<or> X \<subset> f X"
mult_scaleR_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and a :: "real"   and y :: "'b" assumes "class.real_algebra plus zero minus uminus scaleR times" shows "times x (scaleR a y) = scaleR a (times x y)"
fst1: fixes P :: "'a trm"   and P' :: "'a trm" assumes "P >>> P'"   and "\<And>A B. P \<noteq> SimplyTyped.Pair A B" shows "Fst P >>> Fst P'"
UNIV_is_not_in_ZF: fixes R :: "ZF" shows "UNIV \<noteq> explode R"
degree_m_eq: fixes f :: "int poly" assumes "lead_coeff f mod p ^ l \<noteq> 0"   and "1 < p ^ l" shows "pl.degree_m f = degree f"
not_present_alloc: fixes h :: "heap"   and v :: "'a" shows "\<not> Ref.present h (fst (Ref.alloc v h))"
rtrancl_path_imp_rtrancl_tab: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" assumes "rtrancl_path r x xs y"   and "distinct (x # xs)"   and "({x} \<union> set xs) \<inter> set ys = {}" shows "rtrancl_tab r ys x y"
fresh_constants: fixes m :: "nat"   and p :: "('a, 'b) form" assumes "infinite UNIV" shows "\<exists>cs. length cs = m \<and> list_all (\<lambda>c. new c p) cs \<and> distinct cs"
PDQ: shows "(P, D, Q) = diagonalize_2x2 A"
poly_pcompose: fixes p :: "'c poly"   and q :: "'c poly"   and x :: "'c" shows "poly (p \<circ>\<^sub>p q) x = poly p (poly q x)"
map_def: fixes f :: "'a1 \<times> 'a2" shows "local.map f = (if A1xA2.arr f then (snd f, fst f) else A2xA1.null)"
CatIdDomCod_1: fixes X :: "'a" assumes "X \<in> obj" shows "dom (id X) = X"