hcju/isabelleps huggingface dataset:
X3_pow: fixes X :: "X3 set" shows "X \<in> set ` set (subseqs [X3.Xd1, X3.Xd1', X3.Xd2])"
crk_is_preserved: fixes P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec" assumes "invertible P" shows "col_rank A = col_rank (P ** A)"
prems_2: fixes xa :: "nat" assumes "xa \<in> {..<k}"   and "x\<^sub>0 \<le> (ts ! xa) x" shows "f ((ts ! xa) x) \<le> gc1 * f' ((ts ! xa) x)"
compare_GT_iff_LT: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and compare :: "'b \<rightarrow> 'b \<rightarrow> Ordering"   and x :: "'b"   and y :: "'b" assumes "class.Ord_linear eq compare" shows "(compare\<cdot>x\<cdot>y = GT) = (compare\<cdot>y\<cdot>x = LT)"
simps_1: fixes uu :: "'b"   and uv :: "'b"   and F :: "'a" shows "J_C.Map (J_C.MkArr uu uv F) = F"
prefix_imp_subseq: fixes xs :: "'c list"   and ys :: "'c list" assumes "prefix xs ys" shows "subseq xs ys"
bind_option_cong: fixes x :: "'c option"   and y :: "'c option"   and f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" assumes "x = y"   and "\<And>z. z \<in> set_option y \<Longrightarrow> f z = g z" shows "x \<bind> f = y \<bind> g"
denseE: fixes A :: "'a llist set" assumes "A dense" obtains "carrier \<subseteq> closure A"
eq_linorder_class_conv: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res" shows "eq_linorder cmp = class.linorder (comp2le cmp) (comp2lt cmp)"
find_closest_code_eq: fixes ps :: "(int \<times> int) list"   and \<delta> :: "real"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and \<delta>' :: "int"   and p :: "int \<times> int"   and c :: "int \<times> int"   and \<delta>\<^sub>c' :: "int"   and c' :: "int \<times> int" assumes "0 < length ps"   and "\<delta> = dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))"   and "\<delta>' = dist_code c\<^sub>0 c\<^sub>1"   and "sorted_snd (p # ps)"   and "c = find_closest p \<delta> ps"   and "(\<delta>\<^sub>c', c') = find_closest_code p \<delta>' ps" shows "c = c'"
leg_simps_1: shows "C.arr assoc_fgh.dom.leg0"
assms_5: shows "snd ` snd ` set ps \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs)"
simps_1: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set" assumes "getIJ getIJ" shows "New_max.merge_option (None, None) = None"
assms_1: shows "A \<Longrightarrow>\<^sub>A B"
range_adjoint_isometry: fixes U :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "isometry U" shows "U* *\<^sub>S \<top> = \<top>"
connected_Int_frontier: fixes s :: "'b set"   and t :: "'b set" assumes "connected s"   and "s \<inter> t \<noteq> {}"   and "s - t \<noteq> {}" shows "s \<inter> frontier t \<noteq> {}"
LstSeq_append_app1: fixes s :: "hf"   and k :: "hf"   and y :: "hf"   and l :: "hf"   and s' :: "hf" assumes "LstSeq s k y"   and "l \<^bold>\<in> succ k" shows "app (seq_append (succ k) s s') l = app s l"
integer_group_def: shows "integer_group = \<lparr>carrier = UNIV, mult = (+), one = 0\<rparr>"
finite_measure_subadditive_countably: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> sets M"   and "summable (\<lambda>i. Sigma_Algebra.measure M (A i))" shows "Sigma_Algebra.measure M (\<Union> (range A)) \<le> (\<Sum>i. Sigma_Algebra.measure M (A i))"
Skip_is_CSP: shows "is_CSP_process (R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A = alpha_rp.more A'))"
wf_tuple_drop: fixes b :: "nat"   and n :: "nat"   and A :: "nat set"   and xs :: "'a option list"   and B :: "nat set" assumes "wf_tuple (b + n) A xs"   and "{x - b |x. x \<in> A \<and> b \<le> x} = B" shows "wf_tuple n B (drop b xs)"
non_empty_cycle_root_rtc_symmetric: fixes r :: "'a"   and x :: "'a" assumes "non_empty_cycle_root r x" shows "x\<^sup>\<star> ; r = x\<^sup>T\<^sup>\<star> ; r"
\<mu>: shows "local.iso \<mu> \<and> \<guillemotleft>\<mu> : g \<Rightarrow> h\<guillemotright>"
nested_case_prod_simp: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'c"   and x :: "'d \<times> 'e"   and y :: "'f" shows "(case x of (a, b) \<Rightarrow> f a b) y = (case x of (a, b) \<Rightarrow> f a b y)"
assms_3: shows "trg f \<star> \<mu> = (\<i>[trg f] \<star> f) \<cdot> local.inv \<a>[trg f, trg f, f]"
interaction_bounded_by_exec_gpv_bad: fixes gpv :: "('c, 'b, 'd) gpv"   and n :: "nat"   and bad :: "'e \<Rightarrow> bool"   and s :: "'e"   and callee :: "'e \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'e) spmf"   and k :: "real" assumes "interaction_bounded_by' (\<lambda>_. True) gpv n"   and "True"   and "\<not> bad s"   and "\<And>s x. \<lbrakk>True; \<not> bad s; x \<in> outs_\<I> \<I>_full\<rbrakk> \<Longrightarrow> spmf (map_spmf (bad \<circ> snd) (callee s x)) True \<le> k"   and "0 \<le> k"   and "\<I>_full \<turnstile>g gpv \<surd>" shows "spmf (map_spmf (bad \<circ> snd) (exec_gpv callee gpv s)) True \<le> k * real n"
restr_simple_state_rel_combine: fixes R :: "'e set"   and ES :: "('f \<times> 'g) set" shows "\<langle>ES\<rangle>restr_simple_state_rel R = restr_rel R O \<langle>ES\<rangle>simple_state_rel"
inj_map: fixes f :: "'b \<Rightarrow> 'c" assumes "inj f" shows "inj ((\<circ>) f)"
leftmost_append: fixes i :: "nat"   and u :: "symbol list"   and v :: "symbol list" assumes "leftmost i (u @ v)"   and "is_word u" shows "length u \<le> i"
prems_2: shows "(oid, op1) \<in> set (x # op_list)"
inverse_fract: fixes a :: "'b"   and b :: "'b" shows "inverse (Fract a b) = Fract b a"
adm_unlr: fixes R :: "'a admS" shows "adm (\<lambda>x. x \<in> unlr R)"
init_Plus: fixes fsm1 :: "('b, 'c) fsm_scheme"   and fsm2 :: "('b, 'd) fsm_scheme" shows "init (Plus fsm1 fsm2) = init fsm1 + init fsm2"
iptables_bigstep_process_ret_undecided: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>process_ret rs, Undecided\<rangle> \<Rightarrow> t"
simps_15: fixes x21 :: "'f"   and x22 :: "'f list" shows "set (x21 # x22) = insert x21 (set x22)"
list_all2_app_l: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b list"   and r :: "'b list"   and z :: "'b list" assumes "reflp R"   and "list_all2 R l r" shows "list_all2 R (z @ l) (z @ r)"
natural_isomorphism_axioms: shows "natural_isomorphism VVV.comp (\<cdot>) HoHV HoVH \<alpha>.map"
integrable_add: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "integrable M f"   and "integrable M g" shows "integrable M (\<lambda>x. f x + g x)"
mult_left_zero: fixes x :: "'a" shows "bot \<sqinter> x = bot"
subgroupE_4: fixes H :: "'c set"   and a :: "'c"   and b :: "'c" assumes "subgroup H H"   and "a \<in> H"   and "b \<in> H" shows "a \<otimes>\<^bsub>H\<^esub> b \<in> H"
acc_option: assumes "acc {(x, y). x \<sqsubseteq> y}" shows "acc {(x, y). x \<sqsubseteq> y}"
psubset_imp_subset: fixes A :: "'g set"   and B :: "'g set" assumes "A \<subset> B" shows "A \<subseteq> B"
fds_cong: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "\<And>n. 0 < n \<Longrightarrow> f n = g n" shows "fds f = fds g"
ttree_belowI: fixes t :: "'b ttree"   and t' :: "'b ttree" assumes "\<And>xs. xs \<in> paths t \<Longrightarrow> xs \<in> paths t'" shows "t \<sqsubseteq> t'"
s: shows "s = rat_of_int sz / rat_of_int sn"
span_eq_iff: fixes s :: "('b \<Rightarrow>\<^sub>0 'c) set" shows "(phull s = s) = phull.subspace s"
a_insert_node_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and ptr :: "(_) object_ptr"   and new_child :: "(_) node_ptr"   and reference_child_opt :: "(_) node_ptr option" shows "l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_node get_child_nodes set_child_nodes ptr new_child reference_child_opt = Heap_Error_Monad.bind (get_child_nodes ptr) (\<lambda>children. set_child_nodes ptr (insert_before_list new_child reference_child_opt children))"
25: shows "irreflexive (prim_W w v e * (w \<sqinter> - prim_EP w v e)\<^sup>\<star> * e * prim_P w v e\<^sup>T\<^sup>\<star> * (w \<sqinter> - prim_EP w v e)\<^sup>\<star>)"
sum_single_nonzero: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "\<forall>x\<in>A. \<forall>y\<in>A. f x y = (if y = x then g x else (0::'b))" shows "\<forall>x\<in>A. sum (f x) A = g x"
rsp: shows "eq_onp (\<lambda>M. mat_det M \<noteq> 0) id_moebius_cmat id_moebius_cmat"
shows_pl_def: fixes p :: "nat" shows "shows_pl p = apply_if (0 < p) (shows CHR ''('')"
permutation_permutes: fixes p :: "'c \<Rightarrow> 'c" shows "permutation p = (\<exists>S. finite S \<and> p permutes S)"
pseudominimal_in_below_in: fixes P :: "'a set"   and y :: "'a"   and x :: "'a"   and w :: "'a" assumes "has_bottom (P .\<^bold>\<le> y)"   and "x \<in> P"   and "y \<succeq> x"   and "pseudominimal_in (P .\<^bold>\<le> x) w" shows "pseudominimal_in (P .\<^bold>\<le> y) w"
ps\<^sub>R'_def: shows "ps\<^sub>R' = {p \<in> ps\<^sub>R. dist (case case p of (x, y) \<Rightarrow> (real_of_int x, y) of (x, y) \<Rightarrow> (x, real_of_int y)) (case (l, snd (case case p of (x, y) \<Rightarrow> (real_of_int x, y) of (x, y) \<Rightarrow> (x, real_of_int y))) of (x, y) \<Rightarrow> (real_of_int x, y)) < dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))}"
sifum_lang_no_dma_axioms: shows "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"
Quotient_alt_def5: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Abs :: "'d \<Rightarrow> 'e"   and Rep :: "'e \<Rightarrow> 'd"   and T :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "Quotient R Abs Rep T = (T \<le> BNF_Def.Grp UNIV Abs \<and> BNF_Def.Grp UNIV Rep \<le> T\<inverse>\<inverse> \<and> R = T OO T\<inverse>\<inverse>)"
HInv1_def: fixes s :: "state" shows "HInv1 s = (Inv1 s \<and> allInput s \<subseteq> Inputs)"
ide_Ide: fixes a :: "'a \<times> 'a" assumes "ide (fst a) \<and> ide (snd a)" shows "VxV.ide a"
differentiable_scaleR: fixes f :: "'a \<Rightarrow> real"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f differentiable at x within s"   and "g differentiable at x within s" shows "(\<lambda>x. f x *\<^sub>R g x) differentiable at x within s"
arrow_of_spans_hcomp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.arr (span_vertical_category.vcomp C) \<mu>"   and "partial_magma.arr (span_vertical_category.vcomp C) \<nu>"   and "span_vertical_category.src C \<nu> = span_vertical_category.trg C \<mu>" shows "arrow_of_spans C (span_bicategory.hcomp C prj0 prj1 \<nu> \<mu>)"
analz_insert_freshK3_bis: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list"   and authK' :: "nat" assumes "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace>, servTicket\<rbrace> \<in> set evs"   and "authK \<noteq> authK'"   and "authK' \<notin> range shrK"   and "evs \<in> kerbV" shows "(Key servK \<in> analz (insert (Key authK') (knows Spy evs))) = (servK = authK' \<or> Key servK \<in> analz (knows Spy evs))"
insert_not_empty: fixes a :: "'f"   and A :: "'f set" shows "insert a A \<noteq> {}"
id_subst_comp_subst: fixes \<sigma> :: "'s" shows "id_subst \<odot> \<sigma> = \<sigma>"
cupcake_build_conv_preserve: fixes vs :: "v list"   and i :: "char list"   and v :: "v" assumes "list_all is_cupcake_value vs"   and "build_conv static_cenv (Some (Short i)) vs = Some v" shows "is_cupcake_value v"
assms_4: shows "flow0 x t1 \<in> {a<--<b}"
complex_conjugate_order: fixes p :: "complex poly"   and c :: "complex" assumes "set (coeffs p) \<subseteq> \<real>"   and "p \<noteq> 0" shows "order (cnj c) p = order c p"
A: fixes Y' :: "'a set" shows "(xs @ y # ys, Y') \<in> failures P \<longrightarrow> (xs @ ipurge_tr I D (D y) ys, ipurge_ref I D (D y) ys Y') \<in> failures P"
fits_def: fixes G :: "prog"   and s :: "st"   and a' :: "val"   and T :: "ty" shows "G,s\<turnstile>a' fits T = ((\<exists>rt. T = RefT rt) \<longrightarrow> a' = Null \<or> G\<turnstile>obj_ty (lookup_obj s a')\<preceq>T)"
similar_mat_witD2_2: fixes A :: "'a mat"   and n :: "nat"   and m :: "nat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" assumes "A \<in> carrier_mat n m"   and "similar_mat_wit A B P Q" shows "Q * P = 1\<^sub>m n"
prems_1: shows "Broadcast m1 \<in> set (history i)"
rel_transp: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "transp R" shows "transp (llist_all2 R)"
set_\<Phi>o: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<AA> :: "('d \<times> 'd) set"   and \<phi> :: "'b \<Rightarrow> 'c"   and a :: "'b" assumes "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>"   and "partial_magma.ide S a" shows "set_category_data.set S' (set_category.some_img S' \<AA>) (two_set_categories_bij_betw_Univ.\<Phi>o S S' \<AA> \<phi> a) = \<phi> ` set_category_data.set S (set_category.some_img S \<AA>) a"
directed_def: fixes X :: "'a set" shows "ord.directed (\<lambda>x y. y \<le> x) X = (\<forall>Y. finite Y \<and> Y \<subseteq> X \<longrightarrow> (\<exists>x\<in>X. \<forall>y\<in>Y. x \<le> y))"
simps_2: fixes E :: "'a \<Rightarrow> 'm"   and i :: "int" shows "eval E (Const i) = return i"
Graph9: fixes j :: "nat"   and M :: "node list" assumes "j < length M" shows "Blacks M \<subseteq> Blacks (M[j := Black])"
aprimedivisor_pos_nat: fixes n :: "nat" assumes "1 < n" shows "0 < aprimedivisor n"
fls_deriv_def: fixes f :: "'a fls" shows "fls_deriv f = Abs_fls (\<lambda>n. of_int (n + 1) * fls_nth f (n + 1))"
r_inv: fixes x :: "'a" shows "x ** local.inv x = one"
InvariantImpliedLiteralsAfterBackjump: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and p :: "(Literal \<times> bool) list"   and bClause :: "Literal list"   and bLiteral :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantImpliedLiterals F M"   and "isPrefix p M"   and "formulaEntailsClause F bClause"   and "isUnitClause bClause bLiteral (elements p)"   and "M' = p @ [(bLiteral, False)]" shows "InvariantImpliedLiterals F M'"
ereal_eq_0_1: fixes r :: "real" shows "(ereal r = 0) = (r = 0)"
ineq_1: shows "real_of_rat l' \<le> real_of_1 x"
elims_2: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and xa :: "'b literal set set" assumes "validate_clause_set x xa" obtains I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and S :: "'b literal set set" where "x = I" and "xa = S" and "\<forall>C. C \<in> S \<longrightarrow> validate_clause I C"
simps: fixes p :: "bool Heap"   and f :: "'b Heap" shows "while p f = p \<bind> (\<lambda>b. if b then f \<bind> (\<lambda>_. while p f) else return ())"
inducts: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and x :: "'a stream"   and P :: "'a stream \<Rightarrow> bool" assumes "ev \<phi> x"   and "\<And>xs. \<phi> xs \<Longrightarrow> P xs"   and "\<And>xs. \<lbrakk>ev \<phi> (stl xs); P (stl xs)\<rbrakk> \<Longrightarrow> P xs" shows "P x"
mat1: fixes env :: "v sem_env"   and s1 :: "'a state"   and e :: "exp0"   and s2 :: "'a state"   and v1 :: "v"   and pes :: "(pat \<times> exp0) list"   and e' :: "exp0"   and env' :: "(char list \<times> v) list"   and bv :: "'a state \<times> (v, v) result" assumes "Big_Step_Unclocked_Single.evaluate env s1 e (s2, Rval v1)"   and "match_result env s2 v1 pes Bindv = Rval (e', env')"   and "Big_Step_Unclocked_Single.evaluate (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) s2 e' bv" shows "Big_Step_Unclocked_Single.evaluate env s1 (Mat e pes) bv"
S_Min_Lt: shows "S_Min_Lt = {- d - u c |c d. 0 < v c \<and> v c \<le> n \<and> M 0 (v c) = Lt d}"
multpw_split1R: fixes x :: "'a"   and X :: "'a multiset"   and Y :: "'a multiset"   and ns :: "('a \<times> 'a) set" assumes "(add_mset x X, Y) \<in> multpw ns" obtains z :: "'a" and Z :: "'a multiset" where "Y = add_mset z Z" and "(x, z) \<in> ns" and "(X, Z) \<in> multpw ns"
cont_eff_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and cf :: "('c, 'd, 'a) cmd \<times> 'b"   and b :: "nat" assumes "PL cval" shows "PL.cont_eff aval tval cval cf b = snd (PL.wt_cont_eff aval tval cval (fst cf) (snd cf) b)"
ba_lim_exist: shows "convergent (\<lambda>n. real_of_int (b (n + 1) - b n) / real_of_int (a n))"
ih_7: shows "InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state')"
a1: shows "w = (CNOT \<Otimes> Quantum.Id 1) * (\<phi> \<Otimes> |\<beta>\<^sub>0\<^sub>0\<rangle>)"
function_uminus_def: fixes S :: "'d set"   and R :: "('e, 'f) ring_scheme"   and a :: "'d \<Rightarrow> 'e" shows "function_uminus S R a = (\<lambda>x\<in>S. \<ominus>\<^bsub>R\<^esub> a x)"
downset_prop: fixes x :: "'c" shows "\<down> x = {y. y \<le> x}"
real_polynomial_function_compose: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> real" assumes "polynomial_function f"   and "real_polynomial_function g" shows "real_polynomial_function (g \<circ> f)"
TrustCertVerify_lemma: fixes X :: "msg"   and evs :: "event list"   and A :: "agent"   and nb :: "msg"   and B :: "agent"   and pms :: "msg" assumes "X \<in> parts (knows Spy evs)"   and "X = Crypt (priEK A) (Hash \<lbrace>nb, Agent B, pms\<rbrace>)"   and "evs \<in> tls"   and "A \<notin> bad" shows "Says A B X \<in> set evs"
False: shows "fst (memory_read asi address s) \<noteq> None"
iT_Mult_iSince_conv: fixes k :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" assumes "0 < k" shows "(P t1. t1 \<S> t2 I \<otimes> k. Q t2) = (P (t1 * k). t1 \<S> t2 I. Q (t2 * k))"
simps_9: fixes uu :: "'b ltln set" shows "false\<^sub>n[uu]\<^sub>\<mu> = false\<^sub>n"
eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "orda x y"
SUP_union: fixes M :: "'e \<Rightarrow> 'c \<times> 'd"   and A :: "'e set"   and B :: "'e set" shows "Sup_prod (\<P> M (A \<union> B)) = sup_prod (Sup_prod (\<P> M A)) (Sup_prod (\<P> M B))"
sqrt_rat_floor: fixes x :: "rat" shows "sqrt_rat_floor x = \<lfloor>sqrt (real_of_rat x)\<rfloor>"
linorder_neqE: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y" obtains "x < y"   | "y < x"
set_add_atoms: fixes r :: "'c rexp"   and as :: "'c list" shows "set (add_atoms r as) = atoms r \<union> set as"
WT_converter_mono: fixes \<I>1 :: "('h, 'i) \<I>"   and \<I>2 :: "('j, 'k) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter"   and \<I>1' :: "('h, 'i) \<I>"   and \<I>2' :: "('j, 'k) \<I>" assumes "\<I>1, \<I>2 \<turnstile>\<^sub>C conv \<surd>"   and "\<I>1' \<le> \<I>1"   and "\<I>2 \<le> \<I>2'" shows "\<I>1', \<I>2' \<turnstile>\<^sub>C conv \<surd>"
trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
llist_of_stream_def: shows "llist_of_stream = unfold_llist (\<lambda>_. False) shd stl"
encode_problem_has_model_then_also_partial_encodings_2: fixes \<A> :: "sat_plan_variable \<Rightarrow> bool"   and \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "\<A> \<Turnstile> \<Phi> \<Pi> t" shows "\<A> \<Turnstile> (\<Phi>\<^sub>G \<Pi>) t"
divide_quat_def: fixes x :: "quat"   and y :: "quat" shows "x / y = x * inverse y"
linorder_neqE_linordered_idom: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "x \<noteq> y" obtains "less x y"   | "less y x"
asymp_equiv_imp_diff_smallo: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g" shows "(\<lambda>x. f x - g x) \<in> o[F](g)"
path2_in_\<alpha>n: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and l :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "l \<in> set ns" shows "l \<in> set (\<alpha>n g)"
P1: shows "Col C D X \<and> Col C D Y \<and> A B TS X Y"
tminus_zero: fixes x :: "real" assumes "0 \<le> x" shows "x \<ominus> 0 = x"
eventually_happens: fixes P :: "'a \<Rightarrow> bool"   and net :: "'a filter" assumes "eventually P net" shows "net = bot \<or> (\<exists>x. P x)"
enabled_conj2: fixes F :: "state \<times> state \<Rightarrow> bool"   and G :: "state \<times> state \<Rightarrow> bool" shows "\<turnstile> Enabled (F \<and> G) \<longrightarrow> Enabled G"
simps_1: fixes a :: "'b" shows "firstPos a [] = 0"
diff_fun_between_chartsD: fixes k :: "enat"   and charts :: "('c, 'd) chart set"   and f :: "'c \<Rightarrow> 'e"   and c :: "('c, 'd) chart"   and x :: "'c" assumes "diff_fun k charts f"   and "c \<in> c_manifold.atlas charts k"   and "x \<in> domain c" shows "k-smooth_on (codomain c) (f \<circ> inv_chart c)"
type_is_iface: fixes G :: "prog"   and I :: "qtname" assumes "is_type G (Iface I)" shows "is_iface G I"
prod_list_map_hom: fixes xs :: "rat poly list" shows "prod_list (map (map_poly of_rat) xs) = map_poly of_rat (prod_list xs)"
case_5: fixes f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> ((nat \<Rightarrow> nat option) \<Rightarrow> bool) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'a"   and f4 :: "'a"   and f5 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'a"   and x51 :: "nat set"   and x52 :: "nat set"   and x53 :: "nat set"   and x54 :: "nat set" shows "(case Ghost x51 x52 x53 x54 of Read x xa xb \<Rightarrow> f1 x xa xb | Write x xa xb xc xd xe xf \<Rightarrow> f2 x xa xb xc xd xe xf | RMW x xa xb xc xd xe xf xg xh \<Rightarrow> f3 x xa xb xc xd xe xf xg xh | Fence \<Rightarrow> f4 | Ghost x xa xb xc \<Rightarrow> f5 x xa xb xc) = f5 x51 x52 x53 x54"
2137: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - - (x \<squnion> y) = top"
f1: shows "(\<forall>i t. i \<le> n \<and> SEQ i s1 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {}) \<and> (\<forall>i t. i \<le> n \<and> SEQ i s2 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {})"
transE: fixes r :: "('c \<times> 'c) set"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "trans r"   and "(x, y) \<in> r"   and "(y, z) \<in> r" obtains "(x, z) \<in> r"
method_def: fixes sig :: "sig"   and m :: "qtname \<times> methd" shows "method sig m = (declclass m, mdecl (sig, mthd m))"
one_dim_iso_scaleC: fixes c :: "complex"   and \<psi> :: "'b" shows "one_dim_iso (c *\<^sub>C \<psi>) = c *\<^sub>C one_dim_iso \<psi>"
is_tiny_dghm_op: shows "op_dghm \<GG> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> op_dg \<BB>"