hcju/isabelleps huggingface dataset:
subrelI: fixes r :: "('g \<times> 'h) set"   and s :: "('g \<times> 'h) set" assumes "\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s" shows "r \<subseteq> s"
open_union': fixes F :: "'a set set" assumes "\<And>x. x \<in> F \<Longrightarrow> is_open x" shows "is_open (\<Union> F)"
stabilizer_one_closed: fixes x :: "'c" assumes "x \<in> E" shows "\<one> \<in> stabilizer G \<phi> x"
fieldE_1: fixes R :: "('a, 'b) ring_scheme" assumes "field R" shows "cring R"
image_uminus_greaterThanAtMost: 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 x :: "'a"   and y :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "uminus ` ord.greaterThanAtMost less_eq less x y = ord.atLeastLessThan less_eq less (uminus y) (uminus x)"
exp_lower_taylor_even: fixes n :: "nat"   and x :: "real" assumes "even n" shows "(\<Sum>m<n. x ^ m / fact m) \<le> exp x"
induct: fixes x :: "('c, 'd) ipurge_rec"   and X :: "('c, 'd) ipurge_rec"   and P :: "('c, 'd) ipurge_rec \<Rightarrow> bool" assumes "x \<in> ipurge_tr_rev_t_set X"   and "P X"   and "\<And>I D u x xs ys. \<lbrakk>\<lparr>Pol = I, Map = D, Dom = u, In = x # xs, Out = ys\<rparr> \<in> ipurge_tr_rev_t_set X; P \<lparr>Pol = I, Map = D, Dom = u, In = x # xs, Out = ys\<rparr>; D x \<in> sources I D u (x # xs)\<rbrakk> \<Longrightarrow> P \<lparr>Pol = I, Map = D, Dom = u, In = xs, Out = ys @ [x]\<rparr>"   and "\<And>I D u x xs ys. \<lbrakk>\<lparr>Pol = I, Map = D, Dom = u, In = x # xs, Out = ys\<rparr> \<in> ipurge_tr_rev_t_set X; P \<lparr>Pol = I, Map = D, Dom = u, In = x # xs, Out = ys\<rparr>; D x \<notin> sources I D u (x # xs)\<rbrakk> \<Longrightarrow> P \<lparr>Pol = I, Map = D, Dom = u, In = ipurge_tr I D (D x) xs, Out = ys\<rparr>" shows "P x"
elims: fixes x :: "nat"   and xa :: "'b \<Rightarrow> 'b"   and xb :: "'b"   and y :: "'b" assumes "repeat_stabilize x xa xb = y" obtains uu_ :: "'b \<Rightarrow> 'b" and v :: "'b" where "x = 0" and "xa = uu_" and "xb = v__" and "y = v__"   | n :: "nat" and f :: "'b \<Rightarrow> 'b" and v :: "'b" where "x = Suc n" and "xa = f" and "xb = v" and "y = (let v_new = f v in if v = v_new then v else repeat_stabilize n f v_new)"
dom_glb: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "ad (ad x) \<le> ad (ad y)"   and "ad (ad x) \<le> ad (ad z)" shows "ad (ad x) \<le> ad (ad y) \<cdot> ad (ad z)"
transitive_closure_trans_2: fixes x :: "'b"   and y :: "'b"   and r :: "('b \<times> 'b) set"   and z :: "'b" assumes "(x, y) \<in> r\<^sup>+"   and "(y, z) \<in> r\<^sup>+" shows "(x, z) \<in> r\<^sup>+"
less_ord0_LimD: fixes x :: "ord0"   and f :: "nat \<Rightarrow> ord0" assumes "x < ord0_Lim f" shows "\<exists>n. x \<le> f n"
FAssThrow2: fixes 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 e\<^sub>1 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "expr"   and e' :: "expr"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and F :: "char list"   and Cs :: "char list list" assumes "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>e\<^sub>1\<bullet>F{Cs} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>"
left_neutral: fixes a :: "'b" shows "- (1::'b) AND a = a"
bind_def: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'c set" shows "A \<bind> f = {x. \<exists>B\<in>f ` A. x \<in> B}"
assms: fixes i :: "nat"   and j :: "nat" assumes "i < nr"   and "j < nc" shows "f (i, j) = f' (i, j)"
le_supE: fixes a :: "'a multiset"   and b :: "'a multiset"   and x :: "'a multiset" assumes "a \<union># b \<subseteq># x" obtains "a \<subseteq># x" and "b \<subseteq># x"
ex_suma: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "D \<noteq> E"   and "E \<noteq> F" shows "\<exists>G H I. A B C D E F SumA G H I"
simps_1: fixes acc :: "nat" shows "acc_lengths acc [] = []"
isPartition_wrt_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and lo :: "nat"   and hi :: "nat"   and p :: "nat" shows "isPartition_wrt R xs lo hi p \<equiv> (\<forall>i. lo \<le> i \<and> i < p \<longrightarrow> R (xs ! i) (xs ! p)) \<and> (\<forall>j. p < j \<and> j \<le> hi \<longrightarrow> R (xs ! p) (xs ! j))"
Szpilrajn: fixes base_r :: "('a \<times> 'a) set" assumes "strict_partial_order base_r" shows "\<exists>r. strict_linear_order r \<and> base_r \<subseteq> r"
simps: fixes h :: "'c \<Rightarrow> 'b"   and data :: "'c"   and subtrees :: "'c rose_tree list" shows "blind_source_tree h (Tree (data, subtrees)) = Tree\<^sub>m (Blinded (Content (h data, map (hash_source_tree h) subtrees)))"
bind_eq_return_spmf: fixes p :: "'c spmf"   and f :: "'c \<Rightarrow> 'b spmf"   and x :: "'b" shows "(p \<bind> f = return_spmf x) = ((\<forall>y\<in>set_spmf p. f y = return_spmf x) \<and> lossless_spmf p)"
empty_WL_GC: fixes s :: "(('f, 'g, 'h) response, char list, 'e process_name, 'e process_name \<times> ('f, 'e, 'g, 'h) request_op, ('f, 'e, 'g, 'h) Model.local_state) system_state" assumes "atS gc get_roots_UN_get_work_locs s"   and "obj_fields_marked_invL s" shows "gc_ghost_honorary_grey s\<down> = {}"
length_nths': fixes xs :: "'b list"   and I :: "nat set" shows "length (nths xs I) = card {i \<in> I. i < length xs}"
prems_1: shows "Di \<in> set (subseqs (di # D))"
iless_Suc_eq: fixes m :: "int"   and n :: "eint" shows "(eint m < n + 1) = (eint m \<le> n)"
assms_1: shows "M \<equiv> dens_ctxt_measure (shift_var_set V, Suc ` V', case_nat t \<Gamma>, insert_dens V V' f \<delta>) \<rho>"
ring_record_simps_1: fixes carrier :: "'e set"   and more :: "'f" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
transE: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "trans r"   and "x \<succeq>[r] y"   and "y \<succeq>[r] z" obtains "x \<succeq>[r] z"
cases: fixes a :: "'d"   and \<delta> :: "('d, 'e) ta_rule set"   and Q :: "'d set" assumes "a \<in> bacc_step \<delta> Q" obtains r :: "('d, 'e) ta_rule" where "a = lhs r" and "r \<in> \<delta>" and "set (rhsq r) \<subseteq> Q"
l\<^sub>2': shows "l\<^sub>2' = [pns [\<mapsto>] vs]"
of_nat_eq_of_nat_power_cancel_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and b :: "nat"   and w :: "nat"   and x :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(power.power one times (semiring_1.of_nat one plus zero b) w = semiring_1.of_nat one plus zero x) = (b ^ w = x)"
unanimity_def: fixes F :: "(indi \<Rightarrow> (alt \<times> alt) set) \<Rightarrow> (alt \<times> alt) set" shows "unanimity F \<equiv> \<forall>P\<in>Prof. \<forall>a b. (\<forall>i. a <\<^bsub>P i\<^esub> b) \<longrightarrow> a <\<^bsub>F P\<^esub> b"
source_target_relation_impl_indRelR: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and Pred :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool" assumes "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"   and "\<forall>(P, Q)\<in>Rel. Pred (P, Q)" shows "\<forall>(P, Q)\<in>indRelR. Pred (P, Q)"
i: shows "if n = 0 then i = IArray.length tail else i = hd is"
inf_sup_absorb: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.lattice inf less_eq less sup" shows "inf x (sup x y) = x"
bounded_uniformly_continuous_image: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "uniformly_continuous_on S f"   and "bounded S" shows "bounded (f ` S)"
segment_free: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" assumes "Order D"   and "a \<notin> carrier D" shows "segment D a = carrier D"
parts_Un_subset1: fixes G :: "msg set"   and H :: "msg set" shows "parts G \<union> parts H \<subseteq> parts (G \<union> H)"
uint_numeral: fixes b :: "num" shows "uint (numeral b) = numeral b mod 2 ^ LENGTH('a)"
space_empty: fixes M :: "'a measure" assumes "space M = {}" shows "M = count_space {}"
assms_3: fixes x :: "'a vec" assumes "x \<in> sat_dual A c" shows "y \<bullet> c \<le> x \<bullet> c"
valid_dbm_cases: fixes X :: "'d set"   and v :: "'d \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'d"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry" assumes "Regions X v n not_in_X"   and "Regions.valid_dbm X v n M" obtains "[M]\<^bsub>v,n\<^esub> \<subseteq> Regions.V X" and "dbm_int M n"
almost_full_on_imp_reflp_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "almost_full_on P A" shows "reflp_on P A"
simps_8: fixes n :: "nat"   and r :: "'a atom rexp" shows "lang n (rexp.Not r) = lists ((set \<circ> \<sigma> \<Sigma>) n) - lang n r"
add_comm: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" shows "a \<oplus> b = b \<oplus> a"
IH: shows "real (Lcm {1..n}) = exp (\<psi> (real n))"
\<omega>_limit_set_def: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.\<omega>_limit_set f X x = {p. auto_ll_on_open.\<omega>_limit_point f X x p}"
extremum_unique: fixes a :: "'d upred" shows "(false \<sqsubseteq> a) = (a = false)"
set_of_l_bal: fixes l :: "'a tree\<^sub>0"   and r :: "'a tree\<^sub>0"   and x :: "'a" assumes "height l = height r + 2" shows "set_of (l_bal\<^sub>0 x l r) = insert x (set_of l \<union> set_of r)"
fold_def: shows "fold \<equiv> \<lambda>uu uua. rec_list (\<lambda>f. id) (\<lambda>x xs xsa f. (xsa f \<circ>\<circ> f) x) uua uu"
econc_fun_ERETURN: fixes E :: "('c \<times> 'e) set"   and R :: "('d \<times> 'f) set"   and x :: "'f" shows "\<Down>\<^sub>E E R (ERETURN x) = ESPEC (\<lambda>_. False) (\<lambda>xi. (xi, x) \<in> R)"
C1_differentiable_polynomial_function: fixes p :: "real \<Rightarrow> 'a"   and S :: "real set" assumes "polynomial_function p" shows "p C1_differentiable_on S"
InfCard_le_cadd_eq: fixes \<kappa> :: "V"   and \<mu> :: "V" assumes "InfCard \<kappa>"   and "\<mu> \<le> \<kappa>" shows "\<kappa> \<oplus> \<mu> = \<kappa>"
strict_coboundedI2: fixes c :: "'a word"   and b :: "'a word"   and a :: "'a word" assumes "c <s b" shows "c <s signed.max a b"
quadratic_sub_less_helper: fixes var :: "nat"   and L :: "real list"   and Cv :: "real"   and C :: "real mpoly"   and A :: "real mpoly"   and Av :: "real"   and B :: "real mpoly"   and Bv :: "real" assumes "var < length L"   and "0 \<le> Cv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) C = Cv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) A = Av"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) B = Bv" shows "aEval (Less (A + B * Var var)) (L[var := sqrt Cv]) = eval (Or (And (fm.Atom (Less A)) (fm.Atom (Less (B\<^sup>2 * C - A\<^sup>2)))) (And (fm.Atom (Leq B)) (Or (fm.Atom (Less A)) (fm.Atom (Less (A\<^sup>2 - B\<^sup>2 * C)))))) (L[var := sqrt Cv])"
treach_ref_l': fixes l :: "Location"   and s :: "Store" assumes "ref l \<noteq> nullV" shows "treach \<tau> ref l \<tau> s@@l"
crypts_msg_def: fixes X :: "msg" shows "crypts_msg X \<equiv> crypts {X}"
sets_restr_to_subalg: fixes M :: "'b measure"   and F :: "'b measure" assumes "subalgebra M F" shows "sets (restr_to_subalg M F) = sets F"
on_unit_circle_conjugate: fixes z :: "complex_homo" shows "on_circline unit_circle (conjugate z) = on_circline unit_circle z"
pdevs_of_list_Nil: shows "pdevs_of_list [] = zero_pdevs"
univ_poly_not_field: fixes K :: "'a list set" assumes "subring K (K [X])" shows "\<not> field (K [X]\<^bsub>K [X]\<^esub>)"
seq_extract_range: fixes A :: "nat set"   and xs :: "'b list" shows "A \<upharpoonleft>\<^sub>l xs = (A \<inter> dom\<^sub>l xs) \<upharpoonleft>\<^sub>l xs"
balance_left_inv1l: fixes lt :: "('a, 'b) rbt"   and rt :: "('a, 'b) rbt"   and k :: "'a"   and x :: "'b" assumes "inv1l lt"   and "inv1 rt" shows "inv1l (balance_left lt k x rt)"
swap_simp: fixes x :: "'e"   and y :: "'d" shows "prod.swap (x, y) = (y, x)"
Zp_converges_to_def: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" shows "Zp_converges_to s a = ((a \<in> carrier Zp \<and> s \<in> carrier Zp\<^bsup>\<omega>\<^esup>) \<and> (\<forall>n. \<exists>k. \<forall>m>k. eint n < val_Zp (s m \<ominus> a)))"
sup_inf_distrib1: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" shows "x \<union># (y \<inter># z) = x \<union># y \<inter># (x \<union># z)"
filterlim_at_top_powr_real: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real"   and g' :: "real" assumes "filterlim f at_top F"   and "(g \<longlongrightarrow> g') F"   and "1 < g'" shows "LIM x F. g x powr f x :> at_top"
diagonalize_is_functor: shows "functor (\<cdot>) \<F>\<^sub>SC.comp \<F>C.D"
abs_eq: shows "Z = abs_var (PMF (return_pmf (True, undefined)))"
updEnv_preserves_wls: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and xs :: "'varSort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and x :: "'var" assumes "wlsEnv rho"   and "wls (asSort xs) X" shows "wlsEnv (rho [x \<leftarrow> X]_xs)"
rel_def: shows "rel = inv_image finite_psubset (\<lambda>S. {n. is_node n \<and> \<not> memb n S})"
qFresh_qFreshInp: fixes xs :: "'varSort"   and x :: "'var"   and delta :: "'f"   and qinp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and qbinp :: "'e \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs option" shows "qFresh xs x (qOp delta qinp qbinp) = (qFreshInp xs x qinp \<and> qFreshBinp xs x qbinp)"
assms_2: fixes "is" :: "nat list" assumes "is \<lhd> dims A" shows "lookup A is = lookup B is"
assert_simplex': fixes i :: "'a"   and s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" shows "assert_simplex i (Simplex_State s) = map_sum id Simplex_State (assert_simplex' i s)"
simps_1: fixes r :: "'c rexp"   and s :: "'c rexp" shows "Pluss (Plus r s) = zip_with_option merge_distinct (Pluss r) (Pluss s)"
nn_integral_count_space': fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "finite A"   and "\<And>x. \<lbrakk>x \<in> B; x \<notin> A\<rbrakk> \<Longrightarrow> f x = 0"   and "A \<subseteq> B" shows "integral\<^sup>N (count_space B) f = sum f A"
phiArg_tranclp_path_ex: fixes r :: "'val \<Rightarrow> 'val \<Rightarrow> bool"   and p :: "'val"   and q :: "'val"   and g :: "'g" assumes "r\<^sup>+\<^sup>+ p q"   and "p \<in> allVars g"   and "\<And>p q. r p q \<Longrightarrow> phiArg g p q" obtains ns :: "'node list" where "g \<turnstile> defNode g q-ns\<rightarrow>defNode g p" and "1 < length ns" and "\<forall>n\<in>set (butlast ns). \<exists>p q m ns'. r p q \<and> g \<turnstile> defNode g q-ns'\<rightarrow>m \<and> defNode g q \<notin> set (tl ns') \<and> q \<in> phiUses g m \<and> m \<in> set (old.predecessors g (defNode g p)) \<and> n \<in> set ns' \<and> set ns' \<subseteq> set ns \<and> defNode g p \<in> set ns"
Refl_under_in: fixes r :: "('a \<times> 'a) set"   and a :: "'a" assumes "Refl r"   and "a \<in> Field r" shows "a \<in> under r a"
t: shows "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t pair ` setops\<^sub>s\<^sub>s\<^sub>t S"
c1: fixes a :: "real"   and c :: "real" assumes "Dyn_Tab2 a c" shows "1 < c"
not_single_rt_fresher: fixes rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and ip :: "nat"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "\<not> the (rt1 ip) \<sqsubseteq> the (rt2 ip)" shows "\<not> rt1 \<sqsubseteq>\<^bsub>ip\<^esub> rt2"
in_interval_interval_bij: fixes x :: "'a"   and a :: "'a"   and b :: "'a"   and u :: "'a"   and v :: "'a" assumes "x \<in> cbox a b"   and "cbox u v \<noteq> {}" shows "interval_bij (a, b) (u, v) x \<in> cbox u v"
act4: fixes x :: "'a"   and P :: "'b set" shows "\<alpha> x (\<Squnion> P) = \<Squnion> (\<alpha> x ` P)"
AudioPlayer_CTRL_def: shows "AudioPlayer_CTRL = Abs_seqauto (AudioPlayer_CTRL_States, AudioPlayer_CTRL_Init, AudioPlayer_CTRL_Labels, AudioPlayer_CTRL_Delta)"
decomps\<^sub>e\<^sub>s\<^sub>t_preserves_wf: fixes D :: "('fun, 'var) extstrand_step list"   and A :: "('fun, 'var) extstrand_step list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and V :: "'var set" assumes "D \<in> decomps\<^sub>e\<^sub>s\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t A) (assignment_rhs\<^sub>e\<^sub>s\<^sub>t A) \<I>"   and "wf\<^sub>e\<^sub>s\<^sub>t V A" shows "wf\<^sub>e\<^sub>s\<^sub>t V (A @ D)"
assms: shows "wf {(l, l'). pointsTo l' = Link l}"
neg_mask_in_mask_range: fixes ptr :: "'b word"   and bits :: "nat"   and ptr' :: "'b word" assumes "is_aligned ptr bits" shows "(ptr' AND NOT (mask bits) = ptr) = (ptr' \<in> mask_range ptr bits)"
commutative_squares_1: shows "C.commutative_square \<nu>\<pi>.leg0 \<rho>.leg1 \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0\<^sub>1 \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0"
cases: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and V :: "(nat \<times> 'd) set"   and a1 :: "(nat \<times> 'd) stream"   and a2 :: "nat \<times> 'd" assumes "grun A w V a1 a2" obtains a :: "nat \<times> 'd" and p :: "nat \<times> 'd" and r :: "(nat \<times> 'd) stream" where "a1 = a ## r" and "a2 = p" and "case p of (k, p) \<Rightarrow> w !! k \<in> alphabet A \<and> a \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V" and "grun A w V r a"
multpw_split1R: fixes x :: "'b"   and X :: "'b multiset"   and Y :: "'b multiset"   and ns :: "('b \<times> 'b) set" assumes "(add_mset x X, Y) \<in> multpw ns" obtains z :: "'b" and Z :: "'b multiset" where "Y = add_mset z Z" and "(x, z) \<in> ns" and "(X, Z) \<in> multpw ns"
assms_2: shows "\<forall>e\<in>carrier - E - F. \<not> indep (insert e F)"
bigo_sum3: fixes f :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b"   and l :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and k :: "'c \<Rightarrow> 'd \<Rightarrow> 'a"   and A :: "'c \<Rightarrow> 'd set" assumes "f \<in> O(h)" shows "(\<lambda>x. \<Sum>y\<in>A x. l x y * f (k x y)) \<in> O(\<lambda>x. \<Sum>y\<in>A x. \<bar>l x y * h (k x y)\<bar>)"
obj_at_field_on_heap_def: fixes P :: "'h \<Rightarrow> bool"   and r :: "'h"   and f :: "'f" shows "obj_at_field_on_heap P r f \<equiv> \<lambda>s. case map_option obj_fields (sys_heap s r) of None \<Rightarrow> False | Some fs \<Rightarrow> case fs f of None \<Rightarrow> True | Some r' \<Rightarrow> P r'"
of_nat_less_numeral_power_cancel_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "nat"   and i :: "num"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less (semiring_1.of_nat one plus zero x) (power.power one times (numeral.numeral one plus i) n) = (x < numeral i ^ n)"
set_remove1_eq: fixes xs :: "'g list"   and x :: "'g" assumes "distinct xs" shows "set (remove1 x xs) = set xs - {x}"
l: shows "l \<in> B \<rightarrow>\<^sub>M A"
scaleC_left: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and r :: "complex"   and a :: "'a"   and b :: "'b" assumes "bounded_cbilinear prod" shows "prod (r *\<^sub>C a) b = r *\<^sub>C prod a b"
repP_def: fixes w :: "32 Word.word \<times> 32 Word.word"   and r :: "real" shows "w \<equiv>\<^sub>P r \<equiv> let (w1, w2) = w in (w1 \<equiv>\<^sub>L r) \<and> (w2 \<equiv>\<^sub>U r)"
of_nat_le_\<omega>\<^sub>z: fixes n :: "nat" shows "of_nat n \<le> \<omega>\<^sub>z"
leg1_composite: shows "\<pi>\<rho>.leg1 = \<pi>.leg1 \<cdot> \<pi>\<rho>.prj\<^sub>1"
take_several_pers: fixes w :: "'b list"   and n :: "nat"   and m :: "nat" assumes "periodN w n"   and "m * n \<le> \<^bold>|w\<^bold>|" shows "take n w \<^sup>@ m = take (m * n) w"
subst_apply_term_ctxt_apply_distrib: fixes C :: "('d, 'f) ctxt"   and t :: "('d, 'f) Term.term"   and \<mu> :: "'f \<Rightarrow> ('d, 'e) Term.term" shows "C\<langle>t\<rangle> \<cdot> \<mu> = (C \<cdot>\<^sub>c \<mu>)\<langle>t \<cdot> \<mu>\<rangle>"
fps_mult_fps_X_plus_1_nth: fixes a :: "'a fps"   and n :: "nat" shows "((1 + fps_X) * a) $ n = (if n = 0 then a $ n else a $ n + a $ (n - 1))"
Compositionality_Seq: fixes c1 :: "('exp, 'id) MWLfCom"   and c1' :: "('exp, 'id) MWLfCom"   and d :: "'d"   and c2 :: "('exp, 'id) MWLfCom"   and c2' :: "('exp, 'id) MWLfCom" assumes "[c1] \<approx>\<^bsub>d\<^esub> [c1']"   and "[c2] \<approx>\<^bsub>d\<^esub> [c2']" shows "[c1;c2] \<approx>\<^bsub>d\<^esub> [c1';c2']"
wp_equiv: fixes c :: "com"   and c' :: "com" assumes "c \<sim> c'" shows "local.wp c = local.wp c'"
simps_65: fixes x91 :: "stmt"   and x92 :: "qtname"   and x93 :: "vname"   and x94 :: "stmt"   and x31 :: "jump"   and x32 :: "stmt" shows "Try x91 Catch(x92 x93) x94 \<noteq> x31\<bullet> x32"
assms: shows "[\<lbrace>x,F\<rbrace> in v]"
axioms_1: fixes point_set :: "'c set"   and block_collection :: "'c set multiset" assumes "design point_set block_collection" shows "finite_incidence_system point_set block_collection"
ANR_closed_Un_local: fixes S :: "'a set"   and T :: "'a set" assumes "closedin (top_of_set (S \<union> T)) S"   and "closedin (top_of_set (S \<union> T)) T"   and "ANR S"   and "ANR T"   and "ANR (S \<inter> T)" shows "ANR (S \<union> T)"
mono_mono_def: fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd" shows "mono_mono F = (mono F \<and> (\<forall>f. mono f \<longrightarrow> mono (F f)))"
uminus_less_lrv: fixes a :: "'a"   and b :: "'a" shows "(- a < - b) = (b < a)"
order_power_n_n: fixes a :: "'b"   and n :: "nat" shows "order a ([:- a, 1::'b:] ^ n) = n"
inner_prod_with_row_bra_vec: fixes u :: "complex Matrix.vec"   and v :: "complex Matrix.vec" assumes "dim_vec u = dim_vec v" shows "\<langle>u|v\<rangle> = Matrix.row (bra_vec u) 0 \<bullet> v"
simps_8: fixes uv :: "cospan_shape.Arr" shows "cospan_shape.comp cospan_shape.Null uv = cospan_shape.Null"
mult_adjugate_det: fixes A :: "('b, 'c) sq_matrix" shows "A * adjugate A = diag (Square_Matrix.det A)"
mtx_invertibleD_2: fixes A :: "'a sq_mtx" assumes "mtx_invertible A" shows "A * A\<^sup>-\<^sup>1 = 1"
dvd_gcd_list_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and b :: "'b"   and xs :: "'b list" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times b (semiring_gcd.Gcd_fin one zero gcd (set xs)) = (\<forall>a\<in>set xs. dvd.dvd times b a)"
gauss_int_norm_dvd_imp_le: fixes b :: "gauss_int"   and a :: "gauss_int" assumes "b \<noteq> 0" shows "gauss_int_norm a \<le> gauss_int_norm (a * b)"