hcju/isabelleps huggingface dataset:
ring_hom_ringI: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "ring R"   and "ring S"   and "\<And>x. x \<in> carrier R \<Longrightarrow> h x \<in> carrier S"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"   and "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>" shows "ring_hom_ring R S h"
synth_auth_data: fixes x :: "rat"   and y :: "rat"   and H :: "msg set" assumes "Auth_Data x y \<in> synth H" shows "Auth_Data x y \<in> H \<or> Pri_AgrK x \<in> H \<and> Pri_AgrK y \<in> H"
word_less_two_pow_divD: fixes x :: "'b word"   and n :: "nat"   and m :: "nat" assumes "x < 2 ^ n div 2 ^ m" shows "m \<le> n \<and> x < 2 ^ (n - m)"
fc_equals_wfc_rel_ipurge: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and I :: "('b \<times> 'b) set" shows "future_consistent P D (rel_ipurge P I D) = weakly_future_consistent P I D (rel_ipurge P I D)"
gram_schmidt_fs_Rn_axioms: shows "gram_schmidt_fs_Rn n (map of_int_hom.vec_hom fs')"
healthy_equiv: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "equiv_trans t u" shows "healthy t = healthy u"
R_DL_def: shows "R_DL = {(h, w). h = \<^bold>g [^] w}"
distrib_imp1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<And>x y z. x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> z" shows "x \<squnion> y \<sqinter> z = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
Int_emptyI: fixes A :: "'d set"   and B :: "'d set" assumes "\<And>x. \<lbrakk>x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> False" shows "A \<inter> B = {}"
fmpred_iff: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('c, 'd) fmap" shows "fmpred P m = (\<forall>x y. m $$ x = Some y \<longrightarrow> P x y)"
seqI': fixes f :: "('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr"   and b :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr"   and c :: "('a S.E.term, 'a) S.arr" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright>"   and "\<guillemotleft>g : b \<Rightarrow>\<^sub>S c\<guillemotright>" shows "S.seq g f"
distinct_14: fixes x11 :: "'a"   and x12 :: "'b" shows "Stop \<noteq> x11 \<leftarrow> x12"
rep_eq: fixes x :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and xa :: "('b, 'c) RBT.rbt" shows "RBT.fold x xa = RBT_Impl.fold x (impl_of xa)"
symrun_interp_left_commute: fixes \<Gamma>\<^sub>1 :: "'a constr list"   and \<Gamma>\<^sub>2 :: "'a constr list"   and \<Gamma>\<^sub>3 :: "'a constr list" shows "\<lbrakk>\<lbrakk> \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 @ \<Gamma>\<^sub>3 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<Gamma>\<^sub>2 @ \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>3 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
elements_mat_append_rows: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nr2 :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat nr2 n" shows "elements_mat (A @\<^sub>r B) = elements_mat A \<union> elements_mat B"
size_inst_fm: fixes p :: "('b, 'c) fm"   and t :: "'b tm"   and m :: "nat" shows "FOL_Axiomatic.size_fm (p\<langle>t/m\<rangle>) = FOL_Axiomatic.size_fm p"
nn_integral_0_iff: fixes u :: "'k \<Rightarrow> ennreal"   and M :: "'k measure" assumes "u \<in> borel_measurable M" shows "(integral\<^sup>N M u = 0) = (emeasure M {x \<in> space M. u x \<noteq> 0} = 0)"
set_poly_mult: shows "set (poly_mult p q @ [\<zero>]) \<subseteq> carrier R"
fpxs_ext: fixes f :: "'b fpxs"   and g :: "'b fpxs" assumes "\<And>r. fpxs_nth f r = fpxs_nth g r" shows "f = g"
abs_fun_eq3: fixes a :: "'b"   and b :: "'b"   and x :: "'a"   and y :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "a \<noteq> b"   and "x = [(a, b)] \<bullet> y"   and "a \<sharp> y" shows "[a].x = [b].y"
Uvariation_univ: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real" shows "Uvariation \<nu> \<nu>' Denotational_Semantics.allvars"
resInputLeft: fixes x :: "name"   and a :: "name"   and y :: "name"   and Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "x \<noteq> a"   and "x \<noteq> y"   and "eqvt Rel"   and "Id \<subseteq> Rel" shows "<\<nu>x>a<y>.P \<leadsto>[Rel] a<y>.(<\<nu>x>P)"
sset_depth1: shows "i.R depth1 = {\<phi>. max_depth \<phi> = 1}"
Trans: shows "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> M\<lparr>N\<rparr> \<prec> T"
2: shows "sum t_A' {Suc n..<length qs} = int (T (s_A' (Suc n)) (drop (Suc n) qs) (drop (Suc n) acts))"
1: fixes y :: "'c" shows "cont luba orda Sup (\<le>) (\<lambda>x. f x y)"
nop_v11_axioms: shows "nop_v11 A (1\<^sub>\<nat>) f"
resT_map_def: shows "resT_map \<equiv> \<Lambda> a. \<mu> uu. resT_abs oo ssum_map\<cdot>(u_map\<cdot>a)\<cdot>(u_map\<cdot>(fmap\<cdot>uu)) oo resT_rep"
gx3_def: shows "gx3 \<equiv> max x\<^sub>1 (SOME gx0. \<forall>x\<ge>gx0. g x \<le> gc1 * g' (real x))"
CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P_def: fixes A :: "'b set" shows "CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<equiv> \<mu> x. SKIP |-| STOP |-| Mprefix A (\<lambda>xa. x)"
l8_16_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and X :: "'a"   and U :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B X"   and "Tarski_neutral_dimensionless.Col Bet A B U"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C X" shows "\<not> Tarski_neutral_dimensionless.Col Bet A B C \<and> Tarski_neutral_dimensionless.Per Bet Cong C X U"
tendsto_at_left_realI_sequentially: fixes c :: "real"   and f :: "real \<Rightarrow> 'a"   and y :: "'a" assumes "\<And>X. filterlim X (at_left c) sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y" shows "(f \<longlongrightarrow> y) (at_left c)"
bot_Bag\<^sub>b\<^sub>a\<^sub>s\<^sub>e_def: shows "UML_Types.bot_class.bot \<equiv> Abs_Bag\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<bottom>"
uminus: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "L F (\<lambda>x. - g x) = L F g"
l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.known_ptr ShadowRootClass.type_wf Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs ShadowRootClass.known_ptrs Shadow_DOM.get_parent Shadow_DOM.get_parent_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.parent_child_rel get_disconnected_nodes"
intro: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "category S" shows "set_category_data S"
eventually_nonzeroD: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "eventually_nonzero F f" shows "\<forall>\<^sub>F x in F. f x \<noteq> (0::'b)"
preserves_cod: fixes f :: "'a" assumes "A.arr f" shows "A.cod (A.map f) = A.map (A.cod f)"
h_arith_prog: shows "h = arith_prog h_start h_step"
degree_le2: shows "degree (Poly (list_of_vec (row (local.berlekamp_mat u) i))) \<le> degree u - 1"
prod_cont_iff: fixes f :: "'c \<times> 'd \<Rightarrow> 'e" shows "cont f = ((\<forall>y. cont (\<lambda>x. f (x, y))) \<and> (\<forall>x. cont (\<lambda>y. f (x, y))))"
code: fixes \<I> :: "('b, 'c) \<I>"   and gpv :: "('a, 'b, 'c) gpv" shows "restrict_gpv \<I> gpv = GPV (map_pmf (case_option None (case_generat (Some \<circ> Pure) (\<lambda>out c. if out \<in> outs_\<I> \<I> then Some (IO out (\<lambda>input. if input \<in> responses_\<I> \<I> out then restrict_gpv \<I> (c input) else Fail)) else None))) (the_gpv gpv))"
summable: fixes X :: "nat \<Rightarrow> 'a" assumes "summable X" shows "summable (\<lambda>n. inner (X n) (1::'a))"
weak_equivalence_is_weak_bisimulation: shows "is_weak_bisimulation (\<equiv>\<cdot>)"
stepSs_def: shows "stepSs = sdrop (length srs) (konigDown t S)"
isoI: fixes f :: "'a"   and g :: "'a" assumes "B.inverse_arrows f g" shows "B.iso f"
min_less_iff_conj: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less" shows "less z (ord.min less_eq x y) = (less z x \<and> less z y)"
rOfWax_def: fixes \<sigma> :: "'e" shows "rOfWax \<sigma> = Fn (Wit \<sigma>) []"
inv: fixes x :: "'a" assumes "x \<in> carrier A" shows "f' (f x) = x"
frame_thms_1: fixes P :: "assn" shows "P \<Longrightarrow>\<^sub>t P"
underS_notIn: fixes a :: "'d"   and r :: "('d \<times> 'd) set" shows "a \<notin> underS r a"
absch: assumes "qs \<in> pattern" shows "2 / 10 * ts + 8 / 10 * bit \<le> 16 / 10 * opt"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>0 'c"   and xa :: "'b \<Rightarrow>\<^sub>0 'c" shows "lookup (x + xa) = (\<lambda>k. lookup x k + lookup xa k)"
abs_eq: fixes x :: "nonce_t" shows "Nonce x = Ab (cNonce x)"
cone_at_ide_is_cone: fixes x :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and \<chi> :: "'j \<Rightarrow> ('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and a :: "'c" assumes "YoD.cone x \<chi>"   and "Cop.ide a" shows "Limit.cone J (\<cdot>\<^sub>S) (YoD.at a (local.map \<circ> D)) (\<chi>.\<Phi>.Ya.Cop_S.Map x a) (YoD.at a \<chi>)"
simps_1: fixes s :: "'state"   and i :: "nat" shows "wt_cont_eff Done s i = (1, Done, s)"
coprime_dvd_mult_right_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 a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one a c" shows "dvd.dvd times a (times c b) = dvd.dvd times a b"
at_to_infinity: shows "at (0::'a) = filtermap inverse at_infinity"
min_satisfying_def: fixes P :: "'a \<Rightarrow> bool"   and l :: "'a list" shows "min_satisfying P l \<equiv> let xs = filter P l in if xs = [] then None else Some (list_min xs)"
Df_Nil: fixes a :: "'a" shows "Df a 0 H = f a"
vertex_subtree_def: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set"   and v :: "'e" assumes "TreeDecomposition G T bag" shows "TreeDecomposition.vertex_subtree T bag v \<equiv> {t \<in> V\<^bsub>T\<^esub>. v \<in> bag t}"
cset_Collect_def: shows "cset_Collect = acset \<circ> Collect"
wa_refine_axioms: shows "wa_refine WB WA \<alpha>2"
const: fixes name :: "name"   and \<Gamma> :: "(name, value) fmap"   and val :: "value" assumes "name |\<notin>| C"   and "fmlookup \<Gamma> name = Some val" shows "\<Gamma> \<turnstile>\<^sub>v Sconst name \<down> val"
mset_subset_diff_self: fixes c :: "'b"   and B :: "'b multiset" assumes "c \<in># B" shows "B - (c, {#}) \<subset># B"
poly_of_const_def: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "ring.poly_of_const R = (\<lambda>k. ring.normalize R [k])"
landau_product_preprocess_13: fixes m :: "num" shows "m * num.One = m"
simply_connected_def: fixes S :: "'b set" shows "simply_connected S \<equiv> \<forall>p q. path p \<and> pathfinish p = pathstart p \<and> path_image p \<subseteq> S \<and> path q \<and> pathfinish q = pathstart q \<and> path_image q \<subseteq> S \<longrightarrow> homotopic_loops S p q"
Smith_normal_form_upt_k_Suc_eq: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and k :: "nat"   and A :: "(('a, 'b) vec, 'c) vec" assumes "is_bezout_ext bezout"   and "k < min (nrows A) (ncols A)" shows "Smith_normal_form_upt_k (diagonal_to_Smith_aux A [0..<Suc k] bezout) k = Smith_normal_form_upt_k (diagonal_to_Smith_aux A [0..<k] bezout) k"
homeomorphic_ball01_UNIV: shows "ball (0::'c) 1 homeomorphic UNIV"
supported_def: fixes a :: "'b zmultiset"   and t :: "'b" shows "supported a t = (\<exists>s<t. zcount a s < 0)"
Mem_SUCC_E: fixes u :: "tm"   and t :: "tm"   and H :: "fm set"   and C :: "fm" assumes "insert (u IN t) H \<turnstile> C"   and "insert (u EQ t) H \<turnstile> C" shows "insert (u IN SUCC t) H \<turnstile> C"
DirProds_group_iff: fixes G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set" shows "Group.group (DirProds G I) = (\<forall>i\<in>I. Group.group (G i))"
IH: fixes s :: "('a, bool) fmap"   and PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set" assumes "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "last ` state_set (state_list s as) \<subseteq> reachable_s PROB s"
prod_prime_subset: fixes A :: "nat set"   and B :: "nat set" assumes "finite A"   and "finite B"   and "\<And>x. x \<in> A \<Longrightarrow> prime x"   and "\<And>x. x \<in> B \<Longrightarrow> prime x"   and "\<Prod>A dvd \<Prod>B" shows "A \<subseteq> B"
A_def: shows "A \<equiv> \<lambda>p s. (s - 1) * pre_zeta (real p) s - of_nat p / (of_nat p powr s * (of_nat p powr s - 1))"
relfun_S: fixes s :: "'w" assumes "s \<in> S" shows "add_order (s + s) = 1"
intros_2: fixes M :: "atom list"   and T :: "atom set"   and H :: "chmsg set"   and c :: "secprop set"   and A :: "agent"   and B :: "agent"   and U :: "fid_t set" assumes "set M \<subseteq> extr T H"   and "auth \<notin> c \<or> A \<in> bad \<or> B \<in> bad" shows "StatCh c A B (Msg M) \<in> fake T U H"
bij_betw_msubset_of: fixes A :: "'c set"   and B :: "'d set" assumes "finite A" shows "bij_betw (msubset_of A) ({f \<in> A \<rightarrow>\<^sub>E B. f ` A = B} // domain_permutation A B) {M. set_mset M = B \<and> size M = card A}"
HEndPhase0_HInv4a_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv4a s q"   and "p \<noteq> q" shows "HInv4a s' q"
less_trans: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x < y"   and "y < z" shows "x < z"
results_gpv_stop_SomeD: fixes x :: "'a"   and \<I> :: "('b, 'c) \<I>"   and gpv :: "('a, 'b, 'c) gpv" assumes "Some x \<in> results_gpv (stop_\<I> \<I>) (gpv_stop gpv)" shows "x \<in> results_gpv \<I> gpv"
find_closest_pair_code_dist_eq: fixes \<delta> :: "int"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and \<Delta> :: "int"   and C\<^sub>0 :: "int \<times> int"   and C\<^sub>1 :: "int \<times> int"   and ps :: "(int \<times> int) list" assumes "\<delta> = dist_code c\<^sub>0 c\<^sub>1"   and "(\<Delta>, C\<^sub>0, C\<^sub>1) = find_closest_pair_code (\<delta>, c\<^sub>0, c\<^sub>1) ps" shows "\<Delta> = dist_code C\<^sub>0 C\<^sub>1"
verit_minus_simplify_1: fixes a :: "'a" shows "a - a = (0::'a)"
IH_2: assumes "A \<noteq> 0"   and "\<And>i. i \<^bold>\<in> A \<Longrightarrow> Ord i" shows "\<Sqinter>A \<^bold>\<in> A"
assms: fixes A :: "'a tensor" assumes "A \<in> set As" shows "A = tensor0 ds"
multrow_carrier: fixes mul :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "nat"   and a :: "'b"   and A :: "'b mat"   and n :: "nat"   and nc :: "nat" shows "(mat_multrow_gen mul k a A \<in> carrier_mat n nc) = (A \<in> carrier_mat n nc)"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "semigroup_add_hom hom"   and "injective hom" shows "inj_semigroup_add_hom hom"
intersect_segment_xline_None_iff: fixes prec :: "nat"   and p0 :: "real \<times> real"   and p1 :: "real \<times> real"   and x :: "real" shows "(intersect_segment_xline prec (p0, p1) x = None) = (closed_segment p0 p1 \<inter> {p. fst p = x} = {})"
domain_Epigraph_aux: fixes x :: "ereal" assumes "x \<noteq> \<infinity>" shows "\<exists>r. x \<le> ereal r"
assms: shows "\<exists>n w. qty n w \<noteq> 0"
contractR: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> F, F, \<Delta>" shows "\<Gamma> \<Rightarrow> F, \<Delta>"
finite_fmap_ran: fixes F :: "'h \<Rightarrow> 'g option" shows "finite (ran F)"
representant_exists: fixes A :: "('a set set \<times> ('a set \<Rightarrow> ('a \<times> 'a) set)) set" assumes "A \<in> eq.Partition" shows "\<exists>a\<in>Sigma (top.neighborhoods \<pp>) sheaf_spec. a \<in> A \<and> A = eq.Class a"
simps: fixes A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)" shows "local.embed A = (\<lambda>s. return_pmf (fst A s), \<lambda>s r. return_pmf (snd A s r))"
inconsistent: fixes R1 :: "'a \<Rightarrow> 'f \<Rightarrow> bool"   and S1 :: "'b \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and S2 :: "'g \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_var R1 S1 OO rel_var R2 S2 \<le> rel_var (R1 OO R2) (S1 OO S2)"
nf_RRR_11: shows "test (K \<circ> C \<circ> K \<circ> C \<circ> K) = [True, True, False, False, True]"
cross7_components_1: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "(x \<times>\<^sub>7 y) $ 1 = x $ 2 * y $ 4 - x $ 4 * y $ 2 + x $ 3 * y $ 7 - x $ 7 * y $ 3 + x $ 5 * y $ 6 - x $ 6 * y $ 5"
eq1: shows "exH_n * exH_n = 1\<^sub>m N"
inner_axis_axis: fixes i :: "'b"   and x :: "'a"   and j :: "'b"   and y :: "'a" shows "axis i x \<bullet> axis j y = (if i = j then x \<bullet> y else 0)"
simps: fixes a1 :: "('a\<^sub>m, 'a\<^sub>h, 'b\<^sub>m, 'b\<^sub>h) G\<^sub>m"   and a2 :: "('a\<^sub>m, 'a\<^sub>h, 'b\<^sub>m, 'b\<^sub>h) G\<^sub>m" shows "local.blinding_of_G a1 a2 = (\<exists>x y. a1 = G\<^sub>m x \<and> a2 = G\<^sub>m y \<and> blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob x y)"
rank_invar_meld_strong: fixes bq1 :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and bq2 :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.rank_invar bq1"   and "BinomialHeapStruc.rank_invar bq2" shows "BinomialHeapStruc.rank_invar (BinomialHeapStruc.meld bq1 bq2) \<and> min (BinomialHeapStruc.rank (hd bq1)) (BinomialHeapStruc.rank (hd bq2)) \<le> BinomialHeapStruc.rank (hd (BinomialHeapStruc.meld bq1 bq2))"
in_homE: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "Fop.Aop.in_hom f a b" obtains "Fop.Aop.arr f" and "Fop.Aop.dom f = a" and "Fop.Aop.cod f = b"
neg_equal_swap: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "(\<ominus> x = y) = (x = \<ominus> y)"
ker_def: shows "ker = {v \<in> carrier M. f v = \<zero>\<^bsub>N\<^esub>}"
nat_pow_closed: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and n :: "nat" assumes "subgroup H G"   and "a \<in> H" shows "a [^]\<^bsub>G\<^esub> n \<in> H"
deepR_list: fixes n :: "nat"   and m :: "('b, 'c) FingerTreeStruc"   and pr :: "('b, 'c) Digit"   and sf :: "('b, 'c) Node list" assumes "is_leveln_ftree (Suc n) m \<and> is_measured_ftree m"   and "is_leveln_digit n pr \<and> is_measured_digit pr"   and "\<forall>x\<in>set sf. (is_measured_node x \<and> is_leveln_node n x) \<and> length sf \<le> 4" shows "toList (deepR pr m sf) = digitToList pr @ toList m @ nlistToList sf"
rule_charn4: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and r :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "wellformed_policy1 p"   and "allNetsDistinct p"   and "singleCombinators p"   and "wellformed_policy3Pr p"   and "DenyAll \<notin> set p"   and "applied_rule_rev Cp x p = \<lfloor>DenyAllFromTo a b\<rfloor>"   and "r \<in> set p"   and "x \<in> dom (Cp r)" shows "r = DenyAllFromTo a b"
assms_2: shows "x \<in> set (fst (interp_ops ops))"
A_1: shows "((s, c), w, [], c') \<in> trcl (trss fg)"
listset_Rep_Exp_Abs_Exp: fixes Us :: "freeExp list" shows "listset (map Rep_Exp (Abs_ExpList Us)) = listrel exprel `` {Us}"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (\<lambda>f2. F (A1.cod f1', f2)) (\<lambda>f2. F (A1.cod f1', f2)) (\<lambda>f2. F (A1.cod f1', f2))"
lunit_simp: fixes f :: "'a" assumes "ide f" shows "lunit f = B.lunit f"
lbE_2: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" assumes "lb T A j" shows "\<Union> (A ` {1..m}) = {1..j}"
targets_determine_composability: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and \<mu> :: "'b"   and \<tau> :: "'b" assumes "regular_weak_composition V H"   and "b \<in> weak_composition.targets V H \<mu>" shows "(H \<tau> \<mu> \<noteq> partial_magma.null V) = (H \<tau> b \<noteq> partial_magma.null V)"
prv_eql_neg_encF_N: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "prv (eql \<langle>neg \<phi>\<rangle> (encF N \<langle>\<phi>\<rangle>))"
cd_le: fixes x :: "nat"   and m :: "nat"   and n :: "nat" assumes "cd x m n"   and "0 < m"   and "0 < n" shows "x \<le> m \<and> x \<le> n"
prob_reachable_le: fixes m :: "nat"   and n :: "nat"   and K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a"   and w :: "'a" assumes "m \<le> n" shows "MC_syntax.p K x y m * MC_syntax.p K y w (n - m) \<le> MC_syntax.p K x w n"
norms_mus_rat_def: fixes n :: "nat"   and fs :: "rat vec list" shows "norms_mus_rat n fs = (let (n_gs, mus) = norms_mus_rat' n fs [] [] in (rev n_gs, rev mus))"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "lcm (lcm a b) c = lcm a (lcm b c)"
nb\<^sub>6: fixes op\<^sub>1' :: "('a \<times> 'b) strips_operator"   and op\<^sub>2' :: "('a \<times> 'b) strips_operator" assumes "op\<^sub>1' \<in> set (map (sasp_op_to_strips \<Psi>) ops)"   and "op\<^sub>2' \<in> set (map (sasp_op_to_strips \<Psi>) ops)"   and "\<exists>(v, a)\<in>set (add_effects_of op\<^sub>1'). \<exists>(v', a')\<in>set (delete_effects_of op\<^sub>2'). (v, a) = (v', a')" shows "\<exists>op\<^sub>1\<in>set ops. \<exists>op\<^sub>2\<in>set ops. \<not> SAS_Plus_Semantics.are_operator_effects_consistent op\<^sub>1 op\<^sub>2"
bigo_plus_absorb_lemma2: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O(g)" shows "O(g) \<subseteq> f +o O(g)"
icsSp_Append_ddep: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and V :: "'var"   and n' :: "'node SDG_node" assumes "n ics-ns\<rightarrow>\<^sub>d* n''"   and "n'' s-V\<rightarrow>\<^sub>d\<^sub>d n'"   and "n'' \<noteq> n'" shows "n ics-ns @ [n'']\<rightarrow>\<^sub>d* n'"
HDomain_Incl_sf: fixes t :: "tm"   and u :: "tm" shows "Sigma_fm (HDomain_Incl t u)"
metric_CauchyI: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'd" assumes "class.metric_space dist uniformity open"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e" shows "uniform_space.Cauchy uniformity X"
antipar_2: shows "src (tab\<^sub>0 r)\<^sup>* = trg (tab\<^sub>0 r)"