hcju/isabelleps huggingface dataset:
delete_edges_set_edges2: fixes G :: "'b list_graph"   and E :: "('b \<times> 'b) list" shows "set (edgesL (FiniteListGraph.delete_edges G E)) = {(a, b). (a, b) \<in> set (edgesL G) \<and> (a, b) \<notin> set E}"
diff_add_assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b" shows "c + b - a = c + (b - a)"
cong: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "F g A = F h B"
upd_nested_max_tstp_def: fixes m :: "('a, ('b, nat + nat) mapping) mapping"   and d :: "nat + nat"   and X :: "('a \<times> 'b) set" shows "upd_nested_max_tstp m d X = upd_nested m d (max_tstp d) X"
a_Lam: fixes x :: "Syntax.var"   and \<Gamma> :: "(Syntax.var, ty) fmap"   and \<tau>\<^sub>1 :: "ty"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>2 :: "ty" assumes "atom x \<sharp> \<Gamma>"   and "\<Gamma>(x $$:= \<tau>\<^sub>1) \<turnstile> e, eP, eV : \<tau>\<^sub>2" shows "\<Gamma> \<turnstile> Lam x e, Lam x eP, Lam x eV : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"
keys_map_indets_subset: fixes f :: "'f \<Rightarrow> 'd"   and p :: "('f \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'e" shows "keys (map_indets f p) \<subseteq> (\<lambda>t. \<Sum>x\<in>keys t. monomial (lookup t x) (f x)) ` keys p"
nonzero_inverse_inverse_eq: fixes a :: "'a" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>" shows "inv (inv a) = a"
sF': fixes b :: "bexp"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and ps :: "char list \<Rightarrow> int option"   and n :: "nat" shows "wp\<^sub>3 (WHILE b DO c) Q (ps, n) = ((\<lambda>(ps, n). vars b \<subseteq> dom ps \<and> (if pbval b ps then wp\<^sub>3 c (wp\<^sub>3 (WHILE b DO c) Q) (ps, n) else Q (ps, n))) \<and>* $ 1) (ps, n)"
dagger_denest2: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<dagger> = x\<^sup>\<dagger> \<cdot> (y \<cdot> x\<^sup>\<dagger>)\<^sup>\<dagger>"
division_inter: fixes p1 :: "'b set set"   and s1 :: "'b set"   and p2 :: "'b set set"   and s2 :: "'b set" assumes "p1 division_of s1"   and "p2 division_of s2" shows "{k1 \<inter> k2 |k1 k2. k1 \<in> p1 \<and> k2 \<in> p2 \<and> k1 \<inter> k2 \<noteq> {}} division_of s1 \<inter> s2"
subM: fixes t :: "'a set"   and M :: "'a set set" assumes "t \<in> M"   and "M \<subseteq> T" shows "t open"
act_inv_iff: fixes g :: "cltn2"   and b :: "proj2"   and c :: "proj2" assumes "g \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "(apply_cltn2 b (inv\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> g) = c) = (b = apply_cltn2 c g)"
pre_ranks_tokens: fixes q :: "'b"   and n :: "nat" assumes "\<not> sink q" shows "(pre_ranks (\<lambda>q. state_rank q n) (w n) q \<noteq> {}) = (configuration q (Suc n) \<noteq> {})"
jkbpC_init: fixes s :: "'s" shows "(tInit s \<in> jkbpC) = (s \<in> set envInit)"
mono_step'_top: fixes C :: "'a st option acom"   and C' :: "'a st option acom" assumes "C \<le> C'" shows "step' \<top> C \<le> step' \<top> C'"
axioms_2: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b" assumes "diagram J C D" shows "category J"
dlex_fun_refl: fixes s :: "'a \<Rightarrow> 'c" shows "dlex_fun s s"
yun_factorization_main_int: fixes f :: "rat poly"   and p :: "rat poly"   and g :: "rat poly"   and i :: "nat"   and hs :: "(rat poly \<times> nat) list"   and res :: "(rat poly \<times> nat) list"   and F :: "int poly"   and G :: "int poly"   and Hs :: "(int poly \<times> nat) list"   and Res :: "(int poly \<times> nat) list"   and c :: "rat" assumes "f = p div gcd p (pderiv p)"   and "g = pderiv p div gcd p (pderiv p)"   and "monic p"   and "yun_gcd.yun_factorization_main gcd f g i hs = res"   and "yun_gcd.yun_factorization_main gcd F G i Hs = Res"   and "yun_rel F c f"   and "yun_wrel G c g"   and "list_all2 (rel_prod yun_erel (=)) Hs hs" shows "list_all2 (rel_prod yun_erel (=)) Res res"
g1_def: shows "g1 = pr_conv_2_to_1 g"
invh_baliL: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "invh l"   and "invh r"   and "bheight l = bheight r" shows "invh (baliL l a r)"
id_measure_preserving: fixes M :: "'b measure" shows "(\<lambda>x. x) \<in> measure_preserving M M"
simps: fixes s :: "nat"   and lm :: "nat list"   and p :: "abc_inst list" shows "abc_final (s, lm) p = (s = length p)"
measure_le_1: fixes M :: "'b measure"   and X :: "'b set" assumes "prob_space M" shows "emeasure M X \<le> 1"
INFM_mono: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<exists>\<^sub>\<infinity>x. P x"   and "\<And>x. P x \<Longrightarrow> Q x" shows "\<exists>\<^sub>\<infinity>x. Q x"
strong_sum_absorption: fixes y :: "i"   and x :: "i" assumes "y = x \<oplus> y" shows "P x y"
p: shows "p = smult c (\<Prod>a\<leftarrow>as. [:- a, 1:])"
NSDERIV_approx: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and D :: "'a"   and h :: "'a star" assumes "NSDERIV f x :> D"   and "h \<in> Infinitesimal"   and "h \<noteq> 0" shows "(*f* f) (star_of x + h) - star_of (f x) \<approx> 0"
l_def_2: shows "\<S>\<bullet>\<C> \<turnstile> es0 : tls _> t2s"
dnso1: fixes x :: "'a" shows "x + x \<cdot> ar ar x = x \<cdot> ar ar x"
valid_Exit_path_intra_path: fixes n :: "'node"   and as :: "'edge list" assumes "n -as\<rightarrow>\<^sub>\<surd>* (_Exit_)" obtains as' :: "'edge list" and pex :: "'node" where "n -as'\<rightarrow>\<^sub>\<iota>* pex" and "method_exit pex" and "set (sourcenodes as') \<subseteq> set (sourcenodes as)"
hyps_1: shows "cupcake_evaluate_single env e (Rval y) \<and> (is_cupcake_all_env env \<longrightarrow> is_cupcake_exp e \<longrightarrow> (\<forall>s. \<exists>s'. Big_Step_Unclocked_Single.evaluate env s e (s', Rval y)))"
arg_u_lt_etc: assumes "is_App u" shows "{#size (arg u), size t, size s#} < {#size u, size t, size s#}"
prod_constant: fixes y :: "'c"   and A :: "'d set" shows "(\<Prod>x\<in>A. y) = y ^ card A"
assms_1: shows "(x, y) \<in> local.Minimal_Solutions"
assms: shows "\<sigma> \<in> unifiers (fterm_to_iterm ` ts \<times> fterm_to_iterm ` ts)"
l4_19: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and C :: "'a"   and B :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A C B"   and "Cong A C A C'"   and "Cong B C B C'" shows "C = C'"
IN_E: fixes e :: "nat \<times> nat" assumes "SAT_PUSH' e \<in> set p" shows "e \<in> E \<union> E\<inverse>"
coprime_add_one_right: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one a (plus a one)"
newind_all: shows "\<forall>i<r. \<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}"
measurable_PiM_component_rev: fixes i :: "'a"   and I :: "'a set"   and f :: "'b \<Rightarrow> 'c"   and M :: "'a \<Rightarrow> 'b measure"   and N :: "'c measure" assumes "i \<in> I"   and "f \<in> M i \<rightarrow>\<^sub>M N" shows "(\<lambda>x. f (x i)) \<in> Pi\<^sub>M I M \<rightarrow>\<^sub>M N"
power_inverse: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "power.power one times (inverse a) n = inverse (power.power one times a n)"
map_id0: shows "map_blindable\<^sub>m id id = id"
assms_2: shows "is_sig_GB_upt d G (lt q)"
zero_in_smallo: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" shows "(\<lambda>_. 0::'b) \<in> o[F](f)"
poly_divides_conv0: fixes q :: "'b poly"   and p :: "'b poly" assumes "degree q < degree p"   and "p \<noteq> 0" shows "(p dvd q) = (q = 0)"
simps_1: fixes x :: "Preliminaries.var" shows "gT (Var x) = Var x"
ofail_bind: fixes m :: "'f \<Rightarrow> 'd \<Rightarrow> 'e option" shows "ofail |>> m = ofail"
Qring_fix1: fixes a :: "'a"   and I :: "'a set" assumes "a \<in> carrier R"   and "ideal R I"   and "a \<in> I" shows "a \<uplus>\<^bsub>R\<^esub> I = I"
t_wise_balance_axioms: shows "t_wise_balance \<V>\<^sup>+ \<B>\<^sup>+ \<t> (\<Lambda>\<^sub>t + \<Lambda>\<^sub>t')"
lasso_cycle_def: fixes L :: "('e, 'f) lasso_scheme" shows "lasso_cycle L = lasso_va L # lasso_cysfx L"
span_scale: fixes x :: "'b"   and S :: "'b set"   and c :: "'a" assumes "x \<in> m1.span S" shows "c *a x \<in> m1.span S"
privateKey_neq_publicKey: fixes A :: "agent"   and A' :: "agent" shows "priK A \<noteq> pubK A'"
FinDimVectorSpaceI: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set" assumes "VectorSpace smult V"   and "\<exists>ms. set ms \<subseteq> V \<and> R_scalar_mult.RSpan UNIV smult ms = V" shows "FinDimVectorSpace smult V"
15: shows "regular ((w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - forest_components f * selected_edge h j g * top \<sqinter> top * selected_edge h j g\<^sup>T * forest_components f)"
AbGroup_inner_dirsum_el_decompI: fixes Gs :: "'b set list"   and x :: "'b" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "x \<in> (\<Oplus>G\<leftarrow>Gs. G)" shows "\<Oplus>Gs\<leftarrow> x \<in> listset Gs \<and> x = sum_list (\<Oplus>Gs\<leftarrow> x)"
fixpoint_eq_comp_pow: fixes m :: "'d \<Rightarrow> nat"   and f :: "'d \<Rightarrow> 'd option"   and x :: "'d" shows "\<exists>n. fixpoint m f x = option_comp_pow f n x"
A_def: fixes n :: "nat"   and i :: "nat" shows "A n i = {w. real i / 2 ^ n \<le> f w} \<inter> {w. f w < real (Suc i) / 2 ^ n}"
in_cfl_in_cfls: fixes G :: "('a, 'b) cfg"   and \<alpha> :: "('a + 'b) list"   and w :: "'a list" assumes "cfg_wgreibach G"   and "in_cfl G \<alpha> w" shows "in_cfls G {|\<alpha>|} w"
lcs_alt_1: fixes a :: "'a"   and x :: "'a"   and b :: "'a"   and y :: "'a" assumes "a + x = b + y" shows "lcs x y = a + x - gcs a b"
I_qe_pres\<^sub>1: fixes as :: "atom list"   and xs :: "int list" assumes "\<forall>a\<in>set as. divisor a \<noteq> 0"   and "\<forall>a\<in>set as. hd_coeff_is1 a" shows "Z.I (qe_pres\<^sub>1 as) xs = (\<exists>x. \<forall>a\<in>set as. I\<^sub>Z a (x # xs))"
fds_const_of_real: fixes c :: "real" shows "fds_const (of_real c) = of_real c"
split_minD: fixes t :: "'b tree234"   and x :: "'b"   and t' :: "'b up\<^sub>d" assumes "split_min t = (x, t')"   and "bal t"   and "0 < height t" shows "x # inorder (tree\<^sub>d t') = inorder t"
edges_are_in_V_1: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Digraph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w" shows "v \<in> V\<^bsub>G\<^esub>"
replacefacesAt_in: fixes i :: "nat"   and "is" :: "nat list"   and Fss :: "face list list"   and olfF :: "face"   and newFs :: "face list" assumes "i \<in> set is"   and "distinct is"   and "i < |Fss|" shows "replacefacesAt is olfF newFs Fss ! i = replace olfF newFs (Fss ! i)"
nth_Abs_fls: fixes f :: "int \<Rightarrow> 'a"   and n :: "int" assumes "\<forall>\<^sub>\<infinity>n. f (- int n) = (0::'a)" shows "fls_nth (Abs_fls f) n = f n"
map_gpv_parametric': fixes A :: "'m \<Rightarrow> 'o \<Rightarrow> bool"   and A' :: "'n \<Rightarrow> 'p \<Rightarrow> bool"   and C :: "'q \<Rightarrow> 't \<Rightarrow> bool"   and C' :: "'r \<Rightarrow> 'u \<Rightarrow> bool"   and R :: "'s \<Rightarrow> 'v \<Rightarrow> bool" shows "((A ===> A') ===> (C ===> C') ===> rel_gpv'' A C R ===> rel_gpv'' A' C' R) map_gpv map_gpv"
coprime_commute: fixes p :: "'b"   and q :: "'b" shows "comm_monoid_mult_class.coprime p q = comm_monoid_mult_class.coprime q p"
proj_stoch_set_finite_range: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure"   and X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'b"   and n :: "nat" assumes "infinite_cts_filtration p M F"   and "borel_adapt_stoch_proc F X" shows "finite (range (proj_stoch_proc X n))"
polychain_segments_of_aform: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" shows "polychain (segments_of_aform X)"
myss3_axioms_def: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a"   and a :: "char list \<Longrightarrow> 'a"   and z :: "char list \<Longrightarrow> 'a"   and v :: "char list \<Longrightarrow> 'a" shows "myss3_axioms x y a z v \<equiv> ((vwb_lens v \<and> v \<bowtie> x) \<and> x \<bowtie> v \<and> v \<bowtie> y) \<and> (y \<bowtie> v \<and> v \<bowtie> a) \<and> a \<bowtie> v \<and> v \<bowtie> z \<and> z \<bowtie> v"
closed_UNIV: fixes "open" :: "'b set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.closed open UNIV"
fp_cop_F_closed_inv': fixes x :: "'a"   and ds :: "'b set"   and c :: "'a" assumes "x \<in> fp_cop_F ds"   and "c \<in> above (Pd (Xd x)) x" shows "c \<in> fp_cop_F ds"
inf_sup_distrib1: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.distrib_lattice inf less_eq less sup" shows "inf x (sup y z) = sup (inf x y) (inf x z)"
0: shows "\<forall>n\<in>num. prv (neg (subst (cnj fls (eql (Var x) (Var x))) n x))"
assms_2: shows "\<forall>x. has_greatest_fixpoint (f x)"
lUnionM_Cons: fixes x :: "'m"   and M :: "'m list" shows "lUnionM (x # M) = lunionM x (lUnionM M)"
Inf_bool_def: fixes A :: "bool set" shows "\<Squnion>A = (\<not> False \<in> A)"
mono_2: fixes i :: "'b" assumes "i \<in> J" shows "G i \<subseteq> F i"
Super: fixes G :: "prog"   and s :: "st"   and n :: "nat" shows "G\<turnstile>Norm s \<midarrow>Super-\<succ>val_this s\<midarrow>n\<rightarrow> Norm s"
no_eq: fixes n :: "nat" assumes "no \<le> n" shows "norm (sum g {..<n} - x) < r"
dvd_odd_square: fixes a :: "int" assumes "odd a" shows "8 dvd a\<^sup>2 - 1"
roundup_ge_0: fixes emBits :: "nat"   and x :: "nat" assumes "0 < emBits"   and "0 < x" shows "0 \<le> roundup emBits x * x - emBits"
4: shows "(\<otimes>\<^bsub>add_monoid (residue_ring (p ^ k))\<^esub>) = (\<oplus>\<^bsub>residue_ring (p ^ k)\<^esub>)"
ae: shows "AE y in K x. R y (B' y) \<or> lim_stream y = B' y"
is_InSet_def: fixes stateful_strand_step :: "('d, 'e) stateful_strand_step" shows "is_InSet stateful_strand_step = (\<exists>x61 x62 x63. stateful_strand_step = \<langle>x61: x62 \<in> x63\<rangle>)"
int3: shows "integrable M (\<lambda>x. g x * q (X x))"
Compl_greaterThan: fixes k :: "'a" shows "- {k<..} = {..k}"
uv_2: shows "(v1, v2) \<noteq> vec_zero"
strict_sorted_into_WW: fixes x :: "nat"   and ns :: "nat list" shows "strict_sorted (into_WW x ns)"
prems_2: shows "\<forall>i\<le>Suc x. 0 < rec_exec f (xs @ [i])"
ipresSwapAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'f, 'g, 'var, 'h, 'c, 'i) Iteration.model_scheme" shows "ipresSwapAbs hA MOD \<equiv> \<forall>zs z1 z2 us s A. wlsAbs (us, s) A \<longrightarrow> hA (A $[z1 \<and> z2]_zs) = igSwapAbs MOD zs z1 z2 (hA A)"
divide_inverse: fixes x :: "'c"   and y :: "'c" shows "x / y = x * inverse y"
integral_eq_nn_integral: fixes f :: "'d \<Rightarrow> real"   and M :: "'d measure" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x" shows "integral\<^sup>L M f = enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>M)"
assms_4: shows "set (p # ps) = ps\<^sub>L \<union> ps\<^sub>R"
simps: fixes a :: "'a"   and b :: "'c"   and c :: "'b"   and d :: "'d" shows "rel_witness_prod ((a, b), c, d) = ((a, c), b, d)"
assms_1: shows "x \<in> recurrent_subset_infty A"
wt_subst_trm'': fixes \<sigma> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma>" shows "\<Gamma> t = \<Gamma> (t \<cdot> \<sigma>)"
rev_bexI: fixes x :: "'h"   and A :: "'h set"   and P :: "'h \<Rightarrow> bool" assumes "x \<in> A"   and "P x" shows "\<exists>x\<in>A. P x"
lub_Pair: fixes A :: "nat \<Rightarrow> 'b"   and B :: "nat \<Rightarrow> 'c" assumes "chain A"   and "chain B" shows "(\<Squnion>i. (A i, B i)) = (\<Squnion>i. A i, \<Squnion>i. B i)"
pr_algo_maintains_hb_labeling: fixes a :: "pr_operation"   and f' :: "nat \<times> nat \<Rightarrow> 'capacity"   and l' :: "nat \<Rightarrow> nat" assumes "((f'__, l'__), a, f', l') \<in> pr_algo_lts" shows "Height_Bounded_Labeling c s t f' l'"
fun_neq_ext_iff: fixes m :: "'a \<Rightarrow> 'b"   and m' :: "'a \<Rightarrow> 'b" shows "(m \<noteq> m') = (\<exists>x. m x \<noteq> m' x)"
return_measurable: fixes N :: "'c measure" shows "return N \<in> N \<rightarrow>\<^sub>M subprob_algebra N"
homeomorphic_map_relative_homology_iso: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set"   and p :: "int" assumes "homeomorphic_map X Y f"   and "S \<subseteq> topspace X"   and "f ` S = T" shows "hom_induced p X S Y T f \<in> Group.iso (relative_homology_group p X S) (relative_homology_group p Y T)"
translation_isomorphism: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "left_translations_of_monoid M composition unit" shows "monoid_isomorphism (left_translations_of_monoid.translation M composition) M composition unit (left_translations_of_monoid.translation M composition ` M) (compose M) (identity M)"
boundOutputScopeDest: fixes xvec :: "name list"   and M :: "'d"   and P :: "('d, 'e, 'f) psi"   and yvec :: "name list"   and N :: "'d"   and z :: "name"   and Q :: "('d, 'e, 'f) psi" assumes "\<lparr>\<nu>*xvec\<rparr>M \<prec>' P = \<lparr>\<nu>*yvec\<rparr>N \<prec>' \<lparr>\<nu>z\<rparr>Q"   and "z \<sharp> xvec"   and "z \<sharp> yvec" obtains R :: "('d, 'e, 'f) psi" where "P = \<lparr>\<nu>z\<rparr>R" and "\<lparr>\<nu>*xvec\<rparr>M \<prec>' R = \<lparr>\<nu>*yvec\<rparr>N \<prec>' Q"
P4: shows "Bet P Y Q \<and> Cong Y Q Y A"
stereographic_def: shows "stereographic \<equiv> map_fun Rep_riemann_sphere abs_complex_homo stereographic_r3_hcoords"
mono_Max_commute: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "class.linorder less_eq less"   and "order.mono less_eq f"   and "finite A"   and "A \<noteq> {}" shows "f (linorder.Max less_eq A) = Max (f ` A)"
finite_generator_def: fixes R :: "('h, 'i) Ring_scheme"   and M :: "('j, 'h, 'k) Module_scheme"   and H :: "'j set" shows "finite_generator R M H = (finite H \<and> generator R M H)"
nonzero_inverse_eq_divide: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "inverse a = (1::'a) / a"
SUP_UN_eq: fixes r :: "'e \<Rightarrow> 'd set"   and S :: "'e set" shows "(SUP i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> \<Union> (r ` S))"
mat_adjoint_def: fixes A :: "'c mat" shows "adjoint A \<equiv> mat_of_rows (dim_row A) (map conjugate (cols A))"
index_le_lengthD: fixes xs :: "'c list"   and x :: "'c" assumes "index xs x < length xs" shows "x \<in> set xs"
rw2: shows "[0..<Suc (Suc k)] = [0..<Suc k] @ [Suc k]"
rbt_map_entry_is_rbt: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and t :: "('a, 'b) RBT_Impl.rbt" shows "ord.is_rbt less (ord.rbt_map_entry less k f t) = ord.is_rbt less t"
fac_Lyndon_mono: fixes w :: "'a list" shows "Lyndon_mono (LynFac w)"
B_def: fixes A :: "((complex, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" assumes "real_non_neg_mat A" shows "??.Perron_Frobenius.B A \<equiv> \<chi>i j. Re (A $h i $h j)"
distinct_199: fixes x121 :: "expr"   and x122 :: "expr"   and x131 :: "expr"   and x132 :: "expr"   and x133 :: "expr" shows "x121;; x122 \<noteq> if (x131) x132 else x133"
trunc_bound_eucl_float_1: fixes p :: "nat"   and x :: "real" shows "fst (trunc_bound_eucl p x) \<in> float"
eq_GPV_iff: fixes f :: "('d, 'e, 'f) gpv"   and g :: "('d, 'e, ('d, 'e, 'f) rpv) generat spmf" shows "(f = GPV g) = (the_gpv f = g)"
not_prime_elem_one: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "\<not> comm_semiring_1.prime_elem times one zero one"
pick_asTerm: fixes qX :: "('e, 'f, 'varSort, 'var, 'g) qTerm" assumes "qGood qX" shows "pick (asTerm qX) #= qX"
except_keys_Diff: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and U :: "'a set" shows "except p (keys p - U) = except p (- U)"
finite_UN: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'a set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" shows "\<Union> (A ` I) \<in> M"
openInjectAux: fixes xvec1 :: "name list"   and x :: "name"   and xvec2 :: "name list"   and yvec :: "name list" assumes "length (xvec1 @ x # xvec2) = length yvec" shows "\<exists>yvec1 y yvec2. yvec = yvec1 @ y # yvec2 \<and> length xvec1 = length yvec1 \<and> length xvec2 = length yvec2"
triv_def: shows "triv = G\<lparr>carrier := {\<one>}\<rparr>"
exec_step_prealloc_pres: fixes h :: "heap"   and P :: "jvm_prog"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and xp' :: "nat option"   and h' :: "heap"   and frs' :: "frame list"   and sh' :: "sheap" assumes "preallocated h"   and "exec_step P h stk loc C M pc ics frs sh = (xp', h', frs', sh')" shows "preallocated h'"
finite: shows "finite {i. 2 \<le> i \<and> real i \<le> log 2 x}"
simps_2: fixes gi :: "int poly" shows "choose_u [gi] = gi"
Rep_N_inf: fixes a :: "'b N"   and b :: "'b N" shows "Rep_N (a \<sqinter> b) = Rep_N a \<sqinter> Rep_N b"
fds_nth_minus: fixes f :: "'a fds"   and g :: "'a fds" shows "fds_nth (f - g) = (\<lambda>n. fds_nth f n - fds_nth g n)"
equiv_c_def: fixes c :: "com"   and c' :: "com" shows "c \<sim> c' \<equiv> \<forall>\<pi> s t. \<pi>: (c, s) \<Rightarrow> t = \<pi>: (c', s) \<Rightarrow> t"
extf_ext_hd_or_tl: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> 'c"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and f :: "'b" assumes "kbo_std_basis ground_heads_var gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf" shows "ext_hd_or_tl (extf f)"
in_a_I: fixes I :: "nat set"   and i :: "nat" assumes "I \<subseteq> {0..<m}"   and "i < m" shows "(a i \<in> a ` I) = (i \<in> I)"
line_def: fixes a :: "'b"   and b :: "'b"   and u :: "real" shows "{a -- b}\<^bsub>u\<^esub> = a + u *\<^sub>R (b - a)"
comp_of_tangle_dgms: fixes y :: "wall"   and x :: "wall" assumes "is_tangle_diagram y"   and "is_tangle_diagram x \<and> codomain_wall x = domain_wall y" shows "is_tangle_diagram (x \<circ> y)"