hcju/isabelleps huggingface dataset:
rcos_const: fixes h :: "'a" assumes "Group.group G"   and "h \<in> H" shows "H #> h = H"
foo_def: shows "foo \<equiv> let X = Map.empty in ASSERT (1 \<notin> dom X) \<bind> (\<lambda>_. RETURN (X(1 \<mapsto> 2)))"
intro: fixes p :: "real"   and M :: "bool stream measure" assumes "0 \<le> p"   and "p \<le> 1"   and "M = bernoulli_stream p" shows "infinite_coin_toss_space p M"
cSUP_upper2: fixes f :: "'b \<Rightarrow> 'a multiset"   and A :: "'b set"   and x :: "'b"   and u :: "'a multiset" assumes "subset_mset.bdd_above (f ` A)"   and "x \<in> A"   and "u \<subseteq># f x" shows "u \<subseteq># \<Squnion> (f ` A)"
bl_shiftr1: fixes w :: "'b word" shows "to_bl (shiftr1 w) = False # butlast (to_bl w)"
d2_neg: fixes y :: "'a"   and t :: "real" assumes "(y, t) \<in> cball (x, 0) d2"   and "0 < t" shows "blinfun_apply (Ds (flow0 y t)) (f (flow0 y t)) < 0"
cdi_path_swap: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi>' :: "nat \<Rightarrow> 'a"   and j :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>'"   and "IFC_def.is_cdi prob j \<pi> k"   and "IFC_def.eq_up_to \<pi> j \<pi>'" shows "IFC_def.is_cdi prob j \<pi>' k"
set_reorder: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "set (xs @ y # ys) = set (y # xs @ ys)"
arr_iff_in_hom: fixes f :: "'c" shows "C.arr f = \<guillemotleft>f : C.dom f \<rightarrow>\<^sub>C C.cod f\<guillemotright>"
adds_poly_mapping: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "(s adds t) = (lookup s \<le> lookup t)"
inducts: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "linkage \<Gamma> f" obtains "\<And>x. x \<in> A \<Gamma> \<Longrightarrow> d_OUT f x = weight \<Gamma> x"
equiv_class_self: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and a :: "'a" assumes "equiv A r"   and "a \<in> A" shows "a \<in> r `` {a}"
evaluation_map_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d"   and V :: "'b \<Rightarrow> 'd" shows "evaluation_map C D T \<alpha> \<iota> V \<equiv> monoidal_language C \<and> monoidal_category D T \<alpha> \<iota> \<and> functor C D V"
d1_def: shows "st.d1 = prod_list st.dims1"
C_4: shows "\<forall>l s. P l s \<longrightarrow> Q l (postQ C s)"
sum_list_split: fixes xs :: "real list"   and m :: "real" shows "sum_list xs = sum_list (list_neq xs m) + sum_list (list_eq xs m)"
prime_divisors_def: fixes K :: "('c, 'd) Ring_scheme" shows "Pds\<^bsub>K\<^esub> = {P. \<exists>v. valuation K v \<and> P = P\<^bsub> K v\<^esub>}"
assms: shows "\<forall>x\<in>set L. set (F x) = G x"
filter_min_aux_subset: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "set (General.filter_min_aux rel xs ys) \<subseteq> set xs \<union> set ys"
homeomorphic_locally_path_connected_space: fixes X :: "'a topology"   and Y :: "'b topology" assumes "X homeomorphic_space Y" shows "locally_path_connected_space X = locally_path_connected_space Y"
cases: fixes a1 :: "('a, 'b) Signature"   and a2 :: "('a, 'b) Language" assumes "Sig a1 \<triangleright> a2" obtains A :: "'a" and S :: "('a, 'b) Signature" where "a1 = S__" and "a2 = \<turnstile> A__ Type" and "A__ \<in> Ty\<^bsub>S__\<^esub>"   | S :: "('a, 'b) Signature" and A :: "'a" where "a1 = Sa__" and "a2 = Vx : Aa__ \<turnstile> Vx : Aa__" and "Sig Sa__ \<triangleright> \<turnstile> Aa__ Type"   | S :: "('a, 'b) Signature" and A :: "'a" and e :: "('a, 'b) Expression" and B :: "'a" and f :: "'b" and C :: "'a" where "a1 = Sb__" and "a2 = Vx : Ab__ \<turnstile> f E@ e : C" and "Sig Sb__ \<triangleright> Vx : Ab__ \<turnstile> e : B__" and "f \<in> Sig Sb__ : B__ \<rightarrow> C"   | S :: "('a, 'b) Signature" and A :: "'a" and e1 :: "('a, 'b) Expression" and B :: "'a" and e2 :: "('a, 'b) Expression" where "a1 = S" and "a2 = Vx : A \<turnstile> e1\<equiv>e2 : B" and "Sig S \<triangleright> Vx : A \<turnstile> e1 : B" and "Sig S \<triangleright> Vx : A \<turnstile> e2 : B"
H: shows "\<Gamma> \<turnstile> (\<lambda><:T\<^sub>1\<^sub>1. t\<^sub>1\<^sub>2) \<bullet>\<^sub>\<tau> T\<^sub>2 : T"
prems_1: shows "ifex_sat_list (IF v t e) = Some u"
match_list_def: fixes d :: "'d \<Rightarrow> ('e, 'f) Term.term"   and P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) list" shows "match_list d P = map_option (subst_of_map d) (match_term_list P Map.empty)"
nth_to_bl: fixes n :: "nat"   and w :: "'b word" assumes "n < LENGTH('b)" shows "to_bl w ! n = bit w (LENGTH('b) - Suc n)"
h_fact: shows "h \<in> carrier Zp \<and> \<beta> = \<alpha> \<oplus> h"
ProblemIkTpart_TEModel_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) T.efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" shows "ProblemIkTpart_TEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw eintT eintF eintP \<equiv> ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw \<and> CM.Model (ProblemIkTpart.TE_wtFsym wtFsym resOf) wtPsym (ProblemIkTpart.TE_arOf arOf) (ProblemIkTpart.TE_resOf resOf) parOf (ProblemIkTpart.tPB wtFsym arOf resOf \<Phi> prot protFw) eintT eintF eintP"
limitin_topspace: fixes X :: "'d topology"   and f :: "'e \<Rightarrow> 'd"   and l :: "'d"   and F :: "'e filter" assumes "limitin X f l F" shows "l \<in> topspace X"
length_Lev1': fixes kl :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and a :: "'d" assumes "kl \<in> Lev1 s1 s2 n a" shows "kl \<in> Lev1 s1 s2 (length kl) a"
eventually_inside: shows "\<forall>\<^sub>F x in at (poincare_map {x \<in> S. s x = 0} x). s x = 0 \<longrightarrow> x \<in> S"
linear_scaleR: fixes c :: "real" shows "linear ((*\<^sub>R) c)"
seqI': fixes f :: "('a, 'a set) Maps.arr"   and a :: "('a, 'a set) Maps.arr"   and b :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr"   and c :: "('a, 'a set) Maps.arr" assumes "Maps.in_hom f a b"   and "Maps.in_hom g b c" shows "Maps.seq g f"
poly_subst_power: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" shows "poly_subst f (p ^ n) = poly_subst f p ^ n"
C: shows "(Asset n, Crypt (SesK SK) (Num 0)) \<in> s"
bij_imp_bij_inv: fixes f :: "'b \<Rightarrow> 'c" assumes "bij f" shows "bij (inv f)"
Span_strict_incl: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list"   and Vs :: "'a list" assumes "ring R"   and "subfield K R"   and "set Us \<subseteq> carrier R"   and "set Vs \<subseteq> carrier R"   and "ring.Span R K Us \<subset> ring.Span R K Vs" shows "\<exists>v\<in>set Vs. v \<notin> ring.Span R K Us"
moebius_pt_moebius_inv_in_set: fixes M :: "moebius"   and z :: "complex_homo"   and A :: "complex_homo set" assumes "moebius_pt M z \<in> A" shows "z \<in> moebius_pt (moebius_inv M) ` A"
wtl_top: fixes ls :: "'b list"   and c :: "'a list"   and pc :: "nat" shows "wtl ls c pc \<top> = \<top>"
rel_Grp: fixes A1 :: "'d set"   and f1 :: "'d \<Rightarrow> 'f"   and A2 :: "'e set"   and f2 :: "'e \<Rightarrow> 'g" shows "rel_nondetT' (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. phantom_nondetT x \<subseteq> A1 \<and> set_nondetT x \<subseteq> A2} (map_nondetT' f1 f2)"
asn_simps_4: fixes x :: "char list"   and P :: "interface" shows "asn (Exists_int x P) = Exists x (asn P)"
simps: fixes m :: "nat"   and c :: "nat" shows "newconf m c = trpl (newleft (left c) (rght c) (actn m (stat c) (rght c))) (newstat m (stat c) (rght c)) (newrght (left c) (rght c) (actn m (stat c) (rght c)))"
frees_list_comb: fixes t :: "'a"   and xs :: "'a list" shows "frees (list_comb t xs) = frees t |\<union>| freess xs"
inf_less_eq_cases: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "- x \<sqinter> - y \<le> - z"   and "- - x \<sqinter> - y \<le> - z" shows "- y \<le> - z"
approx_SReal_mult_cancel: fixes a :: "real star"   and w :: "real star"   and z :: "real star" assumes "a \<in> \<real>"   and "a \<noteq> 0"   and "a * w \<approx> a * z" shows "w \<approx> z"
src_cod: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "src\<^sub>D (D.cod \<mu>) = src\<^sub>D \<mu>"
inner_right_distrib: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (plus y z) = inner x y + inner x z"
p_id_def: shows "1\<^sub>\<pi> \<equiv> \<Union>a. {(a, {})}"
derived_eq_singleton: fixes H :: "'a set" assumes "H \<subseteq> carrier G" shows "derived G H = {\<one>}"
strong_cases: fixes a1 :: "lam"   and a2 :: "lam"   and x :: "name"   and xa :: "name" assumes "a1 \<rightarrow>\<^sub>\<beta> a2" obtains t :: "lam" and s :: "lam" where "\<lbrakk>x \<sharp> a1; x \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = (Lam [x].s__) \<degree> ta__ \<and> a2 = s__[x::=ta__] \<and> x \<sharp> ta__"   | s :: "lam" and t :: "lam" and u :: "lam" where "a1 = sa__ \<degree> u__" and "a2 = tb__ \<degree> u__" and "sa__ \<rightarrow>\<^sub>\<beta> tb__"   | s :: "lam" and t :: "lam" and u :: "lam" where "a1 = u \<degree> sb__" and "a2 = u \<degree> tc__" and "sb__ \<rightarrow>\<^sub>\<beta> tc__"   | s :: "lam" and t :: "lam" where "\<lbrakk>xa \<sharp> a1; xa \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = (Lam [xa].s) \<and> a2 = (Lam [xa].t) \<and> s \<rightarrow>\<^sub>\<beta> t"
data_in'_def: shows "data_in' \<equiv> append_queue (nt, X) data_in"
conv_complement_0_p: fixes x :: "'a" shows "(- x)\<^sup>T \<squnion> (- - x)\<^sup>T = top"
in_homI: fixes f :: "('o, 'a) arr"   and a :: "('o, 'a) arr"   and b :: "('o, 'a) arr" assumes "arr f"   and "local.dom f = a"   and "cod f = b" shows "in_hom f a b"
llbound_def: shows "llbound = lbound {SOME a. a \<in> I}"
bind_cong: fixes x :: "'g option"   and y :: "'g option"   and f :: "'g \<Rightarrow> 'h option"   and g :: "'g \<Rightarrow> 'h option" assumes "x = y"   and "\<And>a. y = Some a \<Longrightarrow> f a = g a" shows "x \<bind> f = y \<bind> g"
perron_frobenius_spectral_radius_yun: fixes d :: "nat" assumes "\<And>f\<^sub>i i. \<lbrakk>(f\<^sub>i, i) \<in> set fis; \<exists>x. poly (map_poly complex_of_real f\<^sub>i) x = 0 \<and> cmod x = 1\<rbrakk> \<Longrightarrow> Suc i \<le> d" shows "\<exists>c1 c2. \<forall>k a. a \<in> elements_mat (A ^\<^sub>m k) \<longrightarrow> \<bar>a\<bar> \<le> c1 + c2 * real k ^ (d - 1)"
simps: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "(a \<in> orbit f x) = (a = f x \<or> (\<exists>y. a = f y \<and> y \<in> orbit f x))"
seqp_sosp_seqp_sos_eq: fixes \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" shows "seqp_sosp \<Gamma> = (\<lambda>x xa xb xc xd. ((x, xa), xb, xc, xd) \<in> seqp_sos \<Gamma>)"
consequence_unit: fixes P :: "'b \<Rightarrow> bool"   and P' :: "'b \<Rightarrow> bool"   and M :: "'b \<Rightarrow> (unit \<times> 'b) option"   and Q' :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>\<sigma>. P \<sigma> \<longrightarrow> P' \<sigma>"   and "\<lbrace>P'\<rbrace> M \<lbrace>\<lambda>x. Q'\<rbrace>"   and "\<And>\<sigma>. Q' \<sigma> \<longrightarrow> Q \<sigma>" shows "\<lbrace>P\<rbrace> M \<lbrace>\<lambda>x. Q\<rbrace>"
right_ideal_generated_def: fixes S :: "'a set" shows "right_ideal_generated S = \<Inter> {I. right_ideal I \<and> S \<subseteq> I}"
theFT_def: fixes cl :: "('a, 'b, 'c) cmd list" shows "theFT cl \<equiv> {n. n < length cl \<and> finished (cl ! n)}"
NUM_wrt_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "\<psi> \<in> \<R>\<^sup>2" shows "NUM_wrt \<psi> \<equiv> {U. \<forall>f\<in>U. \<exists>i. \<psi> i = f}"
incseq_def: fixes X :: "nat \<Rightarrow> 'a" shows "incseq X = (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n)"
exhaust: fixes y :: "'a type_error" obtains "y = TypeError"   | x2 :: "'a" where "y = Normal x2"
det_swaprows_mat: fixes k :: "nat"   and n :: "nat"   and l :: "nat" assumes "k < n"   and "l < n"   and "k \<noteq> l" shows "Determinant.det (swaprows_mat n k l) = - (1::'b)"
atLeastatMost_empty: fixes b :: "'b word"   and a :: "'b word" assumes "b <s a" shows "signed.atLeastAtMost a b = {}"
game3_fst: fixes x :: "nat"   and y :: "nat" shows "map_spmf fst (game3 (\<lambda>_ _ _ x. x) x y) = coin_spmf"
simps_2: fixes a :: "MIR.num"   and b :: "MIR.num" shows "rsplit0 (Add a b) = (let acs = rsplit0 a; bcs = rsplit0 b in map (\<lambda>((p, n, t), q, m, s). (And p q, n + m, Add t s)) (concat (map (\<lambda>a. map (Pair a) bcs) acs)))"
simps_5: fixes F :: "'a formula" shows "nnf (\<^bold>\<not> (\<^bold>\<not> F)) = nnf F"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "P (Suc 0)"   and "\<And>n. \<lbrakk>P (Suc n); P n\<rbrakk> \<Longrightarrow> P (Suc (Suc n))" shows "P a0"
measurable_g: shows "(\<lambda>(x, y). g x y) \<in> borel_measurable (count_space UNIV \<Otimes>\<^sub>M borel)"
normalI: fixes H :: "'a set" assumes "subgroup H G"   and "\<forall>x\<in>carrier G. H #> x = x <# H" shows "H \<lhd> G"
2: shows "p ; q\<^sup>T \<noteq> 0"
comp_in_homE: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and f :: "'a"   and a :: "'a"   and c :: "'a" assumes "category C"   and "partial_magma.in_hom C (C g f) a c" obtains b :: "'a" where "partial_magma.in_hom C f a b" and "partial_magma.in_hom C g b c"
Min_eq_iff: fixes A :: "'c Word.word set"   and m :: "'c Word.word" assumes "finite A"   and "A \<noteq> {}" shows "(signed.Min A = m) = (m \<in> A \<and> (\<forall>a\<in>A. m \<le>s a))"
ll_on_open_real_axioms: shows "ll_on_open_real T f X"
TameEnum_Archive: shows "fgraph ` TameEnum \<subseteq>\<^sub>\<simeq> Archive"
H': fixes xy :: "'v \<times> 'v" shows "(\<lambda>n. ennreal (h (k n) xy)) \<longlonglongrightarrow> ennreal (H xy)"
Diff_insert2: fixes A :: "'d set"   and a :: "'d"   and B :: "'d set" shows "A - insert a B = A - {a} - B"
inj_comm_monoid_add_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_comm_monoid_add_hom hom \<equiv> comm_monoid_add_hom hom \<and> inj_monoid_add_hom hom"
diamond_left_edges_def: shows "diamond_left_edges = (- (1::'a), \<lambda>y. (- diamond_y (diamond_x y), diamond_x y))"
matrix_vector_mult_add_rdistrib: fixes A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and B :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and x :: "('b, 'd) Finite_Cartesian_Product.vec" shows "(A + B) *v x = A *v x + B *v x"
viewRn_inv: fixes t :: "('a, 'b) FingerTreeStruc"   and n :: "nat"   and nd :: "('a, 'b) Node"   and s :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.is_measured_ftree t"   and "FingerTreeStruc.is_leveln_ftree n t"   and "FingerTreeStruc.viewRn t = Some (nd, s)" shows "FingerTreeStruc.is_measured_ftree s \<and> FingerTreeStruc.is_measured_node nd \<and> FingerTreeStruc.is_leveln_ftree n s \<and> FingerTreeStruc.is_leveln_node n nd"
netlimit_at_left: fixes x :: "real" shows "netlimit (at_left x) = x"
dist_suc: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and v :: "nat"   and d :: "nat"   and v' :: "nat"   and v'' :: "nat" assumes "Graph.dist c v d v'"   and "(v', v'') \<in> Graph.E c" shows "Graph.dist c v (Suc d) v''"
psi_narrow: fixes x' :: "'b sort_key extended \<times> 'b sort_key extended"   and i :: "'b"   and x :: "'b sort_key extended \<times> 'b sort_key extended" assumes "elem \<lbrakk>\<Psi> x' i\<rbrakk> x"   and "Psi.subset x x'" shows "\<Psi> x' i = \<Psi> x i"
WTrt_binop_simps_9: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>Mult\<guillemotright>T2 : T) = (T1 = Integer \<and> T2 = Integer \<and> T = Integer)"
OUTfromChCorrectSYSTEM_def: shows "OUTfromChCorrectSYSTEM \<equiv> \<forall>x. OUTfromChCorrect x"
functions_of_is_surj_on: fixes A :: "'c set"   and B :: "'d set"   and P :: "'c set set" assumes "finite A"   and "finite B"   and "partition_on A P"   and "card P = card B" shows "univ (\<lambda>f. f ` A = B) (functions_of P A B)"
f1: shows "fst instr \<noteq> sreg_type WRY"
case_prod_conv: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and a :: "'d"   and b :: "'e" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
iso_lunit: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and b :: "'b"   and f :: "'b" assumes "left_hom_with_unit V H \<a> \<iota> b"   and "partial_magma.ide (subcategory.comp V (weak_composition.left V H b)) f" shows "category.iso (subcategory.comp V (weak_composition.left V H b)) (left_hom_with_unit.lunit V H \<a> \<iota> b f)"
cblinfun_apply_ccspan_using_vec: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and S :: "'b list" shows "A *\<^sub>S ccspan (set S) = ccspan (basis_enum_of_vec ` set (map ((*\<^sub>v) (mat_of_cblinfun A)) (map vec_of_basis_enum S)))"
inf_compl_bot: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf x (uminus x) = bot"
full_quicksort_correct_sorted: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and h :: "'a \<Rightarrow> 'b"   and xs :: "'a list" assumes "\<And>x y z. \<lbrakk>R (h x) (h y); R (h y) (h z)\<rbrakk> \<Longrightarrow> R (h x) (h z)"   and "\<And>x y. x \<noteq> y \<Longrightarrow> R (h x) (h y) \<or> R (h y) (h x)" shows "full_quicksort R h xs \<le> \<Down> Id (SPEC (\<lambda>xs'. mset xs' = mset xs \<and> sorted_wrt (\<lambda>x y. R (h x) (h y)) xs'))"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G G G"
pdevs_apply_cases: fixes y :: "nat \<Rightarrow> 'b" assumes "y \<in> {x. finite {i. x i \<noteq> (0::'b)}}" obtains x :: "'b pdevs" where "y = pdevs_apply x"
hasw_def: fixes xs :: "'a list"   and S :: "'a set" shows "hasw xs S \<equiv> set xs \<subseteq> S \<and> distinct xs"
mult_commute: fixes a :: "('a \<times> 'a set) set"   and b :: "('a \<times> 'a set) set" shows "a \<parallel> b = b \<parallel> a"
i3: shows "i3 \<in> set (roots_of_3 q)"
measure_measure_pmf_not_zero: shows "measure_pmf.prob p s \<noteq> 0"
exact4_def: fixes R :: "('e, 'f) Ring_scheme"   and L0 :: "('g, 'e, 'h) Module_scheme"   and h0 :: "'g \<Rightarrow> 'i"   and L1 :: "('i, 'e, 'h) Module_scheme"   and h1 :: "'i \<Rightarrow> 'j"   and L2 :: "('j, 'e, 'h) Module_scheme"   and h2 :: "'j \<Rightarrow> 'k"   and L3 :: "('k, 'e, 'h) Module_scheme" shows "exact4 R L0 h0 L1 h1 L2 h2 L3 = (h0 ` carrier L0 = ker\<^bsub>L1,L2\<^esub> h1 \<and> h1 ` carrier L1 = ker\<^bsub>L2,L3\<^esub> h2)"
1: shows "state_succ s1 a = snd a ++ s1"
continuous_on_Polygamma: fixes A :: "'a set"   and n :: "nat" assumes "A \<inter> \<int>\<^sub>\<le>\<^sub>0 = {}" shows "continuous_on A (Polygamma n)"
domI: fixes m :: "'e \<Rightarrow> 'd option"   and a :: "'e"   and b :: "'d" assumes "m a = \<lfloor>b\<rfloor>" shows "a \<in> dom m"
sublist_as_proj_eq_as: fixes as' :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) list"   and as :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) list"   and vs :: "'a set" assumes "subseq as' (as_proj as vs)" shows "as_proj as' vs = as'"
producesD: fixes N :: "'b literal multiset set"   and C :: "'b literal multiset"   and A :: "'b" assumes "production N C = {A}" shows "C \<in> N \<and> C \<noteq> {#} \<and> Pos A = Max_mset C \<and> \<not> interp N C \<Turnstile> C \<and> {#} = {#}"
zadd_int_left: fixes m :: "nat"   and n :: "nat"   and z :: "int" shows "int m + (int n + z) = int (m + n) + z"
grd_def: fixes S :: "'f \<Rightarrow> 'e" shows "grd S = - S \<bottom>"
add_diff_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a + (b - c) = a + b - c"
distinct_1: fixes x21 :: "'b"   and x22 :: "'b bintree"   and x23 :: "'b bintree" shows "Leaf \<noteq> Branch x21 x22 x23"
model_then_all_singleton_clauses_modelled: fixes F :: "'b formula"   and L :: "'b literal"   and \<nu> :: "'b \<Rightarrow> bool" assumes "is_cnf F"   and "{L} \<in> cnf F"   and "\<nu> \<Turnstile> F" shows "lit_semantics \<nu> L"
matT_vec_mult_to_scalar: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list"   and v :: "'c list"   and i :: "nat"   and ze :: "'c"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and ti :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "mat nr nc m"   and "vec nr v"   and "i < nc" shows "matT_vec_multI ze pl ti m v ! i = scalar_prodI ze pl ti (col m i) v"
assms_4: shows "h \<turnstile> attach_shadow_root element_ptr new_mode \<rightarrow>\<^sub>h h'"
simps_2: fixes k :: "'a"   and v :: "'c"   and xs :: "('a \<times> 'c) list"   and x :: "'a" shows "lookup_pair ((k, v) # xs) x = (case comp x k of Eq \<Rightarrow> v | Lt \<Rightarrow> 0::'c | Gt \<Rightarrow> lookup_pair xs x)"
atLeast0_lessThan_Suc: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "sum g {0..<Suc n} = sum g {0..<n} + g n"
le_less_div_conv: fixes m :: "nat"   and k :: "nat"   and n :: "nat" assumes "0 < m" shows "(k * m \<le> n \<and> n < Suc k * m) = (n div m = k)"
pairwise_property_lemma: fixes A :: "('b list \<Rightarrow> 'c pmf) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> ((nat \<times> nat list) \<times> 'c) pmf)" assumes "\<And>init qs x y. \<lbrakk>distinct init; qs \<in> {xs. set xs \<subseteq> set init}; (x, y) \<in> {(x, y). x \<in> set init \<and> y \<in> set init \<and> x \<noteq> y}; x \<noteq> y\<rbrakk> \<Longrightarrow> Pbefore_in x y A qs init = Pbefore_in x y A (Lxy qs {x, y}) (Lxy init {x, y})"   and "\<And>xa r. \<forall>z\<in>set_pmf (snd A xa r). snd (fst z) = []" shows "List_Factoring.pairwise A"
homogenize_alt: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "homogenize x p = (\<Sum>q\<in>hom_components p. MPoly_Type_Class.punit.monom_mult (1::'c) (monomial (poly_deg p - poly_deg q) x) q)"
gram_schmidt_def: fixes n :: "nat"   and ws :: "'a vec list" shows "gram_schmidt n ws = rev (gram_schmidt_sub n [] ws)"
take_bit_of_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit n zero = zero"
nat_pow_pow: fixes x :: "'grp"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier \<G>" shows "(x [^] n) [^] m = x [^] (n * m)"
digraph_map: shows "digraph_map (with_proj (list_digraph G_list)) (to_map' (snd G_list) xss)"