hcju/isabelleps huggingface dataset:
add_le_cancel_right: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and c :: "'e"   and b :: "'e" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less_eq (plus a c) (plus b c) = less_eq a b"
arr_dom: fixes f :: "'a" assumes "B.arr f" shows "B.arr (B.dom f)"
And\<^sub>n_def: fixes \<Phi> :: "'b ltln set" shows "And\<^sub>n \<Phi> \<equiv> SOME \<phi>. fold_graph And_ltln true\<^sub>n \<Phi> \<phi>"
galleryD_adj: fixes xs :: "'a set list" assumes "domain.gallery xs" shows "adjacentchain xs"
map_key_PP_mapping_of: fixes p :: "('d \<Rightarrow>\<^sub>0 'e) \<Rightarrow>\<^sub>0 'f" shows "Poly_Mapping.map_key PP (Poly_Mapping.map_key pp.mapping_of p) = p"
aux_lemma: fixes Ps :: "('c multiset \<Rightarrow> 'a) list" assumes "Ps \<in> carrier (poly_ring L)"   and "indexed_eval Ps i = indexed_const \<zero>" shows "Ps = []"
NE: shows "Rep_BinomialHeap q \<noteq> []"
wellFormedSubst_def: fixes \<sigma> :: "('c list \<times> 'f list) list" shows "wellFormedSubst \<sigma> = (filter (\<lambda>(xvec, Tvec). \<not> (length xvec = length Tvec \<and> distinct xvec)) \<sigma> = [])"
set: fixes x1 :: "'a match_expr"   and x2 :: "action" shows "set_rule (Rule x1 x2) = set_match_expr x1"
simps: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk>\<lbrakk> \<Psi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> \<inter> \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> Suc n\<^esup>"
D_WHILE: fixes b :: "BExpr"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" assumes "BExpr_low b"   and "(G, c, Sec \<Phi>) \<in> Deriv" shows "(G, While b c, Sec (PhiWhile b \<Phi>)) \<in> Deriv"
adjoint_pair_def: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr" shows "S.adjoint_pair f g \<equiv> \<exists>\<eta> \<epsilon>. adjunction_in_bicategory (\<cdot>\<^sub>S) (\<star>\<^sub>S) S.\<a> S.\<i> S.src S.trg f g \<eta> \<epsilon>"
bl_not_aux_bin: fixes n :: "nat"   and w :: "int"   and cs :: "bool list" shows "map Not (bin_to_bl_aux n w cs) = bin_to_bl_aux n (NOT w) (map Not cs)"
opposite_dir_5: fixes a :: "'a"   and b :: "'a option" shows "\<lhd>\<^sub>l\<^sub>b (\<lambda>x y. y < x) a b = \<rhd>\<^sub>u\<^sub>b (<) a b"
stepsn_if_star: fixes cs :: "com \<times> (char list \<Rightarrow> int)"   and cs' :: "com \<times> (char list \<Rightarrow> int)" assumes "cs \<rightarrow>* cs'" shows "\<exists>n. cs \<rightarrow>(n) cs'"
converse_iff: fixes a :: "'b"   and b :: "'c"   and r :: "('c \<times> 'b) set" shows "((a, b) \<in> r\<inverse>) = ((b, a) \<in> r)"
i_3: shows "\<forall>n\<ge>n\<^sub>0. s (0\<^sup>\<infinity> \<triangleright> n) \<down>= Suc i"
g_simps: fixes x :: "'v"   and y :: "'v" shows "g (x, y) = (f \<oplus> (j \<circ> map_prod Some Some \<circ> prod.swap)) (x, y)"
noicd: shows "\<forall>l\<ge>n. \<not> k icd\<^bsup>\<pi>\<^esup>\<rightarrow> l"
fBall_cong: fixes A :: "'c fset"   and B :: "'c fset"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "A = B"   and "\<And>x. x |\<in>| B \<Longrightarrow> P x = Q x" shows "fBall A P = fBall B Q"
IH: assumes "periodic_arithmetic g k" shows "periodic_arithmetic g (k * m)"
wf_tuple_empty: fixes n :: "nat"   and v :: "'b option list" shows "wf_tuple n {} v = (v = replicate n None)"
sel_2: fixes x :: "complex" shows "Im (- x) = - Im x"
le_minus_one_simps_3: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less_eq zero (uminus one)"
simps_1: fixes \<delta> :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'a"   and s :: "'a" shows "pre_RDR.exec \<delta> s [] = s"
simps: fixes v :: "nat \<Rightarrow> 'b"   and lb :: "nat \<Rightarrow> 'b option"   and ub :: "nat \<Rightarrow> 'b option"   and S :: "nat set" shows "v \<Turnstile>\<^sub>b (lb, ub) \<parallel> S = (\<forall>x\<in>S. in_bounds x v (lb, ub))"
closure_update_mono: fixes Z :: "('c \<Rightarrow> real) set"   and r :: "'c list" assumes "Z \<subseteq> V"   and "set r \<subseteq> X" shows "Closure\<^sub>\<alpha> Z\<^bsub>r \<rightarrow> 0\<^esub> \<subseteq> Closure\<^sub>\<alpha> Z\<^bsub>r \<rightarrow> 0\<^esub>"
set_complete: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set" assumes "set_category_given_img S img \<AA>"   and "A \<subseteq> Collect (category.terminal S)"   and "|A| <o \<AA>" shows "\<exists>a. partial_magma.ide S a \<and> set_category_data.set S img a = A"
msubset_mset_set_iff: fixes A :: "'b set"   and M :: "'b multiset" assumes "finite A" shows "(set_mset M \<subseteq> A \<and> (\<forall>x. count M x \<le> 1)) = (M \<subseteq># mset_set A)"
insertI: fixes A :: "'e set"   and a :: "'e" assumes "finite A" shows "finite (insert a A)"
ge_tpoly_refl: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   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 arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and p :: "('a pvar, hmultiset) tpoly" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym p p"
TermP_quot: fixes \<alpha> :: "tm" shows "{} \<turnstile> TermP \<guillemotleft>\<alpha>\<guillemotright>"
elims_2: fixes x :: "kon_node"   and xa :: "(kon_node \<times> kon_bridge \<times> kon_node) list"   and xb :: "kon_node" assumes "kon_graph.is_trail x xa xb" obtains v :: "kon_node" and v' :: "kon_node" where "x = va__" and "xa = []" and "xb = v'a__" and "va__ = v'a__ \<and> v'a__ \<in> kon_graph.V"   | v :: "kon_node" and v1 :: "kon_node" and w :: "kon_bridge" and v2 :: "kon_node" and ps :: "(kon_node \<times> kon_bridge \<times> kon_node) list" and v' :: "kon_node" where "x = v" and "xa = (v1, w, v2) # ps" and "xb = v'" and "v = v1 \<and> (v1, w, v2) \<in> kon_graph.E \<and> (v1, w, v2) \<notin> set ps \<and> (v2, w, v1) \<notin> set ps \<and> kon_graph.is_trail v2 ps v'"
E: fixes i :: "'a" assumes "i \<in> I" shows "E i \<subseteq> Pow (\<Omega> i)"
assms: shows "s \<in> set \<langle>a --\<guillemotright> b\<rangle>\<langle>Fun f T\<rangle>"
Elem_affine_of_ivl_ge: fixes e :: "nat \<Rightarrow> real"   and l :: "'a"   and u :: "'a" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "l \<le> u" shows "aform_val e (aform_of_ivl l u) \<le> u"
build_succ_correct: fixes nds :: "('a, 'b) node_scheme set" assumes "expand_init \<notin> name ` nds"   and "finite nds"   and "\<And>nd. nd \<in> nds \<Longrightarrow> incoming nd \<subseteq> insert expand_init (name ` nds)" shows "build_succ nds \<le> SPEC (\<lambda>r. E_of_succ (\<lambda>qn. the_default {} (r qn)) = rename_E name {(q, q'). q \<in> nds \<and> q' \<in> nds \<and> name q \<in> incoming q'})"
mono: fixes m :: "nat" assumes "0 < m" shows "f (Suc m) \<le> f m"
simps_7: fixes x31 :: "'a convnet"   and x32 :: "'a convnet"   and x1 :: "nat" shows "Pool x31 x32 \<noteq> Input x1"
bypass: fixes z :: "'v" assumes "x \<noteq> y"   and "z \<in> set p" shows "z = x \<or> z \<notin> TER f \<union> TER g"
maps_to_n_drop_res: fixes n :: "nat"   and f :: "nat \<Rightarrow> int" assumes "maps_to_n (int (Suc n)) f" shows "maps_to_n (int n) (drop_res (int (Suc n)) f)"
remove_child_heap_is_wellformed_preserved_1: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'"   and "known_ptrs h"   and "type_wf h" shows "type_wf h'"
scaleR_one_nonzero: fixes x :: "'b nonzero" shows "1 *\<^sub>R x = x"
strategyproofness: shows "pmf (sds R42) c + pmf (sds R42) a \<le> pmf (sds R6) c + pmf (sds R6) a"
floor_one: shows "\<lfloor>1::'b\<rfloor> = 1"
M11: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and B :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'd" assumes "|~ _actrans A f \<longrightarrow> _actrans B g" shows "\<turnstile> \<box>[A]_f \<longrightarrow> \<box>[B]_g"
l_set_disconnected_nodes_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_disconnected_nodes_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf DocumentClass.type_wf set_disconnected_nodes set_disconnected_nodes_locs get_shadow_root get_shadow_root_locs"
abs_eq_0: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "(abs a = zero) = (a = zero)"
isdisj_disjD: fixes f1 :: "'a formula"   and f2 :: "'a formula" assumes "is_disj (f1 \<^bold>\<or> f2)" shows "is_disj f1 \<and> is_disj f2"
upd: shows "bij_betw upd {..<n} {..<n}"
sets: fixes i :: "'a" assumes "i \<in> I" shows "sets (M i) = sets (N i)"
add_le_add_imp_diff_le: 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 i :: "'b"   and k :: "'b"   and n :: "'b"   and j :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq (plus i k) n"   and "less_eq n (plus j k)"   and "less_eq (plus i k) n"   and "less_eq n (plus j k)" shows "less_eq (minus n k) j"
upRules_def: shows "upRules \<equiv> {(xa, x). upRulesp xa x}"
snoc: fixes ns :: "'node list"   and m' :: "'node"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m'"   and "P n ns m'"   and "m' \<in> set (predecessors g m)" shows "P n (ns @ [m]) m"
a_kernel_defs_1: fixes G :: "('c, 'd) ring_scheme"   and H :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" shows "a_kernel G H h = kernel (add_monoid G) (add_monoid H) h"
INF_greatest: fixes A :: "'b set"   and u :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "\<And>i. i \<in> A \<Longrightarrow> u \<le> f i" shows "u \<le> Inf (f ` A)"
Borsukian_imp_unicoherent: fixes U :: "'a set" assumes "Borsukian U" shows "unicoherent U"
leg1_composite_1: shows "HfHgh.dom.leg1 = fgh.\<mu>.leg1 \<cdot> fgh.Prj\<^sub>1"
assms_5: shows "i \<in> ideal_points (poincare_line x1 x2)"
iso_block_in: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "\<pi> ` bl \<in># \<B>'"
p''_grid: shows "p[d := p'' ! d] \<in> grid (b[d := b' ! d]) ds"
map_indets_monomial: fixes f :: "'e \<Rightarrow> 'c"   and t :: "'e \<Rightarrow>\<^sub>0 nat"   and c :: "'d" shows "map_indets f (monomial c t) = monomial c (\<Sum>x\<in>keys t. monomial (lookup t x) (f x))"
coinduct: fixes X :: "'c llist \<Rightarrow> 'c llist \<Rightarrow> bool"   and x1 :: "'c llist"   and x2 :: "'c llist" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> (\<exists>xs. x1 = LNil \<and> x2 = xs) \<or> (\<exists>xs ys x. x1 = LCons x xs \<and> x2 = LCons x ys \<and> (X xs ys \<or> xs \<sqsubseteq> ys))" shows "x1 \<sqsubseteq> x2"
that: assumes "r = fst (alloc (replicate n x) h)"   and "h' = snd (alloc (replicate n x) h)"   and "get h' r = replicate n x"   and "present h' r"   and "\<not> present h r" shows "thesis"
mem1: shows "rotate (n - 1) ls = rotate (n - 1 + |ls|) ls"
simps_5: fixes v :: "'b list"   and va :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) list"   and S :: "('a, 'b) strand_step list" shows "dual\<^sub>s\<^sub>t (\<forall>v\<langle>\<or>\<noteq>: va\<rangle>\<^sub>s\<^sub>t # S) = \<forall>v\<langle>\<or>\<noteq>: va\<rangle>\<^sub>s\<^sub>t # dual\<^sub>s\<^sub>t S"
mHom_mem: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'i"   and m :: "'e" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N"   and "m \<in> carrier M" shows "f m \<in> carrier N"
set_eq_iff_mset_eq_distinct: fixes x :: "'c list"   and y :: "'c list" assumes "distinct x"   and "distinct y" shows "(set x = set y) = (mset x = mset y)"
eval_app: fixes \<rho> :: "(nat \<times> bval) list"   and e1 :: "exp"   and x :: "nat"   and e :: "exp"   and \<rho>' :: "(nat \<times> bval) list"   and e2 :: "exp"   and arg :: "bval"   and v :: "bval" assumes "\<rho> \<turnstile> e1 \<Down> BClos x e \<rho>'"   and "\<rho> \<turnstile> e2 \<Down> arg"   and "(x, arg) # \<rho>' \<turnstile> e \<Down> v" shows "\<rho> \<turnstile> EApp e1 e2 \<Down> v"
order_1_eq_0: fixes x :: "'a" shows "order x 1 = 0"
fps_power_Suc_eqD: fixes f :: "'a fps"   and m :: "nat"   and g :: "'a fps" assumes "f ^ Suc m = g ^ Suc m"   and "fps_nth f 0 = fps_nth g 0"   and "fps_nth f 0 \<noteq> (0::'a)" shows "f = g"
simps_7: fixes sh :: "sheap"   and e :: "'a exp"   and F :: "char list"   and D :: "char list" shows "iconf sh (e\<bullet>F{D}) = iconf sh e"
FL_valid_Pred: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and P :: "'a"   and f :: "'d"   and \<phi> :: "'b" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.FL_valid satisfies transition effect_apply P (Pred f \<phi>) = satisfies (effect_apply f P) \<phi>"
ccInf_union_distrib: fixes A :: "'a set"   and B :: "'a set" assumes "countable A"   and "countable B" shows "Inf (A \<union> B) = inf (Inf A) (Inf B)"
intro: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "\<And>a. partial_magma.ide A a \<Longrightarrow> category.iso B (\<tau> a)" shows "natural_isomorphism_axioms A B \<tau>"
arr_Par_id_ParI: fixes A :: "V" assumes "A \<in>\<^sub>\<circ> Vset \<alpha>" shows "arr_Par \<alpha> (id_Rel A)"
A\<^sub>1_def: shows "A\<^sub>1 = {y \<in> A. y < x}"
inv_fromD: fixes n :: "nat"   and p :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat"   and m :: "nat"   and i :: "nat"   and j :: "nat" assumes "inv_from n p A m"   and "i < n"   and "j < n"   and "m < j" shows "p A i j"
strong_cases: fixes a1 :: "trm"   and a2 :: "trm"   and x :: "name"   and xa :: "name" assumes "a1 \<Down> a2" obtains e :: "trm" where "\<lbrakk>x \<sharp> a1; x \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = Lam [x].ea__ \<and> a2 = Lam [x].ea__"   | e\<^sub>1 :: "trm" and e\<^sub>2 :: "trm" and e' :: "trm" and e :: "trm" and e\<^sub>2' :: "trm" where "\<lbrakk>xa \<sharp> a1; xa \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = App e\<^sub>1 e\<^sub>2 \<and> a2 = e' \<and> xa \<sharp> (e\<^sub>1, e\<^sub>2, e') \<and> e\<^sub>1 \<Down> Lam [xa].e \<and> e\<^sub>2 \<Down> e\<^sub>2' \<and> e[xa::=e\<^sub>2'] \<Down> e'"
adjust_nonzero: fixes us :: "'b vec list"   and n :: "nat"   and w :: "'b vec" assumes "set us \<subseteq> carrier_vec n"   and "distinct us"   and "w \<in> carrier_vec n"   and "w \<notin> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (set us)" shows "adjuster n w us + w \<noteq> 0\<^sub>v n"
ddvd_trans: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times x y \<and> dvd.dvd times y x"   and "dvd.dvd times y z \<and> dvd.dvd times z y" shows "dvd.dvd times x z \<and> dvd.dvd times z x"
assms: fixes gs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd" assumes "is_Groebner_basis (fst ` set gs)"   and "sps \<noteq> []"   and "set sps \<subseteq> set ps"   and "unique_idx (gs @ bs) data" shows "fst ` set (fst (compl gs bs (ps -- sps) sps data)) \<subseteq> pmdl (args_to_set (gs, bs, ps))"
assms_1: shows "seq \<pp>\<^sub>0[b, c] f"
open_prod_def: fixes S :: "('c \<times> 'd) set" shows "open S = (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
Inv2a_def: fixes s :: "state" shows "Inv2a s = (\<forall>p. Inv2a_inner s p)"
scalar_times_polynTr: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier S" shows "\<forall>f. pol_coeff S (n, f) \<longrightarrow> a \<cdot>\<^sub>r polyn_expr R X n (n, f) = polyn_expr R X n (sp_cf S a (n, f))"
bind_duplicate: fixes m :: "'m"   and f :: "'a option \<Rightarrow> 'a option \<Rightarrow> 'm" shows "bind m (\<lambda>x. bind m (f x)) = bind m (\<lambda>x. f x x)"
h1: shows "\<nabla>x = (zero_class.zero::'a)"
nth_default_Nil: fixes dflt :: "'b"   and n :: "nat" shows "nth_default dflt [] n = dflt"
Landau_inequality_mahler_measure: fixes f :: "int poly" shows "mahler_measure f \<le> sqrt (real_of_int (\<Sum>a\<leftarrow>coeffs f. a * a))"
assms: shows "((\<lambda>x. EM_remainder' n f (real_of_int b) (real_of_int x)) \<longlongrightarrow> L) at_top"
treeLess_induct: fixes phi :: "('a, 'b) tree \<Rightarrow> bool"   and T :: "('a, 'b) tree" assumes "\<And>T'. (\<And>T. (T, T') \<in> treeLess \<Longrightarrow> phi T) \<Longrightarrow> phi T'" shows "phi T"
check_proof_sound: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and P :: "proofterm"   and res :: "Core.term" assumes "check_proof thy P res" shows "thy,set (hyps P) \<turnstile> res"
map_sel_1: fixes a :: "'b literal"   and f :: "'b \<Rightarrow> 'c" assumes "is_pos a" shows "atm_of (map_literal f a) = f (atm_of a)"
diff_fun_space_times: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "f \<in> sub.diff_fun_space"   and "g \<in> sub.diff_fun_space" shows "f * g \<in> sub.diff_fun_space"
range_binaryset_eq: fixes A :: "'b set"   and B :: "'b set" shows "range (binaryset A B) = {A, B, {}}"
prime_dvd_power_nat: fixes p :: "nat"   and x :: "nat"   and n :: "nat" assumes "prime p"   and "p dvd x ^ n" shows "p dvd x"
norm_prod_diff: fixes I :: "'a set"   and z :: "'a \<Rightarrow> 'b"   and w :: "'a \<Rightarrow> 'b" assumes "\<And>i. i \<in> I \<Longrightarrow> norm (z i) \<le> 1"   and "\<And>i. i \<in> I \<Longrightarrow> norm (w i) \<le> 1" shows "norm (prod z I - prod w I) \<le> (\<Sum>i\<in>I. norm (z i - w i))"
lprefix_lappend: fixes xs :: "'a llist"   and ys :: "'a llist" shows "lprefix xs (lappend xs ys)"
trans_equiv_greater: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "comparator cmp"   and "cmp a b = Equiv"   and "cmp b c = Greater" shows "cmp a c = Greater"
eg: shows "A *\<^sub>v v = e \<cdot>\<^sub>v v"
make_affine_proj_eq_iff: fixes p :: "'b \<times> 'b \<times> 'b"   and p' :: "'b \<times> 'b \<times> 'b" shows "proj_eq p p' = (make_affine p = make_affine p')"
rel: fixes P :: "'b"   and Q :: "'b"   and Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "(P, Q) \<in> Rel" shows "P \<R>\<mapsto><Rel,Cal> Q"
NewA: fixes G :: "prog"   and s0 :: "st"   and T :: "ty"   and n :: "nat"   and s1 :: "state"   and e :: "expr"   and i' :: "val"   and s2 :: "state"   and a :: "loc"   and s3 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>init_comp_ty T\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>e-\<succ>i'\<midarrow>n\<rightarrow> s2"   and "G\<turnstile>abupd (check_neg i') s2 \<midarrow>halloc Arr T (the_Intg i')\<succ>a\<rightarrow> s3" shows "G\<turnstile>Norm s0 \<midarrow>New T[e]-\<succ>Addr a\<midarrow>n\<rightarrow> s3"
assms_2: shows "(\<Psi>, P, Q) \<in> closeSubst Rel"
initial_abs': fixes s :: "'b" assumes "s \<in> wa_initial WB" shows "\<alpha>2 s \<in> wa_initial WA"
mapmeasure_def: fixes M :: "('a \<Rightarrow> 'd) measure"   and N :: "'b \<Rightarrow> 'd measure" shows "mapmeasure M N = distr M (Pi\<^sub>F (Collect finite) N) fm"
Cesare: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A E B"   and "A Q C" shows "B E C"
Ri''_def: shows "Ri'' = lift_prefs alts' alts Ri'"
finite_UN: fixes I :: "'d set"   and A :: "'d \<Rightarrow> 'e set"   and M :: "'e measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M" shows "\<Union> (A ` I) \<in> sets M"
rwof_cons: fixes m0 :: "'a nres"   and cond :: "'a \<Rightarrow> bool"   and step :: "'a \<Rightarrow> 'a nres"   and I :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "is_rwof_invar m0 cond step I"   and "rwof m0 cond step s" shows "I s"
refl: fixes G :: "prog"   and T :: "ty" shows "G\<turnstile>T\<preceq>T"
assms_1: shows "typ_of1 (uty # Ts) t = Some tty"
pos_oriented_cmat_mult_positive: fixes H1 :: "complex \<times> complex \<times> complex \<times> complex"   and H2 :: "complex \<times> complex \<times> complex \<times> complex" assumes "hermitean H1 \<and> H1 \<noteq> mat_zero"   and "hermitean H2 \<and> H2 \<noteq> mat_zero"   and "\<exists>k>0. H2 = cor k *\<^sub>s\<^sub>m H1" shows "pos_oriented_cmat H1 = pos_oriented_cmat H2"
log2_lm2: fixes u :: "nat" assumes "0 < u" shows "c_in (log2 u) u = 1"
l\<^sub>2_p: shows "l\<^sub>2 \<in> L\<^sub>2"
simulation_steps': fixes a :: "'b"   and as :: "'b list"   and b :: "'a" assumes "B.steps (a # as)"   and "b \<sim> a"   and "PB a"   and "PA b" shows "\<exists>bs. A.steps (b # bs) \<and> list_all2 (\<lambda>a b. b \<sim> a) as bs \<and> list_all PB as \<and> list_all PA bs"
that: fixes a :: "'a"   and b :: "real" assumes "a \<in> S"   and "(a - z) \<bullet> z < b"   and "\<And>x. x \<in> S \<Longrightarrow> b < (a - z) \<bullet> x" shows "thesis"
union_equiv: fixes V :: "'b set"   and R :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "equiv V R" shows "equiv V (per_union R a b)"
zero_fds_def: shows "0 = fds (\<lambda>_. 0::'b)"