hcju/isabelleps huggingface dataset:
unit_mult_left_cancel: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "(times a b = times a c) = (b = c)"
null_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "free_monoidal_category C" shows "partial_magma.null (free_monoidal_category.comp C) = {}"
update_valuation_nonlhs: fixes update :: "nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x :: "nat"   and x' :: "nat"   and v :: "'a" assumes "Update update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "x' \<notin> lvars (\<T> s) \<longrightarrow> Mapping.lookup (\<V> (update x v s)) x' = (if x = x' then Some v else Mapping.lookup (\<V> s) x')"
aarden_aux: fixes y :: "'a"   and x :: "'a"   and z :: "'a"   and n :: "nat" assumes "y \<le> y \<cdot> x + z" shows "y \<le> y \<cdot> x\<^bsup>Suc n\<^esup> + z \<cdot> x\<^sup>\<star>"
Ft: shows "\<not> length X < length Y \<and> \<not> length Y < length Z"
lessThan_eq_iff: fixes x :: "'a"   and y :: "'a" shows "({..<x} = {..<y}) = (x = y)"
split_le_Suc: fixes k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<forall>x\<le>Suc k. P x) = (P (Suc k) \<and> (\<forall>x\<le>k. P x))"
word_wi_log_defs_1: fixes a :: "int" shows "NOT (word_of_int a) = word_of_int (NOT a)"
le_cases: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" obtains "x \<preceq> y"   | "y \<preceq> x"
mat_less_eq_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "mat_less_eq f g = (f \<oplus> g = g)"
iso_dghm_ArrMap_vrange: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) = \<BB>\<lparr>Arr\<rparr>"
sets_bind': fixes A :: "'d measure"   and M :: "'d measure"   and B :: "'d \<Rightarrow> 'e measure"   and N :: "'e measure" assumes "A \<in> space (prob_algebra M)"   and "B \<in> M \<rightarrow>\<^sub>M prob_algebra N" shows "sets (A \<bind> B) = sets N"
bind_eq_None_conv: fixes a :: "'e option"   and f :: "'e \<Rightarrow> 'd option" shows "(a \<bind> f = None) = (a = None \<or> f (the a) = None)"
path_decomp_1: fixes G :: "('a, 'b) Graph_scheme"   and xs :: "'a list"   and ys :: "'a list" assumes "Digraph G"   and "Digraph.walk G (xs @ ys) \<and> distinct (xs @ ys)" shows "Digraph.walk G xs \<and> distinct xs"
Gimage_def: fixes F :: "('g, 'h) Group_scheme"   and G :: "('e, 'f) Group_scheme"   and f :: "'g \<Rightarrow> 'e" shows "Img\<^bsub>F,G\<^esub> f = \<natural>\<^bsub>G\<^esub>f ` carrier F"
Lyndon_rotate_rel_iff: fixes w :: "'b list" shows "acyclic (rotate_rel w) = (\<exists>r. strict_linear_order r \<and> rotate_rel w \<subseteq> r)"
preserves_section_retraction: fixes e :: "'a"   and m :: "'a" assumes "Adj.C.ide (e \<cdot>\<^sub>C m)" shows "Adj.D.ide (G' e \<cdot>\<^sub>D G' m)"
long_division_closed_2: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "UP.pmod p q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"
mv: fixes i :: "nat" assumes "i < length fas" shows "max_Var_floatarith (fas ! i) \<le> length vs"
congruent_triangle_sss_5: fixes a1 :: "'b"   and b1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c1 :: "'b"   and c2 :: "'c" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "dist a1 c1 = dist a2 c2" shows "dist c1 a1 = dist a2 c2"
rebase_poly_id: fixes p :: "'q mod_ring poly" shows "##p = p"
last_prefixes: fixes i :: "nat"   and p :: "'b list" assumes "i < length p" shows "last (tl (prefixes p) ! i) = p ! i"
find_dep: fixes m' :: "('a, 'b) insert_message"   and i :: "'a" assumes "Insert m' \<in> M \<union> {Insert m}"   and "i \<in> deps (Insert m')" shows "\<lbrakk>i\<rbrakk> \<in> set (ext_ids s)"
Co: fixes i :: "'a" assumes "i \<in> I" shows "F \<in> A i Co A' i"
enabled_iff_snth: fixes s :: "'s"   and \<omega> :: "'s stream" shows "enabled s \<omega> = (\<forall>i. \<omega> !! i \<in> set_pmf (K ((s ## \<omega>) !! i)))"
wf_operators_1: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" assumes "wf_ast_problem problem" shows "distinct (map fst (ast_problem.ast\<delta> problem))"
inverse_equality: fixes u :: "'a \<Rightarrow> 'a"   and v :: "'a \<Rightarrow> 'a" assumes "compose S u v = identity S"   and "compose S v u = identity S"   and "u \<in> S \<rightarrow>\<^sub>E S"   and "v \<in> S \<rightarrow>\<^sub>E S" shows "inverse u = v"
distinct_delete_aux: fixes ps :: "('a \<times> 'b) list"   and k :: "'a" assumes "distinct (map fst ps)" shows "distinct (map fst (AList.delete_aux k ps))"
l'_def: shows "l' \<equiv> gap_relabel_effect f l u"
\<delta>3: shows "\<delta>3 \<in> conv ars"
aform_vals_def: fixes e :: "nat \<Rightarrow> real"   and X :: "('a \<times> 'a pdevs) list" shows "aform_vals e X = map (aform_val e) X"
v_S\<^sub>e: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set"   and cfg :: "'a cfg" assumes "Reachability_Problem K S S1 S2"   and "cfg \<in> Finite_Markov_Decision_Process.valid_cfg K S"   and "state cfg \<in> Reachability_Problem.S\<^sub>e K S S1 S2" shows "Reachability_Problem.v K S S1 S2 cfg = 0"
ltake_0: fixes xs :: "'b llist" shows "ltake 0 xs = LNil"
naturally_isomorphic_transitive: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and H :: "'b \<Rightarrow> 'c" assumes "naturally_isomorphic A B F G"   and "naturally_isomorphic A B G H" shows "naturally_isomorphic A B F H"
not_in_iprev_fix: fixes n :: "nat"   and I :: "nat set" assumes "n \<notin> I" shows "iprev n I = n"
antisym_conv3: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d"   and x :: "'d" assumes "class.linorder less_eq less"   and "\<not> less y x" shows "(\<not> less x y) = (x = y)"
prj_tuple_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "category.commutative_square C f g h k" shows "C (prj1 f g) (elementary_category_with_pullbacks.tuple C prj0 prj1 h f g k) = h"
w_simps_3: shows "src cmp.chine = src (tab\<^sub>0 s \<star> p\<^sub>0)"
pair_graph_impl_def: fixes G :: "'b list \<times> ('b \<times> 'b) list" shows "pair_graph_impl G \<equiv> case G of (V, A) \<Rightarrow> \<forall>(u, v)\<in>set A. u \<noteq> v \<and> u \<in> set V \<and> v \<in> set V \<and> (v, u) \<in> set A"
AE_eq_cst: fixes c :: "'d"   and d :: "'d"   and M :: "'c measure" assumes "AEeq M (\<lambda>w. c) (\<lambda>w. d)"   and "emeasure M (space M) \<noteq> 0" shows "c = d"
bin_to_bl_cat: fixes nv :: "nat"   and nw :: "nat"   and w :: "int"   and v :: "int" shows "bin_to_bl (nv + nw) (concat_bit nw w v) = bin_to_bl_aux nv v (bin_to_bl nw w)"
depthMin: fixes P :: "pi" shows "0 \<le> depth P"
weakStatImpMonotonic: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and A :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and B :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> P \<lessapprox><A> Q"   and "A \<subseteq> B" shows "\<Psi> \<rhd> P \<lessapprox><B> Q"
basis: fixes T :: "'a set"   and x :: "'a" assumes "open T"   and "x \<in> T"   and "x \<in> U" shows "\<exists>b\<in>B. x \<in> b \<and> b \<inter> U \<subseteq> T"
iptables_bigstep_rw_Reject: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rw_Reject rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
USUP_mem_UNIV: fixes P :: "'c \<Rightarrow> ('a, 'b) uexpr" shows "(\<Squnion> x \<in> UNIV \<bullet> P x) = (\<Squnion> x \<bullet> P x)"
arg_poss_def: fixes p :: "lambda" shows "arg_poss p = Discr ` (fstate_poss p <+> cstate_poss p)"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vrat_inverse"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vrat_inverse" shows "b = c"
transformation_by_components_axioms: shows "transformation_by_components hom.comp hom.comp hom.map GF.map (\<lambda>f. (d\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[e\<^sub>1, f \<star> d\<^sub>0, e\<^sub>0]) \<cdot> \<a>[d\<^sub>1, e\<^sub>1, (f \<star> d\<^sub>0) \<star> e\<^sub>0] \<cdot> ((d\<^sub>1 \<star> e\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[f, d\<^sub>0, e\<^sub>0]) \<cdot> (\<eta>\<^sub>1 \<star> f \<star> \<eta>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[f \<star> src e\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[f])"
lessThan_minus_lessThan: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "'b"   and m :: "'b" assumes "class.linorder less_eq less" shows "ord.lessThan less n - ord.lessThan less m = ord.atLeastLessThan less_eq less m n"
62434: fixes x :: "'a"   and y :: "'a" shows "- (- - x \<sqinter> y) \<sqinter> x = - y \<sqinter> x"
sgn_power_root: fixes n :: "nat"   and x :: "real" assumes "0 < n" shows "sgn (root n x) * \<bar>root n x\<bar> ^ n = x"
assms_1: shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"
Diff_insert_absorb: fixes x :: "'a"   and A :: "'a set" assumes "x \<notin> A" shows "insert x A - {x} = A"
sd_prod_def: fixes act :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" assumes "unital_quantale_module act" shows "unital_quantale_module.sd_prod act x y = (fst x \<cdot> fst y, snd x \<squnion> act (fst x) (snd y))"
FAssThrow\<^sub>2\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e\<^sub>2 :: "nat exp"   and e' :: "nat exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>"
vwalk_join_last: fixes p :: "'a list"   and q :: "'a list" assumes "joinable p q" shows "last (p \<oplus> q) = last q"
trivial_relative_homology_group_contractible_spaces: fixes X :: "'a topology"   and S :: "'a set"   and p :: "int" assumes "contractible_space X"   and "contractible_space (subtopology X S)"   and "topspace X \<inter> S \<noteq> {}" shows "trivial_group (relative_homology_group p X S)"
par_commute: fixes par :: "'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice"   and skip :: "'a::refinement_lattice"   and a :: "'a::refinement_lattice"   and b :: "'a::refinement_lattice" assumes "parallel (par::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) (skip::'a::refinement_lattice)" shows "(par::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) (a::'a::refinement_lattice) (b::'a::refinement_lattice) = par b a"
Extend_commute_unsafe: fixes k :: "'k"   and k' :: "'k"   and j :: "nat"   and i :: "nat"   and \<AA> :: "'i"   and P :: "'v"   and Q :: "'v" assumes "k \<noteq> k'" shows "Extend k j (Extend k' i \<AA> P) Q = Extend k' i (Extend k j \<AA> Q) P"
cindex_eq_cindex_poly: fixes a :: "real"   and b :: "real"   and q :: "real poly"   and p :: "real poly" shows "cindex a b (\<lambda>x. poly q x / poly p x) = cindex_poly a b q p"
simps_2: fixes A :: "'a set" assumes "A \<noteq> {}"   and "A \<subseteq> carrier" shows "weak_ranking_aux A = Max_wrt_among le A # weak_ranking_aux (A - Max_wrt_among le A)"
Collect_cong: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "{x. P x} = {x. Q x}"
that: fixes e :: "real" assumes "e \<in> {- 1..1}"   and "pdevs_val f (pdevs_of_list xs) = e *\<^sub>R sum_list xs" shows "thesis"
ar_eq: fixes x :: "'a" shows "antirange_semiring.ars_r ar x = r x"
IH: fixes i :: "nat" assumes "i < length T" shows "term_variants_pred P (T ! i \<cdot> \<delta>) (S ! i \<cdot> \<delta>)"
assms_2: fixes r :: "'a" assumes "r \<in> carrier R" shows "P r"
init_simplex'_def: shows "init_simplex' = init_cs init_nsc_code to_ns"
tom_Cons: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as' :: "'edge list"   and ms'' :: "'node list"   and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow> (ms',s')"   and "S,f \<turnstile> (ms',s') =as'\<Rightarrow>* (ms'',s'')" shows "S,f \<turnstile> (ms,s) =last as # as'\<Rightarrow>* (ms'',s'')"
typeI_regions_integral: shows "integral s (\<lambda>x. - partial_vector_derivative (\<lambda>x. F x \<bullet> i) j x) = one_chain_line_integral F {i} one_chain_typeI"
notin_range_Some: fixes x :: "'c option" shows "(x \<notin> range Some) = (x = None)"
map_poly_homo_polyadd_eq_zero_iff: fixes f :: "'a \<Rightarrow> 'b"   and p :: "'a poly"   and q :: "'a poly" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>x. (f x = (0::'b)) = (x = (0::'a))" shows "(map_poly f (p +\<^sub>p q) = 0\<^sub>p) = (p +\<^sub>p q = 0\<^sub>p)"
update_convs_2: fixes dcl' :: "(nat \<times> nat \<times> nat \<Rightarrow> nat option) \<Rightarrow> nat \<times> nat \<times> nat \<Rightarrow> nat option"   and dpm :: "(nat \<times> nat \<times> nat) pointermap"   and dcl :: "nat \<times> nat \<times> nat \<Rightarrow> nat option"   and more :: "'a" shows "dcl_update dcl' \<lparr>dpm = dpm, dcl = dcl, \<dots> = more\<rparr> = \<lparr>dpm = dpm, dcl = dcl' dcl, \<dots> = more\<rparr>"
rank_older_seniors_bounded: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and y :: "nat"   and x :: "nat"   and n :: "nat"   and j :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "y \<in> semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n"   and "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = Some j" shows "\<exists>j'<j. semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w y n = Some j'"
to_hma_from_hma\<^sub>v: fixes v :: "('b, 'c) Finite_Cartesian_Product.vec" shows "to_hma\<^sub>v (from_hma\<^sub>v v) = v"
bottom_left_bottom: fixes a :: "'a" shows "gcd (1::'a) a = (1::'a)"
intros_5: fixes cl :: "cl"   and j :: "nat"   and t1s :: "t list"   and t2s :: "t list"   and ts :: "t list"   and es :: "b_e list"   and ves :: "e list"   and vcs :: "v list"   and n :: "nat"   and k :: "nat"   and m :: "nat"   and zs :: "v list"   and s :: "s"   and vs :: "v list"   and i :: "nat" assumes "cl = Func_native j (t1s _> t2s) ts es"   and "ves = $$* vcs"   and "length vcs = n"   and "length ts = k"   and "length t1s = n"   and "length t2s = m"   and "n_zeros ts = zs" shows "\<lparr>s;vs;ves @ [Callcl cl]\<rparr> \<leadsto>_ i \<lparr>s;vs;[Local m j (vcs @ zs) [$Block ([] _> t2s) es]]\<rparr>"
simps_1: fixes p :: "'a node ref option"   and s :: "'a node ref option" shows "lseg [] p s = \<up> (p = s)"
well_ordered_set_def: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "well_ordered_set A less_eq \<equiv> antisymmetric A less_eq \<and> well_related_set A less_eq"
distinct_19: fixes x2 :: "'b"   and x6 :: "'b ptrm" shows "PVar x2 \<noteq> PFst x6"
bounded_clinear_const_scaleC: fixes g :: "'b \<Rightarrow> 'c"   and c :: "complex" assumes "bounded_clinear g" shows "bounded_clinear (\<lambda>x. c *\<^sub>C g x)"
fmranE: fixes y :: "'c"   and m :: "('d, 'c) fmap" assumes "y |\<in>| fmran m" obtains x :: "'d" where "fmlookup m x = Some y"
seqE: fixes g :: "'a"   and f :: "'a" assumes "Left_a.seq g f" obtains "Left_a.arr f" and "Left_a.arr g" and "Left_a.dom g = Left_a.cod f"
trimonoD_flatf_ge: fixes B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" assumes "trimono B" shows "monotone flatf_ge flatf_ge B"
extreme_bound_singleton: fixes x :: "'a" assumes "x \<in> A" shows "extreme_bound A (\<sqsubseteq>) {x} x"
Sup_bot_conv_1: fixes A :: "('c \<times> 'd) set" shows "(Sup_prod A = bot_prod) = (\<forall>x\<in>A. x = bot_prod)"
rreq_rrep_sn_simps_5: fixes d :: "nat"   and dip :: "nat"   and sip :: "nat" shows "rreq_rrep_sn (Pkt d dip sip) = True"
filter_def: fixes P :: "'g \<Rightarrow> bool"   and A :: "'g set" shows "Set.filter P A = {a \<in> A. P a}"
map_drop_delete: fixes k :: "'a"   and ps :: "('a \<times> 'b) list" shows "map_drop k (map_of ps) = map_of (AList.delete k ps)"
simps_2: fixes x :: "nat"   and xs :: "nat list"   and n :: "nat" shows "godel_code' (x # xs) n = Pi n ^ x * godel_code' xs (Suc n)"
pred_set: fixes P :: "'a \<Rightarrow> bool" shows "pred_tttree P = (\<lambda>x. Ball (set_tttree x) P)"
reduced_word_Cons_reduce_contra: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set"   and as :: "'b list"   and a :: "'b" assumes "class.monoid_add plus zero"   and "\<not> monoid_add.reduced_word_for plus zero A (monoid_add.sum_list plus zero as) as" shows "\<not> monoid_add.reduced_word_for plus zero A (monoid_add.sum_list plus zero (a # as)) (a # as)"
clinearI: fixes f :: "'d \<Rightarrow> 'e" assumes "\<And>b1 b2. f (b1 + b2) = f b1 + f b2"   and "\<And>r b. f (r *\<^sub>C b) = r *\<^sub>C f b" shows "clinear f"
subdegree_div_unit: fixes q :: "'b fps"   and p :: "'b fps" assumes "q $ 0 \<noteq> (0::'b)" shows "subdegree (p / q) = subdegree p"
hyps: shows "pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\<Prod>k = 0..2 * n + 1. z + of_nat k / (2::'a))"
int_pow_def2: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and z :: "int" shows "a [^]\<^bsub>G\<^esub> z = (if z < 0 then inv\<^bsub>G\<^esub> (a [^]\<^bsub>G\<^esub> nat (- z)) else a [^]\<^bsub>G\<^esub> nat z)"
cases: fixes a1 :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and a2 :: "typ"   and a3 :: "String.literal set" assumes "has_sort a1 a2 a3" obtains sub :: "(String.literal \<times> String.literal) set" and S :: "String.literal set" and S' :: "String.literal set" and tcs :: "String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option" and a :: "variable" where "a1 = (sub__, tcs__)" and "a2 = Tv a S__" and "a3 = S'" and "sort_leq sub__ S__ S'"   | tcs :: "String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option" and \<kappa> :: "String.literal" and dm :: "String.literal \<Rightarrow> String.literal set list option" and S :: "String.literal set" and sub :: "(String.literal \<times> String.literal) set" and Ts :: "typ list" where "a1 = (sub, tcs)" and "a2 = Ty \<kappa> Ts" and "a3 = S" and "tcs \<kappa> = Some dm" and "\<forall>c\<in>S. \<exists>Ss. dm c = Some Ss \<and> list_all2 (has_sort (sub, tcs)) Ts Ss"
Yi: shows "\<forall>i. \<exists>y. Y i = f y \<and> P y"
braking_movement_axioms: shows "braking_movement a\<^sub>e v\<^sub>e"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a') F \<Phi>'"
simps_1: fixes n :: "nat" shows "good (VNat n) = {BNat n}"
eventually_nonzero_def: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" shows "eventually_nonzero F f = (\<forall>\<^sub>F x in F. f x \<noteq> (0::'b))"
eq_iff_3: fixes fmla :: "fmla"   and fmlaa :: "fmla" shows "(neg fmla = neg fmlaa) = (fmla = fmlaa)"
arc_def: fixes e :: "'b"   and uv :: "'a \<times> 'a" shows "arc e uv \<equiv> e \<in> pre_digraph.arcs G \<and> tail G e = fst uv \<and> head G e = snd uv"
topological_tendstoD: fixes f :: "'c \<Rightarrow> 'a"   and l :: "'a"   and F :: "'c filter"   and S :: "'a set" assumes "(f \<longlongrightarrow> l) F"   and "open S"   and "l \<in> S" shows "\<forall>\<^sub>F x in F. f x \<in> S"
Eq1: shows "VC q t = IC q (i + 1) t"
vectorize_minus: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "local.punit.vectorize_poly (p - q) = local.punit.vectorize_poly p - local.punit.vectorize_poly q"
completeness': fixes s :: "(nat \<times> form) list" assumes "infinite (deriv s)"   and "init s" shows "\<forall>mA\<in>set s. \<not> FEval (model s) ntou (snd mA)"
OclForall_def: fixes S :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and P :: "('d state \<times> 'd state \<Rightarrow> 'e) \<Rightarrow> 'd state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "UML_Set.OclForall S P = (\<lambda>\<tau>. if (\<delta> S) \<tau> = true \<tau> then if \<exists>x\<in>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (S \<tau>)\<rceil>\<rceil>. P (\<lambda>_. x) \<tau> = false \<tau> then false \<tau> else if \<exists>x\<in>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (S \<tau>)\<rceil>\<rceil>. P (\<lambda>_. x) \<tau> = invalid \<tau> then invalid \<tau> else if \<exists>x\<in>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (S \<tau>)\<rceil>\<rceil>. P (\<lambda>_. x) \<tau> = null \<tau> then null \<tau> else true \<tau> else \<bottom>)"
f_last_message_hold_drop_mult: fixes n :: "nat"   and k :: "nat"   and xs :: "'b message_af list" shows "xs \<up> (n * k) \<longmapsto> k = xs \<longmapsto> k \<up> (n * k)"
resolution_class_size1: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and partition :: "'b set multiset multiset"   and u_block_size :: "int"   and S :: "'b set multiset" assumes "r_block_design point_set block_collection partition u_block_size"   and "S \<in># partition" shows "int (card point_set) = u_block_size * int (size S)"