hcju/isabelleps huggingface dataset:
assms_2: shows "((\<lambda>x. ln (b x)) expands_to L) basis"
limit_in_range: fixes r :: "nat \<Rightarrow> 'b" shows "limit r \<subseteq> range r"
confTs_Err: fixes P :: "'a prog"   and h :: "heap"   and n :: "nat"   and v :: "val" shows "P,h \<turnstile> replicate n v [:\<le>\<^sub>\<top>] replicate n Err"
dnf_prop_atoms: fixes \<Phi> :: "'b ltln fset"   and \<phi> :: "'b ltln" assumes "\<Phi> \<in> dnf \<phi>" shows "fset \<Phi> \<subseteq> prop_atoms \<phi>"
matrix_to_iarray_rank: fixes A :: "(('b, 'c) vec, 'd) vec" shows "rank A = rank_iarray (matrix_to_iarray A)"
true_cls_mset_mono: fixes DD :: "'a literal multiset multiset"   and CC :: "'a literal multiset multiset"   and I :: "'a set" assumes "set_mset DD \<subseteq> set_mset CC"   and "I \<Turnstile>m CC" shows "I \<Turnstile>m DD"
in_join: fixes x :: "Vars.var"   and y :: "Vars.var"   and G :: "CoCalls"   and G' :: "CoCalls" shows "x--y\<in>(G \<squnion> G') = (x--y\<in>G \<or> x--y\<in>G')"
chainPar: fixes P :: "pi"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "Q \<Longrightarrow>\<^sub>\<tau> Q'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>\<tau> P' \<parallel> Q'"
assms_1: shows "x \<notin> env_dom (e\<lparr>a:X\<rparr>)"
out_arcs: shows "out_arcs (with_proj G) w \<subseteq> {(w, u), (w, v)}"
trans: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" assumes "xs <~~> ys"   and "ys <~~> zs" shows "xs <~~> zs"
map_poly_inj_comm_ring_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "map_poly_inj_comm_ring_hom hom \<equiv> inj_comm_ring_hom hom"
DERIV_floatarith: fixes n :: "nat"   and vs :: "real list"   and f :: "floatarith"   and x :: "real" assumes "n < length vs"   and "isDERIV n f (vs[n := x])" shows "((\<lambda>x'. interpret_floatarith f (vs[n := x'])) has_real_derivative interpret_floatarith (DERIV_floatarith n f) (vs[n := x])) (at x)"
strict_implies_order: fixes b :: "'d"   and a :: "'d" assumes "b < a" shows "b \<le> a"
assms_5: shows "h (u (l (\<nu> h))) \<le> u (g (l (\<nu> h)))"
neg3: shows "Negligible.negligible (\<lambda>n. local.ddh.advantage n (P1_DDH_mal_adv_\<sigma>_false n M z \<A> D))"
admissible_kbo_axioms: shows "admissible_kbo w w0 pr_strict pr_weak least scf"
assms_2: fixes u :: "'t" assumes "lookup p u \<noteq> (0::'b)" shows "u \<preceq>\<^sub>t v"
maxim_isMaxim: fixes A :: "'b set" assumes "finite A"   and "A \<noteq> {}"   and "A \<subseteq> Field s" shows "s.isMaxim A (s.maxim A)"
add_mset_remove_trivial: fixes x :: "'c"   and M :: "'c multiset" shows "add_mset x M - {#x#} = M"
elims_3: fixes x :: "arctic"   and xa :: "arctic" assumes "\<not> x \<le> xa" obtains uu_ :: "int" where "x = Num_arc uu_" and "xa = MinInfty"   | y :: "int" and xb :: "int" where "x = Num_arc y" and "xa = Num_arc xb" and "\<not> y \<le> xb"
a_def: shows "a = (\<lambda>i. P $$ (j, i - (m - 1)) * u $v i)"
satisfies_properties_def: fixes p :: "real poly"   and qs :: "real poly list"   and subsets :: "nat list list"   and signs :: "rat list list"   and matrix :: "rat mat" shows "satisfies_properties p qs subsets signs matrix = (all_list_constr subsets (length qs) \<and> well_def_signs (length qs) signs \<and> distinct signs \<and> satisfy_equation p qs subsets signs \<and> invertible_mat matrix \<and> matrix = M_mat signs subsets \<and> set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs)"
sup_zmultiset_def: fixes A :: "'a zmultiset"   and B :: "'a zmultiset" shows "sup A B = (if A < B then B else A)"
commute: fixes a :: "'b"   and b :: "'b" shows "normalization_euclidean_semiring_class.gcd a b = normalization_euclidean_semiring_class.gcd b a"
has_field_derivativeD: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and S :: "'a set" assumes "(f has_field_derivative D) (at x within S)" shows "((\<lambda>y. (f y - f x) / (y - x)) \<longlongrightarrow> D) (at x within S)"
fs_gs_N_N': assumes "m \<noteq> 0" shows "fs.gs.N = rat_of_nat N"
assms: fixes i :: "nat" assumes "Suc i \<le> n" shows "w !! i = x !! i"
fashoda_unit: fixes f :: "real \<Rightarrow> (real, 2) vec"   and g :: "real \<Rightarrow> (real, 2) vec" assumes "f ` {- 1..1} \<subseteq> cbox (- 1) 1"   and "g ` {- 1..1} \<subseteq> cbox (- 1) 1"   and "continuous_on {- 1..1} f"   and "continuous_on {- 1..1} g"   and "f (- 1) $ 1 = - 1"   and "f 1 $ 1 = 1"   and "g (- 1) $ 2 = - 1"   and "g 1 $ 2 = 1" shows "\<exists>s\<in>{- 1..1}. \<exists>t\<in>{- 1..1}. f s = g t"
False: shows "f * inverse (unit_factor g) \<noteq> 0"
simps_1: fixes A :: "'e set"   and uu :: "'e set"   and uv :: "'e \<Rightarrow> 'e" shows "Dom (MkArr A uu uv) = A"
Inf_eqI: fixes A :: "'a set"   and x :: "'a" assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> i"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x" shows "\<Sqinter>A = x"
empty_ne_singleton_zmset: fixes x :: "'b" shows "{#}\<^sub>z \<noteq> {#x#}\<^sub>z"
ID_distr_store_2: fixes \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and l :: "'a"   and v :: "('b, 'a, 'c) val" shows "LID\<^sub>S (\<tau>(l \<mapsto> v)) = insert l (LID\<^sub>S (\<tau>(l := None)) \<union> LID\<^sub>V v)"
representant_exists: fixes A :: "'a set" assumes "A \<in> L.Partition" shows "\<exists>a\<in>Ker. a \<in> A \<and> A = L.Class a"
Inl_transfer: fixes S :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and T :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "(S ===> rel_sum S T) Inl Inl"
dom_inv: fixes f :: "'a" assumes "local.iso f" shows "local.dom (local.inv f) = cod f"
assms_2: fixes a' :: "nat \<Rightarrow> nat" assumes "a' \<in> s" shows "a \<le> a'"
inner_zero_left: 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" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner zero x = 0"
High: shows "high_proc\<^sub>2 s\<^sub>H"
untime_empty_conv: fixes s :: "'b message_af list" shows "(untime s = []) = (\<forall>n<length s. s ! n = NoMsg)"
nat_pow_closed: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "x [^]\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> n \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)"
that_2: fixes \<delta> :: "V" assumes "Ord \<delta>"   and "\<mu> = \<omega> \<up> \<delta>" shows "thesis"
sel_rews_2: shows "left\<cdot>\<bottom> = \<bottom>"
insert_noteq_member: fixes b :: "'c"   and B :: "'c multiset"   and c :: "'c"   and C :: "'c multiset" assumes "add_mset b B = add_mset c C"   and "b \<noteq> c" shows "c \<in># B"
is_natural_1: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "H.HoVH f \<cdot> \<alpha> (VVV.dom f) = \<alpha> f"
weakBisimulation_def: shows "weakBisimulation \<equiv> {(xa, x). weakBisimulationp xa x}"
simple_distributed_sum_space: fixes X :: "'a \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> real" assumes "simple_distributed M X f" shows "sum f (X ` space M) = 1"
integrable_indicator: fixes A :: "'c set"   and M :: "'c measure"   and c :: "'d" assumes "A \<in> sets M"   and "emeasure M A < \<infinity>" shows "integrable M (\<lambda>x. indicat_real A x *\<^sub>R c)"
hyps: assumes "trace init t final"   and "channel cid = Some (p, q)"   and "\<not> ps (S t n) p \<noteq> None" shows "Marker \<notin> set (msgs (S t n) cid)"
N: shows "lnorm (lebesgue_on S) p (g N - h) < e / 2"
gcsort_sort_input: fixes key :: "'c \<Rightarrow> 'd"   and xs :: "'c list" shows "sort_inv key (0, [length xs], xs)"
complex_quadratic_equation_only_two_roots: fixes a :: "complex"   and qf :: "complex \<Rightarrow> complex"   and b :: "complex"   and c :: "complex"   and x1 :: "complex"   and x2 :: "complex"   and x :: "complex" assumes "a \<noteq> 0"   and "qf = (\<lambda>x. a * x\<^sup>2 + b * x + c)"   and "qf x1 = 0"   and "qf x2 = 0"   and "x1 \<noteq> x2"   and "qf x = 0" shows "x = x1 \<or> x = x2"
induct: fixes P :: "bool \<Rightarrow> 'a word \<Rightarrow> nat \<Rightarrow> 'a word \<Rightarrow> bool"   and a0 :: "bool"   and a1 :: "'a word"   and a2 :: "nat"   and a3 :: "'a word" assumes "\<And>lc base ml w. (\<lbrakk>\<not> (base < 2 \<or> LENGTH('a) < 2); \<not> (w < base \<and> ml = 0)\<rbrakk> \<Longrightarrow> P lc base (ml - 1) (w div base)) \<Longrightarrow> P lc base ml w" shows "P a0 a1 a2 a3"
IK_nf_set: shows "{f. IK_nf f} = {id, I, K, I \<circ> K, K \<circ> I, I \<circ> K \<circ> I, K \<circ> I \<circ> K}"
gso_cong: fixes x :: "nat"   and f1 :: "'a vec list"   and f2 :: "'a vec list" assumes "\<And>i. i \<le> x \<Longrightarrow> f1 ! i = f2 ! i" shows "gram_schmidt_fs.gso n f1 x = gram_schmidt_fs.gso n f2 x"
q_def: shows "q \<equiv> \<lambda>(x, y). (x, \<lambda>i. y i + g' x i)"
map_le_refl: fixes f :: "'d \<Rightarrow> 'e option" shows "f \<subseteq>\<^sub>m f"
Rep_linear_poly_cases: fixes y :: "nat \<Rightarrow> rat" assumes "y \<in> {c. Abstract_Linear_Poly.inv c}" obtains x :: "linear_poly" where "y = Rep_linear_poly x"
induct: fixes x1 :: "('b com \<times> 'b set \<times> ('b \<times> 'b) set \<times> ('b \<times> 'b) set \<times> 'b set) list"   and x2 :: "'b set"   and x3 :: "('b \<times> 'b) set"   and x4 :: "('b \<times> 'b) set"   and x5 :: "'b set"   and P :: "('b com \<times> 'b set \<times> ('b \<times> 'b) set \<times> ('b \<times> 'b) set \<times> 'b set) list \<Rightarrow> 'b set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> 'b set \<Rightarrow> bool" assumes "\<turnstile> x1 SAT [x2, x3, x4, x5]"   and "\<And>xs rely guar pre post. \<lbrakk>\<forall>i<length xs. rely \<union> (\<Union>j\<in>{j. j < length xs \<and> j \<noteq> i}. Guar (xs ! j)) \<subseteq> Rely (xs ! i); (\<Union>j\<in>{j. j < length xs}. Guar (xs ! j)) \<subseteq> guar; pre \<subseteq> (\<Inter>i\<in>{i. i < length xs}. Pre (xs ! i)); (\<Inter>i\<in>{i. i < length xs}. Post (xs ! i)) \<subseteq> post; \<forall>i<length xs. \<turnstile> Com (xs ! i) sat [Pre (xs ! i), Rely (xs ! i), Guar (xs ! i), Post (xs ! i)]\<rbrakk> \<Longrightarrow> P xs pre rely guar post" shows "P x1 x2 x3 x4 x5"
augment_cf_impl_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'capacity"   and p :: "(nat \<times> nat) list"   and x :: "'capacity" shows "augment_cf_impl cf p x \<equiv> (rec\<^sub>T D. (\<lambda>xaa. case xaa of ([], cf) \<Rightarrow> return cf | (e # p, cf) \<Rightarrow> augment_edge_impl cf e x \<bind> (\<lambda>cf. D (p, cf)))) (p, cf)"
nonmeets3: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and u :: "'a"   and r :: "'a" assumes "class.interval meets \<I>" shows "\<not> (meets u r \<and> (\<exists>p. meets u p \<and> meets p r))"
if_eq_same: fixes a :: "'d"   and b :: "'d"   and f :: "'d \<Rightarrow> 'c" shows "(if a = b then f b else f a) = f a"
monoI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. y \<le> x \<Longrightarrow> f x \<le> f y" shows "local.mono f"
in_outs'_restrict_gpvD: fixes x :: "'b"   and \<I> :: "('b, 'e) \<I>"   and gpv :: "('d, 'b, 'e) gpv" assumes "x \<in> outs'_gpv (restrict_gpv \<I> gpv)" shows "x \<in> outs_\<I> \<I>"
mset_ran_empty: fixes a :: "int \<Rightarrow> 'b" shows "mset_ran a {} = {#}"
inputCases: fixes a :: "name"   and x :: "name"   and P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and Prop :: "freeRes \<Rightarrow> pi \<Rightarrow> bool" assumes "a<x>.P \<longmapsto> \<alpha> \<prec> P'"   and "\<And>u. Prop (a<u>) (P[x::=u])" shows "Prop \<alpha> P'"
nb: shows "set (List_Supplement.embed \<pi>) = {[op] |op. op \<in> set \<pi>}"
is_adj_map_def: fixes ps :: "nat \<Rightarrow> nat list" shows "cf.is_adj_map ps \<equiv> \<forall>u. distinct (ps u) \<and> set (ps u) = cf.E `` {u} \<union> cf.E\<inverse> `` {u}"
reciprocal_smult: fixes P :: "'a poly"   and p :: "nat"   and n :: "'a" assumes "degree P \<le> p" shows "reciprocal_poly p (smult n P) = smult n (reciprocal_poly p P)"
mod_mod_le: fixes a :: "int"   and b :: "int"   and c :: "int" assumes "a \<le> b"   and "0 < a"   and "0 \<le> c" shows "c mod a mod b = c mod a"
inject_1: fixes x1 :: "nat"   and y1 :: "nat" shows "(CA x1 = CA y1) = (x1 = y1)"
rel_with_target_impl_reflC_transC_TRel_is_strong_reduction_bisimulation: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "strong_reduction_simulation Rel (STCal Source Target) \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> (P', Q') \<in> Rel))"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>*" shows "strong_reduction_simulation (TRel\<^sup>*) Target \<and> (\<forall>P Q Q'. (P, Q) \<in> TRel\<^sup>* \<and> Q \<longmapsto>Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Target P' \<and> (P', Q') \<in> TRel\<^sup>*))"
as_2: shows "bounded_linear f'' \<and> ((\<lambda>y. (f y - f (netlimit (at x within S)) - f'' (y - netlimit (at x within S))) /\<^sub>R norm (y - netlimit (at x within S))) \<longlongrightarrow> (0::'b)) (at x within S)"
lunit_tensor_gen: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "(a \<otimes> \<l>[b \<otimes> c]) \<cdot> (a \<otimes> \<a>[\<I>, b, c]) = a \<otimes> \<l>[b] \<otimes> c"
fold_congs_2: fixes r :: "('a, 'b, 'c) automaton_scheme"   and r' :: "('a, 'b, 'c) automaton_scheme"   and v' :: "('a \<times> 'b \<times> 'a) set"   and f :: "('a \<times> 'b \<times> 'a) set \<Rightarrow> ('a \<times> 'b \<times> 'a) set"   and f' :: "('a \<times> 'b \<times> 'a) set \<Rightarrow> ('a \<times> 'b \<times> 'a) set" assumes "r = r'"   and "automaton.trans r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "trans_update f r = trans_update f' r'"
bound_def: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c" shows "bound X r b \<equiv> \<forall>x\<in>X. r x b"
Gr_domain_2: fixes A :: "'e set"   and R :: "('e \<times> 'f) set" shows "Domain (BNF_Def.Gr A id O R) = A \<inter> Domain R"
substT2_fresh_switch: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and zer :: "'b"   and suc :: "'b \<Rightarrow> 'b"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tms :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and r :: "'b"   and t :: "'b"   and s :: "'b"   and x :: "'a"   and y :: "'a" assumes "Syntax_Arith var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"   and "r \<in> Syntax_Arith.atrm var Var num suc pls tms"   and "t \<in> trm"   and "s \<in> trm"   and "x \<in> var"   and "y \<in> var"   and "x \<noteq> y"   and "x \<notin> FvarsT s"   and "y \<notin> FvarsT t" shows "substT (substT r s y) t x = substT (substT r t x) s y"
img_is_subring: fixes K :: "'a set" assumes "subring K R" shows "subring (h ` K) S"
configuration_token_succeeds: fixes x :: "nat"   and q :: "'b"   and n :: "nat"   and y :: "nat" assumes "x \<in> configuration q n"   and "y \<in> configuration q n" shows "token_succeeds x = token_succeeds y"
smult_distrib_left_diff: fixes r :: "('f, 'g) aezfun"   and m :: "'v"   and n :: "'v" assumes "r \<in> FG"   and "m \<in> V"   and "n \<in> V" shows "r \<cdot> (m - n) = r \<cdot> m - r \<cdot> n"
numbering_translation: fixes psi :: "recf" assumes "recfn 2 psi" obtains c :: "recf" where "recfn 1 c" and "Partial_Recursive.total c" and "\<forall>i x. eval psi [i, x] = eval r_phi [the (eval c [i]), x]"
lhs_step_SPEC: fixes \<Phi> :: "'e \<Rightarrow> bool"   and m :: "'e nres" assumes "\<And>x. \<Phi> x \<Longrightarrow> RETURN x \<le> m" shows "SPEC \<Phi> \<le> m"
liminf_subseq_lim: fixes u :: "nat \<Rightarrow> 'a" shows "\<exists>r. strict_mono r \<and> (u \<circ> r) \<longlonglongrightarrow> liminf u"
L: fixes x :: "'oid" assumes "x \<in> set (a # xs)" shows "x < oid"
compI_2: fixes CC :: "('b, 'c, 'd) category_scheme"   and f :: "'c"   and g :: "'c" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>"   and "g \<in> Ar\<^bsub>CC\<^esub>"   and "Cod\<^bsub>CC\<^esub> f = Dom\<^bsub>CC\<^esub> g" shows "Dom\<^bsub>CC\<^esub> (g \<bullet>\<^bsub>CC\<^esub> f) = Dom\<^bsub>CC\<^esub> f"
cts_wp_SetPC_const: fixes p :: "'a \<Rightarrow> real"   and a :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>x. x \<in> supp p \<Longrightarrow> bd_cts (wp (a x))"   and "\<And>x. x \<in> supp p \<Longrightarrow> healthy (wp (a x))"   and "unitary p"   and "sum p (supp p) \<le> 1"   and "finite (supp p)" shows "bd_cts (wp (SetPC a (\<lambda>_. p)))"
le_zero_eq: fixes n :: "'d multiset" shows "(n \<subseteq># {#}) = (n = {#})"
semInt_prFresh: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prFresh (semInt SEM) SEM"
dominoes_tile_row: fixes i :: "nat"   and n :: "nat" shows "{i} \<times> below (2 * n) \<in> tiling domino"
all_j_zero: shows "\<forall>j\<in>{i + 1..<n}. S $$ (0, j) = (0::'a)"
dfs_eq_rtrancl: fixes ks :: "int list"   and l :: "int"   and y :: "int"   and x :: "int"   and n :: "nat" assumes "dioph_is_node ks l y"   and "dioph_is_node ks l x" shows "dioph_memb y (dioph_ineq_dfs.dfs n ks (dioph_empt ks l) [x]) = ((x, y) \<in> (succsr (dioph_ineq_succs n ks))\<^sup>*)"
joinable_le_rtscl: fixes R :: "('b \<times> 'b) set" shows "joinable (R\<^sup>*) \<subseteq> (R \<union> R\<inverse>)\<^sup>*"
happen_implies_can_occur: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration" assumes "c \<turnstile> ev \<mapsto> c'" shows "can_occur ev c"
iMODb_I: fixes x :: "nat"   and m :: "nat"   and r :: "nat"   and c :: "nat" assumes "x mod m = r mod m"   and "r \<le> x"   and "x \<le> r + m * c" shows "x \<in> [ r, mod m, c ]"
is_serial_solution_for_problem_def: fixes \<Pi> :: "('c, 'd) strips_problem_scheme"   and \<pi> :: "'c strips_operator list" shows "STRIPS_Semantics.is_serial_solution_for_problem \<Pi> \<pi> \<equiv> \<Pi>\<^sub>G \<subseteq>\<^sub>m execute_serial_plan (\<Pi>\<^sub>I) \<pi> \<and> list_all (\<lambda>op. ListMem op (\<Pi>\<^sub>\<O>)) \<pi>"
Cauchy_altdef2: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'a" assumes "class.metric_space dist uniformity open" shows "uniform_space.Cauchy uniformity s = (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (s n) (s N) < e)"
safe_max_Var_le: fixes x :: "'c"   and z :: "nat" assumes "x \<in> Csafe"   and "DIM('c) \<le> z" shows "max_Var_floatariths ode_e \<le> z"
cmp_simps'_1: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.arr (\<Phi> (\<mu>, \<nu>))"
div_def: fixes x :: "'a mpoly"   and y :: "'a mpoly" shows "x div y = (THE q'. \<exists>a q r. pseudo_divmod_rel a x y q r \<and> q' = smult (inverse a) q)"
admissible_is_TNil: shows "ccpo.admissible local.tSup local.tllist_ord is_TNil"
cod_null: shows "B.VVV.cod B.VVV.null = B.VVV.null"
simps_12: fixes i :: "int"   and a :: "MIR.num" shows "simpfm (Dvd i a) = (if i = 0 then simpfm (Eq a) else if \<bar>i\<bar> = 1 \<and> check_int a then T else let a' = simpnum a in case a' of C v \<Rightarrow> if i dvd v then T else F | _ \<Rightarrow> let (x, y) = simpdvd i a' in Dvd x y)"
berlekamp_zassenhaus_factorization_def: fixes f :: "int poly" shows "berlekamp_zassenhaus_factorization f = (let p = suitable_prime_bz f; (uu_, fs) = finite_field_factorization_int p f; max_deg = degree_bound fs; bnd = 2 * \<bar>lead_coeff f\<bar> * factor_bound f max_deg; k = find_exponent p bnd; vs = hensel_lifting p k f fs in zassenhaus_reconstruction vs p k f)"
clock_repSim_simps_2: fixes t :: "'s Trace" shows "clock_repSim (clock_sim t) = tLast t"
reduction_subsets_R_def: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list"   and matr :: "rat mat" shows "reduction_subsets_R p qs signs subsets matr = take_indices subsets (rows_to_keep (reduce_mat_cols matr (solve_for_lhs_R p qs subsets matr)))"
fBall_alt_def: fixes S :: "'c fset"   and P :: "'c \<Rightarrow> bool" shows "fBall S P = (\<forall>x. x |\<in>| S \<longrightarrow> P x)"
entWeaken: fixes \<Psi> :: "'b"   and \<phi> :: "'c"   and \<Psi>' :: "'b" assumes "\<Psi> \<turnstile> \<phi>" shows "\<Psi> \<otimes> \<Psi>' \<turnstile> \<phi>"
assms_3: fixes x :: "'a" assumes "x \<in> A" shows "g x dvd d"
validOutputTransition: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "P \<longmapsto> a[b] \<prec> P'"   and "valid P" shows "valid P'"
LFO_pre_refine: fixes I :: "'b set \<Rightarrow> 'c \<Rightarrow> bool" assumes "(li, l) \<in> \<langle>A\<rangle>list_set_rel"   and "(ci, c) \<in> R \<rightarrow> bool_rel"   and "(fi, f) \<in> A \<rightarrow> R \<rightarrow> \<langle>R\<rangle>nres_rel"   and "(s0i, s0) \<in> R" shows "LIST_FOREACH' (RETURN li) ci fi s0i \<le> \<Down> R (FOREACH\<^sub>C\<^bsup>I\<^esup> l c f s0)"
dvd_times_right_cancel_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<noteq> (0::'a)" shows "(b * a dvd c * a) = (b dvd c)"
bind_sample2: fixes m :: "'m"   and p :: "int pmf"   and f :: "int \<Rightarrow> int \<Rightarrow> 'm" shows "bind m (\<lambda>y. sample p (f y)) = sample p (\<lambda>x. bind m (\<lambda>y. f y x))"
init_fin_bisim_simps_6: fixes final1 :: "'a \<Rightarrow> bool"   and final2 :: "'b \<Rightarrow> bool"   and bisim :: "'c \<Rightarrow> 'a \<times> 'd \<Rightarrow> 'b \<times> 'e \<Rightarrow> bool"   and t :: "'c"   and s1 :: "status"   and x1 :: "'a"   and m1 :: "'d"   and x2 :: "'b"   and m2 :: "'e" shows "FWbisimulation_base.init_fin_bisim final1 final2 bisim t ((s1, x1), m1) ((Finished, x2), m2) = (s1 = Finished \<and> bisim t (x1, m1) (x2, m2) \<and> final1 x1 \<and> final2 x2)"
comm_diagrams: fixes U :: "'a set"   and V :: "'a set"   and x :: "'b" assumes "is_open U"   and "is_open V"   and "V \<subseteq> U"   and "x \<in> \<FF> U" shows "(\<rho>' U V \<circ> fam_morphisms U) x = (fam_morphisms V \<circ> \<rho> U V) x"
axioms_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
reach_alloc: fixes s :: "Store"   and t :: "New"   and l :: "Location"   and x :: "Value" shows "s\<langle>t\<rangle>\<turnstile> l reachable_from x = s\<turnstile> l reachable_from x"
lex_uminus0: fixes a :: "real \<times> real" shows "Counterclockwise_2D_Arbitrary.lex (- a) 0 = Counterclockwise_2D_Arbitrary.lex 0 a"
order_iff_strict: fixes b :: "'b"   and a :: "'b" shows "(b \<le> a) = (b < a \<or> a = b)"