hcju/isabelleps huggingface dataset:
vector_derivative_at_within_ivl: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and a :: "real"   and b :: "real" assumes "(f has_vector_derivative f') (at x)"   and "a \<le> x"   and "x \<le> b"   and "a < b" shows "vector_derivative f (at x within {a..b}) = f'"
sum_SDG_return_edge: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and fs :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and n :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>fs"   and "n = CFG_node (sourcenode a)"   and "n' = CFG_node (targetnode a)" shows "n s-p\<rightarrow>\<^bsub>ret\<^esub> n'"
Val_semilattice_axioms: shows "Val_semilattice \<gamma>_const Const plus_const"
maintained_def: fixes R :: "('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph"   and G :: "('d, 'f) labeled_graph" shows "maintained R G \<equiv> \<forall>f. graph_homomorphism (fst R) G f \<longrightarrow> extensible R G f"
MOST_nat_add: fixes P :: "nat \<Rightarrow> bool"   and j :: "nat" shows "(\<forall>\<^sub>\<infinity>i. P i) = (\<forall>\<^sub>\<infinity>i. P (i + j))"
set_id_right: fixes f :: "'b set_arrow"   and U :: "'b set" assumes "f \<in> Ar\<^bsub>set_cat U\<^esub>" shows "f \<odot> set_id U (set_dom f) = f"
gfp_to_lfp: shows "gfp = \<partial> \<circ> lfp \<circ> \<partial>\<^sub>F"
set_4: fixes x21 :: "'d"   and x22 :: "('d, 'c) Term.term list" shows "vars_term (Fun x21 x22) = \<Union> (vars_term ` set x22)"
cring_pow_deg_bound: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P" shows "deg R (p [^]\<^bsub>P\<^esub> n) \<le> n * deg R p"
bad_def: shows "bad \<equiv> SOME bad. Spy \<in> bad \<and> Server \<notin> bad"
absolutely_integrable_onI: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "f integrable_on S"   and "(\<lambda>x. norm (f x)) integrable_on S" shows "f absolutely_integrable_on S"
sublist_refl: fixes l :: "'c list" shows "subseq l l"
P: shows "P \<subseteq> {p. irreducible p \<and> monic p}"
comp_cod_arr: fixes f :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VV.arr f"   and "VV.cod f = b" shows "VV.comp b f = f"
min_maxsimpchain_adj: fixes x :: "'a set"   and y :: "'a set" assumes "chamber x"   and "chamber y"   and "x \<sim> y"   and "x \<noteq> y" shows "min_gallery [x, y]"
Upper_def: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" shows "Upper L A = {u. \<forall>x. x \<in> A \<inter> carrier L \<longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> u} \<inter> carrier L"
cinner_commute': fixes y :: "'c"   and x :: "'c" shows "cnj \<langle>y, x\<rangle> = \<langle>x, y\<rangle>"
IN_DISJ_DEP_IMP_DEP_DIFF: fixes v :: "'d"   and vs' :: "'d set"   and v' :: "'d"   and vs :: "'d set"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" assumes "v \<in> vs'"   and "v' \<in> vs'"   and "disjnt vs vs'" shows "dep PROB v v' \<longrightarrow> dep (prob_proj PROB (prob_dom PROB - vs)) v v'"
diff_add: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<subseteq># b" shows "b - a + a = b"
translation_self: fixes e :: "'b allegorical_term" shows "(0, 1) \<in> :translation e:\<lbrakk>e\<rbrakk>"
I_minWeightBasisD_6: fixes T :: "'a set"   and E :: "'a set" assumes "I_minWeightBasis (T, E)" shows "no_smallest_element_skipped E T"
ys'_1: shows "ys = ys' @ x # ys''"
convergentI: fixes F :: "'a set set"   and x :: "'a" assumes "F \<longlongrightarrow> x"   and "x \<in> carrier" shows "F convergent"
converse_converse: fixes r :: "('b \<times> 'c) set" shows "(r\<inverse>)\<inverse> = r"
simps_3: fixes f1 :: "'c"   and f2 :: "'c" shows "(case assign of assign \<Rightarrow> f1 | check \<Rightarrow> f2) = f1"
simps_2: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s :: "'a list \<times> 'b"   and r :: "'a"   and rs :: "'a list" shows "Partial_Cost_Model.T_on' A s (r # rs) = t\<^sub>p (fst s) r (fst (snd A s r)) + Partial_Cost_Model.T_on' A (Partial_Cost_Model.Step A s r) rs"
plus_oracle_split: fixes P :: "(('b + 'd) \<times> 's) spmf \<Rightarrow> bool"   and lr :: "'a + 'c" shows "P (local.plus_oracle lr) = ((\<forall>x. lr = Inl x \<longrightarrow> P (map_spmf (apfst Inl) (left s x))) \<and> (\<forall>y. lr = Inr y \<longrightarrow> P (map_spmf (apfst Inr) (right s y))))"
minimal_element_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "minimal_element P A \<equiv> po_on P A \<and> wfp_on P A"
hyps: shows "fst (echelon_form_of_upt_k_det bezout (1::'a, A) k) * det A = det (snd (echelon_form_of_upt_k_det bezout (1::'a, A) k))"
rem_condless_act_cons: fixes s :: "('c, 'd) fmap"   and h' :: "('c, 'd) fmap \<times> ('c, 'd) fmap"   and pfx :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "rem_condless_act s (h' # pfx) as = h' # rem_condless_act (state_succ s h') pfx as"
shows_pr_def: fixes p :: "nat" shows "shows_pr p = apply_if (0 < p) (shows CHR '')'')"
splice_take_nths: fixes xs :: "'b list" shows "splice (take_nths 0 (Suc 0) xs) (take_nths (Suc 0) (Suc 0) xs) = xs"
lprefix_trans: fixes xs :: "'b llist"   and ys :: "'b llist"   and zs :: "'b llist" assumes "lprefix xs ys"   and "lprefix ys zs" shows "lprefix xs zs"
path_state_containment: fixes M :: "('d, 'e, 'f, 'g) FSM_scheme"   and p :: "(('d \<times> 'e) \<times> 'f) list"   and q :: "'f" assumes "path M p q"   and "q \<in> nodes M" shows "set (map snd p) \<subseteq> nodes M"
\<epsilon>o_def: fixes x :: "'a" shows "Adj'.\<epsilon>o x = \<psi>' x (G' x)"
intros_1: fixes c :: "('c, 'd) conf" shows "se c Skip c"
preserves_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "trg\<^sub>D (F \<mu>) = map\<^sub>0 (trg\<^sub>C \<mu>)"
mono_inf: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b"   and B :: "'b" assumes "class.semilattice_inf inf less_eq less"   and "order.mono less_eq f" shows "f (inf A B) \<le> inf (f A) (f B)"
VPow_top: fixes A :: "V" shows "A \<in>\<^sub>\<circ> VPow A"
minim_Under_under: fixes A :: "'a set" assumes "A \<noteq> {}"   and "A \<subseteq> Field r" shows "local.Under A = local.under (local.minim A)"
intros: fixes W :: "'d list"   and \<Sigma>i :: "'d set" assumes "distinct W"   and "set W = \<Sigma>i" shows "(\<Sigma>i, W) \<in> dfs_initial \<Sigma>i"
coboundedI2: fixes c :: "'b"   and b :: "'b"   and a :: "'b" assumes "c \<le> b" shows "c \<le> max a b"
spoly: shows "spoly g1 g2 = (\<Sum>g\<in>G. q' g \<odot> g)"
distinct_mset_union: fixes A :: "'b multiset"   and B :: "'b multiset" assumes "distinct_mset (A + B)" shows "distinct_mset A"
is_ortho_set_def: fixes S :: "'a set" shows "is_ortho_set S = ((\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> is_orthogonal x y) \<and> (0::'a) \<notin> S)"
smullyan_double_fixed_point: fixes g :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "g \<in> \<R>\<^sup>2"   and "h \<in> \<R>\<^sup>2" obtains m :: "nat" and n :: "nat" where "\<phi> m = \<phi> (the (g m n))" and "\<phi> n = \<phi> (the (h m n))"
state_succ_fixpoint_if: fixes a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) set"   and s :: "('d, 'e) fmap" assumes "a \<in> PROB"   and "s \<in> valid_states PROB"   and "fst a \<subseteq>\<^sub>f s"   and "agree (snd a) s" shows "state_succ s a = s"
DomFunctor: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and f :: "'f" assumes "Functor F"   and "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "dom\<^bsub>CatCod F\<^esub> (F ## f) = F @@ dom\<^bsub>CatDom F\<^esub> f"
le_add1_hmset: fixes n :: "hmultiset"   and m :: "hmultiset" shows "n \<le> n + m"
ElemClassEquiv: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and C :: "'a set" assumes "equiv A r"   and "C \<in> A // r" shows "\<forall>x\<in>C. \<forall>y\<in>C. (x, y) \<in> r"
arden_conv1: fixes x :: "'a" assumes "\<forall>y z. z + x \<cdot> y = y \<longrightarrow> x\<^sup>\<star> \<cdot> z = y" shows "\<not> ewp x"
pred_pmf_of_set: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "finite A"   and "A \<noteq> {}" shows "pred_pmf P (pmf_of_set A) = Ball A P"
swap_eq_on: fixes arr1 :: "'b \<Rightarrow> 'c"   and arr2 :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "arr1 = arr2 on S"   and "x \<notin> S"   and "y \<notin> S" shows "arr1 = swap arr2 x y on S"
omega_sum_unfold_ind: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<star> \<cdot> y)\<^sup>\<star> \<cdot> x\<^sup>\<omega> \<le> (x + y)\<^sup>\<omega>"
x': shows "MPoly_Type.monom (monomial (Suc 0) x') (1::'a) = Var x'"
sqrt_int_ceiling_def: fixes x :: "int" shows "sqrt_int_ceiling x = root_int_ceiling 2 x"
assms_1: shows "injective (w \<sqinter> - d \<squnion> e)"
Eq\<^sub>2S: fixes v :: "i"   and F :: "\<Pi>\<^sub>2"   and G :: "\<Pi>\<^sub>2" shows "[F \<^bold>=\<^sub>2 G in v] = (F = G)"
CSP_is_CSP1: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "is_CSP_process P" shows "P is CSP1 healthy"
dvd_minus_iff: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "dvd.dvd times x (uminus y) = dvd.dvd times x y"
is_faithful: fixes f :: "'d"   and f' :: "'d" assumes "D.par f f'"   and "GF.map f = GF.map f'" shows "f = f'"
contour_integrable_sum: fixes s :: "'a set"   and f :: "'a \<Rightarrow> complex \<Rightarrow> complex"   and p :: "real \<Rightarrow> complex" assumes "finite s"   and "\<And>a. a \<in> s \<Longrightarrow> f a contour_integrable_on p" shows "(\<lambda>x. \<Sum>a\<in>s. f a x) contour_integrable_on p"
leg_simps_3: shows "C.arr \<mu>.cod.leg1"
s_def: fixes n :: "nat"   and i :: "nat" shows "s n i = (SOME j. i < j \<and> f j \<in> A (Suc n))"
bin_rest_shiftr: fixes x :: "int"   and n :: "nat" shows "(x >> n) div 2 = x >> Suc n"
oalist_abstract_axioms: shows "oalist_abstract rep_key_order list_of_oalist oalist_of_list"
list2policy_def: fixes l :: "('b \<Rightarrow> 'c decision option) list" shows "list2policy l = foldr (\<lambda>x y. x \<Oplus> y) l \<emptyset>"
PChain: shows "P \<Longrightarrow>\<^sub>\<tau> P'''"
left_respects_isomorphic: fixes f :: "'a"   and g :: "'a" assumes "f \<cong> g" shows "left f = left g"
sub_list_comp: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and ps :: "('b, 'e) fm list" shows "sub_list f (sub_list g ps) = sub_list (f \<circ> g) ps"
gate_is_inv: fixes n :: "nat"   and A :: "complex mat" assumes "gate n A" shows "invertible_mat A"
Call: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and a' :: "val"   and s1 :: "state"   and args :: "expr list"   and vs :: "val list"   and s2 :: "state"   and D :: "qtname"   and mode :: "inv_mode"   and statT :: "ref_ty"   and mn :: "mname"   and pTs :: "ty list"   and s3 :: "state"   and s3' :: "state"   and accC :: "qtname"   and v :: "val"   and s4 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<rightarrow> s2"   and "D = invocation_declclass G mode (snd s2) a' statT \<lparr>name = mn, parTs = pTs\<rparr>"   and "s3 = init_lvars G D \<lparr>name = mn, parTs = pTs\<rparr> mode a' vs s2"   and "s3' = check_method_access G accC statT mode \<lparr>name = mn, parTs = pTs\<rparr> a' s3"   and "G\<turnstile>s3' \<midarrow>Methd D \<lparr>name = mn, parTs = pTs\<rparr>-\<succ>v\<rightarrow> s4" shows "G\<turnstile>Norm s0 \<midarrow>{accC,statT,mode}e\<cdot>mn( {pTs}args)-\<succ>v\<rightarrow> set_lvars (locals (snd s2)) s4"
mset_le_subtract_left: fixes A :: "'a multiset"   and B :: "'a multiset"   and X :: "'a multiset" assumes "A + B \<subseteq># X" shows "B \<subseteq># X - A \<and> A \<subseteq># X"
bdd_belowI2: fixes A :: "'b set"   and m :: "'c multiset"   and f :: "'b \<Rightarrow> 'c multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> m \<subseteq># f x" shows "subset_mset.bdd_below (f ` A)"
isin_sorted_split_right: fixes ts :: "('a btree \<times> 'a) list"   and x :: "'a"   and ls :: "('a btree \<times> 'a) list"   and sub :: "'a btree"   and sep :: "'a"   and rs :: "('a btree \<times> 'a) list"   and t :: "'a btree" assumes "split ts x = (ls, (sub, sep) # rs)"   and "sorted_less (inorder (Node ts t))"   and "sep \<noteq> x" shows "(x \<in> set (inorder_list ((sub, sep) # rs) @ inorder t)) = (x \<in> set (inorder sub))"
lfinite_LConsI: fixes xs :: "'a llist"   and x :: "'a" assumes "lfinite xs" shows "lfinite (LCons x xs)"
OrdP_SUCC_I: fixes H :: "fm set"   and k :: "tm" assumes "H \<turnstile> OrdP k" shows "H \<turnstile> OrdP (SUCC k)"
init'_atoms_imply_bounds_index: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and t :: "(nat \<times> linear_poly) list" assumes "Init' init"   and "\<triangle> t" shows "{} \<Turnstile>\<^sub>i \<B>\<I> (init t)"
finite_favorites: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a" assumes "pref_profile_wf agents alts R" shows "finite (favorites R i)"
rat_number_collapse_3: fixes w :: "num" shows "Fract (numeral w) 1 = numeral w"
oc: shows "open_cover_of_open_subset S' is_open' U I V"
isSp_Append_ddep: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and V :: "'var"   and n' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n''"   and "n'' s-V\<rightarrow>\<^sub>d\<^sub>d n'"   and "n'' \<noteq> n'" shows "n is-ns @ [n'']\<rightarrow>\<^sub>d* n'"
intro: fixes \<eta> :: "'b \<Rightarrow> 'c"   and G :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and G' :: "'c set"   and composition' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit' :: "'c" assumes "monoid_homomorphism \<eta> G composition unit G' composition' unit'"   and "Group_Theory.group G composition unit"   and "Group_Theory.group G' composition' unit'" shows "group_homomorphism \<eta> G composition unit G' composition' unit'"
deg: shows "degree (poly_real_alg_1 rai) = 0"
eta_norm_reduces_depth: fixes t :: "Core.term" assumes "eta_norm_dom t" shows "depth (eta_norm t) \<le> depth t"
ennreal_minus_eq_top: fixes a :: "ennreal"   and b :: "ennreal" shows "(a - b = \<top>) = (a = \<top>)"
Loss_L3: fixes t :: "nat"   and a :: "nat \<Rightarrow> aType list"   and lose :: "nat \<Rightarrow> bool list"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and y :: "nat \<Rightarrow> aType list"   and i :: "nat \<Rightarrow> ECall_Info list"   and t2 :: "nat" assumes "\<forall>t2<t. a t2 = []"   and "Loss lose a i2 y i"   and "t2 < t"   and "ts lose" shows "y t2 = []"
simps_8: fixes \<sigma> :: "(char list \<times> 'b list) Trace.trace"   and v :: "'b list"   and i :: "nat"   and \<phi> :: "'b formula"   and I :: "\<I>"   and \<psi> :: "'b formula" shows "sat \<sigma> v i (Since \<phi> I \<psi>) = (\<exists>j\<le>i. mem (\<tau> \<sigma> i - \<tau> \<sigma> j) I \<and> sat \<sigma> v j \<psi> \<and> (\<forall>k\<in>{j<..i}. sat \<sigma> v k \<phi>))"
f_bnd_mono: fixes outside :: "bool" shows "f_bnd outside \<le> f_bnd False"
add_strict_left_mono: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subset># b" shows "c + a \<subset># c + b"
is_shortest_pathD2: fixes G :: "graph"   and m :: "nat"   and n :: "nat"   and p :: "nat list"   and p' :: "nat list" assumes "is_shortest_path G m n p"   and "p' \<in> path_set G m n" shows "path_weight G p \<le> path_weight G p'"
det30_uminus2: fixes a :: "real \<times> real"   and b :: "real \<times> real" shows "coll 0 (- a) b = coll 0 a b"
atom_eq_iff: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and atom :: "'b \<Rightarrow> atom"   and a :: "'b"   and b :: "'b" assumes "class.at_base permute atom" shows "(atom a = atom b) = (a = b)"
summable_zeta_real: fixes x :: "real" assumes "1 < x" shows "summable (\<lambda>n. real (Suc n) powr - x)"
Ballot_disj: shows "\<forall>p q. p \<noteq> q \<longrightarrow> Ballot p \<inter> Ballot q = {}"
assms: shows "(\<Prod>x\<leftarrow>xs. [:- x, 1:]) = monom 1 n - 1"
divide_left_mono: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<le> a"   and "(0::'a) \<le> c"   and "(0::'a) < a * b" shows "c / a \<le> c / b"
ceq_fun_def: shows "CEQ('b \<Rightarrow> 'c) \<equiv> None"
strong_fermat_liar_bounded: shows "card {a \<in> {1..<n}. strong_fermat_liar a n} < (n - 1) div 2"
step_j: shows "S t j \<turnstile> t ! j \<mapsto> S t (j + 1)"
acc_modi_accmodi_def: fixes a :: "acc_modi" shows "accmodi a = a"
I: shows "SkewBinomialHeapStruc.invar (map bsmapt q)"
fmember_def: shows "(|\<in>|) \<equiv> map_fun id (map_fun fset id) (\<in>)"
extremum: fixes a :: "'d multiset" shows "{#} \<subseteq># a"
hyps_2: fixes s :: "'a"   and c :: "('a, 'b, 'c) com" assumes "\<Gamma>\<turnstile>c2' \<down> Normal s"   and "c \<subseteq>\<^sub>g c2'"   and "\<Gamma>\<turnstile> \<langle>c2',Normal s\<rangle> \<Rightarrow>\<notin>range Fault" shows "\<Gamma>\<turnstile>c \<down> Normal s"
complete_latticeI: assumes "\<And>A. A \<subseteq> carrier L \<Longrightarrow> \<exists>s. is_lub L s A"   and "\<And>A. A \<subseteq> carrier L \<Longrightarrow> \<exists>i. is_glb L i A" shows "complete_lattice L"
sim_1: shows "trace_callee_eq callee1 callee2 A p q"
gfp_def: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and f :: "'d \<Rightarrow> 'd" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "complete_lattice.gfp Sup less_eq f = Sup {u. less_eq u (f u)}"
Cons_5: assumes "f \<noteq> 0"   and "\<not> lt f adds\<^sub>t u" shows "thesis"
bound: fixes c_o :: "'c_o"   and c_a :: "'c_a"   and s :: "'s" assumes "(c_o, c_a, s) \<in> set_spmf init" shows "interaction_bounded_by' (Not \<circ> isl) (\<A> c_a) q"
reflects_right_adjoint: fixes g :: "'a" assumes "ide g"   and "S.is_right_adjoint (S.UP g)" shows "is_right_adjoint g"
flrestriction_vsingleton_in: fixes a :: "V"   and A :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "set {[a, b]\<^sub>\<circ>} \<restriction>\<^sup>l\<^sub>\<bullet> A = set {[a, b]\<^sub>\<circ>}"
Map_ide_2: fixes a :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "ide a" shows "Map a = Cod a"
mono: fixes c :: "'h"   and a :: "'h"   and d :: "'h"   and b :: "'h" assumes "c \<le> a"   and "d \<le> b" shows "max c d \<le> max a b"
C_step: fixes n :: "nat"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat" assumes "0 < n" shows "C M2 M1 \<Omega> V m (Suc n) \<subseteq> append_set (C M2 M1 \<Omega> V m n) (inputs M2) - C M2 M1 \<Omega> V m n"
sub: fixes a :: "'a" assumes "a \<in> R\<setminus>\<pp>" shows "a \<in> R"
walk_decomp_1: fixes xs :: "'a list"   and ys :: "'a list" assumes "Q.walk (xs @ ys)" shows "Q.walk xs"
min_key_val_raw_minimal: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and z :: "'a \<times> 'c"   and ko :: "'o" assumes "oalist_inv xs"   and "z \<in> set (fst xs)" shows "le ko (fst (min_key_val_raw ko xs)) (fst z)"
timpls_transformable_to'_inv_3: fixes TI :: "('e set \<times> 'e set) list"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and g :: "('f, 'g, 'e) prot_fun"   and S :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list" assumes "timpls_transformable_to' TI (Fun f T) (Fun g S)"   and "f \<noteq> g" shows "is_Abs f \<and> is_Abs g \<and> in_trancl TI (the_Abs f) (the_Abs g)"
translate_plus: fixes a :: "'a"   and x :: "'a"   and b :: "'a"   and c :: "'a" shows "ccw (a + x) (b + x) (c + x) = ccw a b c"
FreeUltrafilterNat_HInfinite: fixes X :: "nat \<Rightarrow> 'b" assumes "\<And>u. \<forall>\<^sub>F n in \<U>. u < norm (X n)" shows "star_n X \<in> HInfinite"
axioms_1: fixes G :: "('c, 'd) ring_scheme" assumes "abelian_group G" shows "abelian_monoid G"
ssum_map_sinr: fixes x :: "'e"   and f :: "'h \<rightarrow> 'f"   and g :: "'e \<rightarrow> 'g" assumes "x \<noteq> \<bottom>" shows "ssum_map\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>x) = sinr\<cdot>(g\<cdot>x)"
assms: shows "(as, x) \<approx>res (=) supp p' (as', x')"
scaleR_minus_left: fixes a :: "real"   and x :: "'a" shows "- a *\<^sub>R x = - (a *\<^sub>R x)"
valid_refs_invD_5: fixes w :: "('a, 'b, 'c) mem_store_action"   and s :: "'d process_name \<Rightarrow> ('a, 'd, 'b, 'c) Model.local_state"   and m :: "'d"   and x :: "'c"   and y :: "'c" assumes "w \<in> set (sys_mem_store_buffers (mutator m) s)"   and "x \<in> store_refs w"   and "(x reaches y) s"   and "valid_refs_inv s" shows "obj_at (\<lambda>s. True) y s"
relation_of_Det: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "action.relation_of (P \<box> Q) = R (\<not> action.relation_of P\<^sup>f\<^sub>f \<and> \<not> action.relation_of Q\<^sup>f\<^sub>f \<turnstile> action.relation_of P\<^sup>t\<^sub>f \<and> action.relation_of Q\<^sup>t\<^sub>f \<triangleleft> \<lambda>(A, A'). tr A = tr A' \<and> wait A' \<triangleright>  action.relation_of P\<^sup>t\<^sub>f \<or> action.relation_of Q\<^sup>t\<^sub>f)"
relabel_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" shows "relabel f l u \<equiv> assert (Height_Bounded_Labeling c s t f l) \<bind> (\<lambda>_. assert (relabel_precond f l u) \<bind> (\<lambda>_. assert (u \<in> V - {s, t}) \<bind> (\<lambda>_. return (relabel_effect f l u))))"
eval_trm_rename: fixes k' :: "name"   and t :: "trm"   and e :: "(name, nat) finfun"   and k :: "name"   and x :: "nat" assumes "atom k' \<sharp> t" shows "eval_trm (finfun_update e k x) t = eval_trm (finfun_update e k' x) ((k' \<leftrightarrow> k) \<bullet> t)"
\<Delta>_def: fixes f :: "'b \<Rightarrow> 'a"   and k :: "int" shows "\<Delta> f k = f (of_int (k + 1)) - f (of_int k)"
length_dec_interp: fixes n :: "nat"   and FO :: "nat set"   and x :: "('a \<times> bool list) list" shows "length (dec_interp n FO x) = n"
is_parser_exactly: fixes xs :: "char list" shows "is_parser (exactly xs)"
inject_2: fixes x2 :: "(char list \<times> pat) list"   and y2 :: "(char list \<times> pat) list" shows "(PRcd x2 = PRcd y2) = (x2 = y2)"
card_of_Times_same_infinite: fixes A :: "'b set" assumes "infinite A" shows "|A \<times> A| =o |A|"
lemma_A1: shows "\<exists>M\<in>calM // RelM. \<not> p ^ Suc (multiplicity p m) dvd card M"
pinf_8: fixes t :: "'a" shows "\<exists>z. \<forall>x>z. (t \<le> x) = True"
alphaAbs_refl: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodAbs A" shows "A $= A"
weak_ring_morphism_def: fixes f :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and R :: "('d, 'f) ring_scheme" shows "weak_ring_morphism f I R \<equiv> ideal I R \<and> weak_ring_morphism_axioms f I R"
assms: shows "valid_path (\<lambda>x. (f x, g x))"
orderE: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<preceq> b" obtains "a = linordered_monoid.min a b"
squarefree_prod_coprime: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A; a \<noteq> b\<rbrakk> \<Longrightarrow> algebraic_semidom_class.coprime (f a) (f b)"   and "\<And>a. a \<in> A \<Longrightarrow> squarefree (f a)" shows "squarefree (prod f A)"
A4': fixes q :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" shows "\<exists>x. B q a x \<and> a x \<equiv> b c"