hcju/isabelleps huggingface dataset:
Diff: fixes a :: "('i \<Rightarrow> 'a) set"   and b :: "('i \<Rightarrow> 'a) set" assumes "a \<in> generator"   and "b \<in> generator" shows "a - b \<in> generator"
ldropWhile_eq_LNil_iff: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "(ldropWhile P xs = LNil) = (\<forall>x\<in>lset xs. P x)"
assms_8: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and S :: "stack_elem list" assumes "map_of \<Gamma> x = Some e"   and "isVal e"   and "I (\<Gamma>, Var x, S)" shows "P (\<Gamma>, Var x, S) ((x, e) # delete x \<Gamma>, e, S)"
step: shows "revision_step r s\<^sub>1 s\<^sub>2"
extend_state_oracle_transfer': fixes S :: "'h \<Rightarrow> 'k \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'l \<Rightarrow> bool"   and R :: "'j \<Rightarrow> 'm \<Rightarrow> bool"   and s :: "'n" shows "rel_fun (rel_fun S (rel_fun C (rel_spmf (rel_prod R S)))) (rel_fun (cr_prod2 s S) (rel_fun C (rel_spmf (rel_prod R (cr_prod2 s S))))) (\<lambda>oracle. oracle) extend_state_oracle"
valid_SDG_node_parent_Exit: fixes n :: "'node SDG_node" assumes "valid_SDG_node n"   and "parent_node n = (_Exit_)" shows "n = CFG_node (_Exit_)"
connected_component_sym_eq: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" shows "connected_component S x y = connected_component S y x"
map_iterator_genord_rule_insert_P: fixes it :: "('e \<Rightarrow> bool) \<Rightarrow> ('f \<times> 'g \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and m :: "'f \<Rightarrow> 'g option"   and R :: "'f \<times> 'g \<Rightarrow> 'f \<times> 'g \<Rightarrow> bool"   and I :: "'f set \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma>0 :: "'e"   and c :: "'e \<Rightarrow> bool"   and f :: "'f \<times> 'g \<Rightarrow> 'e \<Rightarrow> 'e"   and P :: "'e \<Rightarrow> bool" assumes "map_iterator_genord it m R"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> dom m - it; m k = Some v; it \<subseteq> dom m; I it \<sigma>; \<forall>k' v'. k' \<in> dom m - it - {k} \<and> m k' = Some v' \<longrightarrow> R (k, v) (k', v'); \<forall>k' v'. k' \<in> it \<and> m k' = Some v' \<longrightarrow> R (k', v') (k, v)\<rbrakk> \<Longrightarrow> I (insert k it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (dom m) \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> dom m; \<not> c \<sigma>; I it \<sigma>; \<forall>k v k' v'. k \<in> it \<and> m k = Some v \<and> k' \<notin> it \<and> m k' = Some v' \<longrightarrow> R (k, v) (k', v')\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
dual_galois: fixes B :: "('d, 'f) gorder_scheme"   and A :: "('e, 'g) gorder_scheme"   and f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'd" shows "galois_connection \<lparr>orderA = inv_gorder B, orderB = inv_gorder A, lower = f, upper = g\<rparr> = galois_connection \<lparr>orderA = A, orderB = B, lower = g, upper = f\<rparr>"
assms_3: assumes "x \<in> set (local.filter_min_append xs ys)" shows "thesis"
abs_wpt_def: fixes x :: "'b" shows "abs_wpt x = Abs_Assertion (wpt x)"
minGraphProps7': fixes g :: "graph"   and f :: "face"   and v :: "nat" assumes "minGraphProps g"   and "f \<in> \<F> g"   and "v \<in> \<V> f" shows "f \<in> set (faceListAt g ! v)"
tm_round_floats_of_normed_range: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t :: "taylor_model"   and a :: "float interval list"   and I :: "float interval list"   and prec :: "nat" assumes "x \<in>\<^sub>i range_tm e t"   and "develops_at_within e a I"   and "num_params (tm_poly t) \<le> length I" shows "x \<in>\<^sub>i range_tm e (tm_round_floats_of_normed prec I a t)"
distinct_Ex1: fixes xs :: "'b list"   and x :: "'b" assumes "distinct xs"   and "x \<in> set xs" shows "\<exists>!i. i < length xs \<and> xs ! i = x"
vimage_INT: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "f -` \<Inter> (B ` A) = (\<Inter>x\<in>A. f -` B x)"
rem_rem_commute: fixes i1 :: "'d set"   and e1 :: "'c"   and i2 :: "'d set"   and e2 :: "'c" shows "\<langle>Rem i1 e1\<rangle> \<rhd> \<langle>Rem i2 e2\<rangle> = \<langle>Rem i2 e2\<rangle> \<rhd> \<langle>Rem i1 e1\<rangle>"
strict_suffix_reflclp_conv: shows "strict_suffix\<^sup>=\<^sup>= = suffix"
not_UINF: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'a upred" shows "(\<not> (\<Sqinter> i \<in> A \<bullet> P i)) = (\<Squnion> i \<in> A \<bullet> \<not> P i)"
edge_to_Nil: fixes c :: "'form" shows "edge_to c [] = ((c, []), plain_ant c)"
xor_compl_left: fixes x :: "'a"   and y :: "'a" shows "\<sim> x \<oplus> y = \<sim> (x \<oplus> y)"
Red_I_def: fixes N :: "('f \<times> 'l) set" shows "Red_I N = (\<Inter>q\<in>Q. Red_I_\<G>_L_q q N)"
is_singletonI: fixes x :: "'g" shows "is_singleton {x}"
lt_zero: shows "punit.lt 0 = punit.min_term"
cop_nts__os: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Coplanar A B C D"   and "\<not> Col C A B"   and "\<not> Col D A B"   and "\<not> A B TS C D" shows "A B OS C D"
trancl_impl_def: fixes gen_succ :: "('b \<times> 'b) list \<Rightarrow> 'b list \<Rightarrow> 'b list"   and un :: "'b list \<Rightarrow> 'c \<Rightarrow> 'c"   and memb :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and emp :: "'c"   and rel :: "('b \<times> 'b) list" shows "trancl_impl gen_succ un memb emp rel = (let succ = gen_succ rel; n = length rel in (\<lambda>as. relpow_impl succ un memb (succ as) emp n))"
connected_disjoint_Union_open_pick: fixes B :: "'a set set"   and A :: "'a set set"   and S :: "'a set" assumes "disjoint B"   and "\<And>S. S \<in> A \<Longrightarrow> connected S \<and> S \<noteq> {}"   and "\<And>S. S \<in> B \<Longrightarrow> open S"   and "\<Union> A \<subseteq> \<Union> B"   and "S \<in> A" obtains T :: "'a set" where "T \<in> B" and "S \<subseteq> T" and "S \<inter> \<Union> (B - {T}) = {}"
star_n_export2: fixes x :: "'a"   and y :: "'a" shows "(!x \<cdot> y)\<^sup>\<star> \<cdot> !x \<le> y\<^sup>\<star> \<cdot> !x"
map_filter_def: fixes P :: "'d \<Rightarrow> bool"   and m :: "'d \<Rightarrow> 'e option" shows "map_filter P m = (\<lambda>x. if P x then m x else None)"
inorder_delete: fixes t :: "'b tree234"   and x :: "'b" assumes "bal t"   and "Sorted_Less.sorted (inorder t)" shows "inorder (delete x t) = del_list x (inorder t)"
rely_quotient: fixes i :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice" assumes "chaos \<sqsubseteq> (i::'a::refinement_lattice)" shows "(c::'a::refinement_lattice) \<sqsubseteq> c // (i::'a::refinement_lattice) \<parallel> i"
sort_lt_var_imp_varSort_lt_var: fixes Delta :: "('a, 'b, 'c, 'd, 'e) signature" assumes "varSortAsSort_inj Delta"   and "sort_lt_var undefined undefined" shows "varSort_lt_var undefined undefined"
fresh_constant: fixes cs :: "'a list"   and p :: "('a, 'b) form" assumes "infinite UNIV" shows "\<exists>c. c \<notin> set cs \<and> new c p"
abs_eq: fixes x :: "'a" shows "prod_right x = CBlinfun ((**) x)"
loop: fixes tf :: "tf"   and tn :: "t list"   and tm :: "t list"   and \<C> :: "t_context"   and es :: "b_e list" assumes "tf = tn _> tm"   and "\<C>\<lparr>label := [tn] @ label \<C>\<rparr> \<turnstile> es : tn _> tm" shows "\<C> \<turnstile> [Loop tf es] : tn _> tm"
Pd_shuffle_to_top_def: fixes ds' :: "'b set"   and Y :: "'a set" shows "Pd_shuffle_to_top ds' Y = override_on Pd (shuffle_to_top Y) ds'"
prj_simps_6: shows "C.cod \<pi>\<rho>.prj\<^sub>0 = \<rho>.apex"
map_nth_append1: fixes xs :: "'e list"   and d :: "nat"   and ys :: "'e list" assumes "length xs = d" shows "map ((!) (xs @ ys)) [0..<d] = xs"
push_forward_id: fixes f :: "'a \<Rightarrow> 'a"   and X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a" assumes "\<And>x. x \<in> sub.carrier \<Longrightarrow> f x = x"   and "X \<in> sub.tangent_space p"   and "p \<in> sub.carrier" shows "diff.push_forward k (charts_submanifold S) (charts_submanifold S) f X = X"
simps: fixes a :: "'var"   and S :: "'node SDG_node set"   and n :: "'node SDG_node" shows "(a \<in> rv S n) = (\<exists>as n' V. a = V \<and> parent_node n -as\<rightarrow>\<^sub>\<iota>* parent_node n' \<and> n' \<in> HRB_slice S \<and> V \<in> Use\<^bsub>SDG\<^esub> n' \<and> (\<forall>n''. valid_SDG_node n'' \<and> parent_node n'' \<in> set (sourcenodes as) \<longrightarrow> V \<notin> Def\<^bsub>SDG\<^esub> n''))"
rev_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "rev (map f xs) = map f (rev xs)"
update_convs_2: fixes bar2' :: "'b \<Rightarrow> 'b"   and bar1 :: "'a"   and bar2 :: "'b"   and bar3 :: "'c"   and bar21 :: "'b \<times> 'a"   and bar32 :: "'c \<times> 'b"   and bar31 :: "'c \<times> 'a"   and more :: "'d" shows "bar2_update bar2' \<lparr>bar1 = bar1, bar2 = bar2, bar3 = bar3, bar21 = bar21, bar32 = bar32, bar31 = bar31, \<dots> = more\<rparr> = \<lparr>bar1 = bar1, bar2 = bar2' bar2, bar3 = bar3, bar21 = bar21, bar32 = bar32, bar31 = bar31, \<dots> = more\<rparr>"
monom_mult_dist_left: fixes c :: "'c"   and d :: "'c"   and t :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "MPoly_Type_Class.punit.monom_mult (c + d) t p = MPoly_Type_Class.punit.monom_mult c t p + MPoly_Type_Class.punit.monom_mult d t p"
inj_on_image_Int: fixes f :: "'a \<Rightarrow> 'b"   and C :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "inj_on f C"   and "A \<subseteq> C"   and "B \<subseteq> C" shows "f ` (A \<inter> B) = f ` A \<inter> f ` B"
isMaxFlow_alt: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity" shows "isMaxFlow f = (NFlow c s t f \<and> (\<forall>f'. NFlow c s t f' \<longrightarrow> Flow.val c s f' \<le> Flow.val c s f))"
ESem_fresh_cong: fixes e :: "'a"   and \<rho> :: "'b \<Rightarrow> 'c"   and \<rho>' :: "'b \<Rightarrow> 'c" assumes "\<rho> f|` fv e = \<rho>' f|` fv e" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> e \<rbrakk>\<^bsub>\<rho>'\<^esub>"
lins2D_3: fixes i :: "nat"   and j :: "nat"   and v :: "node"   and e :: "node fset \<times> command_gadget \<times> node fset" assumes "i < length (Inr e # \<pi>)"   and "j < length (Inr e # \<pi>)"   and "(Inr e # \<pi>) ! i = Inl v"   and "(Inr e # \<pi>) ! j = Inr e"   and "v |\<in>| fst3 e" shows "i < j"
add_le_cancel_right: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a + c \<le> b + c) = (a \<le> b)"
filter_eq_Cons_iff: fixes P :: "'g \<Rightarrow> bool"   and ys :: "'g list"   and x :: "'g"   and xs :: "'g list" shows "(filter P ys = x # xs) = (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
sem: shows "(\<nu>, \<mu>) \<in> prog_sem I \<alpha>"
right_idem: fixes a :: "'a list"   and b :: "'a list" shows "gcp (gcp a b) b = gcp a b"
that: fixes xs' :: "'node list"   and ys' :: "'node list"   and z' :: "'node" assumes "old.pathsConverge g x xs' y ys' z'"   and "prefix xs' xs"   and "prefix ys' ys" shows "thesis"
composition_conseq2ts: fixes \<sigma>\<^sub>2 :: "char list \<Rightarrow> fterm"   and \<sigma>\<^sub>1 :: "char list \<Rightarrow> fterm"   and ts :: "fterm list" shows "ts \<cdot>\<^sub>t\<^sub>s \<sigma>\<^sub>1 \<cdot>\<^sub>t\<^sub>s \<sigma>\<^sub>2 = ts \<cdot>\<^sub>t\<^sub>s (\<sigma>\<^sub>1 \<cdot> \<sigma>\<^sub>2)"
dom_mult_closed: fixes domain_op :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.domain_semigroup domain_op times" shows "domain_op (times (domain_op x) (domain_op y)) = times (domain_op x) (domain_op y)"
ovalidNF_is_triple: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b option"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "ovalidNF P f Q = triple_judgement P f (ovalidNF_property Q (\<lambda>s f. f s))"
H: shows "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x"
subterms_subst': fixes t :: "('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "\<forall>x\<in>fv t. (\<exists>f. \<theta> x = Fun f []) \<or> (\<exists>y. \<theta> x = Var y)" shows "subterms (t \<cdot> \<theta>) = subterms t \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
objI_trg: fixes a :: "'a" assumes "C.arr a"   and "trg\<^sub>C a = a" shows "C.obj a"
pm_of_poly_monom: fixes x :: "'a"   and c :: "'b"   and d :: "nat" shows "pm_of_poly x (Polynomial.monom c d) = monomial c (monomial d x)"
assms_1: shows "unit_counit_adjunction C D F G \<eta> \<epsilon>"
atom_eqvt: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and atom :: "'e \<Rightarrow> atom"   and p :: "perm"   and a :: "'e" assumes "class.at_base permute atom" shows "p \<bullet> atom a = atom (permute p a)"
inj_idom_hom_axioms: shows "inj_idom_hom (map_poly lift_poly)"
Join_project_stable: fixes F :: "'a program"   and G :: "'c program"   and C :: "'c set"   and A :: "'a set" assumes "extend h F \<squnion> G \<in> stable C" shows "(F \<squnion> project h C G \<in> stable A) = (extend h F \<squnion> G \<in> stable (C \<inter> extend_set h A) \<and> F \<in> stable A)"
in_topspace_nsphere: fixes n :: "nat" shows "(\<lambda>n. if n = 0 then 1 else 0) \<in> topspace (nsphere n)"
countable_Un_iff: fixes A :: "'a set"   and B :: "'a set" shows "countable (A \<union> B) = (countable A \<and> countable B)"
distrib_right': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.ab_near_semiring plus times" shows "times (plus x y) z = plus (times x z) (times y z)"
word_div_1: fixes n :: "'b word" shows "n div 1 = n"
subst_rawpsubst_fresh_switch: fixes \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list"   and s :: "'trm"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "\<forall>x\<in>snd ` set txs. x \<notin> FvarsT s"   and "\<forall>t\<in>fst ` set txs. y \<notin> FvarsT t"   and "distinct (map snd txs)"   and "s \<in> trm"   and "y \<in> var"   and "y \<notin> snd ` set txs" shows "subst (rawpsubst \<phi> txs) s y = rawpsubst \<phi> ((s, y) # txs)"
integrable_cong: fixes M :: "'d measure"   and N :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "M = N"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "integrable M f = integrable N g"
ProcDynModifyReturnNoAbrSameFaults: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P dynCall init p return' c Q,A"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s\<in>P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{\<sigma>} Call (p s) (Modif \<sigma>),{}" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P dynCall init p return c Q,A"
arr_Par_ArrVal_vrange: shows "\<R>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) \<subseteq>\<^sub>\<circ> T\<lparr>ArrCod\<rparr>"
l_get_parent_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_parent_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs known_ptrs get_parent"
check_return_def: fixes chk :: "'c + unit"   and res :: "'d" shows "check_return chk res = chk \<bind> (\<lambda>_. Inr res)"
pderiv_add: fixes p :: "'c poly"   and q :: "'c poly" shows "pderiv (p + q) = pderiv p + pderiv q"
assms_2: shows "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"
finite_V: fixes V :: "name set" assumes "V \<subseteq> Vs" shows "finite V"
sgn_1_neg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(sgn a = uminus one) = less a zero"
eq_h: shows "At = EqUni (a, b, c) \<longrightarrow> (\<exists>w\<ge>x. w \<le> y \<and> a * w\<^sup>2 + b * w + c = 0)"
satisfies_zero_le_constraint: fixes v :: "nat \<Rightarrow> 'a" shows "v \<Turnstile>\<^sub>l\<^sub>e 0"
P': shows "P' \<in> carrier_mat (dim_row (Mod_Type_Connect.from_hma\<^sub>m A)) (dim_row (Mod_Type_Connect.from_hma\<^sub>m A))"
cmp_set_def: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp_res" shows "cmp_set cmp \<equiv> cmp_extend (Collect finite) (cmp_img (linorder.sorted_list_of_set (comp2le cmp)) (cmp_lex cmp))"
cong': fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "prod' g A = prod' h B"
UNIV: fixes S :: "'k set set" shows "generate_topology S UNIV"
fmsubset_restrict_set_mono: fixes m :: "('d, 'e) fmap"   and n :: "('d, 'e) fmap"   and A :: "'d set" assumes "m \<subseteq>\<^sub>f n" shows "fmrestrict_set A m \<subseteq>\<^sub>f fmrestrict_set A n"
width_def: shows "width \<equiv> if V\<^bsub>T\<^esub> = {} then 0 else max_bag_card - 1"
has_integral_cmul_iff: fixes c :: "real"   and f :: "'b \<Rightarrow> 'c"   and I :: "'c"   and A :: "'b set" assumes "c \<noteq> 0" shows "((\<lambda>x. c *\<^sub>R f x) has_integral c *\<^sub>R I) A = (f has_integral I) A"
conv1: fixes x :: "'a"   and y :: "'a" shows "x \<rhd> y = (x \<rhd> (1::'a)) \<cdot> y"
mult_left_le_one_le: fixes 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less_eq zero x"   and "less_eq zero y"   and "less_eq y one" shows "less_eq (times y x) x"
\<iota>_mapsto: shows "UP \<in> UNIV \<rightarrow> Univ"
superunary: shows "1 < arity_hd\<^sub>h (head s)"
fmap_distinct_2: fixes k :: "'c"   and v :: "'d"   and m :: "('c, 'd) fmap" shows "m(k $$:= v) \<noteq> {$$}"
opaque_dist_cong: fixes x :: "nat itrm"   and y :: "nat itrm"   and vs :: "nat list" assumes "x \<simeq>\<^sup>+ y" shows "opaque_dist vs x \<simeq>\<^sup>+ opaque_dist vs y"
div_dvd_div: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a b"   and "dvd.dvd times a c" shows "dvd.dvd times (divide b a) (divide c a) = dvd.dvd times b c"
less_eq_complexI: fixes x :: "complex"   and y :: "complex" assumes "Re x \<le> Re y"   and "Im x = Im y" shows "x \<le> y"
cJ1Def: shows "cJ1 = \<lparr>states = start, msgs = initMsg (Suc j)\<rparr>"
assms_3: shows "\<forall>\<^sup>\<infinity>n. f n \<le> g n"
induct: fixes x :: "'c \<Rightarrow> ('d, 'c) Term.term"   and P :: "('c \<Rightarrow> ('d, 'c) Term.term) \<Rightarrow> bool" assumes "x \<in> fsubst"   and "P Var"   and "\<And>\<theta> v t. \<lbrakk>\<theta> \<in> fsubst; P \<theta>; v \<notin> subst_domain \<theta>; t \<noteq> Var v\<rbrakk> \<Longrightarrow> P (\<theta>(v := t))" shows "P x"
maddux_3_5: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> x = x \<sqinter> - (y \<sqinter> - y)"
dropWhile_append3: fixes P :: "'b \<Rightarrow> bool"   and y :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "\<not> P y" shows "dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys"
assms: shows "\<forall>l\<in>labels \<Gamma> p. \<forall>l'\<in>labels \<Gamma> p'. P ((\<xi>, l), a, \<xi>', l')"
nat_filtration_info: fixes f :: "bool stream \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> borel_measurable (nat_filtration n)" shows "f \<circ> pseudo_proj_True n = f"
elims: fixes x :: "('a, 'rule) graph_node"   and xa :: "('form, 'var) out_port"   and y :: "('form, 'var) antecedent option" assumes "hyps x xa = y" obtains r :: "'rule" and h :: "'form" and a :: "('form, 'var) antecedent" where "x = Rule r" and "xa = Hyp h a" and "y = (if a |\<in>| f_antecedent r \<and> h |\<in>| a_hyps a then Some a else None)"   | v :: "'a" and uv_ :: "('form, 'var) out_port" where "x = Assumption v__" and "xa = uv___" and "y = None"   | v :: "'a" and uv_ :: "('form, 'var) out_port" where "x = Conclusion va__" and "xa = uva___" and "y = None"   | uv_ :: "('form, 'var) out_port" where "x = Helper" and "xa = uv_" and "y = None"   | uu_ :: "('a, 'rule) graph_node" and v :: "'form" where "x = uu_" and "xa = Reg v" and "y = None"
l3_inv5I: fixes x :: "l3_state" assumes "synth (analz (ik x)) \<subseteq> dy_fake_msg (l3_state.bad x) (ik x \<inter> payload) (local.abs (ik x)) \<union> - payload" shows "x \<in> l3_inv5"
uminus1: fixes ccw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and a :: "'b"   and b :: "'b" assumes "ccw_vector_space ccw S" shows "ccw (0::'b) (- a) b = ccw (0::'b) b a"
eventuallyE: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "(\<diamond>P) \<sigma>" obtains i :: "nat" where "P (\<sigma> |\<^sub>s i)"
wset_Suspend_okD1: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set" assumes "s \<in> wset_Suspend_ok I" shows "s \<in> I"
Inf_ow_def: fixes U :: "'a set"   and Inf :: "'a set \<Rightarrow> 'a" shows "Inf_ow U Inf \<equiv> Inf ` Pow U \<subseteq> U"
perm_bij: fixes pi :: "(atom \<times> atom) list"   and x :: "'a"   and y :: "'a" shows "(pi \<bullet> x = pi \<bullet> y) = (x = y)"
MWLsSteps_det_cases_1: fixes \<iota> :: "nat"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m' :: "'id \<Rightarrow> 'val" assumes "SRabbr (skip\<^bsub>\<iota>\<^esub>) m \<alpha> p m'" obtains "\<alpha> = []" and "p = None" and "m' = m"
ceiling_less_numeral: fixes x :: "'b"   and v :: "num" shows "(\<lceil>x\<rceil> < numeral v) = (x \<le> numeral v - (1::'b))"
mult_scalar_def: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'b \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.mult_scalar pair_of_term term_of_pair p = term_powerprod.lift_poly_fun pair_of_term term_of_pair ((*) p)"
simple_function_compose1: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e" assumes "simple_function M f" shows "simple_function M (\<lambda>x. g (f x))"
fact_in_Ints: fixes n :: "nat" shows "fact n \<in> \<int>"