hcju/isabelleps huggingface dataset:
A0_def: shows "A0 = {x. jumpF f (at_right x) \<noteq> 0 \<and> a \<le> x \<and> x < c}"
sel_4: fixes z :: "quat" shows "Im3 (Quaternions.cnj z) = - Im3 z"
left_neutral: fixes a :: "('a \<times> 'a set) set" shows "1\<^sub>\<pi> \<parallel> a = a"
swap_rassocl_def: shows "swap_rassocl \<equiv> lassocr\<^sub>C \<odot> (1\<^sub>C |\<^sub>= swap\<^sub>C) \<odot> rassocl\<^sub>C"
tab_simps_5: shows "cod ((r \<star> \<phi>) \<cdot> \<rho>) = r \<star> f'"
\<psi>s: shows "list_all2 (wf_mformula \<sigma> j P V n R) \<phi>s \<psi>s"
fin: fixes t :: "'t" assumes "r1.final_thread s1 t" shows "r2.final_thread s2' t"
simps_2: fixes t :: "dB"   and u :: "dB"   and s :: "dB"   and k :: "nat" shows "(t \<degree> u)[s/k] = t[s/k] \<degree> u[s/k]"
assms: shows "t \<in> timpl_closure (Var x) TI"
convex_hull_sum: fixes A :: "'a vec set"   and B :: "'a vec set" assumes "A \<subseteq> carrier_vec n"   and "B \<subseteq> carrier_vec n" shows "convex_hull (A + B) = convex_hull A + convex_hull B"
v11_injective: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "b \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "r\<lparr>a\<rparr> = r\<lparr>b\<rparr>" shows "a = b"
less_tag: fixes x :: "'a tag_const"   and y :: "'a tag_const" shows "(x < y) = (x \<le> y \<and> x \<noteq> y)"
borel_measurable_integrable: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "integrable M f" shows "f \<in> borel_measurable M"
map_add_dom_app_simps_3: fixes m :: "'b"   and l2 :: "'b \<Rightarrow> 'c option"   and l1 :: "'b \<Rightarrow> 'c option" assumes "m \<notin> dom l2" shows "(l1 ++ l2) m = l1 m"
exhaust_sel: fixes tree :: "'b tree" obtains "tree = \<langle>\<rangle>"   | "tree = \<langle>un_Node1 tree, value tree, un_Node3 tree\<rangle>"
sep_disj_addI1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj x (plus y z)"   and "sep_disj y z" shows "sep_disj (plus x y) z"
unlabel_mono: fixes A :: "('d \<times> 'e) list"   and B :: "('d \<times> 'e) list" assumes "set A \<subseteq> set B" shows "set (unlabel A) \<subseteq> set (unlabel B)"
global_policy'_def: fixes I :: "infrastructure"   and a :: "char list" shows "global_policy' I a \<equiv> a \<notin> gdpr_actors \<longrightarrow> \<not> enables I cloud (Actor a) get"
subset_segment: fixes T :: "'a set"   and a :: "'a" assumes "T \<subseteq> carrier D"   and "\<forall>b\<in>T. \<forall>x. x \<prec> b \<and> x \<in> carrier D \<longrightarrow> x \<in> T"   and "minimum_elem D (carrier D - T) a" shows "T = segment D a"
list_slice2_append_mod: fixes xs :: "'b list"   and k :: "nat"   and ys :: "'b list" assumes "length xs mod k = 0" shows "list_slice2 (xs @ ys) k = list_slice2 xs k @ list_slice2 ys k"
simps_2: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'e trm"   and at :: "('e, 'f) atm" shows "substL \<pi> (Neg at) = Neg (substA \<pi> at)"
rrr: shows "round_robin_reschedule t0 queue n0 s = \<lfloor>(t, \<lfloor>(ta, x', m')\<rfloor>, \<sigma>')\<rfloor>"
P''Chain: shows "P'' \<Longrightarrow>\<^sub>\<tau> P'"
structure_is_trivial: shows "\<phi> = FoT\<^sub>C.map"
simps_6: fixes i :: "nat"   and t :: "expr"   and U :: "Unary_PCF.type"   and e :: "expr" shows "open i t (\<Lambda>\<langle>U\<rangle> e) = \<Lambda>\<langle>U\<rangle> open (i + 1) t e"
assms_3: fixes b :: "'a" assumes "b \<in> B" shows "\<exists>a\<in>A. a \<le> b"
fls_mult_const_nth_1: fixes x :: "'a"   and f :: "'a fls"   and n :: "int" shows "fls_nth (fls_const x * f) n = x * fls_nth f n"
bi_totalI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "left_total R"   and "right_total R" shows "bi_total R"
ResNil: fixes x :: "name" shows "<\<nu>x>\<zero> \<equiv>\<^sub>e \<zero>"
vec_of_scal_dim_1: fixes x :: "'b" shows "dim_vec (vec_of_scal x) = 1"
simps_21: fixes x21 :: "'g"   and x22 :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "results (Resumption.resumption.Pause x21 x22) = \<Union> (results ` range x22)"
zero_not_mem_HNatInfinite: shows "0 \<notin> HNatInfinite"
mdqd2: shows "(\<exists>x. Ifm bbs (x # bs) p) = Ifm bbs bs (decr (Cooper.disj (evaldjf (\<lambda>j. simpfm (subst0 (C j) (simpfm (minusinf (fst (unit p)))))) [1..snd (snd (unit p))]) (evaldjf (\<lambda>(b, j). simpfm (subst0 (Add b (C j)) (fst (unit p)))) (concat (map (\<lambda>b. map (Pair b) [1..snd (snd (unit p))]) (fst (snd (unit p))))))))"
nth_sequence_of: fixes i :: "nat"   and A :: "'b set"   and enum :: "nat \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "i < card A" shows "sequence_of A enum f ! i = f (enum i)"
hindered: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "hindrance \<Gamma> f"   and "current \<Gamma> f"   and "wave \<Gamma> f" shows "hindered \<Gamma>"
set_butlast: shows "(c, d) \<in> set (butlast (polychain_of x0 xs))"
fv: fixes \<phi> :: "'fmla" assumes "set (xs @ us @ vs) \<subseteq> var"   and "set ts \<subseteq> trm"   and "set [\<phi>] \<subseteq> fmla"   and "\<phi> \<in> set [\<phi>]" shows "set (getFrN (xs @ us @ vs) ts [\<phi>] (length xs)) \<inter> Fvars \<phi> = {}"
matrix_vector_right_distrib: fixes M :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and v :: "('b, 'd) Finite_Cartesian_Product.vec"   and w :: "('b, 'd) Finite_Cartesian_Product.vec" shows "M *v (v + w) = M *v v + M *v w"
l_set_val_get_tag_name_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_val_get_tag_name type_wf set_val set_val_locs get_tag_name get_tag_name_locs \<equiv> l_set_val type_wf set_val set_val_locs \<and> l_get_tag_name type_wf get_tag_name get_tag_name_locs \<and> l_set_val_get_tag_name_axioms set_val_locs get_tag_name_locs"
1: shows "shd (v ## gtrace r v) \<in> gunodes A w"
Schema_def: fixes sc :: "'d \<times> 'd \<Rightarrow> bool" shows "Schema sc \<equiv> action_of (R (\<lambda>(A, A'). Pre sc (alpha_rp.more A) \<turnstile> \<lambda>(A, A'). sc (alpha_rp.more A, alpha_rp.more A') \<and> \<not> wait A' \<and> tr A = tr A'))"
fst_lens_plus: fixes y :: "'c \<Longrightarrow> 'd"   and x :: "'e \<Longrightarrow> 'd" assumes "wb_lens y" shows "fst\<^sub>L ;\<^sub>L (x +\<^sub>L y) = x"
p_rpd_annir: fixes x :: "'a" shows "x \<parallel> (0::'a) = (0::'a)"
two_add_def: shows "two_add = (\<Lambda> x y. if x \<sqsubseteq> none then y else if y \<sqsubseteq> none then x else many)"
g_def: shows "g \<equiv> \<lambda>x. if poly (gcd p q) x = 0 then 0 else 1"
induct: fixes P :: "'b rexp \<Rightarrow> 'b rexp \<Rightarrow> bool"   and a0 :: "'b rexp"   and a1 :: "'b rexp" assumes "\<And>uu_. P Zero uu_"   and "P One Zero"   and "\<And>v. P (Atom v) Zero"   and "\<And>v va. P (Plus v va) Zero"   and "\<And>v va. P (Times v va) Zero"   and "\<And>v. P (Star v) Zero"   and "P One One"   and "\<And>v. P One (Atom v)"   and "\<And>v va. P One (Plus v va)"   and "\<And>v va. P One (Times v va)"   and "\<And>v. P One (Star v)"   and "\<And>v. P (Atom v) One"   and "\<And>v va. P (Plus v va) One"   and "\<And>v va. P (Times v va) One"   and "\<And>v. P (Star v) One"   and "\<And>a b. P (Atom a) (Atom b)"   and "\<And>uy_ v va. P (Atom uy_) (Plus v va)"   and "\<And>uy_ v va. P (Atom uy_) (Times v va)"   and "\<And>uy_ v. P (Atom uy_) (Star v)"   and "\<And>v vc vb_. P (Plus v vc) (Atom vb_)"   and "\<And>v vc vb_. P (Times v vc) (Atom vb_)"   and "\<And>v vb_. P (Star v) (Atom vb_)"   and "\<And>r s. P r s \<Longrightarrow> P (Star r) (Star s)"   and "\<And>vc_ v va. P (Star vc_) (Plus v va)"   and "\<And>vc_ v va. P (Star vc_) (Times v va)"   and "\<And>v va vf_. P (Plus v va) (Star vf_)"   and "\<And>v va vf_. P (Times v va) (Star vf_)"   and "\<And>r r' s s'. \<lbrakk>r = s \<Longrightarrow> P r' s'; r \<noteq> s \<Longrightarrow> P r s\<rbrakk> \<Longrightarrow> P (Plus r r') (Plus s s')"   and "\<And>vg_ vh_ v va. P (Plus vg_ vh_) (Times v va)"   and "\<And>v va vk_ vl_. P (Times v va) (Plus vk_ vl_)"   and "\<And>r r' s s'. \<lbrakk>r = s \<Longrightarrow> P r' s'; r \<noteq> s \<Longrightarrow> P r s\<rbrakk> \<Longrightarrow> P (Times r r') (Times s s')" shows "P a0 a1"
distinct_1: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c"   and x23 :: "'d" assumes "concrete_category Obj Hom Id Comp" shows "concrete_category.Null \<noteq> concrete_category.MkArr x21 x22 x23"
simps_3: fixes d :: "int"   and i :: "int"   and ks :: "int list" shows "divisor (NDvd d i ks) = d"
indicator_vimage: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'c set"   and x :: "'b" shows "indicator (f -` A) x = indicator A (f x)"
valid_path_negatepath: fixes \<gamma> :: "real \<Rightarrow> 'a" assumes "valid_path \<gamma>" shows "valid_path (uminus \<circ> \<gamma>)"
that: fixes s' :: "('t \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'b" assumes "s' \<in> pmdl.syzygy_module (set bs)"   and "s = atomize_poly (idx_pm_of_pm bs s')" shows "thesis"
arden_conv2: fixes x :: "'a" assumes "\<forall>y z. z + x \<cdot> y = y \<longrightarrow> x\<^sup>\<star> \<cdot> z = y" shows "x\<^sup>\<omega> = (0::'a)"
4: fixes f :: "('fun, 'var) Term.term \<times> ('fun, 'var) Term.term"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "f \<in> set F'"   and "subst_domain \<delta> = set X \<and> ground (subst_range \<delta>)" shows "f \<cdot>\<^sub>p \<delta> \<circ>\<^sub>s \<I> \<notin> set (map snd D) - set (map snd (dbproj i D)) \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>"
assn_fn_def: fixes assn_raw :: "assn_raw" shows "assn_fn assn_raw = (case assn_raw of Assn x \<Rightarrow> x)"
map_1: fixes f :: "'s \<Rightarrow> 'r" shows "map_option f None = None"
d\<^sub>1_inject: fixes x :: "\<Pi>\<^sub>1"   and y :: "\<Pi>\<^sub>1" assumes "d\<^sub>1 x = d\<^sub>1 y" shows "x = y"
set_of_distrib_left: fixes B :: "'b interval"   and A1 :: "'b interval"   and A2 :: "'b interval" shows "set_of (B * (A1 + A2)) \<subseteq> set_of (B * A1 + B * A2)"
Normal_def: shows "image.induced.Ker = image.induced.kernel.Class (kernel.Class \<one>)"
nil_ssubseqs: fixes xs :: "'b list" shows "[] \<in> ssubseqs xs"
progress_unop_testop_i: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and cs :: "e list"   and t :: "t"   and e :: "b_e"   and iop :: "unop_i"   and testop :: "testop"   and s :: "s"   and vs :: "v list"   and i :: "nat" assumes "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [t]"   and "is_int_t t"   and "const_list cs"   and "e = Unop_i t iop \<or> e = Testop t testop" shows "\<exists>a s' vs' es'. \<lparr>s;vs;cs @ [$e]\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
separatedin_imp_disjoint: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "separatedin X S T" shows "disjnt S T"
sdrop_simps_1: fixes n :: "nat"   and s :: "'a stream" shows "shd (sdrop n s) = s !! n"
gcd_dvd_1: fixes a :: "'b"   and b :: "'b" shows "(some_gcd a b dvd (1::'b)) = comm_monoid_mult_class.coprime a b"
ThinChamberComplexFolding_g: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "ThinChamberComplexFolding X g"
interfree_def: fixes Ts :: "('a ann_com option \<times> 'a set) list" shows "interfree Ts \<equiv> \<forall>i j. i < length Ts \<and> j < length Ts \<and> i \<noteq> j \<longrightarrow> interfree_aux (OG_Hoare.com (Ts ! i), OG_Hoare.post (Ts ! i), OG_Hoare.com (Ts ! j))"
arr_dom_iff_arr: fixes f :: "'a \<times> 'a" shows "CC.arr (CC.dom f) = CC.arr f"
Pi_pmf_bind: fixes A :: "'b set"   and d :: "'c"   and p :: "'b \<Rightarrow> 'd pmf"   and q :: "'b \<Rightarrow> 'd \<Rightarrow> 'c pmf"   and d' :: "'d" assumes "finite A" shows "Pi_pmf A d (\<lambda>x. p x \<bind> q x) = Pi_pmf A d' p \<bind> (\<lambda>f. Pi_pmf A d (\<lambda>x. q x (f x)))"
path_split_snoc: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "as \<noteq> []" obtains a' :: "'a" and as' :: "'a list" where "as = as' @ [a']" and "CFG.path sourcenode targetnode valid_edge n as' (sourcenode a')" and "valid_edge a'" and "n' = targetnode a'"
sum_pos2: fixes I :: "'c set"   and i :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "finite I"   and "i \<in> I"   and "(0::'d) < f i"   and "\<And>i. i \<in> I \<Longrightarrow> (0::'d) \<le> f i" shows "(0::'d) < sum f I"
is_expansion_length: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a"   and basis :: "(real \<Rightarrow> real) list" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "is_expansion F basis" shows "length basis = expansion_level TYPE('a)"
mkIde_as_mkArr: fixes A :: "'s set" assumes "A \<subseteq> S.Univ" shows "S.mkArr A A (\<lambda>x. x) = S.mkIde A"
singular_simplex_subtopology: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c" shows "singular_simplex p (subtopology X S) f = (singular_simplex p X f \<and> f ` standard_simplex p \<subseteq> S)"
newpkt_def: shows "newpkt \<equiv> \<lambda>(d, did). Newpkt d did"
10: shows "length w\<^sub>1 = length w\<^sub>2"
cont2cont_LAM': fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "cont (\<lambda>p. f (fst p) (snd p))" shows "cont (\<lambda>x. \<Lambda> y. f x y)"
more_temp_simps3_6: fixes F :: "'a \<Rightarrow> bool"   and sigma :: "behavior" shows "\<not> (sigma \<Turnstile> \<box>F) \<equiv> sigma \<Turnstile> \<diamond>\<not> F"
nnegD: fixes P :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "nneg P" shows "(0::'c) \<le> P x"
antimonoI: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f y \<le> f x" shows "order.antimono less_eq f"
ev: shows "\<forall>\<^sub>F s in nhds s. s \<in> {s. conv_abscissa f < ereal (s \<bullet> (1::'a))}"
transfer: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "bi_total A" shows "rel_fun (pcr_mapping A B) (rel_set A) dom Mapping.keys"
outputPres: fixes P :: "pi"   and Q :: "pi"   and a :: "name"   and b :: "name" assumes "P \<approx> Q" shows "a{b}.P \<approx> a{b}.Q"
assms_2: fixes z :: "'a" assumes "z \<in> ball a r" shows "f z \<noteq> 0"
lossless: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>"   and "I s" shows "colossless_gpv \<I>' (callee s x)"
set_iterator_linord_rule_P: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and it :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and S0 :: "'a set"   and I :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma>0 :: "'b"   and c :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "linorder.set_iterator_linord less_eq it S0"   and "I S0 \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; x \<in> S; I S \<sigma>; S \<subseteq> S0; \<And>x'. x' \<in> S0 - S \<Longrightarrow> less_eq x' x; \<And>x'. x' \<in> S \<Longrightarrow> less_eq x x'\<rbrakk> \<Longrightarrow> I (S - {x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> {}; \<And>x x'. \<lbrakk>x \<in> S; x' \<in> S0 - S\<rbrakk> \<Longrightarrow> less_eq x' x; \<not> c \<sigma>; I S \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
injective_imp_isometric: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "closed s"   and "subspace s"   and "bounded_linear f"   and "\<forall>x\<in>s. f x = (0::'c) \<longrightarrow> x = (0::'b)" shows "\<exists>e>0. \<forall>x\<in>s. e * norm x \<le> norm (f x)"
tuple_simps_1: 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 "partial_magma.arr C (elementary_category_with_pullbacks.tuple C prj0 prj1 h f g k)"
2: shows "c \<otimes> b [^] 2 = c \<otimes> \<ominus> b [^] 2"
induct: fixes x :: "'a"   and R :: "('a, 'b) ring_scheme"   and H :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> generate_ring R H"   and "P \<one>\<^bsub>R\<^esub>"   and "\<And>h. h \<in> H \<Longrightarrow> P h"   and "\<And>h. \<lbrakk>h \<in> generate_ring R H; P h\<rbrakk> \<Longrightarrow> P (\<ominus>\<^bsub>R\<^esub> h)"   and "\<And>h1 h2. \<lbrakk>h1 \<in> generate_ring R H; P h1; h2 \<in> generate_ring R H; P h2\<rbrakk> \<Longrightarrow> P (h1 \<oplus>\<^bsub>R\<^esub> h2)"   and "\<And>h1 h2. \<lbrakk>h1 \<in> generate_ring R H; P h1; h2 \<in> generate_ring R H; P h2\<rbrakk> \<Longrightarrow> P (h1 \<otimes>\<^bsub>R\<^esub> h2)" shows "P x"
rep_eq: fixes x :: "'f \<Rightarrow>\<^sub>U 'd"   and xa :: "'c['e, 'f]" shows "fromQ (qconv x xa) = \<lparr>mag = of_rat (dconvfactor x (Quantity.dim (fromQ xa))) \<cdot> mag (fromQ xa), dim = Quantity.dim (fromQ xa), unit_sys = unit\<rparr>"
0: shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"
exec: shows "exec_moves ci P t es h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')"
pdevs_val_scaleR_pdevs: fixes e :: "nat \<Rightarrow> real"   and x :: "real"   and Y :: "'a pdevs" shows "pdevs_val e (scaleR_pdevs x Y) = x *\<^sub>R pdevs_val e Y"
arr_eval: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d"   and V :: "'b \<Rightarrow> 'd"   and f :: "'b local.term" assumes "evaluation_map C D T \<alpha> \<iota> V"   and "monoidal_language.Arr C f" shows "partial_magma.arr D (evaluation_map.eval D T \<alpha> \<iota> V f)"
justification_well_formed_def: fixes P :: "'a prog"   and J :: "nat \<Rightarrow> ('b, 'c) justifying_execution" shows "justification_well_formed P J = (\<forall>n. P \<turnstile> (justifying_exec (J n), justifying_ws (J n)) \<surd>)"
abs_eq: fixes x :: "\<nu>" shows "x\<^sup>P = make\<kappa> (Some x)"
bex_imageD: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>x\<in>f ` A. P x" shows "\<exists>x\<in>A. P (f x)"
cases: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and \<psi> :: "'c stream \<Rightarrow> bool"   and a :: "'c stream" assumes "(\<phi> until \<psi>) a" obtains xs :: "'c stream" where "a = xsa__" and "\<psi> xsa__"   | xs :: "'c stream" where "a = xs" and "\<phi> xs" and "(\<phi> until \<psi>) (stl xs)"
bottom_left_bottom: fixes a :: "'c" shows "gcd (1::'c) a = (1::'c)"
is_unit_gcd_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) one = (gcd a b = one)"
Ball_set_list_all: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool" shows "Ball (set xs) P = list_all P xs"
cong: assumes "A - {x} = A - {x}"   and "\<And>xa. xa \<in> A - {x} \<Longrightarrow> g xa = h xa" shows "sum g (A - {x}) = sum h (A - {x})"
box_empty_imp: fixes a :: "real"   and b :: "real" assumes "{a<..<b} = {}" shows "g {a..b} = \<^bold>1"
mod_0_imp_mod_factor_0_left: fixes n :: "nat"   and m :: "nat"   and m' :: "nat" assumes "n mod (m * m') = 0" shows "n mod m = 0"
diff_add_eq_diff_diff_swap: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.group_add minus plus zero uminus" shows "minus a (plus b c) = minus (minus a c) b"
misomorphic_def: fixes R :: "('e, 'f) Ring_scheme"   and M :: "('g, 'e, 'h) Module_scheme"   and N :: "('i, 'e, 'j) Module_scheme" shows "M \<cong>\<^bsub>R\<^esub> N = (\<exists>f. f \<in> mHom R M N \<and> bijec\<^bsub>M,N\<^esub> f)"
dvd_unit_imp_unit: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a b"   and "dvd.dvd times b one" shows "dvd.dvd times a one"
that: fixes us :: "'a list list" assumes "1 < \<^bold>|us\<^bold>|"   and "us \<noteq> \<epsilon>"   and "us \<in> lists \<BB> G"   and "hd us \<noteq> \<epsilon>"   and "hd us \<in> \<langle>G\<rangle>"   and "concat (tl us) \<noteq> \<epsilon>"   and "concat (tl us) \<in> \<langle>G\<rangle>"   and "w = hd us \<cdot> concat (tl us)" shows "thesis"
none_empty: shows "i \<noteq> bot \<and> j \<noteq> bot \<and> k \<noteq> bot"
argmax_2: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "f (fst (argmax f xs)) = snd (argmax f xs)"
stock_pf_vp_expand: fixes pf :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and n :: "nat"   and w :: "bool stream" assumes "stock_portfolio Mkt pf" shows "val_process Mkt pf n w = geom_proc n w * pf stk (Suc n) w + disc_rfr_proc r n w * pf risk_free_asset (Suc n) w"
dom_def: fixes m :: "'b \<Rightarrow> 'c option" shows "dom m = _Collect a (m a \<noteq> None)"
fps_deriv_def: fixes f :: "'a fps" shows "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f $ (n + 1))"
IH: fixes i :: "nat" assumes "i < length T" shows "\<exists>u. (T ! i, u) \<in> timpl_closure' c \<and> (S ! i, u) \<in> timpl_closure' c"
comp_in_homE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and c :: "'a \<times> 'a" assumes "VoV.in_hom (VoV.comp g f) a c" obtains b :: "'a \<times> 'a" where "VoV.in_hom f a b" and "VoV.in_hom g b c"