hcju/isabelleps huggingface dataset:
gama2_cont_comp: shows "continuous_on {0..1} (\<lambda>x. \<gamma>2 x \<bullet> b)"
succ_le_iff: fixes i :: "V"   and j :: "V" assumes "Ord i"   and "Ord j" shows "(ZFC_in_HOL.succ i \<subseteq>\<^sub>\<circ> j) = (i \<subset>\<^sub>\<circ> j)"
take_0: fixes xs :: "'b list" shows "take 0 xs = \<epsilon>"
that: fixes sr' :: "'a option" assumes "(Some r, sr') \<in> option_while' C B" shows "thesis"
IH_2: shows "(\<Sum>d = 1..2 * p + 1. (- 1) ^ (d + 1) * psi (n div d)) \<le> (\<Sum>d = 1..2 * Suc k + 1. (- 1) ^ (d + 1) * psi (n div d))"
intros: fixes x :: "Predicate_Compile_Tests.char" assumes "x = F \<or> x = G" shows "is_F_or_G x"
glbound_in_ofD_glbound: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a set"   and x :: "'a"   and y :: "'a"   and b :: "'a"   and a :: "'a" assumes "ordering less_eq less"   and "ordering.glbound_in_of less_eq P x y b"   and "a \<in> P"   and "ordering.lbound_of less_eq x y a" shows "less_eq a b"
FW_1: shows "fw m n n n n i j = D m i j n"
literalNotInEarlierLevelsThanItsLevel: fixes level :: "nat"   and e :: "'b"   and M :: "('b \<times> bool) list" assumes "level < elementLevel e M" shows "e \<notin> set (elements (prefixToLevel level M))"
tauActTauStepChain: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'" shows "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'"
distinct_prime_factorization_nat': fixes n :: "nat" shows "distinct (prime_factorization_nat' n)"
simps_6: fixes b :: "'a set"   and c :: "('a, 'b, 'c) com" shows "redexes (While b c) = {While b c}"
end_LE_def: shows "end_LE = measures [end_state, end_stage, end_step]"
ucast16_ucast128_masks_highest_bits0: shows "UCAST(16 \<rightarrow> 128) (UCAST(128 \<rightarrow> 16) (ip && 65535)) = ip && 65535"
conversion_mono: fixes A :: "('b \<times> 'b) set"   and B :: "('b \<times> 'b) set" assumes "A \<subseteq> B" shows "A\<^sup>\<leftrightarrow>\<^sup>* \<subseteq> B\<^sup>\<leftrightarrow>\<^sup>*"
intro: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" assumes "ChamberComplexWithApartmentSystem X \<A>"   and "ChamberComplexApartmentSystemTriangle_axioms X \<A> A B B' C D E z" shows "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z"
subdegree_geI: fixes f :: "'a fps"   and n :: "nat" assumes "f \<noteq> 0"   and "\<And>i. i < n \<Longrightarrow> fps_nth f i = (0::'a)" shows "n \<le> subdegree f"
latestActEq: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and nid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n' :: "nat"   and n :: "nat" assumes "active nid (t n')"   and "\<not> (\<exists>n''>n'. n'' < n \<and> active nid (t n'))"   and "n' < n" shows "dynamic_component.latestAct active nid t n = n'"
reindex_cong: fixes l :: "'d \<Rightarrow> 'e"   and B :: "'d set"   and A :: "'e set"   and g :: "'e \<Rightarrow> 'f"   and h :: "'d \<Rightarrow> 'f" assumes "inj_on l B"   and "A = l ` B"   and "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" shows "sum g A = sum h B"
ground_subst_\<sigma>_\<eta>: shows "is_ground_subst (\<sigma> \<odot> \<eta>)"
ldropn_ldropn: fixes n :: "nat"   and m :: "nat"   and xs :: "'b llist" shows "ldropn n (ldropn m xs) = ldropn (n + m) xs"
less_iff_diff_less_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less a b = less (minus a b) zero"
ring_record_simps_1: fixes carrier :: "'c set"   and more :: "'d" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
induces_extended_monoidal_category: shows "extended_monoidal_category (\<cdot>) T \<alpha> \<iota>"
minusinf_nb: fixes p :: "fm" assumes "islin p" shows "bound0 (minusinf p)"
contour_integrable_join: fixes g1 :: "real \<Rightarrow> complex"   and g2 :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" assumes "valid_path g1"   and "valid_path g2" shows "(f contour_integrable_on g1 +++ g2) = (f contour_integrable_on g1 \<and> f contour_integrable_on g2)"
pn_map_correct: fixes nds :: "('a, 'b) node_scheme set" assumes "finite nds"   and "\<And>nd. nd \<in> nds \<Longrightarrow> finite (old nd)"   and "inj_on name nds" shows "pn_map nds \<le> SPEC (\<lambda>r. \<forall>qn. case r qn of None \<Rightarrow> qn \<notin> name ` nds | Some (P, N) \<Rightarrow> qn \<in> name ` nds \<and> P = {p. prop\<^sub>r(p) \<in> old (the_inv_into nds name qn)} \<and> N = {p. nprop\<^sub>r(p) \<in> old (the_inv_into nds name qn)})"
equiv_test: fixes ifex_of :: "'a \<Rightarrow> 'b ifex"   and val :: "'a \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> bool"   and b1 :: "'a"   and b2 :: "'a" assumes "reduced_bdt_checkers ifex_of val" shows "equiv_test ifex_of b1 b2 = (\<forall>s. val b1 s = val b2 s)"
sp_def: shows "sp = (\<lambda>w. spick w n x)"
partition_on_msetI: fixes P :: "'a multiset multiset"   and A :: "'a multiset" assumes "\<Sum>\<^sub># P = A"   and "{#} \<notin># P" shows "partition_on_mset A P"
avl_insert_auto: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "avl t" shows "avl (AVL_Set_Code.insert x t) \<and> height (AVL_Set_Code.insert x t) \<in> {height t, height t + 1}"
no_id: fixes v :: "'a"   and w :: "'b" shows "(v, w, v) \<notin> G.E"
nn_integral_measure_pmf_finite: fixes M :: "'c pmf"   and f :: "'c \<Rightarrow> ennreal" assumes "finite (set_pmf M)"   and "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x" shows "integral\<^sup>N (measure_pmf M) f = (\<Sum>x\<in>set_pmf M. f x * ennreal (pmf M x))"
bit_nat_iff: fixes k :: "int"   and n :: "nat" shows "bit (nat k) n = (0 \<le> k \<and> bit k n)"
conseqS: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>a {P} c { Q}"   and "\<And>s n. P' (s, n) \<Longrightarrow> P (s, n)"   and "\<And>s n. Q (s, n) \<Longrightarrow> Q' (s, n)" shows "\<turnstile>\<^sub>3\<^sub>a {P'} c { Q'}"
dests_vD_inc_sqn: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PAodv-:15, PPkt-:7, PRreq-:9, PRreq-:21, PRrep-:9} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>) \<and> the (dests \<xi> ip) = inc (sqn (rt \<xi>) ip))) \<and> (l = PRerr-:1 \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>) \<and> sqn (rt \<xi>) ip < the (dests \<xi> ip))))"
2: shows "set w\<^sub>2 \<subseteq> invisible"
take_bit_Suc: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit (Suc n) a = plus (times (take_bit n (divide a (numeral.numeral one plus (num.Bit0 num.One)))) (numeral.numeral one plus (num.Bit0 num.One))) (modulo a (numeral.numeral one plus (num.Bit0 num.One)))"
seqSubst_def: fixes subst :: "'c \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'c"   and M :: "'c"   and \<sigma> :: "(name list \<times> 'd list) list" assumes "substType subst" shows "substType.seqSubst subst M \<sigma> \<equiv> foldl (\<lambda>N (xvec, Tvec). subst N xvec Tvec) M \<sigma>"
zmset_of_le: fixes M :: "'a multiset"   and N :: "'a multiset" shows "(zmset_of M \<le> zmset_of N) = (M \<le> N)"
Well_order_Restr: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "Well_order r" shows "Well_order (Restr r A)"
fresh_star_prod_set: fixes xs :: "'a set"   and a :: "'b"   and b :: "'c" shows "xs \<sharp>* (a, b) = (xs \<sharp>* a \<and> xs \<sharp>* b)"
log: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Log], Undecided\<rangle> \<Rightarrow> Undecided"
EWHILET_rule: fixes s\<^sub>0 :: "int \<times> 'a"   and b :: "int \<times> 'a \<Rightarrow> bool"   and f :: "int \<times> 'a \<Rightarrow> ('c + int \<times> 'a) nres"   and E :: "'c \<Rightarrow> bool"   and \<Phi> :: "int \<times> 'a \<Rightarrow> bool" assumes "wf (measure (\<lambda>(i, uu_). nat (ub - i)))"   and "case s\<^sub>0 of (i, \<sigma>) \<Rightarrow> lb \<le> i \<and> i \<le> ub \<and> I i \<sigma>"   and "\<And>s. \<lbrakk>case s of (i, \<sigma>) \<Rightarrow> lb \<le> i \<and> i \<le> ub \<and> I i \<sigma>; b s; (s, s\<^sub>0) \<in> (measure (\<lambda>(i, uu_). nat (ub - i)))\<^sup>*\<rbrakk> \<Longrightarrow> f s \<le> ESPEC E (\<lambda>s'. (case s' of (i, \<sigma>) \<Rightarrow> lb \<le> i \<and> i \<le> ub \<and> I i \<sigma>) \<and> (s', s) \<in> measure (\<lambda>(i, uu_). nat (ub - i)))"   and "\<And>s. \<lbrakk>case s of (i, \<sigma>) \<Rightarrow> lb \<le> i \<and> i \<le> ub \<and> I i \<sigma>; \<not> b s; (s, s\<^sub>0) \<in> (measure (\<lambda>(i, uu_). nat (ub - i)))\<^sup>*\<rbrakk> \<Longrightarrow> \<Phi> s" shows "EWHILET b f s\<^sub>0 \<le> ESPEC E \<Phi>"
prefix_snocD: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" assumes "prefix (xs @ [x]) ys" shows "strict_prefix xs ys"
pi'_dom: fixes it :: "'f \<Rightarrow> ('g \<Rightarrow> bool) \<Rightarrow> ('h \<times> 'i \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and it' :: "'f \<Rightarrow> ('j \<Rightarrow> bool) \<Rightarrow> ('h \<times> 'i \<Rightarrow> 'j \<Rightarrow> 'j) \<Rightarrow> 'j \<Rightarrow> 'j" assumes "proper_it' it it'" shows "proper_it' (map_iterator_dom \<circ> it) (map_iterator_dom \<circ> it')"
subst_lconsts_subst_renameLCs: fixes p :: "'var \<Rightarrow> 'var"   and s :: "'subst" shows "subst_lconsts (subst_renameLCs p s) = p ` subst_lconsts s"
ctr_transfer_1: fixes R2 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun R2 (tllist_all2 R1 R2) TNil TNil"
finite_imp_not_open: fixes S :: "'a set" assumes "finite S"   and "open S" shows "S = {}"
l_get_scdom_component_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_scdom_component_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs get_scdom_component is_strongly_scdom_component_safe is_weakly_scdom_component_safe get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_disconnected_nodes to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_element_by_id get_elements_by_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
wtL_grdLit_unprot: fixes x :: "Preliminaries.var" assumes "\<not> unprot (GE.tpOfV x)" shows "GE.wtL (grdLit x)"
arith_extra_simps_7: shows "- (0::'a) = (0::'a)"
eval_def: fixes P :: "'a fset"   and x :: "'a" shows "eval P x = (x |\<in>| P)"
in_CompI: fixes F :: "'a set"   and G :: "'a set"   and f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.is_iso_class F"   and "B.is_iso_class G"   and "f \<in> F"   and "g \<in> G"   and "g \<star> f \<cong>\<^sub>B h" shows "h \<in> Comp G F"
distr_restrict: fixes J :: "'i set"   and K :: "'i set" assumes "J \<subseteq> K"   and "finite K" shows "Pi\<^sub>M J M = distr (Pi\<^sub>M K M) (Pi\<^sub>M J M) (\<lambda>f. restrict f J)"
not_essential_def: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" shows "not_essential f z = (\<exists>x. f \<midarrow>z\<rightarrow> x \<or> is_pole f z)"
minus_mod_eq_mult_div: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "minus a (modulo a b) = times b (divide a b)"
new_last_neq_v0: shows "new_last \<noteq> v0"
F: fixes Q :: "(char list \<Rightarrow> int option) \<Rightarrow> bool"   and n :: "nat"   and P :: "(char list \<Rightarrow> int option) \<Rightarrow> bool"   and m :: "nat" assumes "(embP Q \<and>* $ n) = (embP P \<and>* $ m)" obtains "Q = P" and "n = m"   | "Q = (\<lambda>s. False)" and "P = (\<lambda>s. False)"
leq: shows "(lang r = lang s) = (\<forall>(r', s')\<in>{((r0, s0), nderiv a r0, nderiv a s0) |r0 s0 a. a \<in> set as}\<^sup>* `` {(r, s)}. nullable r' = nullable s')"
eq2: shows "(\<Sum>v\<in>keys r \<union> keys s. lookup s v *s v) = (\<Sum>v\<in>keys s. lookup s v *s v)"
sigma_sets_into_sp: fixes A :: "'b set set"   and sp :: "'b set"   and x :: "'b set" assumes "A \<subseteq> Pow sp"   and "x \<in> sigma_sets sp A" shows "x \<subseteq> sp"
sorted_list_of_set_Max_snoc: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "sorted_list_of_set A = sorted_list_of_set (A - {Max A}) @ [Max A]"
neutr: fixes x :: "'b DBMEntry" shows "dbm_add x (Le (0::'b)) = x"
aext_or: fixes P :: "'c upred"   and Q :: "'c upred"   and x :: "'c \<Longrightarrow> 'b" shows "(P \<or> Q) \<oplus>\<^sub>p x = (P \<oplus>\<^sub>p x \<or> Q \<oplus>\<^sub>p x)"
boundOutputChainAlpha: fixes p :: "(name \<times> name) list"   and xvec :: "name list"   and B :: "('d, 'e, 'f) boundOutput"   and yvec :: "name list" assumes "(p \<bullet> xvec) \<sharp>* B"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "set xvec \<subseteq> set yvec" shows "\<lparr>\<nu>*yvec\<rparr>B = \<lparr>\<nu>*p \<bullet> yvec\<rparr>p \<bullet> B"
test_bit_eq_bit: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.semiring_bit_syntax plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_bit_syntax.test_bit bit = bit"
disjoint: fixes x :: "'b"   and y :: "'b" assumes "x \<in> s"   and "y \<in> s"   and "x \<noteq> y" shows "f x \<inter> f y = {}"
a_r_coset_def: fixes G :: "('e, 'f) ring_scheme" shows "(+>\<^bsub>G\<^esub>) = (#>\<^bsub>add_monoid G\<^esub>)"
linorder_le_cases: fixes x :: "'b"   and y :: "'b" obtains "x \<le> y"   | "y \<le> x"
hadamard_on_i_dim: fixes k :: "nat" assumes "k < n" shows "hadamard_on_i k \<in> carrier_mat N N"
not_reduced_word_not_min_gallery: fixes ss :: "'a permutation list" assumes "ss \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "\<not> reduced_word (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) ss" shows "\<not> min_gallery (map (\<lambda>w. w `\<rightarrow> C0) (sums ss))"
finfun_rec_wf_aux_axioms: fixes g :: "('b, 'c) finfun" shows "finfun_rec_wf_aux (\<lambda>b. finfun_comp (Pair b) g) (\<lambda>a b c. finfun_update c a (b, finfun_apply g a))"
wf_plan_act_is_sound: fixes n :: "char list"   and args :: "object list" assumes "wf_plan_action (PAction n args)" shows "sound_opr (instantiate_action_schema (the (resolve_action_schema n)) args) (pddl_opr_to_act (instantiate_action_schema (the (resolve_action_schema n)) args))"
dim_ty_sem_Amount_def: fixes uu :: "N itself" shows "dim_ty_sem uu = \<^bold>N"
iterates_above_ge: fixes y :: "'a"   and a :: "'a" assumes "y \<in> iterates_above a"   and "a \<in> Field leq" shows "(a, y) \<in> leq"
is_associative_weak_composition: shows "associative_weak_composition (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B"
constructive_security_obsf'D: fixes \<eta> :: "nat" shows "constructive_security_obsf (real_resource \<eta>) (ideal_resource \<eta>) (sim \<eta>) (\<I>_real \<eta>) (\<I>_ideal \<eta>) (\<I>_common \<eta>) (absorb (\<A> \<eta>) (obsf_converter (1\<^sub>C |\<^sub>= conv \<eta>))) (advantage (absorb (\<A> \<eta>) (obsf_converter (1\<^sub>C |\<^sub>= conv \<eta>))) (obsf_resource (sim \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal_resource \<eta>)) (obsf_resource (real_resource \<eta>)))"
pullback_commutes: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g" shows "category.commutative_square C f g (prj1 f g) (prj0 f g)"
mem_WWa_sub_carrier: fixes W :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "W \<in> WWa D g a" shows "W \<subseteq> carrier D"
measurable_stretch: fixes S :: "(real, 'a) vec set"   and m :: "'a \<Rightarrow> real" assumes "S \<in> lmeasurable" shows "(\<lambda>x. \<chi>k. m k * x $ k) ` S \<in> lmeasurable"
acute_ang_def: fixes \<alpha> :: "real" shows "acute_ang \<alpha> = (if pi / 2 < \<alpha> then pi - \<alpha> else \<alpha>)"
init_rev_wrap_7: fixes A :: "'b list set"   and P :: "'b list \<Rightarrow> bool" shows "\<exists>x\<in>A. P x \<equiv> Bex_rev_wrap A P"
components_snd_sym_preproc_subset: fixes gs :: "('t \<Rightarrow>\<^sub>0 'd) list"   and fs :: "('t \<Rightarrow>\<^sub>0 'd) list" shows "component_of_term ` Keys (set (snd (sym_preproc gs fs))) \<subseteq> component_of_term ` Keys (set gs \<union> set fs)"
card_order_csum: fixes r1 :: "('c \<times> 'c) set"   and r2 :: "('d \<times> 'd) set" assumes "card_order r1"   and "card_order r2" shows "card_order (r1 +c r2)"
ag_carrier_carrier: fixes A :: "('c, 'd) aGroup_scheme" assumes "aGroup A" shows "carrier (b_ag A) = carrier A"
subset_imp_less_mset: fixes A :: "'b multiset"   and B :: "'b multiset" assumes "A \<subset># B" shows "A < B"
plus_mpoly_def: shows "(+) \<equiv> map_fun mapping_of (map_fun mapping_of MPoly) (+)"
simps_6: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<alpha> :: "game"   and \<phi> :: "fml" shows "usubstappf \<sigma> U (\<langle> \<alpha> \<rangle> \<phi>) = (let V\<alpha> = usubstappp \<sigma> U \<alpha> in Diamondo (snd V\<alpha>) (usubstappf \<sigma> (fst V\<alpha>) \<phi>))"
mig_lv_nth: fixes i :: "nat"   and a :: "'b list"   and b :: "'b list" assumes "i < length a"   and "i < length b" shows "mig_real (a ! i) (b ! i) = mig_lv a b ! i"
finalGraph_def: fixes g :: "graph" shows "final g \<equiv> nonFinals g = []"
dagger_ext: fixes x :: "('a \<times> 'a) set" shows "x \<subseteq> x\<^sup>*"
finite_Diff2: fixes B :: "'a set"   and A :: "'a set" assumes "finite B" shows "finite (A - B) = finite A"
degree_sum_list_le: fixes ps :: "'b poly list"   and n :: "nat" assumes "\<And>p. p \<in> set ps \<Longrightarrow> degree p \<le> n" shows "degree (sum_list ps) \<le> n"
iso_tuple_cons_def: fixes isom :: "('c, 'a, 'b) tuple_isomorphism" shows "iso_tuple_cons isom = curry (abst isom)"
eval_fds_newman_conv_infsetsum_1: fixes s :: "complex" assumes "1 < Re s" shows "eval_fds fds_newman s = (\<Sum>\<^sub>ap | prime p. complex_of_real (ln (real p) / real p) * hurwitz_zeta (real p) s)"
map_simp: fixes f :: "J.Arr" assumes "J.arr f" shows "\<chi>.A.map f = a"
fold_Cons: fixes f :: "'l \<Rightarrow> 'k \<Rightarrow> 'k"   and x :: "'l"   and xs :: "'l list" shows "fold f (x # xs) = fold f xs \<circ> f x"
digraph_op: shows "digraph \<alpha> (op_dg \<BB>)"
complete_imp_full: fixes t :: "'b tree23" assumes "complete t" shows "full (height t) t"
finer_nchotomy: fixes I0 :: "'a set"   and P :: "'a set set"   and Q :: "'a set set"   and I :: "'a set"   and II :: "'a set" assumes "part I0 P"   and "part I0 Q"   and "finer P Q"   and "I \<in> P"   and "II \<in> Q" shows "I \<subseteq> II \<or> I \<inter> II = {}"
Cauchy_converges_subseq: fixes u :: "nat \<Rightarrow> 'a"   and r :: "nat \<Rightarrow> nat"   and l :: "'a" assumes "Cauchy u"   and "strict_mono r"   and "(u \<circ> r) \<longlonglongrightarrow> l" shows "u \<longlonglongrightarrow> l"
Domain_rtrancl: fixes R :: "('b \<times> 'b) set" shows "Domain (R\<^sup>*) = UNIV"
Pivot'_axioms: shows "Pivot' TYPE('a) eq_idx_for_lvar pivot_eq subst_var"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_same a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> wt t \<ge>\<^sub>p wt s \<and> head t = head s \<and> (\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)))"
simps_4: fixes h :: "'a \<Rightarrow> 'b list \<Rightarrow> bool"   and \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm"   and xs :: "'b list" shows "interpret h (And \<phi>\<^sub>1 \<phi>\<^sub>2) xs = (interpret h \<phi>\<^sub>1 xs \<and> interpret h \<phi>\<^sub>2 xs)"
assms_1: shows "1 / p + 1 / q = 1 / r"
term_order_rel_alt_def: shows "term_order_rel = lexord (p2rel char.lexordp)"
assms_1: shows "\<langle>x\<rangle> \<Turnstile>\<^sub>c\<^sub>s set (mat_x_leq_vec A b)"
enum_enumerator: fixes i :: "nat"   and j :: "nat" shows "Enumerator.enum i j = enumerator i j"
build_invar: fixes n :: "nat"   and seed :: "natural \<times> natural" shows "invar (fst (local.build n seed))"
mnhelper3: shows "of_bl (to_bl b) * 2 ^ n < 2 ^ m"
dim_col: fixes A :: "'a mat"   and i :: "nat" shows "dim_vec (col A i) = dim_row A"
simps_1: fixes x1 :: "nat"   and y1 :: "nat" shows "(Const x1 = Const y1) = (x1 = y1)"
f'I: shows "f' (x, y) o\<^sub>L embed2_blinfun \<in> invertibles_blinfun"
prefixes_set: fixes w :: "'b llist" shows "{v. v \<le> w} = {ltake k w |k. k \<le> llength w}"
sees_methods_decl_mono: fixes P :: "'a prog"   and C' :: "String.literal"   and C :: "String.literal"   and Mm :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "P \<turnstile> C sees_methods Mm" shows "\<exists>Mm' Mm\<^sub>2. P \<turnstile> C' sees_methods Mm' \<and> Mm' = Mm ++ Mm\<^sub>2 \<and> (\<forall>M m D. Mm\<^sub>2 M = \<lfloor>(m, D)\<rfloor> \<longrightarrow> P \<turnstile> D \<preceq>\<^sup>* C)"
Inv_HcompNml: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Can t"   and "Can u"   and "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Inv (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = Inv t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Inv u"
exhaust_disc: fixes prot_atom :: "'e prot_atom" obtains "prot_atom.is_Atom prot_atom"   | "prot_atom = Value"   | "prot_atom = SetType"   | "prot_atom = AttackType"   | "prot_atom = Bottom"   | "prot_atom = OccursSecType"
simps_2: fixes g :: "'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" shows "iter g a (Suc n) = g (iter g a n)"