hcju/isabelleps huggingface dataset:
cong: fixes init :: "('e \<times> 'a \<times> 'f) spmf"   and "oracle" :: "'e \<Rightarrow> 'f \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'f) spmf"   and eval :: "'e \<Rightarrow> 'a \<Rightarrow> 'f \<Rightarrow> 'c \<Rightarrow> bool spmf" shows "guessing_many_one.advantage_multi init oracle eval \<equiv> guessing_many_one.advantage_multi init oracle eval"
S_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b" shows "??.Applicative.S f g x \<equiv> f x (g x)"
SIGMAF_2: shows "\<Sigma>' = (Q', W', rcm')"
star_sim2: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "\<lparr>z\<rparr>x\<lparr>y \<cdot> z\<rparr>" shows "\<lparr>z\<rparr>x\<^sup>\<star>\<lparr>y\<^sup>\<star> \<cdot> z\<rparr>"
exhaust: fixes y :: "('a, 'b) S_action" obtains x1 :: "'a" where "y = S_action.Act x1"   | x2 :: "'b" where "y = S_action.Pred x2"
ChiRel_OneAncestor: fixes S1 :: "'a"   and U :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and S2 :: "'a" assumes "(S1, U) \<in> ChiRel A"   and "(S2, U) \<in> ChiRel A" shows "S1 = S2"
od_fm_fun_inPr0: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and C :: "('a set \<times> ('a \<Rightarrow> 'g)) set"   and y :: "'a set"   and z :: "'a" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "Chain (od_fm_fun R M N f H) C"   and "C \<noteq> {}"   and "\<exists>b. (y, b) \<in> C"   and "z \<in> y" shows "z \<in> fgs R M (\<Union> {a. \<exists>b. (a, b) \<in> C})"
assms_1: shows "rewrite_prog f n p1 \<Sigma>1 = Ok (p2, \<Sigma>2)"
H_def: fixes p :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set" shows "rel_antidomain_kleene_algebra.H p x q = (rel_antidomain_kleene_algebra.t p ; x \<subseteq> x ; rel_antidomain_kleene_algebra.t q)"
g_1_lemma: fixes p :: "nat"   and q :: "nat"   and res :: "int \<times> int" assumes "QR p q"   and "0 \<le> fst res"   and "fst res < int p"   and "0 \<le> snd res"   and "snd res < int q" shows "QR.P_1 p q res (QR.g_1 p q res)"
left_idem: fixes a :: "'c"   and b :: "'c" shows "a \<sqinter> (a \<sqinter> b) = a \<sqinter> b"
add_mset_remove_trivial_eq: fixes N :: "'b multiset"   and a :: "'b" shows "(N = a, N - (a, {#})) = (a \<in># N)"
sd_path_id: fixes x :: "'b"   and y :: "'b"   and p :: "('b \<times> 'b) list"   and w :: "'b" assumes "(x, y) \<notin> set p"   and "(y, x) \<notin> set p" shows "sd_path (x, y) w p = p"
subprob_indicator: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr" assumes "subprob_cexpr V V' \<Gamma> e1"   and "nonneg_cexpr (V \<union> V') \<Gamma> e1"   and "\<Gamma> \<turnstile>\<^sub>c e1 : REAL"   and "\<Gamma> \<turnstile>\<^sub>c e2 : BOOL"   and "free_vars e1 \<subseteq> V \<union> V'"   and "free_vars e2 \<subseteq> V \<union> V'" shows "subprob_cexpr V V' \<Gamma> (e1 *\<^sub>c \<langle>e2\<rangle>\<^sub>c)"
poly_inf_0_iff_1: fixes p :: "'a poly" shows "(poly_inf p = (0::'a)) = (p = 0)"
vars_iff_subtermeq: fixes x :: "'a"   and t :: "('b, 'a) Term.term" shows "(x \<in> fv t) = (Var x \<sqsubseteq> t)"
residual_countable_network: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "countable_network \<Delta>"   and "countable_network.wf_residual_network \<Delta>"   and "flow \<Delta> f" shows "countable_network (countable_network.residual_network \<Delta> f)"
normalizability_def: fixes A :: "('b \<times> 'b) set" shows "A\<^sup>! = {(a, b). (a, b) \<in> A\<^sup>* \<and> b \<in> NF A}"
assms_2: shows "op\<^sub>1 \<in> set (\<Pi>\<^sub>\<O>)"
execl_last: fixes P :: "instr list"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list"   and k :: "nat" assumes "P \<Turnstile> cfs"   and "k < length cfs"   and "fst (cfs ! k) \<notin> {0..<size P}" shows "length cfs - 1 = k"
wf_C_mdecl_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list" shows "wf_C_mdecl P C \<equiv> \<lambda>(M, Ts, T, pns, body). length Ts = length pns \<and> distinct pns \<and> this \<notin> set pns \<and> P,[this \<mapsto> Class C, pns [\<mapsto>] Ts] \<turnstile> body :: T \<and> \<D> body \<lfloor>{this} \<union> set pns\<rfloor>"
cospan_\<nu>\<pi>: shows "C.cospan g.leg0 f.leg1"
fst3_simp: fixes a :: "'a"   and b :: "'b"   and c :: "'c" shows "fst3 (a, b, c) = a"
wqo_on_Sigma: fixes P1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A1 :: "'a set"   and P2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A2 :: "'b set" assumes "wqo_on P1 A1"   and "wqo_on P2 A2" shows "wqo_on (prod_le P1 P2) (A1 \<times> A2)"
simps_1: fixes x :: "char list"   and ya :: "char list" shows "(Iface x = Iface ya) = (x = ya)"
Tr'_def: shows "Tr' = {t. \<exists>t'\<in>Tr. t \<le> t'}"
that: fixes p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "p \<in> F"   and "is_red {p} q" shows "thesis"
eigen_vector_norm_sr: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and v :: "(complex, 'a) Finite_Cartesian_Product.vec"   and \<alpha> :: "complex" assumes "perron_frobenius A"   and "eigen_vector (HMA_Connect.map_matrix complex_of_real A) v \<alpha>" shows "cmod \<alpha> \<le> perron_frobenius.r A (perron_frobenius.z A)"
simps_2: fixes n :: "nat" shows "Nat_Bijection.list_decode (Suc n) = (case prod_decode n of (x, y) \<Rightarrow> x # Nat_Bijection.list_decode y)"
nres_relI: fixes c :: "'a nres"   and R :: "('a \<times> 'b) set"   and a :: "'b nres" assumes "c \<le> \<Down> R a" shows "(c, a) \<in> \<langle>R\<rangle>nres_rel"
stream_enc_enc: fixes I :: "(nat + nat set) list"   and v :: "('a \<times> bool list) list"   and w :: "'a list" assumes "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P"   and "v \<in> enc (w, I)" shows "stream_enc (w, I) = v @- sconst (any, replicate (length I) False)"
fn_ref: fixes n :: "nat" shows "f^n \<bottom> \<sqsubseteq> f^n \<bottom>"
assms: fixes j :: "'a" assumes "j \<in> Basis" shows "((\<lambda>n. blinfun_apply (b n) j) \<longlongrightarrow> blinfun_apply a j) F"
preserves_inverse_arrows: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "CC.inverse_arrows f g" shows "CC.inverse_arrows (CC.map f) (CC.map g)"
Dup: fixes A :: "'b set"   and n :: "nat"   and p :: "('c, 'b) fm"   and ps :: "('c, 'b) fm list"   and a :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list" assumes "A, n \<turnstile> (p # ps, a) # branch"   and "\<not> new p a ((ps, a) # branch)" shows "A, n \<turnstile> (ps, a) # branch"
word_ctz_le: fixes w :: "'b word" shows "word_ctz w \<le> LENGTH('b)"
AE_pair: fixes Q :: "'a \<times> 'b \<Rightarrow> bool" assumes "almost_everywhere (M1 \<Otimes>\<^sub>M M2) Q" shows "AE x in M1. AE y in M2. Q (x, y)"
updated_call: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and c :: "char list"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>(c \<mapsto> rs),\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t"   and "matches \<gamma> m p" shows "\<Gamma>(c \<mapsto> rs),\<gamma>,p\<turnstile> [Rule m (Call c)] \<Rightarrow>\<^sub>z t"
rel_compp_Grp: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_option R = (BNF_Def.Grp {x. set_option x \<subseteq> {(x, y). R x y}} (map_option fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. set_option x \<subseteq> {(x, y). R x y}} (map_option snd)"
simps_20: fixes vb :: "nat"   and va :: "(val \<times> val) fset" shows "VNat vb \<sqsubseteq> val.Fun va = False"
strict_trans2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < b"   and "b \<le> c" shows "a < c"
prime_elem_multiplicity_power_distrib: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and x :: "'b"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "x \<noteq> zero" shows "factorial_semiring.multiplicity times one p (power.power one times x n) = n * factorial_semiring.multiplicity times one p x"
case_nat_unfold: fixes x :: "'d"   and f :: "nat \<Rightarrow> 'd"   and n :: "nat" shows "case n of 0 \<Rightarrow> x | Suc x \<Rightarrow> f x \<equiv> if n = 0 then x else f (n - 1)"
o_bij: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "g \<circ> f = id"   and "f \<circ> g = id" shows "bij f"
properD_2: fixes P :: "heap \<times> nat set \<Rightarrow> bool"   and h :: "heap"   and as :: "nat set"   and h' :: "heap" assumes "proper P"   and "P (h, as)"   and "relH as h h'"   and "in_range (h', as)" shows "P (h', as)"
nat_power_nat_def: fixes a :: "nat"   and b :: "nat" shows "nat_power a b = a ^ b"
pre_get_pending_def: fixes s :: "'s" shows "pre_get_pending s \<equiv> gen_rwof s \<and> \<not> gds_is_empty_stack gds s \<and> \<not> gds_is_break gds s"
set_intros_2: fixes x2 :: "'a" shows "x2 \<in> set2_sum3 (In2 x2)"
long_division_closed_2: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "q \<in> carrier (K [X])" shows "p pmod q \<in> carrier (K [X])"
range_Some: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and a :: "'a" assumes "MaxR r" shows "MaxR.MaxR_opt r A = Some a \<longrightarrow> a \<in> A \<inter> Field r"
g: fixes u :: "'a"   and v :: "'a" shows "\<exists>w z. g ` cbox u v = cbox w z"
finite_enum_subset: fixes X :: "'b set"   and Y :: "'b set" assumes "\<And>i. i < card X \<Longrightarrow> wellorder_class.enumerate X i = wellorder_class.enumerate Y i"   and "finite X"   and "finite Y"   and "card X \<le> card Y" shows "X \<subseteq> Y"
prime_dvd_power: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a"   and n :: "nat" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "dvd.dvd times p (power.power one times x n)" shows "dvd.dvd times p x"
f: shows "\<forall>l. l \<in> L \<longrightarrow> vars\<^sub>l l \<subseteq> vars\<^sub>l\<^sub>s L"
H'_1: shows "M = X \<and> hmac \<langle>AuthTag, M\<rangle> (shrK A B) = Y"
M3_def: shows "M3 = (\<lambda>i. (f 0, f (Suc i))) ` {..<m}"
edge_preserving_subset: fixes R\<^sub>1 :: "('e \<times> 'f) set"   and R\<^sub>2 :: "('e \<times> 'f) set"   and E\<^sub>1 :: "('g \<times> 'e \<times> 'e) set"   and E\<^sub>2 :: "('g \<times> 'e \<times> 'e) set"   and G :: "('g \<times> 'f \<times> 'f) set" assumes "R\<^sub>1 \<subseteq> R\<^sub>2"   and "E\<^sub>1 \<subseteq> E\<^sub>2"   and "edge_preserving R\<^sub>2 E\<^sub>2 G" shows "edge_preserving R\<^sub>1 E\<^sub>1 G"
path_image_subset_topspace: fixes X :: "'b topology"   and g :: "real \<Rightarrow> 'b" assumes "pathin X g" shows "g ` {0..1} \<subseteq> topspace X"
gselect_edge_code_def: fixes node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "gselect_edge_code node_eq_impl \<equiv> \<lambda>x. let y' = x in case y' of (xa, xb) \<Rightarrow> select_edge_code node_eq_impl xb \<bind> (\<lambda>(xc, xd). RETURN (xc, xa, xd))"
A1_DAcc_level0: shows "DAcc level0 sA1 = {sA2}"
qstar_def: fixes x :: "'b \<times> 'c" shows "prod.qstar x = Sup_prod (range (prod.power x))"
create_character_data_is_weakly_dom_component_safe_step: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and "text" :: "char list"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'a" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>h h'"   and "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)|\<^sub>r"   and "ptr \<noteq> cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |h \<turnstile> create_character_data document_ptr text|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
assms_1: shows "\<I> \<turnstile>res res \<surd>"
listrel_refl_on: fixes A :: "'g set"   and r :: "('g \<times> 'g) set" assumes "refl_on A r" shows "refl_on (lists A) (listrel r)"
aint: fixes b :: "real"   and s :: "complex" shows "complex_set_integrable lebesgue {0..b} (laplace_integrand f s)"
powr_times_real: fixes x :: "complex"   and y :: "complex"   and z :: "complex" assumes "x \<in> \<real>"   and "y \<in> \<real>"   and "0 \<le> Re x"   and "0 \<le> Re y" shows "(x * y) powr z = x powr z * y powr z"
simple_syntactic_or_def: fixes P :: "'b \<Rightarrow> bool" shows "simple_syntactic_or P \<equiv> \<forall>t u. P (app t u) = (P t \<or> P u)"
IH: shows "reverse\<cdot>(reverse\<cdot>xs) \<sqsubseteq> xs"
IntD1: fixes c :: "'f"   and A :: "'f set"   and B :: "'f set" assumes "c \<in> A \<inter> B" shows "c \<in> A"
Call_lemma_switch_Call_body: 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 \<sigma> :: "'e"   and p :: "'d"   and Z :: "'e" assumes "\<forall>q\<in>dom \<Gamma>. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>Call q,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>Call q \<down> Normal s \<and> ((s, q), \<sigma>, p) \<in> termi_call_steps \<Gamma>} Call q {t. \<Gamma>\<turnstile> \<langle>Call q,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>Call q,Normal Z\<rangle> \<Rightarrow> Abrupt t}"   and "p \<in> dom \<Gamma>" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({\<sigma>} \<inter> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>Call p \<down> Normal s}) the (\<Gamma> p) {t. \<Gamma>\<turnstile> \<langle>Call p,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>Call p,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
uniquely_geodesic: fixes G :: "'a set"   and x :: "'a"   and y :: "'a"   and H :: "'a set" assumes "geodesic_segment_between G x y"   and "geodesic_segment_between H x y" shows "G = H"
dvd_minus_eq_mod: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<noteq> (0::'a)"   and "c dvd a - b" shows "a mod c = b mod c"
r_block_design_axioms: shows "r_block_design \<V> \<B> \<P> \<k>"
simps: fixes A_pos :: "nat set list"   and A_neg :: "nat set list"   and L :: "'b option list set list" shows "mmulti_join' A_pos A_neg L = (let Q = set (zip A_pos L) in Let (set (zip A_neg (drop (length A_pos) L))) (New_max_getIJ_wrapperGenericJoin Q))"
conga_os__out: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and C' :: "'p" assumes "A B C CongA A B C'"   and "A B OS C C'" shows "B Out C C'"
inject_1: fixes x1 :: "'e"   and y1 :: "'e" shows "(broadcast x1 = broadcast y1) = (x1 = y1)"
comm_monoid_def: fixes G :: "('b, 'c) monoid_scheme" shows "Group.comm_monoid G \<equiv> Group.monoid G \<and> Group.comm_monoid_axioms G"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_diff a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> vars_mset s \<subseteq># vars_mset t \<and> wt t = wt s \<and> head t >\<^sub>h\<^sub>d head s)"
Spy_knows_Spy_bad: fixes A :: "agent"   and evs :: "event list" assumes "A \<in> bad" shows "Key (shrK A) \<in> knows Spy evs"
dma_\<C>_vars: fixes y :: "'Var\<^sub>A"   and mem\<^sub>1 :: "'Var\<^sub>A \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var\<^sub>A \<Rightarrow> 'Val" assumes "\<forall>x\<in>\<C>_vars\<^sub>A y. mem\<^sub>1 x = mem\<^sub>2 x" shows "dma\<^sub>A mem\<^sub>1 y = dma\<^sub>A mem\<^sub>2 y"
map_poly_hom_smult: fixes c :: "'a"   and p :: "'a poly" shows "map_poly hom (Polynomial.smult c p) = Polynomial.smult (hom c) (map_poly hom p)"
singleton_matrix_add: fixes j :: "nat"   and i :: "nat"   and a :: "'a"   and b :: "'a" shows "singleton_matrix j i (a + b) = singleton_matrix j i a + singleton_matrix j i b"
borel_measurable_vimage_halfspace_component_gt: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. a < f x \<bullet> i} \<in> sets (lebesgue_on S))"
distinct_17: fixes x2 :: "('e, 'f) Term.term"   and x51 :: "('e, 'f) Term.term"   and x52 :: "('e, 'f) Term.term" shows "receive\<langle>x2\<rangle> \<noteq> delete\<langle>x51,x52\<rangle>"
Nml_HcompD_4: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml (t \<^bold>\<star> u)" shows "Nml u"
convexI: fixes S :: "'a vec set" assumes "S \<subseteq> carrier_vec n"   and "\<And>a x y. \<lbrakk>x \<in> S; y \<in> S; (0::'a) \<le> a; a \<le> (1::'a)\<rbrakk> \<Longrightarrow> a \<cdot>\<^sub>v x + ((1::'a) - a) \<cdot>\<^sub>v y \<in> S" shows "convex S"
M: shows "m \<le>\<^sub>n \<Down> R m'"
Upper_memD: fixes u :: "'a"   and L :: "('a, 'b) gorder_scheme"   and A :: "'a set"   and x :: "'a" assumes "u \<in> Upper L A"   and "x \<in> A"   and "A \<subseteq> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> u \<and> u \<in> carrier L"
simps: fixes l0 :: "'a"   and r0 :: "'b"   and l1 :: "'a"   and r1 :: "'b" shows "local.size (Bound (l0, r0) (l1, r1)) = size0 (Bound l0 l1) * size1 (Bound r0 r1)"
rep_eq: fixes x :: "'a"   and xa :: "'a sq_mtx" shows "\<c>\<o>\<l> x xa = column x (to_vec xa)"
sup_eq_bot_iff: fixes x :: "'b multiset"   and y :: "'b multiset" shows "(x \<union># y = {#}) = (x = {#} \<and> y = {#})"
antisym: fixes b :: "'c"   and a :: "'c" assumes "b \<le>\<^sub>\<K> a"   and "a \<le>\<^sub>\<K> b" shows "a = b"
integral_on_image_ubound: fixes S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> real"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec" assumes "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f (g x)"   and "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"   and "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) integrable_on S" shows "integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x))"
simps_4: fixes i :: "char list \<Rightarrow> tv"   and p :: "fm"   and q :: "fm" shows "eval i (Con' p q) = (if eval i p = eval i q then eval i p else if eval i p = Det True then eval i q else if eval i q = Det True then eval i p else Det False)"
of_complex_sub_of_complex: fixes x :: "complex"   and y :: "complex" shows "of_complex x -\<^sub>h of_complex y = of_complex (x - y)"
1: fixes a :: "('a, 'b) fmap" assumes "a \<in> A" shows "(fmrestrict_set X \<circ> fmrestrict_set Y) a = fmrestrict_set (X \<inter> Y) a"
AB: shows "\<forall>a\<in>A. \<forall>b\<in>B. a \<sqsubseteq> b \<longrightarrow> b \<sqsubseteq> a \<longrightarrow> a = b"
sup_same_context: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "y \<sqinter> z \<le> x"   and "x \<sqinter> z \<le> y" shows "x \<sqinter> z = y \<sqinter> z"
has_parametrized_subprob_densityD_2: fixes M :: "'a measure"   and N :: "'a \<Rightarrow> 'b measure"   and R :: "'b measure"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal"   and x :: "'a" assumes "has_parametrized_subprob_density M N R f"   and "x \<in> space M" shows "subprob_space (N x)"
nonneg_of_real_in_complex_cone: fixes x :: "real"   and a :: "real"   and b :: "real" assumes "0 \<le> x"   and "a \<le> 0"   and "0 \<le> b" shows "complex_of_real x \<in> complex_cone a b"
simps: fixes a :: "'form"   and i :: "'var itself" shows "(a \<in> global_assms' i) = (\<exists>v p. a = labelAtOut v (Reg p) \<and> v |\<in>| vertices \<and> nodeOf v = Assumption p)"
lm001: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set"   and X :: "'a set" shows "(P \<union> Q) || X = P || X \<union> Q || X"
distinct_set_remove_last: fixes xs :: "'c list"   and x :: "'c" assumes "distinct (xs @ [x])" shows "set xs = set (xs @ [x]) - {x}"
test_eq_conv: fixes x :: "'a" assumes "is_test x" shows "x\<^sup>\<smile> = x"
dual_inv2: shows "\<partial> \<circ> \<partial>\<^sup>- = id"
le_cases: fixes x :: "'a"   and y :: "'a" obtains "x \<preceq> y"   | "y \<preceq> x"
eval_unop_type: fixes dt :: "loc \<Rightarrow> ty option"   and unop :: "unop"   and v :: "val" shows "typeof dt (eval_unop unop v) = Some (PrimT (unop_type unop))"
ts_ord_alt: fixes x :: "'a"   and y :: "'a" shows "(!x \<sqsubseteq> !y) = (!x \<oplus> !y = !y)"
ptrm_prv_exi: fixes \<sigma> :: "'fmla"   and n :: "nat" assumes "\<sigma> \<in> ptrm n" shows "prv (exi out \<sigma>)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "integer \<Rightarrow> varType list \<Rightarrow> integer list list \<Rightarrow> 'a"   and f2 :: "varType list \<Rightarrow> 'a"   and f3 :: "'a"   and channel :: "channel" shows "P (case channel of Channel x xa xb \<Rightarrow> f1 x xa xb | HSChannel x \<Rightarrow> f2 x | InvChannel \<Rightarrow> f3) = ((\<forall>x11 x12 x13. channel = Channel x11 x12 x13 \<longrightarrow> P (f1 x11 x12 x13)) \<and> (\<forall>x2. channel = HSChannel x2 \<longrightarrow> P (f2 x2)) \<and> (channel = InvChannel \<longrightarrow> P f3))"
inf_assn_def: fixes P :: "assn"   and Q :: "assn" shows "P \<and>\<^sub>A Q \<equiv> Abs_assn (\<lambda>h. h \<Turnstile> P \<and> h \<Turnstile> Q)"
carrier_vecI: fixes v :: "'b Matrix.vec"   and n :: "nat" assumes "dim_vec v = n" shows "v \<in> carrier_vec n"
smooth_on_const: fixes k :: "enat"   and S :: "'c set"   and c :: "'d" shows "k-smooth_on S (\<lambda>x. c)"
circ_sup_one_left_unfold: fixes x :: "'a" assumes "reflexive x" shows "x\<^sup>+ = x\<^sup>\<star>"
sprefD2: fixes x :: "'b list"   and y :: "'b list" assumes "x <p y" shows "x \<noteq> y"
WT_1: shows "\<I>' \<turnstile>g run_converter conv a \<surd>"
dom_map_of_zip: fixes xs :: "'c list"   and ys :: "'d list" assumes "length xs = length ys" shows "dom (map_of (xs || ys)) = set xs"
rbt_max_opt_is_max: fixes t :: "('b, unit) RBT_Impl.rbt"   and y :: "'b" assumes "rbt_sorted t"   and "t \<noteq> rbt.Empty"   and "y \<in> set (RBT_Impl.keys t)" shows "y \<le> rbt_max_opt t"
set_product: fixes xs :: "'a list"   and ys :: "'b list" shows "set (List.product xs ys) = set xs \<times> set ys"
star_assnI: fixes P :: "assn"   and h :: "heap"   and as :: "nat set"   and Q :: "assn"   and as' :: "nat set" assumes "(h, as) \<Turnstile> P"   and "(h, as') \<Turnstile> Q"   and "as \<inter> as' = {}" shows "(h, as \<union> as') \<Turnstile> P * Q"
primeidealI: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "ideal I R"   and "cring R"   and "carrier R \<noteq> I"   and "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes>\<^bsub>R\<^esub> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I" shows "primeideal I R"
key: fixes \<eta> :: "nat" shows "callee_invariant_on (CIPHER.KEY.key_oracle \<eta> \<oplus>\<^sub>O CIPHER.KEY.key_oracle \<eta>) (pred_option (\<lambda>x. length x = \<eta>)) (\<I>_uniform UNIV (nlists UNIV \<eta>) \<oplus>\<^sub>\<I> \<I>_uniform UNIV (nlists UNIV \<eta>))"