hcju/isabelleps huggingface dataset:
simps_16: fixes V :: "'a"   and T :: "ty"   and vo :: "'b val option"   and e :: "('a, 'a, 'b) exp" shows "syncvars {V:T=vo; e} = syncvars e"
prems_3: fixes x :: "complex" assumes "x \<in> interior (convex hull {a, b, c}) - S" shows "f field_differentiable at x"
lift_SD_strict_iff: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and agents' :: "'a set"   and alts' :: "'b set"   and p :: "'b pmf"   and q :: "'b pmf"   and i :: "'a" assumes "pref_profile_wf agents alts R"   and "agents \<subseteq> agents'"   and "alts \<subseteq> alts'"   and "alts \<noteq> {}"   and "agents \<noteq> {}"   and "finite alts'"   and "p \<in> lotteries_on alts"   and "q \<in> lotteries_on alts"   and "i \<in> agents" shows "p \<prec>[SD (lift_pref_profile agents alts agents' alts' R i)] q = p \<prec>[SD (R i)] q"
fun_upd_other: fixes z :: "'g"   and x :: "'g"   and f :: "'g \<Rightarrow> 'h"   and y :: "'h" assumes "z \<noteq> x" shows "(f(x := y)) z = f z"
distinct_1: fixes x1 :: "nat" shows "pnat x1 \<noteq> pnat.PInfty"
fps_exp_ii_sin_cos: fixes c :: "complex" shows "fps_exp (\<i> * c) = fps_cos c + fps_const \<i> * fps_sin c"
bind_strict_mono_strong2: fixes m :: "('b, 'c) state"   and f :: "'c \<Rightarrow> ('b, 'd) state" assumes "strict_mono_state m"   and "\<And>x s s'. run_state m s = (x, s') \<Longrightarrow> mono_state (f x)" shows "strict_mono_state (m \<bind> f)"
subgroup_generated_group_carrier: shows "subgroup_generated (Z 1) (carrier (Z 1)) = Z 1"
arrE: fixes f :: "'c \<times> 'c" assumes "CopxC.arr f" obtains "Cop.arr (fst f) \<and> C.arr (snd f)"
codomain_unique: fixes b :: "'b"   and f :: "'b"   and b' :: "'b" assumes "b \<in> B.codomains f"   and "b' \<in> B.codomains f" shows "b = b'"
span_subsetI: fixes ws :: "'a vec set"   and us :: "'a vec set" assumes "ws \<subseteq> carrier_vec (n + m)"   and "us \<subseteq> MN.span ws" shows "MN.span us \<subseteq> MN.span ws"
t_reflD_1: fixes x :: "'a"   and y :: "'a" assumes "t x y" shows "x \<in> A"
simps_3: fixes x1 :: "int" shows "Any \<noteq> Const x1"
rhs_eq_val: fixes v :: "(nat, 'a) mapping"   and e :: "nat \<times> linear_poly"   and x :: "nat"   and c :: "'a" assumes "\<langle>v\<rangle> \<Turnstile>\<^sub>e e" shows "rhs_eq_val v x c e = rhs e \<lbrace> \<langle>v\<rangle>(x := c) \<rbrace>"
f_Exec_N_eq_f_Exec_Stream_Init_nth: fixes n :: "nat"   and xs :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "n \<le> length xs" shows "f_Exec_Comp_N trans_fun n xs c = f_Exec_Comp_Stream_Init trans_fun xs c ! n"
suminf_cmult_indicator: fixes A :: "nat \<Rightarrow> 'b set"   and x :: "'b"   and i :: "nat"   and f :: "nat \<Rightarrow> ennreal" assumes "disjoint_family A"   and "x \<in> A i" shows "(\<Sum>n. f n * indicator (A n) x) = f i"
fst_set_def: fixes s :: "('b \<times> 'c) set" shows "fst_set s = {a. \<exists>b. (a, b) \<in> s}"
dickson_less_pD1: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and d :: "'d \<Rightarrow> nat"   and m :: "nat"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and q :: "'c \<Rightarrow>\<^sub>0 'f" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "gd_term.dickson_less_p pair_of_term ord_term_strict d m p q" shows "p \<in> gd_term.dgrad_p_set pair_of_term d m"
bij_betw_mset_set_card: fixes A :: "'b set"   and k :: "nat" assumes "finite A" shows "bij_betw mset_set {X. X \<subseteq> A \<and> card X = k} {M. M \<subseteq># mset_set A \<and> size M = k}"
simps_1: fixes e :: "'c"   and a :: "'d" shows "nodeToList (Tip e a) = [(e, a)]"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "\<And>uu_ a ts. (\<And>z. z \<in># mset ts \<Longrightarrow> P z) \<Longrightarrow> P (Node uu_ a ts)" shows "P a0"
inf_mono: fixes a :: "'a::type"   and c :: "'a::type"   and b :: "'a::type"   and d :: "'a::type" assumes "(a::'a::type) \<sqsubseteq> (c::'a::type)"   and "(b::'a::type) \<sqsubseteq> (d::'a::type)" shows "(a::'a::type) \<sqinter> (b::'a::type) \<sqsubseteq> (c::'a::type) \<sqinter> (d::'a::type)"
point_is_point: fixes x :: "'a" shows "point x = is_point x"
Basic: fixes H :: "('a, 'b) fm set"   and P :: "'b"   and ts :: "'a tm list" assumes "Hintikka H"   and "\<^bold>\<ddagger> P ts \<in> H" shows "(\<^bold>\<not> \<^bold>\<ddagger> P ts) \<notin> H"
idinfo_mpair: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and X :: "msg"   and Y :: "msg" assumes "s\<^sub>0 \<Turnstile> s"   and "\<langle>n, \<lbrace>X, Y\<rbrace>\<rangle> \<in> spied s" shows "key_sets \<lbrace>X, Y\<rbrace> (crypts (Log -` spied s)) \<noteq> {}"
distinct_3: fixes x11 :: "CAttId"   and x12 :: "nat"   and x31 :: "Arraytype"   and x32 :: "nat" shows "objLoc x11 x12 \<noteq> arrLenLoc x31 x32"
map_M_pure_I: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> ('b, 'c, 'd) prog"   and h :: "'b" assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (f x) h" shows "pure (map_M f xs) h"
R_def: shows "R \<equiv> \<lambda>z. 3 * exp (pi * exp (pi * (2 + 2 * cmod z + 12)))"
LEX_eq_2: fixes dg :: "bool"   and ps :: "bool" shows "nat_term_order_eq Term_Order.LEX DRLEX dg ps = False"
prod_topology_euclidean: shows "prod_topology euclidean euclidean = euclidean"
eigenvector_def: fixes A :: "'b mat"   and v :: "'b vec"   and k :: "'b" shows "eigenvector A v k = (v \<in> carrier_vec (dim_row A) \<and> v \<noteq> 0\<^sub>v (dim_row A) \<and> A *\<^sub>v v = k \<cdot>\<^sub>v v)"
scalar_prod_smult_left: fixes w :: "'b vec"   and v :: "'b vec"   and k :: "'b" assumes "dim_vec w = dim_vec v" shows "(k \<cdot>\<^sub>v w) \<bullet> v = k * (w \<bullet> v)"
learn_totalI: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "environment \<psi> U s"   and "\<And>f. f \<in> U \<Longrightarrow> \<exists>i. \<psi> i = f \<and> (\<exists>n\<^sub>0. \<forall>n\<ge>n\<^sub>0. s (f \<triangleright> n) \<down>= i)"   and "\<And>f n. f \<in> U \<Longrightarrow> \<psi> (the (s (f \<triangleright> n))) \<in> \<R>" shows "learn_total \<psi> U s"
remove_all_cycles_subs: fixes xs :: "'b list"   and ys :: "'b list" shows "set (remove_all_cycles xs ys) \<subseteq> set ys"
simps_2: fixes et :: "('var, 'val, 'ret, 'pname) edge_kind"   and ets :: "('var, 'val, 'ret, 'pname) edge_kind list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" shows "transfers (et # ets) s = transfers ets (transfer et s)"
ex_f: shows "\<exists>f. u = #v * #w + Polynomial.smult (of_nat q) (#f)"
shift_snth: fixes xs :: "'c list"   and s :: "'c stream"   and n :: "nat" shows "(xs @- s) !! n = (if n < length xs then xs ! n else s !! (n - length xs))"
assms_2: fixes x :: "complex"   and n :: "nat" assumes "x \<in> S" shows "(f n has_field_derivative f' n x) (at x)"
alice_def: fixes \<phi> :: "complex Matrix.mat" shows "alice \<phi> \<equiv> (H \<Otimes> Quantum.Id 2) * ((CNOT \<Otimes> Quantum.Id 1) * (\<phi> \<Otimes> |\<beta>\<^sub>0\<^sub>0\<rangle>))"
image_Pow_surj: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'b set" assumes "f ` A = B" shows "(`) f ` Pow A = Pow B"
has_path_has: 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 D :: "char list"   and C :: "char list"   and Ds :: "char list list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "char list list \<times> expr"   and Cs :: "char list list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> Path D to C via Ds "   and "P \<turnstile> C has M = (Ts, T, m) via Cs"   and "wf_prog wf_md P" shows "P \<turnstile> D has M = (Ts, T, m) via Ds @\<^sub>p Cs"
fds_divisor_count: shows "fds divisor_count = fds_zeta\<^sup>2"
nth_weak_ranking_index_2: fixes x :: "'a" assumes "x \<in> carrier" shows "x \<in> weak_ranking le ! weak_ranking_index x"
create_character_data_preserves_wellformedness_1: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and "text" :: "char list"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>h h'"   and "type_wf h"   and "known_ptrs h" shows "heap_is_wellformed h'"
HVZK_unfold1: assumes "\<Sigma>1.\<Sigma>_protocol" shows "\<forall>h w e. (h, w) \<in> Rel1 \<longrightarrow> e \<in> carrier L \<longrightarrow> \<Sigma>1.R h w e = \<Sigma>1.S h e"
bin_nth_take_bit_iff: fixes n :: "nat"   and c :: "int"   and k :: "nat" shows "bit (take_bit n c) k = (k < n \<and> bit c k)"
sum: shows "M.sumlist (map (\<lambda>j. \<mu> i j \<cdot>\<^sub>v gso j) [0..<i]) \<in> carrier_vec n"
simps: fixes p :: "uint64"   and x :: "uint64"   and n :: "uint64" shows "power_p64 p x n = (if n = 0 then 1 else let rec = power_p64 p (mult_p64 p x x) (n >> 1) in if n AND 1 = 0 then rec else mult_p64 p rec x)"
compactin_subspace: fixes X :: "'d topology"   and S :: "'d set" shows "compactin X S = (S \<subseteq> topspace X \<and> compact_space (subtopology X S))"
add_neg_neg: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subset># {#}"   and "b \<subset># {#}" shows "a + b \<subset># {#}"
inducedhomgsurjec: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "gsurj\<^bsub>F,G\<^esub> f" shows "gsurj\<^bsub>(F / gker\<^bsub>F,G\<^esub> f),G\<^esub> (f\<dieresis>\<^bsub>F,G\<^esub>)"
sec_id_pubk_less_rel: fixes m :: "nat"   and m' :: "nat"   and n :: "nat" assumes "{m, m'} \<subseteq> bad_sec_prik"   and "(n, m) \<notin> bad_id"   and "(n, m') \<in> bad_id" shows "s\<^sub>0 \<Turnstile> sec_id_pubk_less n"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "int \<Rightarrow> 'a"   and f2 :: "char list \<Rightarrow> 'a"   and f3 :: "aexp \<Rightarrow> aexp \<Rightarrow> 'a"   and aexp :: "aexp" shows "P (case aexp of N x \<Rightarrow> f1 x | V x \<Rightarrow> f2 x | Plus x xa \<Rightarrow> f3 x xa) = ((\<forall>x1. aexp = N x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. aexp = V x2 \<longrightarrow> P (f2 x2)) \<and> (\<forall>x31 x32. aexp = Plus x31 x32 \<longrightarrow> P (f3 x31 x32)))"
IFSC_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.IFSC Bet Cong A B C D A' B' C' D' \<equiv> Bet A B C \<and> Bet A' B' C' \<and> Cong A C A' C' \<and> Cong B C B' C' \<and> Cong A D A' D' \<and> Cong C D C' D'"
nat_mod_eq: fixes b :: "nat"   and n :: "nat"   and a :: "nat" assumes "b < n"   and "a mod n = b mod n" shows "a mod n = b"
Le_Le_dbm_lt_D: fixes a :: "'c"   and b :: "'c" assumes "Le a \<prec> Lt b" shows "a < b"
swapidseq_ext_extension: fixes A :: "'a set"   and n :: "nat"   and p :: "'a \<Rightarrow> 'a"   and B :: "'a set"   and m :: "nat"   and q :: "'a \<Rightarrow> 'a" assumes "swapidseq_ext A n p"   and "swapidseq_ext B m q"   and "A \<inter> B = {}" shows "swapidseq_ext (A \<union> B) (n + m) (p \<circ> q)"
Sup_bot_conv_1: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Sup A = bot) = (\<forall>x\<in>A. x = bot)"
indexed_eval_is_weak_ring_morphism: fixes L :: "('d multiset \<Rightarrow> 'c multiset \<Rightarrow> 'a) ring"   and i :: "'d"   and q :: "('d multiset \<Rightarrow> 'c multiset \<Rightarrow> 'a) list" assumes "field L"   and "\<And>P. P \<in> carrier L \<Longrightarrow> Hom.carrier_coeff P"   and "\<And>P. P \<in> carrier L \<Longrightarrow> Hom.index_free P i"   and "\<zero>\<^bsub>L\<^esub> = Hom.indexed_const \<zero>\<^bsub>image_ring (eval_pmod q) (poly_ring L)\<^esub>"   and "q \<in> carrier (poly_ring L)" shows "weak_ring_morphism (Hom.eval_pmod L i q) (PIdl\<^bsub>poly_ring L\<^esub> q) (poly_ring L)"
mono_def: fixes f :: "'a \<Rightarrow> 'b" shows "mono f = (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
one_element_vec_carrier: fixes n :: "nat"   and e :: "'a" shows "one_element_vec n e \<in> carrier_vec n"
Rep_preal_inject: fixes x :: "preal"   and y :: "preal" shows "(Rep_preal x = Rep_preal y) = (x = y)"
extend1_to_2: fixes S :: "('a, 'b) sequent"   and r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent" shows "extendRule2 S S r = extendRule S r"
rb_spp_aux_3: fixes fs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and rword_strict :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and dgrad :: "'a \<Rightarrow> nat"   and z :: "nat" assumes "is_strict_rewrite_ord fs rword_strict"   and "dickson_grading dgrad"   and "distinct fs"   and "0 \<notin> set fs" shows "set (map snd (fst (fst (rb_spp_aux fs rword_strict (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), z))))) \<subseteq> punit.dgrad_p_set dgrad (qpm_inf_term.dgrad_max fs dgrad)"
bpE_2: fixes P :: "'a set set" assumes "bp P" shows "{} \<notin> P"
comp_simp: fixes g :: "'a"   and f :: "'a" assumes "S.seq g f" shows "g \<cdot>\<^sub>S f = g \<cdot> f"
simps_2: fixes delta :: "'e"   and inp :: "'a \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm option"   and binp :: "'b \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs option" shows "qSkel (qOp delta inp binp) = Branch (lift qSkel inp) (lift qSkelAbs binp)"
fls_X_intpow_times_fls_X_intpow: fixes i :: "int"   and j :: "int" shows "fls_X_intpow i * fls_X_intpow j = fls_X_intpow (i + j)"
key_map_is_coherent: fixes V :: "'a set set"   and U :: "'a set set"   and s :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "V \<subseteq> U"   and "is_zariski_open U"   and "is_zariski_open V"   and "\<pp> \<in> V"   and "s \<in> \<O> U" shows "(local.key_map V \<circ> sheaf_spec_morphisms U V) s = local.key_map U s"
intro: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "Group.group G"   and "Group.group H"   and "group_hom_axioms G H h" shows "group_hom G H h"
ir_valid_While3: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c :: "com"   and k :: "nat"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_valid (get_back P b c k) (WHILE b DO c) c' Q" shows "ir_valid (\<lambda>s s'. get_back P b c k s s' \<and> bval b s) c c' (\<lambda>t t'. \<exists>s'. (c', s') \<Rightarrow> t' \<and> get_back P b c (Suc k) t s' \<and> (\<exists>u. (WHILE b DO c, t) \<Rightarrow> u \<and> Q u t'))"
addpreRT_partly_welldefined: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PRreq-:16..PRreq-:18} \<union> {PRrep-:2..PRrep-:6} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>)) \<and> (l \<in> {PRreq-:3..PRreq-:17} \<longrightarrow> oip \<xi> \<in> kD (rt \<xi>)))"
sum_of_two_squares: fixes n :: "nat" shows "is_sum2sq_nat n = (\<forall>k. prime (4 * k + 3) \<longrightarrow> even (multiplicity (4 * k + 3) n))"
empty_continuous_imp_countably_additive: fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive M f"   and "Measure_Space.additive M f"   and "\<forall>A\<in>M. f A \<noteq> \<infinity>"   and "\<And>A. \<lbrakk>range A \<subseteq> M; decseq A; \<Inter> (range A) = {}\<rbrakk> \<Longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0" shows "countably_additive M f"
OclAllInstances_at_pre_def: shows "OclAllInstances_at_pre = OclAllInstances_generic fst"
plusi_def: fixes e :: "complex" shows "plusi e \<equiv> e + inverse e"
dom_uniq_2: fixes \<sigma> :: "'Var \<Rightarrow> 'Val option" assumes "dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k" shows "dom_g2 = dom (g2 \<sigma>)"
mono_implies_arr: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "local.mono f" shows "arr f"
approx: shows "P,\<Phi> |- (None, h, (stk, loc, C, M, pc) # frs) [ok]"
index_add_mat_1: fixes i :: "nat"   and B :: "'d mat"   and j :: "nat"   and A :: "'d mat" assumes "i < dim_row B"   and "j < dim_col B" shows "(A + B) $$ (i, j) = A $$ (i, j) + B $$ (i, j)"
abs_eq: fixes y :: "'b list set" assumes "eq_onp (\<lambda>xss. [] \<in> xss \<and> downset xss) y y" shows "paths (Abs_ttree y) = y"
ldropn_lappend2: fixes xs :: "'c llist"   and n :: "nat"   and ys :: "'c llist" assumes "llength xs \<le> enat n" shows "ldropn n (lappend xs ys) = ldropn (n - the_enat (llength xs)) ys"
radj_adj: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "f \<stileturn> g" shows "g = radj f"
i_set_mod_0_mult_in_i_set_mult: fixes I :: "nat set"   and m :: "nat"   and k :: "nat" assumes "I \<in> i_set"   and "m mod k = 0" shows "I \<otimes> m \<in> i_set_mult k"
store_typing_imp_mem_length_eq: fixes s :: "s"   and \<S> :: "s_context" assumes "store_typing s \<S>" shows "length (s.mem s) = length (s_mem \<S>)"
zero_le_double_add_iff_zero_le_single_add: 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 a :: "'c" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq zero (plus a a) = less_eq zero a"
sgn_sgn: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "sgn (sgn a) = sgn a"
naturality: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>"   and "C.ide f" shows "\<sigma>\<^sub>1 f \<cdot>\<^sub>D (G f \<star>\<^sub>D \<Gamma> a) = (\<Gamma> b \<star>\<^sub>D F f) \<cdot>\<^sub>D \<rho>\<^sub>1 f"
pcr_fset_def: fixes T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "pcr_fset T \<equiv> rel_set T OO cr_fset"
add_sum_orient: fixes f :: "'d \<Rightarrow> 'c"   and k :: "'d"   and j :: "'d"   and l :: "'d" shows "sum f {k..<j} + sum f {l..<k} = sum f {l..<k} + sum f {k..<j}"
converse3pI: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'a"   and b :: "'b"   and c :: "'c" assumes "r a b c" shows "converse3p r c b a"
get_dom_component_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_dom_component ptr"
is_array_list_def: fixes l :: "'a list" shows "is_array_list l \<equiv> \<lambda>(a, n). \<exists>\<^sub>Al'. a \<mapsto>\<^sub>a l' * \<up> (n \<le> length l' \<and> l = take n l' \<and> 0 < length l')"
face_distance_eq_chamber_distance_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set"   and D :: "'b set"   and z :: "'b set" assumes "ChamberComplexEndomorphism X f"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<noteq> D"   and "order.greater_eq C z"   and "ChamberComplex.face_distance X (f ` z) (f ` D) = ChamberComplex.face_distance X z D"   and "ChamberComplex.face_distance X z D = ChamberComplex.chamber_distance X C D" shows "ChamberComplex.face_distance X (f ` z) (f ` D) = ChamberComplex.chamber_distance X (f ` C) (f ` D)"
Brouwer_degree2_id: fixes p :: "nat" shows "Brouwer_degree2 p id = 1"
WT_oracle1: fixes s1 :: "'s1" shows "\<I> \<turnstile>c oracle1 s1 \<surd>"
ln3: shows "ln 3 \<in> {1..10 / 9}"
tllist_of_llist_eq_lappendt_conv: fixes a :: "'d"   and xs :: "'c llist"   and ys :: "'c llist"   and zs :: "('c, 'd) tllist" shows "(tllist_of_llist a xs = lappendt ys zs) = (\<exists>xs' a'. xs = lappend ys xs' \<and> zs = tllist_of_llist a' xs' \<and> (lfinite ys \<longrightarrow> a = a'))"
Sup_unit_def: fixes uu :: "unit set" shows "Sup uu = ()"
iterator_to_ordered_iterator_quicksort_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and it :: "('d list \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'd list \<Rightarrow> 'd list) \<Rightarrow> 'd list \<Rightarrow> 'd list" shows "iterator_to_ordered_iterator_quicksort R it = iterator_to_ordered_iterator (quicksort_by_rel R []) it"
select_convs_2: fixes nodes :: "'a set"   and edges :: "('a \<times> 'b \<times> 'a) set"   and more :: "'c" shows "edges \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = edges"
assms_2: shows "f \<in> borel_measurable (lebesgue_on S)"
in_lset_conv_lnth: fixes x :: "'b"   and xs :: "'b llist" shows "(x \<in> lset xs) = (\<exists>n. enat n < llength xs \<and> lnth xs n = x)"
bot_less: fixes a :: "'a" shows "(a \<noteq> bot) = (bot < a)"
monoid_axioms: shows "Group_Theory.monoid carrier_local_ring_at add_local_ring_at zero_local_ring_at"
degree_sum_less: fixes n :: "nat"   and f :: "nat \<Rightarrow> real mpoly"   and var :: "nat"   and x :: "nat" assumes "\<forall>i\<in>{0..n}. MPoly_Type.degree (f i) var < x" shows "MPoly_Type.degree (sum f {0..n}) var < x"
ivl_not_trivial_limit_within: fixes s :: "real set"   and a :: "real"   and b :: "real"   and x :: "real" assumes "finite s"   and "a < b"   and "x \<in> {a..b} - s" shows "at x within {a..b} - s \<noteq> bot"
le_refl: fixes a :: "'a" assumes "a \<in> carrier D" shows "a \<preceq> a"
substd_simplex: fixes d :: "'a set" assumes "d \<subseteq> Basis" shows "convex hull insert (0::'a) d = {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) d \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)}"
enn2real_mult: fixes a :: "ennreal"   and b :: "ennreal" shows "enn2real (a * b) = enn2real a * enn2real b"
prems_3: shows "Span K (u # Us) \<inter> Span K Vs = {\<zero>}"
pref_pow_conv: fixes x :: "'b list"   and r :: "'b list" shows "(\<exists>k. x \<le>p r \<^sup>@ k) = (\<exists>k z. r \<^sup>@ k \<cdot> z = x \<and> z \<le>p r)"
F6: shows "\<forall>x\<^sub>1. analz (synth (analz x\<^sub>1)) = synth (analz x\<^sub>1)"
one_side_symmetry: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.OS Bet P Q A B" shows "Tarski_neutral_dimensionless.OS Bet P Q B A"
sep_implD: fixes x :: "'a \<Rightarrow> bool"   and y :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "(x \<longrightarrow>\<^sup>* y) h" shows "\<forall>h' h''. h \<oplus> h' = Some h'' \<and> x h' \<longrightarrow> y h''"