hcju/isabelleps huggingface dataset:
add_rel_class_of: fixes U :: "'c set"   and x :: "'d"   and V :: "'c set"   and y :: "'d"   and W :: "'c set" assumes "(U, x) \<in> Sigma (target.neighborhoods (f x)) \<O>\<^sub>Y"   and "(V, y) \<in> Sigma (target.neighborhoods (f x)) \<O>\<^sub>Y"   and "W \<in> target.neighborhoods (f x)"   and "W \<subseteq> U"   and "W \<subseteq> V" shows "dlY.add_rel (dlY.class_of U x) (dlY.class_of V y) = dlY.class_of W (add_str\<^sub>Y W (\<rho>\<^sub>Y U W x) (\<rho>\<^sub>Y V W y))"
ftv_Ctxt: fixes \<Gamma> :: "(W.var \<times> tyS) list" shows "supp \<Gamma> = set (ftv \<Gamma>)"
Suc_1: obtains x :: "'a" where "x \<in> C" and "enat n \<le> esize x"
generatorE: fixes x :: "'a" assumes "x \<in> carrier G" obtains n :: "nat" where "x = \<^bold>g [^] n"
block_choices: shows "int (size (filter_mset ((\<in>) x) \<B>)) = \<B> rep x"
arr_cod_iff_arr: fixes f :: "'b" shows "C'.arr (C'.cod f) = C'.arr f"
X_def: shows "X \<equiv> mat 2 2 (\<lambda>(i, j). if i = j then 0 else 1)"
bisimilar_symp: shows "symp (\<sim>\<cdot>)"
integrableI_bounded_set_indicator: fixes A :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and B :: "real" assumes "A \<in> sets M"   and "f \<in> borel_measurable M"   and "emeasure M A < \<infinity>"   and "AE x\<in>A in M. norm (f x) \<le> B" shows "integrable M (\<lambda>x. indicat_real A x *\<^sub>R f x)"
Lcm_upto_real_conv_\<psi>: fixes x :: "real" shows "real (Lcm {1..nat \<lfloor>x\<rfloor>}) = exp (\<psi> x)"
sumResRight: fixes Rel :: "(pi \<times> pi) set"   and x :: "name"   and P :: "pi"   and Q :: "pi" assumes "Id \<subseteq> Rel"   and "eqvt Rel" shows "<\<nu>x>(P \<oplus> Q) \<leadsto>[Rel] <\<nu>x>P \<oplus> <\<nu>x>Q"
intersects_x_axis_positive_iff: fixes H :: "circline" assumes "is_poincare_line H"   and "H \<noteq> x_axis" shows "intersects_x_axis_positive H = (\<exists>x. x \<in> unit_disc \<and> x \<in> circline_set H \<inter> positive_x_axis)"
unique_row_col_1: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zer :: "'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inver :: "'b \<Rightarrow> 'b"   and nr1 :: "nat"   and nc1 :: "nat"   and M :: "'c list list"   and nr2 :: "nat"   and nc2 :: "nat" assumes "plus_mult id f zer g inver"   and "mat nr1 nc1 M"   and "mat nr2 nc2 M"   and "M \<noteq> []" shows "nr1 = nr2"
Gramian_determinant_1: fixes n :: "nat"   and m :: "nat"   and fs_init :: "int vec list"   and fs :: "int vec list"   and k :: "nat" assumes "LLL.LLL_invariant_weak n m fs_init fs"   and "k \<le> m" shows "rat_of_int (gram_schmidt.Gramian_determinant n fs k) = (\<Prod>j<k. \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) j\<parallel>\<^sup>2)"
proj_ltake: fixes t :: "cnf llist"   and n :: "nat"   and c :: "'id" assumes "\<forall>n'. enat n' \<le> llength t \<longrightarrow> n \<le> n' \<longrightarrow> \<not> \<parallel>c\<parallel>\<^bsub>lnth t n'\<^esub>" shows "\<pi>\<^bsub>c\<^esub>t = \<pi>\<^bsub>c\<^esub>ltake (enat n) t"
a1: shows "s' = snd (fst (sethi_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
coprime_def': fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and p :: "'a"   and q :: "'a" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.coprime times one p q \<equiv> \<forall>r. dvd.dvd times r p \<longrightarrow> dvd.dvd times r q \<longrightarrow> dvd.dvd times r one"
simps_232: fixes f :: "'c \<Rightarrow> 'b"   and x10 :: "'c ltlc" shows "map_ltlc f (G\<^sub>c x10) = G\<^sub>c map_ltlc f x10"
stabilizer_def: fixes m :: "'a set" shows "conj.stabilizer m = {g \<in> carrier G. conjugation_action (p ^ a) g m = m}"
intros_3: fixes t :: "('e, 'f, 'g) Tree"   and \<alpha> :: "'g" shows "(t, tAct \<alpha> t) \<in> Tree_wf"
\<Phi>_def: fixes f :: "'s" shows "\<Phi> f = (if S.arr f then S'.mkArr (S'.set (\<Phi>o (S.dom f))) (S'.set (\<Phi>o (S.cod f))) (\<Phi>a f) else S'.null)"
coprime_cong_prod_poly: fixes A :: "'c set"   and m :: "'c \<Rightarrow> 'd poly"   and x :: "'d poly"   and y :: "'d poly" assumes "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> coprime (m i) (m j)"   and "\<forall>i\<in>A. [x = y] (mod m i)" shows "[x = y] (mod prod m A)"
iMOD_div_ge: fixes m :: "nat"   and k :: "nat"   and r :: "nat" assumes "0 < m"   and "m \<le> k" shows "[ r, mod m ] \<oslash> k = [r div k\<dots>]"
IH: assumes "p \<in> setops\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>)" shows "\<exists>s\<in>setops\<^sub>s\<^sub>s\<^sub>t S. \<exists>X. set X \<subseteq> bvars\<^sub>s\<^sub>s\<^sub>t S \<and> p = s \<cdot>\<^sub>p rm_vars (set X) \<delta>"
RPCIPSpec_def: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "RPCIPSpec send rcv rst p \<equiv> Init RPCInit rcv rst p \<and> \<box>[RPCNext send rcv rst p]_(rst!p, rtrner send!p, caller rcv!p) \<and> WF(RPCNext send rcv rst p)_(rst!p, rtrner send!p, caller rcv!p)"
cos_double_sin: fixes w :: "'b" shows "cos ((2::'b) * w) = (1::'b) - (2::'b) * (sin w)\<^sup>2"
tensor\<^sub>F\<^sub>M\<^sub>C_def: fixes f :: "'c local.term set"   and g :: "'c local.term set" shows "f \<otimes> g \<equiv> if arr f \<and> arr g then mkarr (rep f \<^bold>\<otimes> rep g) else null"
distinct_vars_list: fixes p :: "linear_poly" shows "distinct (vars_list p)"
term_to_nterm_vars0: fixes \<Gamma> :: "name list"   and t :: "Term_Class.term"   and s :: "name" assumes "wellformed' (length \<Gamma>) t" shows "frees (fst (run_state (term_to_nterm \<Gamma> t) s)) |\<subseteq>| frees t |\<union>| fset_of_list \<Gamma>"
ss_nil: fixes CT :: "nat \<Rightarrow> classDef option" shows "CT \<turnstile>+ [] <: []"
sum_projr_param: fixes s :: "'a + 'b"   and s' :: "'c + 'd"   and Ra :: "('c \<times> 'a) set"   and Rb :: "('d \<times> 'b) set" assumes "is_Inr s"   and "(s', s) \<in> \<langle>Ra, Rb\<rangle>sum_rel" shows "(projr s', projr s) \<in> Rb"
causal_program_history_Ghost: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and is\<^sub>s\<^sub>b :: "instr list"   and sb :: "'b memref list" assumes "causal_program_history (Ghost A L R W # is\<^sub>s\<^sub>b) sb" shows "causal_program_history is\<^sub>s\<^sub>b (sb @ [Ghost\<^sub>s\<^sub>b A L R W])"
le: fixes n :: "nat" shows "emeasure M {x \<in> space M. 0 < F n x} \<le> emeasure M (Y n)"
pred_def: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r" shows "pred r i \<equiv> SOME j. j \<in> Field r \<and> aboveS r j \<noteq> {} \<and> succ r j = i"
ConstFtor: fixes A :: "('a, 'b, 'c) Category_scheme"   and B :: "('d, 'e, 'f) Category_scheme"   and b :: "'d" assumes "Category A"   and "Category B"   and "b \<in> obj\<^bsub>B\<^esub>" shows "Functor (ConstFunctor A B b)"
ev_shift: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and xs :: "'d stream"   and xl :: "'d list" assumes "ev \<phi> xs" shows "ev \<phi> (xl @- xs)"
index_matrix_prod: fixes i :: "nat"   and A :: "'a Matrix.mat"   and j :: "nat"   and B :: "'a Matrix.mat" assumes "i < dim_row A"   and "j < dim_col B"   and "dim_col A = dim_row B" shows "(A * B) $$ (i, j) = (\<Sum>k<dim_row B. A $$ (i, k) * B $$ (k, j))"
outside_list_coeff0: fixes xs :: "rat vec"   and i :: "nat" assumes "dim_vec xs \<le> i" shows "Abstract_Linear_Poly.coeff (vec_to_lpoly xs) i = 0"
nn_integral_measurable_subprob_algebra: fixes f :: "'c \<Rightarrow> ennreal"   and N :: "'c measure" assumes "f \<in> borel_measurable N" shows "(\<lambda>M. integral\<^sup>N M f) \<in> borel_measurable (subprob_algebra N)"
fst_diag_snd: shows "fst \<circ> ((\<lambda>x. (x, x)) \<circ> snd) = snd"
ngr_t_or_s: shows "vars t \<noteq> {} \<or> vars s \<noteq> {}"
Limsup_bot: fixes f :: "'d \<Rightarrow> 'b" shows "Limsup bot f = bot"
multi_member_skip: fixes x :: "'d"   and XS :: "'d multiset"   and y :: "'d" assumes "x \<in># XS" shows "x \<in># {#y#} + XS"
assms_3: shows "\<exists>k>0. H2 = cor k *\<^sub>s\<^sub>m H1"
inverse_unique: fixes f :: "'a1"   and g :: "'a1" assumes "C1.inverse_arrows f g" shows "C1.inv f = g"
homom4_2: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set"   and H :: "'e set" assumes "Group G"   and "G \<triangleright> N"   and "G \<guillemotright> H " shows "Group (\<natural>\<^bsub>G\<^esub>H / (H \<inter> N))"
preservation1_eq: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x \<le> x \<cdot> (y \<cdot> x)"   and "x \<cdot> y \<le> y" shows "x \<cdot> (y \<cdot> x)\<^sup>\<star> = y\<^sup>\<star> \<cdot> x"
gt_hd_irrefl: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and ground_heads_var :: "'b \<Rightarrow> 'a set"   and \<zeta> :: "('a, 'b) hd" assumes "Lambda_Free_Term.ground_heads gt_sym arity_sym arity_var ground_heads_var" shows "\<not> ground_heads.gt_hd gt_sym ground_heads_var \<zeta> \<zeta>"
midpoint_in_open_segment: fixes a :: "'a"   and b :: "'a" shows "(midpoint a b \<in> open_segment a b) = (a \<noteq> b)"
finprod_reindex: fixes f :: "'c \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "f \<in> h ` A \<rightarrow> carrier R"   and "inj_on h A" shows "finsum R f (h ` A) = (\<Oplus>x\<in>A. f (h x))"
DDH_advantage_def: fixes \<G> :: "'b cyclic_group"   and \<A> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool spmf" shows "ddh.DDH_advantage \<G> \<A> = \<bar>spmf (ddh.DDH0 \<G> \<A>) True - spmf (ddh.DDH1 \<G> \<A>) True\<bar>"
cut_greater_in_imp: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<in> I \<down>> t" shows "x \<in> I"
assms_2: shows "(Var x \<lhd> \<sigma>, Var x \<lhd> \<eta>) \<in> trm_ord"
exhaust: fixes y :: "('c, 'd) Tagged_Prod_Sum.sum" obtains x11 :: "char list option" and x12 :: "'c" where "y = sum.Inl x11 x12"   | x21 :: "char list option" and x22 :: "'d" where "y = sum.Inr x21 x22"
right_inverse: fixes a :: "'b perm" shows "a * inverse a = 1"
map_iterator_rule_P: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<times> 'e \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "'d \<Rightarrow> 'e option"   and I :: "'d set \<Rightarrow> 'c \<Rightarrow> bool"   and \<sigma>0 :: "'c"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<times> 'e \<Rightarrow> 'c \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> bool" assumes "map_iterator it m"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; m k = Some v; it \<subseteq> dom m; I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
emeasure_real: fixes A :: "'b set" shows "\<exists>r\<ge>0. emeasure (M \<bind> f) A = ennreal r"
assms_5: shows "val_Zp b < val_Zp (c \<ominus> a)"
l8_10: 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 A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Cong3 Cong A B C A' B' C'" shows "Tarski_neutral_dimensionless.Per Bet Cong A' B' C'"
round_up_diff_round_down: fixes prec :: "int"   and x :: "real" shows "round_up prec x - round_down prec x \<le> 2 powr - real_of_int prec"
measurable_nested_Union: fixes S :: "nat \<Rightarrow> 'c set"   and B :: "real" assumes "\<And>n. S n \<in> lmeasurable"   and "\<And>n. Sigma_Algebra.measure lebesgue (S n) \<le> B"   and "\<And>n. S n \<subseteq> S (Suc n)" shows "\<Union> (range S) \<in> lmeasurable"
representsI: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and r :: "'d \<Rightarrow>\<^sub>0 'c"   and B :: "'d set"   and x :: "'d" assumes "module scale"   and "keys r \<subseteq> B"   and "module.rep scale r = x" shows "module.represents scale B r x"
assms_3: shows "c \<in>\<^sub>\<circ> dg_prod J \<AA>\<lparr>Arr\<rparr>"
assms: shows "(\<box>(\<lambda>s. enabled s \<longrightarrow> (\<circle>(\<lambda>s. enabled s \<or> taken s)) s)) \<sigma>"
valid_modif: shows "\<forall>s\<in>P. \<forall>\<sigma> n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> {\<sigma>} Call (p s) (Modif \<sigma>),(ModifAbr \<sigma>)"
IN_plus_flow: fixes \<Delta> :: "('k, 'l) graph_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "\<And>e. e \<notin> \<^bold>E\<^bsub>\<Delta>\<^esub> \<Longrightarrow> f e = 0"   and "\<And>x y. edge \<Delta> x y \<Longrightarrow> g (y, x) \<le> f (x, y)"   and "d_OUT g x \<noteq> \<top>" shows "d_IN (f \<oplus>\<^bsub>\<Delta>\<^esub> g) x = d_IN f x + (\<Sum>\<^sup>+ y. g (y, x) * indicator \<^bold>E\<^bsub>\<Delta>\<^esub> (y, x)) - (\<Sum>\<^sup>+ y. g (x, y) * indicator \<^bold>E\<^bsub>\<Delta>\<^esub> (y, x))"
iptables_bigstep_MatchAnd_comm: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m1 :: "'c match_expr"   and m2 :: "'c match_expr"   and a :: "action"   and s :: "state"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule (MatchAnd m1 m2) a], s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule (MatchAnd m2 m1) a], s\<rangle> \<Rightarrow> t"
wf_bounded_if_total: fixes gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "nat" assumes "\<And>z. \<not> gt z z"   and "\<And>z y x. \<lbrakk>gt z y; gt y x\<rbrakk> \<Longrightarrow> gt z x"   and "\<And>y x. gt y x \<or> gt x y \<or> y = x"   and "wfP (\<lambda>x y. gt y x)" shows "wfP (\<lambda>xs ys. length ys \<le> n \<and> length xs \<le> n \<and> ext gt ys xs)"
\<sigma>': shows "(fst \<sigma>1, snd \<sigma>1 @ snd \<sigma>2 @ snd \<sigma>3) \<in> seq ars"
leftmost_append_suffix: fixes i :: "nat"   and v :: "symbol list"   and u :: "symbol list" assumes "leftmost i v" shows "leftmost i (v @ u)"
quant_equiv_iff: fixes x :: "'b['c, 'd]"   and y :: "'b['e, 'd]" shows "(x \<cong>\<^sub>Q y) = (\<lbrakk>x\<rbrakk>\<^sub>Q = \<lbrakk>y\<rbrakk>\<^sub>Q \<and> QD('c) = QD('e))"
card_of_ordLeqI: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "inj_on f A"   and "\<And>a. a \<in> A \<Longrightarrow> f a \<in> B" shows "|A| \<le>o |B|"
ldrop_ltl: fixes n :: "enat"   and xs :: "'c llist" shows "ldrop n (ltl xs) = ldrop (eSuc n) xs"
less_trans: fixes y :: "'d"   and x :: "'d"   and z :: "'d" assumes "y < x"   and "z < y" shows "z < x"
F: shows "F = {f \<in> {0..<n} \<rightarrow> {0..<m}. \<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i}"
facet_of_irrefl: fixes S :: "'b set" shows "\<not> S facet_of S"
simps_5: fixes F :: "'a formula"   and G :: "'a formula" shows "to_mini_formula (F \<^bold>\<and> G) = (to_mini_formula F \<^bold>\<rightarrow> to_mini_formula G \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> \<bottom>"
Pd'_linear: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d"   and X :: "'d set"   and ds :: "'e set"   and Pd' :: "'e \<Rightarrow> ('d \<times> 'd) set"   and d :: "'e" assumes "TruePrefs Xd Xh Pd Ch x X ds Pd'" shows "Linear_order (Pd' d)"
always_eventually_within: fixes S :: "'b set"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<forall>x\<in>S. P x" shows "eventually P (at x within S)"
strategy_def: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "ParityGame G" shows "ParityGame.strategy G p \<sigma> \<equiv> \<forall>v\<in>if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>. \<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w) \<longrightarrow> v \<rightarrow>\<^bsub>G\<^esub> \<sigma> v"
a_comm: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier (UP INTEG)"   and "y \<in> carrier (UP INTEG)" shows "x \<oplus>\<^bsub>UP INTEG\<^esub> y = y \<oplus>\<^bsub>UP INTEG\<^esub> x"
le_Suc_eq': fixes x :: "nat"   and y :: "nat" shows "(x \<le> Suc y) = (x = 0 \<or> (\<exists>x'. x = Suc x' \<and> x' \<le> y))"
A'_wf: shows "list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) A'"
rep_eq: fixes x :: "perm"   and xa :: "('a, 'b) finfun" shows "finfun_apply (x \<bullet> xa) = x \<bullet> finfun_apply xa"
wf_liftB: fixes \<Gamma> :: "binding list"   and i :: "nat"   and T :: "POPLmark.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma>\<langle>i\<rangle> = \<lfloor>VarB T\<rfloor>" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f \<up>\<^sub>\<tau> (Suc i) 0 T"
glb_lower: fixes S :: "'a set"   and x :: "'a" assumes "S \<subseteq> A"   and "x \<in> S" shows "(glb S cl, x) \<in> r"
a: shows "a \<sharp> (\<theta>_n, \<theta>_c)"
LID\<^sub>LI_2: fixes s :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and e :: "('b, 'a, 'c) expr"   and r :: "'a" assumes "s = (\<sigma>, \<tau>, e)"   and "r \<in> LID\<^sub>S \<tau>" shows "r \<in> LID\<^sub>L s"
CompDiag_preserves_Ide: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Ide t"   and "Ide u"   and "Diag t"   and "Diag u"   and "Dom t = Cod u" shows "Ide (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u)"
empty_path_is_elem: fixes k :: "nat"   and u :: "nat" shows "[] \<in> \<P> k u"
design_blocks_nempty: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "proper_design point_set block_collection" shows "block_collection \<noteq> {#}"
cos_vangle: fixes u :: "'b"   and v :: "'b" shows "cos (vangle u v) = u \<bullet> v / (norm u * norm v)"
Refl: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" shows "(X, X) \<in> swapped"
red_wf_finite: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite F" shows "wfP (punit.red F)\<inverse>\<inverse>"
piecewise_differentiable_on_def: fixes f :: "'b \<Rightarrow> 'c"   and i :: "'b set" shows "f piecewise_differentiable_on i \<equiv> continuous_on i f \<and> (\<exists>S. finite S \<and> (\<forall>x\<in>i - S. f differentiable at x within i))"
alphaAbs_qAbs_imp_alphaAbs_all_equal_or_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X"   and "qAbs xs x X $= qAbs xs' x' X'" shows "alphaAbs_all_equal_or_qAFresh xs x X xs' x' X'"
add_ac_1: fixes a :: "'g"   and b :: "'g"   and c :: "'g" shows "a + b + c = a + (b + c)"
is_faithful_functor: shows "faithful_functor (\<cdot>) Cop_S.comp local.map"
P_kk: fixes i :: "nat"   and r :: "nat \<Rightarrow> 'a" shows "P i (r \<circ> kk i r)"
Rats_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<rat>" shows "sum f A \<in> \<rat>"
mat_inverse_simps_1: fixes c :: "'c" assumes "c \<noteq> (0::'c)" shows "mat (inverse c) ** mat c = mat (1::'c)"
bounded_iff: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf"   and c :: "'b::semilattice_inf" shows "((a::'b::semilattice_inf) \<le> (b::'b::semilattice_inf) \<sqinter> (c::'b::semilattice_inf)) = (a \<le> b \<and> a \<le> c)"
prob0_fst_qubits_def: fixes n :: "nat"   and v :: "complex Matrix.mat" shows "prob0_fst_qubits n v \<equiv> \<Sum>j\<in>{k |k. k < 2 ^ (n + 1) \<and> (\<forall>i\<in>{0..<n}. \<not> select_index (n + 1) i k)}. (cmod (v $$ (j, 0)))\<^sup>2"
half_gt_zero_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide a (numeral.numeral one plus (num.Bit0 num.One))) = less zero a"
hsnd: shows "continuous_map (prod_topology (top_of_set {0..1}) X1) X2 (h \<circ> snd)"
mset_insort: fixes k :: "'c \<Rightarrow> 'd"   and x :: "'c"   and xs :: "'c list" shows "mset (insort_key k x xs) = add_mset x (mset xs)"
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" assumes "l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs"   and "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes" shows "l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed parent_child_rel get_disconnected_nodes"
ideal_ring_multiple: fixes I :: "'a set"   and x :: "'a"   and r :: "'a" assumes "ideal R I"   and "x \<in> I"   and "r \<in> carrier R" shows "r \<cdot>\<^sub>r x \<in> I"
assms_5: fixes K :: "complex set" assumes "compact K"   and "K \<subseteq> S" shows "uniform_limit K \<F> g sequentially"
alpha_l_Suc_l_derive: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and l :: "nat" assumes "grover_state n f" shows "2 * (grover_state.alpha_l n f l * complex_of_real (cos (grover_state.\<theta> n f / 2)) - grover_state.beta_l n f l * complex_of_real (sin (grover_state.\<theta> n f / 2))) * complex_of_real (cos (grover_state.\<theta> n f / 2)) - grover_state.alpha_l n f l = grover_state.alpha_l n f (l + 1)"
card_fset: fixes xs :: "'c fset" shows "card_fset xs = card (fset xs)"
listset_appendI: fixes ys1 :: "'b list"   and xs1 :: "'b set list"   and ys2 :: "'b list"   and xs2 :: "'b set list"   and ys :: "'b list" assumes "ys1 \<in> listset xs1"   and "ys2 \<in> listset xs2"   and "ys = ys1 @ ys2" shows "ys \<in> listset (xs1 @ xs2)"
assms_4: shows "f piecewise_C1_differentiable_on s"
length_induct: fixes P :: "'d list \<Rightarrow> bool"   and xs :: "'d list" assumes "\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs" shows "P xs"
less: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "\<phi> i < \<phi> j"
II: fixes it :: "'u set"   and \<sigma> :: "'\<sigma>" assumes "it \<subseteq> dom (\<alpha> m)"   and "it \<noteq> {}"   and "\<not> c \<sigma>"   and "I it \<sigma>" shows "P \<sigma>"
runit'_simps_4: fixes f :: "'a" assumes "ide f" shows "local.dom \<r>\<^sup>-\<^sup>1[f] = f"