hcju/isabelleps huggingface dataset:
cont_While_False: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and tst :: "'c"   and s :: "'b"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "('c, 'd, 'a) cmd"   and i :: "nat" assumes "PL cval"   and "tval tst s = False" shows "PL.cont aval tval cval (While tst c) s i = Done"
id_stake_snth_suffix: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "\<sigma> = (stake i \<sigma> @ [\<sigma> i]) @- (\<sigma> |\<^sub>s Suc i)"
R\<^sub>0_def: shows "R\<^sub>0 = \<lbrakk>\<lbrakk>tab\<^sub>0 r\<rbrakk>\<rbrakk>"
cases: fixes P :: "'a prog"   and a1 :: "ty"   and a2 :: "ty" assumes "P \<turnstile> a1 \<le> a2" obtains T :: "ty" where "a1 = T" and "a2 = T"   | C :: "char list" and D :: "char list" where "a1 = Class C__" and "a2 = Class D" and "P \<turnstile> C__ \<preceq>\<^sup>* D"   | C :: "char list" where "a1 = NT" and "a2 = Class C"
fv_subst_bvs1_upper_bound: fixes t :: "Core.term"   and lev :: "nat"   and us :: "Core.term list" shows "fv (subst_bvs1 t lev us) \<subseteq> fv t \<union> \<Union> (fv ` set us)"
diff_union_cancelR: fixes M :: "'d multiset"   and N :: "'d multiset" shows "M + N - N = M"
reads_consistent_append_first: fixes pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat"   and xs :: "'b memref list"   and ys :: "'b memref list" assumes "reads_consistent pending_write \<O> m (xs @ ys)" shows "reads_consistent pending_write \<O> m xs"
polymul_normh: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
\<Omega>_def: shows "\<Omega> = (\<Union>(n, m)\<in>F. set_pmf (p n m))"
WTrt_binop_fun: fixes P :: "'b prog"   and T1 :: "ty"   and bop :: "bop"   and T2 :: "ty"   and T :: "ty"   and T' :: "ty" assumes "P \<turnstile> T1\<guillemotleft>bop\<guillemotright>T2 : T"   and "P \<turnstile> T1\<guillemotleft>bop\<guillemotright>T2 : T'" shows "T = T'"
unitary_const: fixes c :: "real" assumes "0 \<le> c"   and "c \<le> 1" shows "unitary (\<lambda>s. c)"
min_dep_in_imp_ex1_supset_circuit_in: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> \<E>"   and "indep_in \<E> X"   and "\<not> indep_in \<E> (insert x X)" shows "\<exists>!C. circuit_in \<E> C \<and> C \<subseteq> insert x X"
simps_24: fixes g1 :: "'b gexp"   and g2 :: "'b gexp"   and v :: "'b aexp"   and va :: "'b aexp" shows "GExp_Lexorder.less_gexp_aux (Nor g1 g2) (Gt v va) = False"
Regularity: fixes A :: "ZF" shows "A \<noteq> Empty \<longrightarrow> (\<exists>x. Elem x A \<and> (\<forall>y. Elem y x \<longrightarrow> \<not> Elem y A))"
lub_spmf_upper: fixes Y :: "'c spmf set"   and p :: "'c spmf" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y"   and "p \<in> Y" shows "ord_spmf (=) p (lub_spmf Y)"
Fun_param_is_subterm: fixes x :: "('a, 'b) Term.term"   and X :: "('a, 'b) Term.term list"   and f :: "'a" assumes "x \<in> set X" shows "x \<sqsubset> Fun f X"
section_retraction_of_iso_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "category C"   and "category.iso C f"   and "partial_magma.ide C (C f g)" shows "category.inverse_arrows C f g"
mono: fixes f :: "'a \<Rightarrow> 'a" shows "mono (\<lambda>p x. (\<exists>xa. x = f xa \<and> p xa) \<or> (\<exists>M. x = \<Squnion> M \<and> Complete_Partial_Order.chain (\<le>) M \<and> (\<forall>x\<in>M. p x)))"
7: shows "forest_components f * forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e) \<le> forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e)"
ran_distinct: fixes al :: "('b \<times> 'c) list" assumes "distinct (map fst al)" shows "ran (map_of al) = snd ` set al"
underS_ofilter: fixes a :: "'a" shows "local.ofilter (local.underS a)"
eq_fst_iff: fixes a :: "'b"   and p :: "'b \<times> 'c" shows "(a = fst p) = (\<exists>b. p = (a, b))"
WR_Quote: fixes i :: "nat"   and y :: "hf" assumes "WR (ord_of i) y" shows "Quote (ord_of i) y"
Formal_out_SDG_Use: fixes n :: "'node SDG_node"   and m :: "'node"   and x :: "nat"   and p :: "'pname"   and ins :: "'var list"   and outs :: "'var list"   and V :: "'var" assumes "valid_SDG_node n"   and "n = Formal_out (m, x)"   and "get_proc m = p"   and "(p, ins, outs) \<in> set procs"   and "V = outs ! x" shows "V \<in> Use\<^bsub>SDG\<^esub> n"
1156: fixes x :: "'a"   and y :: "'a" shows "- - x \<sqinter> - (- x \<sqinter> y) = - - x"
mat_row_first_dim_1: fixes A :: "'b mat"   and n :: "nat" shows "dim_row (mat_row_first A n) = n"
chamber_retraction2: fixes C :: "'a set" assumes "C \<in> g \<turnstile> folding_g.\<C>" shows "g ` C = C"
coeff_def: fixes p :: "'b mpoly" shows "MPoly_Type.coeff p = lookup (mapping_of p)"
simps: fixes lineA :: "'a Line"   and lineB :: "'a Line" shows "lineA \<parallel>  lineB = (direction lineA = 0 \<or> direction lineB = 0 \<or> (\<exists>k. k \<noteq> (0::'a) \<and> direction lineB = k ** direction lineA))"
eventually_ball_finite: fixes A :: "'e set"   and P :: "'f \<Rightarrow> 'e \<Rightarrow> bool"   and net :: "'f filter" assumes "finite A"   and "\<forall>y\<in>A. \<forall>\<^sub>F x in net. P x y" shows "\<forall>\<^sub>F x in net. \<forall>y\<in>A. P x y"
preserves_dom: fixes f :: "'j" assumes "J.arr f" shows "C.dom (A.map f) = A.map (J.dom f)"
\<gamma>: shows "\<guillemotleft>THE \<gamma>. \<guillemotleft>\<gamma> : src f \<Rightarrow> w' \<star> w\<guillemotright> \<and> (\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g] = local.dom \<rho> \<star> \<gamma> \<and> \<r>[f] = (\<theta> \<cdot> (\<theta>' \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[f, w', w]) \<cdot> (f \<star> \<gamma>) : src f \<rightarrow> src f\<guillemotright> \<and> \<guillemotleft>THE \<gamma>. \<guillemotleft>\<gamma> : src f \<Rightarrow> w' \<star> w\<guillemotright> \<and> (\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g] = local.dom \<rho> \<star> \<gamma> \<and> \<r>[f] = (\<theta> \<cdot> (\<theta>' \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[f, w', w]) \<cdot> (f \<star> \<gamma>) : src f \<Rightarrow> w' \<star> w\<guillemotright> \<and> (\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g] = local.dom \<rho> \<star> (THE \<gamma>. \<guillemotleft>\<gamma> : src f \<Rightarrow> w' \<star> w\<guillemotright> \<and> (\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g] = local.dom \<rho> \<star> \<gamma> \<and> \<r>[f] = (\<theta> \<cdot> (\<theta>' \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[f, w', w]) \<cdot> (f \<star> \<gamma>)) \<and> \<r>[f] = (\<theta> \<cdot> (\<theta>' \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[f, w', w]) \<cdot> (f \<star> (THE \<gamma>. \<guillemotleft>\<gamma> : src f \<Rightarrow> w' \<star> w\<guillemotright> \<and> (\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g] = local.dom \<rho> \<star> \<gamma> \<and> \<r>[f] = (\<theta> \<cdot> (\<theta>' \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[f, w', w]) \<cdot> (f \<star> \<gamma>)))"
preal_Ex_mem: fixes A :: "rat set" assumes "Dedekind_Real.cut A" shows "\<exists>x. x \<in> A"
wellformed_transaction_strand_unlabel_memberD_8: fixes T :: "('e, 'f, 'g, 'h) prot_transaction"   and x :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step"   and t :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) Term.term" assumes "wellformed_transaction T"   and "x \<in> set (unlabel (transaction_strand T))"   and "x = send\<langle>t\<rangle>" shows "x \<in> set (unlabel (transaction_send T))"
assms: shows "homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S f g"
u_1: shows "u \<in> subterms s \<union> subterms t - range Var"
tan_periodic_pi: fixes x :: "real" shows "tan (x + pi) = tan x"
algebraic_int_of_real_iff: fixes x :: "real" shows "algebraic_int (of_real x) = algebraic_int x"
truncate_up_eq_truncate_down: fixes p :: "nat"   and x :: "real" shows "truncate_up p x = - truncate_down p (- x)"
less_multiset\<^sub>H\<^sub>O_def: fixes M :: "'a multiset"   and N :: "'a multiset" shows "less_multiset\<^sub>H\<^sub>O M N = (M \<noteq> N \<and> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x>y. count M x < count N x)))"
exhaust_sel: fixes taylor_model :: "taylor_model" obtains "taylor_model = TaylorModel (tm_poly taylor_model) (tm_bound taylor_model)"
horner_eval_cong: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and m :: "nat" assumes "\<And>i. i < n \<Longrightarrow> f i = g i"   and "x = y"   and "n = m" shows "horner_eval f x n = horner_eval g y m"
completeD: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier"   and "x \<noteq> y" shows "x \<succeq> y \<or> y \<succeq> x"
vconn_implies_hpar_2: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>B g\<guillemotright>" shows "trg\<^sub>B \<mu> = trg\<^sub>B f"
assms: shows "\<forall>p\<in>set ps. bound n p"
to_int: fixes ff_ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c mod_ring \<Rightarrow> bool"   and p :: "int"   and y :: "'b"   and z :: "'c mod_ring" assumes "mod_ring_gen ff_ops R p"   and "R y z" shows "arith_ops_record.to_int ff_ops y = to_int_mod_ring z"
lookup_numeral: fixes n :: "num"   and k :: "'b" shows "poly_mapping.lookup (numeral n) k = (numeral n when k = (0::'b))"
globally_consistent_writable: fixes x :: "'Var"   and A :: "'Var \<Rightarrow> ('Val \<times> 'Val) option"   and mds :: "Mode \<Rightarrow> 'Var set" assumes "x \<in> dom A"   and "globally_consistent A mds" shows "x \<notin> mds AsmNoWrite"
wfzf_def: fixes R :: "(ZF \<times> ZF) set" shows "wfzf R \<equiv> regular R \<and> set_like R"
assms_1: shows "lookup p s \<noteq> (0::'b)"
l_delete_shadow_root_get_shadow_root_get_shadow_root_locs: shows "l_delete_shadow_root_get_shadow_root get_shadow_root_locs"
exhaust: fixes y :: "'b exp" obtains x1 :: "'b" where "y = Var x1"   | x2 :: "int" where "y = Const x2"   | x31 :: "'b exp" and x32 :: "'b exp" where "y = Plus x31 x32"   | x41 :: "'b exp" and x42 :: "'b exp" where "y = Div x41 x42"
mpt_skew_product: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and N :: "'c measure"   and U :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "mpt M T"   and "AE x in M. mpt N (U x)"   and "(\<lambda>(x, y). U x y) \<in> M \<Otimes>\<^sub>M N \<rightarrow>\<^sub>M N" shows "mpt (M \<Otimes>\<^sub>M N) (\<lambda>(x, y). (T x, U x y))"
hyperpow_def: fixes R :: "'b star"   and N :: "nat star" shows "R pow N = (*f2* (^)) R N"
vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_NegCheck: fixes X :: "'a list"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and G :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> = set X \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G"
1: shows "ARR f \<and> ARR g \<and> ARR h \<and> DOM h = COD g \<and> DOM g = COD f"
PiE: fixes f :: "'ag \<Rightarrow> 'ah"   and A :: "'ag set"   and B :: "'ag \<Rightarrow> 'ah set"   and x :: "'ag" assumes "f \<in> Pi A B" obtains "f x \<in> B x"   | "x \<notin> A"
ipd_exists: fixes x :: "'n" assumes "x \<in> local.nodes"   and "x \<noteq> local.return" shows "\<exists>y. y ipd\<rightarrow> x"
IH_8: assumes "\<langle>t != s\<rangle> \<in> set A" shows "\<langle>t \<cdot> \<theta> != s \<cdot> \<theta>\<rangle> \<in> set (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
freshAbs_vsubstAbs: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodAbs A" shows "freshAbs zs z (A $[y1 // y]_ys) = ((zs = ys \<and> z = y \<or> freshAbs zs z A) \<and> (freshAbs ys y A \<or> zs \<noteq> ys \<or> z \<noteq> y1))"
linorder_not_le: fixes x :: "'d"   and y :: "'d" shows "(\<not> x \<le> y) = (y < x)"
neq_le_trans: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b"   and "a \<le> b" shows "a < b"
assms: shows "while_option fst (rderiv_and_add as) (True, [\<guillemotleft>r\<guillemotright>]) = Some (b, rs)"
surj_plus: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.ab_group_add plus zero minus uminus" shows "surj (plus a)"
convergentI: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" assumes "X \<longlonglongrightarrow> L" shows "convergent X"
Rep_Store_cases: fixes y :: "StoreImpl" assumes "y \<in> Store" obtains x :: "Store" where "y = Rep_Store x"
has_integral_spike_interior_eq: fixes a :: "'a"   and b :: "'a"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "\<And>x. x \<in> box a b \<Longrightarrow> g x = f x" shows "(f has_integral y) (cbox a b) = (g has_integral y) (cbox a b)"
assms_2: shows "B \<subseteq>\<^sub>\<circ> C"
simplicial_chain_diff: fixes p :: "nat"   and S :: "('c \<Rightarrow> real) set"   and c1 :: "((nat \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real) \<Rightarrow>\<^sub>0 'd"   and c2 :: "((nat \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real) \<Rightarrow>\<^sub>0 'd" assumes "simplicial_chain p S c1"   and "simplicial_chain p S c2" shows "simplicial_chain p S (c1 - c2)"
parallel_def: shows "(\<parallel>) \<equiv> \<lambda>uu uua. corec_dds (\<lambda>(S1, S2) x. case x of Inl a \<Rightarrow> (id (Inl (fst (run S1 a))), Inr (snd (run S1 a), S2)) | Inr c \<Rightarrow> (id (Inr (fst (run S2 c))), Inr (S1, snd (run S2 c)))) (uu, uua)"
count_replicate_mset: fixes n :: "nat"   and x :: "'a"   and y :: "'a" shows "count (replicate_mset n x) y = (if y = x then n else 0)"
inj_on_def: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set" shows "inj_on f A = (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
diff_fun_add: fixes a :: "'a \<Rightarrow> 'd"   and b :: "'a \<Rightarrow> 'd" assumes "diff_fun k charts a"   and "diff_fun k charts b" shows "diff_fun k charts (a + b)"
l2_inv3D: fixes B :: "agent"   and A :: "agent"   and K :: "msg"   and Ra :: "fid_t"   and x :: "l2_state" assumes "Auth B A (Aenc K (epubKF (Ra $ kE))) \<in> chan x"   and "A \<notin> bad x \<and> B \<notin> bad x"   and "x \<in> l2_inv3" shows "\<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress x Rb = Some {xpkE, xsk} \<and> guessed_frame Rb xpkE = Some (epubKF (Ra $ kE)) \<and> K = NonceF (Rb $ sk)"
real_cond_exp_abs: fixes f :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable M" shows "AE x in M. ennreal \<bar>real_cond_exp M F f x\<bar> \<le> nn_cond_exp M F (\<lambda>x. ennreal \<bar>f x\<bar>) x"
assms_1: shows "term_powerprod pair_of_term term_of_pair"
T_map_eq: fixes T_f :: "'a \<Rightarrow> nat"   and xs :: "'a list" shows "T_map T_f xs = sum_list (map T_f xs) + length xs + 1"
deterministic: fixes lc :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)"   and lc' :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)"   and lc'' :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)" assumes "eval_abv\<^sub>C lc lc'"   and "eval_abv\<^sub>C lc lc''" shows "lc' = lc''"
ordIso_equivalence_3: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "r =o r'" shows "r' =o r"
rank_nullity_main_1: assumes "V.fin_dim" shows "vectorspace.dim K (W.vs imT) + vectorspace.dim K (V.vs kerT) = V.dim"
simps_1: fixes f :: "'f \<Rightarrow> 'e option" shows "mapM f [] = Some []"
height_exp_approx_def: fixes n :: "nat" shows "height_exp_approx n = log 2 (real (n + 3 choose 3) / 4) + 1"
rev_iffD1: fixes Q :: "bool"   and P :: "bool" assumes "Q"   and "Q = P" shows "P"
rep_eq: fixes x :: "linear_poly" shows "fmlookup (linear_poly_map x) = (\<lambda>x. if Rep_linear_poly x x = 0 then None else Some (Rep_linear_poly x x))"
subst_free: fixes y :: "'a"   and M :: "'a trm"   and N :: "'a trm" assumes "y \<notin> fvs M" shows "M[y ::= N] = M"
domA_def: fixes h :: "('a \<times> 'b) list" shows "domA h = fst ` set h"
assms_8: shows "val_Zp (pderiv f \<bullet> a) < val_Zp (a \<ominus> \<alpha>)"
beta_bound1: fixes p :: "nat"   and i :: "nat"   and q :: "nat" assumes "correct p (te p (i + 1))"   and "correct q (te p (i + 1))" shows "0 \<le> te p (i + 1) - te q i"
finite_eq_tag_rel: fixes tag :: "'b list \<Rightarrow> 'a" assumes "finite (range tag)" shows "finite (UNIV // =tag=)"
det_not_0_imp_lin_indpt_rows: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "det A \<noteq> (0::'b)" shows "\<not> module.lin_dep class_ring (module_vec TYPE('b) n) (set (rows A))"
snd_mismatch_def: shows "c\<^sub>1 = (u\<^sub>1 \<cdot> u\<^sub>0) ! \<^bold>|\<alpha>\<^bold>|"
middle_rel_b: fixes b :: "'a Bonk_Schramm_extension_unfolded"   and a :: "'a Bonk_Schramm_extension_unfolded" shows "(map_aux b, map_aux (middle a b)) \<in> wo_aux - Id"
simps: fixes \<pi> :: "'b ltl \<Rightarrow> nat option" shows "M_fin \<pi> = {((\<phi>', m), \<nu>, p). \<not> (\<forall>S. (\<forall>\<chi>\<in>dom \<pi>. S \<up>\<Turnstile>\<^sub>P Abs \<chi> \<and> (\<forall>q. (\<exists>j\<ge>the (\<pi> \<chi>). the (m \<chi>) q = Some j) \<longrightarrow> S \<up>\<Turnstile>\<^sub>P \<up>eval\<^sub>G (dom \<pi>) q)) \<longrightarrow> S \<up>\<Turnstile>\<^sub>P \<phi>')}"
timpl_apply_Abs: fixes T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and S :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and a :: "'c set"   and b :: "'c set" assumes "\<And>i. i < length T \<Longrightarrow> S ! i \<in> set \<langle>a --\<guillemotright> b\<rangle>\<langle>T ! i\<rangle>"   and "length T = length S" shows "Fun (Abs b) S \<in> set \<langle>a --\<guillemotright> b\<rangle>\<langle>Fun (Abs a) T\<rangle>"
\<phi>'_def: fixes i :: "nat" shows "af \<phi> (prefix i w) \<equiv>\<^sub>P Rep (fst (fst (r i)))"
abs_add_abs: fixes abs :: "'c \<Rightarrow> 'c"   and 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"   and b :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (plus (abs a) (abs b)) = plus (abs a) (abs b)"
wf_digraph_axioms: shows "wf_digraph (G \<restriction> S)"
image_gen_preserves_inter: fixes A :: "'p"   and A' :: "'p"   and X :: "'p"   and Y :: "'p"   and B :: "'p"   and B' :: "'p"   and C :: "'p"   and C' :: "'p"   and D :: "'p"   and D' :: "'p"   and I :: "'p"   and I' :: "'p" assumes "A A' Reflect X Y"   and "B B' Reflect X Y"   and "C C' Reflect X Y"   and "D D' Reflect X Y"   and "\<not> Col A B C"   and "C \<noteq> D"   and "Col A B I"   and "Col C D I"   and "Col A' B' I'"   and "Col C' D' I'" shows "I I' Reflect X Y"
castmsg_simps_3: fixes P :: "'d \<Rightarrow> bool"   and ii :: "nat set"   and ni :: "nat set"   and m :: "'d" shows "castmsg P (ii\<not>ni:arrive(m)) = True"
finite_pdigraphI: fixes \<alpha> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V" assumes "pdigraph_base \<alpha> I \<AA>"   and "vfinite I" shows "finite_pdigraph \<alpha> I \<AA>"
icard_image: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "icard (f ` A) = icard A"
seq_lident: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and seq :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seqstar :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.cka plus less_eq less one zero times star seq seqstar" shows "seq one x = x"
cont_minus1: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a" assumes "class.minus_cpo below" shows "cont (\<lambda>x. x - y)"
exp_minus: fixes x :: "'a" shows "exp (- x) = inverse (exp x)"
fune_mono: fixes A :: "char list set"   and B :: "char list set"   and C :: "acom" assumes "A \<subseteq> B" shows "fune C A \<subseteq> fune C B"
dis_log2_def: fixes \<A> :: "'a \<Rightarrow> (nat \<times> nat) spmf" shows "local.dis_log_alt.dis_log2 \<A> = TRY sample_uniform (order \<G>) \<bind> (\<lambda>w. let h = \<^bold>g [^] w in \<A> h \<bind> (\<lambda>(w1', w2'). return_spmf [w = w1' + x * w2'] (mod order \<G>))) ELSE return_spmf False"
matrix_to_iarray_adjugate: fixes A :: "(('b, 'c) vec, 'c) vec" shows "matrix_to_iarray (adjugate A) = adjugate_iarrays (matrix_to_iarray A)"
sets_pair_swap: fixes A :: "('c \<times> 'd) set"   and M1 :: "'c measure"   and M2 :: "'d measure" assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
OF_lm_noa_none_iff: fixes \<gamma> :: "'b set \<Rightarrow> 'c \<Rightarrow> bool"   and ft :: "('b, 'a) flow_entry_match list"   and p :: "'c" shows "(OF_match_linear \<gamma> ft p = NoAction) = (\<forall>e\<in>set ft. \<not> \<gamma> (ofe_fields e) p)"
powr_less_mono2_neg: fixes a :: "real"   and x :: "real"   and y :: "real" assumes "a < 0"   and "0 < x"   and "x < y" shows "y powr a < x powr a"
left_moving_Oc_Oc: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and list :: "cell list" assumes "wadjust_goon_left_moving_O m rs (c, Oc # list)"   and "hd c = Oc" shows "wadjust_goon_left_moving_O m rs (tl c, Oc # Oc # list)"
ge_kde2: fixes x :: "complex" assumes "x \<in> closed_segment (a - complex_of_real d) (a - complex_of_real kde)" shows "\<exists>y. x = a + complex_of_real y \<and> y \<le> - kde"
boundedBy_def: fixes N :: "nat"   and A :: "(nat \<times> 'b) set" shows "boundedBy N A = (\<forall>(n, delta)\<in>A. n < N)"
prob_space_axioms: fixes M :: "'a pmf" shows "prob_space (measure_pmf M)"
preprocess'_Tableau_Poly_Mapping_Some: fixes cs :: "('a \<times> QDelta ns_constraint) list"   and start :: "nat"   and p :: "linear_poly"   and v :: "nat" assumes "Poly_Mapping (preprocess' cs start) p = Some v" shows "linear_poly_to_eq p v \<in> set (Tableau (preprocess' cs start))"
principal_ideal_n_pow1: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "R \<diamondsuit>\<^sub>p a \<^bsup>\<diamondsuit>R n\<^esup> = R \<diamondsuit>\<^sub>p (a^\<^bsup>R n\<^esup>)"
l4_ports_negate_one_not_has_disc_negated_2: fixes ports :: "ipt_l4_ports" shows "\<not> has_disc_negated is_Dst_Ports False (l4_ports_negate_one Dst_Ports ports)"
G_transpose: fixes A :: "(('b, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" shows "fixed_mat.G (Finite_Cartesian_Product.transpose A) = (fixed_mat.G A)\<inverse>"
bot_acom_Com: fixes c :: "com"   and X :: "char list set" shows "\<bottom>\<^sub>c c \<in> Com X"
psubstEnv_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" shows "rho' &[rho] \<equiv> \<lambda>xs x. case rho' xs x of None \<Rightarrow> rho xs x | Some X \<Rightarrow> Some X #[rho]"
elims: fixes key :: "'a \<Rightarrow> 'b"   and x :: "'a"   and xa :: "'a list \<Rightarrow> 'a list"   and xb :: "'a list"   and y :: "'a list list" assumes "asc key x xa xb = y" obtains a :: "'a" and as :: "'a list \<Rightarrow> 'a list" and b :: "'a" and bs :: "'a list" where "x = a__" and "xa = as__" and "xb = b # bs" and "y = (if key a__ \<le> key b then asc key b (\<lambda>ys. as__ (a__ # ys)) bs else as__ [a__] # sequences key (b # bs))"   | a :: "'a" and as :: "'a list \<Rightarrow> 'a list" where "x = a" and "xa = as" and "xb = []" and "y = [as [a]]"
single_Diff_lessThan: fixes k :: "'c" shows "{k} - {..<k} = {k}"