hcju/isabelleps huggingface dataset:
redT_thread_not_disappear: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t' :: "'t" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "thr s' t' = None" shows "thr s t' = None"
simps: fixes x :: "'b"   and y :: "'b" shows "neq\<cdot>x\<cdot>y = neg\<cdot>(eq\<cdot>x\<cdot>y)"
OclAllInstances_at_pre_defined: fixes \<tau> :: "'d state \<times> 'd state"   and H :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>" shows "\<tau> \<Turnstile> \<delta> H .allInstances@pre()"
rep_eq: fixes x :: "'c \<rightharpoonup>\<^sub>f 'd"   and xa :: "'c \<rightharpoonup>\<^sub>f 'd" shows "fmlookup (x ++\<^sub>f xa) = fmlookup x ++ fmlookup xa"
cr_spmf_gpv_def: fixes p :: "'g spmf"   and gpv :: "('g, 'h, 'i) gpv" shows "cr_spmf_gpv p gpv = (gpv = lift_spmf p)"
leI: fixes x :: "'a"   and y :: "'a" assumes "\<not> lt x y" shows "le y x"
bchoice: fixes S :: "'b set"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<forall>x\<in>S. \<exists>y. Q x y" shows "\<exists>f. \<forall>x\<in>S. Q x (f x)"
SUP_bot_conv_2: fixes Inf :: "'h set \<Rightarrow> 'h"   and Sup :: "'h set \<Rightarrow> 'h"   and inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and bot :: "'h"   and top :: "'h"   and B :: "'i \<Rightarrow> 'h"   and A :: "'i set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(bot = Sup (B ` A)) = (\<forall>x\<in>A. B x = bot)"
assms_5: shows "Fvars \<phi> \<inter> snd ` set txs = {}"
51: fixes x :: "'a" shows "top \<squnion> - - x = top"
gallery_def: fixes xs :: "'a set list" shows "gallery xs \<equiv> Ball (set xs) chamber \<and> adjacentchain xs"
A: shows "(xps @ yps @ [yp], Y) \<in> c_failures step out s\<^sub>0"
star_of_0_eq: fixes y :: "'b" shows "(0 = star_of y) = ((0::'b) = y)"
exp1_approx: fixes n :: "nat" assumes "0 < n" shows "exp 1 - (\<Sum>k<n. 1 / fact k) \<in> {0..2 / fact n}"
abs_eq: fixes xa :: "'b"   and x :: "nat" shows "monom xa x = Abs_poly (\<lambda>n. if x = n then xa else (0::'b))"
smult_l_distr: fixes x :: "rat vec"   and a :: "rat"   and b :: "rat" assumes "True"   and "True"   and "x \<in> Rn" shows "(a + b) \<cdot>\<^sub>v x = a \<cdot>\<^sub>v x + b \<cdot>\<^sub>v x"
less_divide_eq: 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"   and b :: "'b"   and c :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less a (divide b c) = (if less zero c then less (times a c) b else if less c zero then less b (times a c) else less a zero)"
berlekamp_basis_basis: shows "Berlekamp_subspace.basis (set (berlekamp_basis u))"
CombineStrip: 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 P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/{} \<^esub>P strip_guards (- F) c UNIV,UNIV" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/{} \<^esub>P c Q,A"
convex_linear_vimage: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'d set" assumes "linear f"   and "convex S" shows "convex (f -` S)"
from_mregex_shift: fixes r :: "mregex"   and xs :: "Formula.formula list"   and ys :: "Formula.formula list" shows "from_mregex (Monitor.shift r (length xs)) (xs @ ys) = from_mregex r ys"
lookup_eq_value: fixes v :: "'b"   and xs :: "('c, 'b) oalist_ntm"   and k :: "'c" assumes "v \<noteq> (0::'b)" shows "(OAlist_lookup_ntm xs k = v) = ((k, v) \<in> set (fst (list_of_oalist_ntm xs)))"
prems_3: shows "\<forall>l s. PProbs s \<and> PProbl l \<and> list_all P1 l \<longrightarrow> (\<exists>l'. f s l' = f s l \<and> length (filter P2 l') \<le> k1 \<and> length (filter P3 l') \<le> length (filter P3 l) \<and> list_all P1 l' \<and> subseq l' l)"
countable_Un_iff: fixes A :: "'b set"   and B :: "'b set" shows "countable (A \<union> B) = (countable A \<and> countable B)"
biderivs_def: fixes w1 :: "'a list"   and w2 :: "'a list" shows "biderivs w1 w2 = rderivs w2 \<circ> lderivs w1"
range_to_fract_embed_poly: fixes p :: "'b fract poly" assumes "set (coeffs p) \<subseteq> range to_fract" shows "p = map_poly to_fract (map_poly inv_embed p)"
scale_bind_spmf: fixes r :: "real"   and p :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf" assumes "r \<le> 1" shows "scale_spmf r (p \<bind> f) = p \<bind> (\<lambda>x. scale_spmf r (f x))"
divide_eq2: fixes h :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. h x \<noteq> (0::'b)" shows "((\<lambda>x. f x / h x) \<in> L F g) = (f \<in> L F (\<lambda>x. g x * h x))"
euler_step_def: fixes ode_ops :: "ode_ops"   and X0 :: "'c set"   and h :: "real" shows "euler_step ode_ops X0 h = one_step ode_ops X0 h (\<lambda>X0 hl hu CX. let H = lv_ivl [min hl hu] [max hl hu] in mk_safe ode_ops CX \<bind> (\<lambda>_. let env = concat ` listset [list_of_eucl ` X0, list_of_eucl ` CX, H] in approx_slp_spec (euler_fas' ode_ops DIM('c)) (2 * DIM('c)) (euler_slp ode_ops) env \<bind> (\<lambda>env. case env of None \<Rightarrow> RETURN None | Some env \<Rightarrow> let res' = take DIM('c) ` env in ASSERT (env_len res' DIM('c)) \<bind> (\<lambda>_. let res = eucl_of_list ` res' in ASSUME (ncc res) \<bind> (\<lambda>_. let err' = drop DIM('c) ` take (DIM('c) * 2) ` env in ASSERT (env_len err' DIM('c)) \<bind> (\<lambda>_. let err = eucl_of_list ` err' in default_reduce_argument_spec \<bind> (\<lambda>ra. reduce_spec ra res \<bind> (\<lambda>res. ASSUME (ncc res) \<bind> (\<lambda>_. safe_set ode_ops res \<bind> (\<lambda>s. if s then mk_safe ode_ops res \<bind> (\<lambda>res. RETURN (Some (res, err))) else RETURN None))))))))))"
intro: fixes r1 :: "('d, 'a, 'b, 'c, 'e, 'f) semantics"   and r2 :: "('d, 'a, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'a, 'b, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'a, 'g, 'h, 'e, 'f) \<tau>moves"   and bisim_wait :: "'b \<Rightarrow> 'g \<Rightarrow> bool"   and final1 :: "'b \<Rightarrow> bool"   and final2 :: "'g \<Rightarrow> bool" assumes "\<And>t. delay_bisimulation_obs (r1 t) (r2 t) (bisim t) (ta_bisim bisim) \<tau>move1 \<tau>move2"   and "\<And>t' x m1 xx m2 t x1 x2 x1' ta1 x1'' m1' x2' ta2 x2'' m2'. \<lbrakk>bisim t' (x, m1) (xx, m2); bisim t (x1, m1) (x2, m2); (\<tau>trsys.silent_move (r1 t) \<tau>move1)\<^sup>*\<^sup>* (x1, m1) (x1', m1); r1 t (x1', m1) ta1 (x1'', m1'); \<not> \<tau>move1 (x1', m1) ta1 (x1'', m1'); (\<tau>trsys.silent_move (r2 t) \<tau>move2)\<^sup>*\<^sup>* (x2, m2) (x2', m2); r2 t (x2', m2) ta2 (x2'', m2'); \<not> \<tau>move2 (x2', m2) ta2 (x2'', m2'); bisim t (x1'', m1') (x2'', m2'); ta_bisim bisim ta1 ta2\<rbrakk> \<Longrightarrow> bisim t' (x, m1') (xx, m2')"   and "\<And>t x1 m1 x2 m2 x1' ta1 x1'' m1' x2' ta2 x2'' m2' w. \<lbrakk>bisim t (x1, m1) (x2, m2); (\<tau>trsys.silent_move (r1 t) \<tau>move1)\<^sup>*\<^sup>* (x1, m1) (x1', m1); r1 t (x1', m1) ta1 (x1'', m1'); \<not> \<tau>move1 (x1', m1) ta1 (x1'', m1'); (\<tau>trsys.silent_move (r2 t) \<tau>move2)\<^sup>*\<^sup>* (x2, m2) (x2', m2); r2 t (x2', m2) ta2 (x2'', m2'); \<not> \<tau>move2 (x2', m2) ta2 (x2'', m2'); bisim t (x1'', m1') (x2'', m2'); ta_bisim bisim ta1 ta2; Suspend w \<in> set \<lbrace>ta1\<rbrace>\<^bsub>w\<^esub>; Suspend w \<in> set \<lbrace>ta2\<rbrace>\<^bsub>w\<^esub>\<rbrakk> \<Longrightarrow> bisim_wait x1'' x2''"   and "\<And>t x1 m1 x2 m2 ta1 x1' m1'. \<lbrakk>bisim t (x1, m1) (x2, m2); bisim_wait x1 x2; r1 t (x1, m1) ta1 (x1', m1'); Notified \<in> set \<lbrace>ta1\<rbrace>\<^bsub>w\<^esub> \<or> WokenUp \<in> set \<lbrace>ta1\<rbrace>\<^bsub>w\<^esub>\<rbrakk> \<Longrightarrow> \<exists>ta2 x2' m2'. r2 t (x2, m2) ta2 (x2', m2') \<and> bisim t (x1', m1') (x2', m2') \<and> ta_bisim bisim ta1 ta2"   and "\<And>t x1 m1 x2 m2 ta2 x2' m2'. \<lbrakk>bisim t (x1, m1) (x2, m2); bisim_wait x1 x2; r2 t (x2, m2) ta2 (x2', m2'); Notified \<in> set \<lbrace>ta2\<rbrace>\<^bsub>w\<^esub> \<or> WokenUp \<in> set \<lbrace>ta2\<rbrace>\<^bsub>w\<^esub>\<rbrakk> \<Longrightarrow> \<exists>ta1 x1' m1'. r1 t (x1, m1) ta1 (x1', m1') \<and> bisim t (x1', m1') (x2', m2') \<and> ta_bisim bisim ta1 ta2"   and "(\<exists>x1. final1 x1) = (\<exists>x2. final2 x2)" shows "FWdelay_bisimulation_obs_axioms final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"
sup_commute: fixes x :: "'b upred"   and y :: "'b upred" shows "(x \<or> y) = (y \<or> x)"
biessentially_surjective_on_objects: fixes a' :: "'d" assumes "D.obj a'" shows "\<exists>a. C.obj a \<and> D.equivalent_objects (F.map\<^sub>0 a) a'"
T1': fixes y :: "'a"   and x :: "'a" assumes "R1 y x" shows "T1 x (Abs1 y)"
aug_flows_finite: fixes p :: "(nat \<times> nat) list" shows "finite {cf e |e. e \<in> set p}"
quotient_is_empty2: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "({} = A // r) = (A = {})"
is_rat: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_rat :: "'a \<Rightarrow> bool"   and to_rat :: "'a \<Rightarrow> rat"   and x :: "'a" assumes "class.is_rat divide inverse times one plus zero minus uminus is_rat to_rat" shows "is_rat x = (x \<in> field_char_0.Rats divide one plus zero minus)"
commutative_monoid_axioms: shows "commutative_monoid G (\<cdot>) \<one>"
rt_strictly_fresherI': fixes i :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "rt1 \<sqsubseteq>\<^bsub>i\<^esub> rt2"   and "\<not> rt2 \<sqsubseteq>\<^bsub>i\<^esub> rt1" shows "rt1 \<sqsubset>\<^bsub>i\<^esub> rt2"
elims_3: fixes x :: "'b trm" assumes "\<not> is_a_constant x" obtains xa :: "'b" where "x = Var xa__"   | xa :: "'b trm" and y :: "'b trm" where "x = xa \<cdot> y"
goodEnv_imp_qGoodEnv_pickE: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option" assumes "goodEnv rho" shows "qGoodEnv (pickE rho)"
assms_1: shows "cupcake_match_result cenv v0 pes err_v = Rval (e, p, env')"
Lambert_W_eqI: fixes w :: "real"   and x :: "real" assumes "- 1 \<le> w"   and "w * exp w = x" shows "Lambert_W x = w"
intros: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d" assumes "P x"   and "R x y" shows "Imagep R P y"
Eventually_ltlp_def: fixes \<phi> :: "'b pltl" shows "F\<^sub>p \<phi> \<equiv> true\<^sub>p U\<^sub>p \<phi>"
set_rotate1: fixes xs :: "'g list" shows "set (rotate1 xs) = set xs"
cases: fixes x :: "'c list \<times> 'c list" obtains ys :: "'c list" where "x = ([], ys__)"   | xs :: "'c list" where "x = (xs__, [])"   | xa :: "'c" and xs :: "'c list" and y :: "'c" and ys :: "'c list" where "x = (xa # xs, y # ys)"
of_int_add: fixes one :: "'c"   and times :: "'c \<Rightarrow> '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 w :: "int"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (w + z) = plus (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z)"
least_map_elemD: fixes c :: "'a"   and f :: "'a \<Rightarrow> 'b"   and B :: "'a set" assumes "c \<in> least_map f B" shows "c \<in> B"
\<iota>_triangle_1: shows "MC.tensor \<iota> MC.unity = MC.tensor MC.unity \<iota> \<cdot> MC.assoc MC.unity MC.unity MC.unity"
comp_substs_def: fixes \<sigma>s :: "'s list"   and \<tau>s :: "'s list" shows "\<sigma>s \<odot>s \<tau>s = Map2.map2 (\<odot>) \<sigma>s \<tau>s"
id_deriv: shows "((\<lambda>v. v) has_derivative (\<lambda>v'. v')) (at \<nu>)"
mult_scalar_left_diff_distrib: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" shows "(p - q) \<odot> r = p \<odot> r - q \<odot> r"
sin_npi_int: fixes n :: "int" shows "sin (pi * real_of_int n) = 0"
mono_Sync: fixes P :: "'c process"   and Q :: "'c process"   and A :: "'c set"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "(P \<lbrakk>A\<rbrakk> S) \<sqsubseteq> (Q \<lbrakk>A\<rbrakk> S)"
evaln_AbruptIs_4: fixes G :: "prog"   and xc :: "abrupt"   and s :: "st"   and x :: "stmt"   and n :: "nat" shows "G\<turnstile>(Some xc, s) \<midarrow>x\<midarrow>n\<rightarrow> (Some xc, s)"
set_intros_4: fixes x13 :: "'f"   and x11 :: "'g"   and x12 :: "'h" shows "x13 \<in> set3_new_thread_action (NewThread x11 x12 x13)"
wf_fmla_atom_mono: fixes tys :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and tys' :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and a :: "'a atom formula" assumes "tys \<subseteq>\<^sub>m tys'"   and "wf_fmla_atom tys a" shows "wf_fmla_atom tys' a"
poly_y_x_def: fixes p :: "'b poly poly" shows "poly_y_x p \<equiv> \<Sum>i\<le>degree p. \<Sum>j\<le>degree (coeff p i). monom (monom (coeff (coeff p i) j) i) j"
src_def: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.src \<mu> \<equiv> if Span.arr \<mu> then Span.mkObj (Maps.cod (Leg0 (Dom \<mu>))) else Span.null"
tauCases': fixes P :: "pi"   and Rs :: "residual" assumes "\<tau>.P \<longmapsto> Rs" obtains P :: "pi" where "\<tau>.Pa__ = \<tau>.P" and "Rs = \<tau> \<prec> P"
extf_mono_strong: fixes ys :: "('s, 'v) tm list"   and xs :: "('s, 'v) tm list"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and gt' :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and f :: "'s" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt' y x"   and "extf f gt ys xs" shows "extf f gt' ys xs"
induct: fixes P :: "op \<Rightarrow> dynamic list \<Rightarrow> bool"   and a0 :: "op"   and a1 :: "dynamic list" assumes "\<And>uu_ uv_. P OpAdd [DNum uu_, DNum uv_]"   and "\<And>uw_ ux_. P OpMul [DNum uw_, DNum ux_]"   and "\<And>uz_. P OpNeg uz_"   and "P OpMul []"   and "\<And>va. P OpMul (DNil # va)"   and "\<And>vb va. P OpMul (DBool vb # va)"   and "\<And>v. P OpMul [v]"   and "\<And>v vc. P OpMul (v # DNil # vc)"   and "\<And>v vd vc. P OpMul (v # DBool vd # vc)"   and "\<And>v vb vd ve. P OpMul (v # vb # vd # ve)"   and "\<And>uy_. P uy_ []"   and "\<And>uy_ va. P uy_ (DNil # va)"   and "\<And>uy_ vb va. P uy_ (DBool vb # va)"   and "\<And>uy_ v. P uy_ [v]"   and "\<And>uy_ v vc. P uy_ (v # DNil # vc)"   and "\<And>uy_ v vd vc. P uy_ (v # DBool vd # vc)"   and "\<And>uy_ v vb vd ve. P uy_ (v # vb # vd # ve)" shows "P a0 a1"
simps_1: fixes x1 :: "'a form multiset"   and x2 :: "'a form"   and y1 :: "'a form multiset"   and y2 :: "'a form" shows "(( x1 \<Rightarrow>* x2) = ( y1 \<Rightarrow>* y2)) = (x1 = y1 \<and> x2 = y2)"
apath_Nil_iff: fixes u :: "'a"   and v :: "'a" shows "S.apath u [] v = (u = v \<and> u \<in> pverts (subdivide G e w))"
mod_ring_minus: shows "(mod_ring_rel ===> mod_ring_rel ===> mod_ring_rel) (minus_p p) (-)"
zero_order_5: fixes n :: "'a multiset" shows "(\<not> {#} \<subset># n) = (n = {#})"
ltake_Suc_conv_snoc_lnth: fixes m :: "nat"   and xs :: "'c llist" assumes "enat m < llength xs" shows "ltake (enat (Suc m)) xs = lappend (ltake (enat m) xs) (LCons (lnth xs m) LNil)"
uniform_limit_flow: fixes a :: "real"   and t0 :: "real"   and x0 :: "'a"   and b :: "real" assumes "a - t0 \<in> existence_ivl0 x0"   and "b - t0 \<in> existence_ivl0 x0"   and "a \<le> b" shows "uniform_limit {a..b} (\<lambda>x0 t. flow0 x0 (t - t0)) (\<lambda>t. flow0 x0 (t - t0)) (at x0)"
exist_nj: shows "\<forall>j\<in>{0..<m}. \<exists>nj. enat (Suc nj) < llength Ns \<and> prems_of \<iota> ! j \<notin> active_subset (lnth Ns nj) \<and> (\<forall>k>nj. enat k < llength Ns \<longrightarrow> prems_of \<iota> ! j \<in> active_subset (lnth Ns k))"
to_nat_on_infinite: fixes S :: "'f set" assumes "countable S"   and "infinite S" shows "bij_betw (to_nat_on S) S UNIV"
pqr_simps_1: shows "sturm p ! i = sturm_squarefree' p ! i * d"
deadlocked_thread_exists: fixes t :: "'t"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "t \<in> deadlocked s" obtains x :: "'x" and ln :: "'l \<Rightarrow>f nat" where "thr s t = \<lfloor>(x, ln)\<rfloor>"
length_signs_at: fixes qs :: "real poly list"   and x :: "real" shows "length (signs_at qs x) = length qs"
disc_2: fixes x2 :: "'b" shows "is_neg (Neg x2)"
factgroup_subgroup_union_char: fixes A :: "'a set set" assumes "subgroup A (G Mod H)" shows "\<Union> A = {x \<in> carrier G. H #> x \<in> A}"
inversion_def: fixes n :: "nat"   and dc :: "nat option \<times> bool list" assumes "dining_cryptographers_space n" shows "dining_cryptographers_space.inversion n dc = map (\<lambda>c. (payer dc = Some c) \<noteq> (dining_cryptographers_space.coin n dc c \<noteq> dining_cryptographers_space.coin n dc (c + 1))) [0..<n]"
has_codomain_iff_arr: fixes f :: "'a" shows "(V.codomains f \<noteq> {}) = V.arr f"
LAssRedsVal: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and V :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>Val v,(h', l')\<rangle>" shows "P \<turnstile> \<langle>V:=e,s\<rangle> \<rightarrow>* \<langle>unit,(h', l'(V \<mapsto> v))\<rangle>"
nullstrict0: fixes X :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2)" shows "(case X of (a, b) \<Rightarrow> f null a b) = invalid"
congruenceSubstDef_1: fixes P :: "pi"   and Q :: "pi" shows "P \<simeq>\<^sup>s Q \<equiv> (P, Q) \<in> substClosed congruence"
start_heap_def: fixes G :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list" shows "start_heap G \<equiv> [addr_of_sys_xcpt NullPointer \<mapsto> blank G NullPointer, addr_of_sys_xcpt ClassCast \<mapsto> blank G ClassCast, addr_of_sys_xcpt OutOfMemory \<mapsto> blank G OutOfMemory]"
conclusion_is_smaller_than_premisses: fixes C :: "'a eclause"   and P :: "'a eclause set"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and C' :: "'a literal set" assumes "derivable C P S \<sigma> Ground C'"   and "\<forall>x\<in>S. finite (cl_ecl x)"   and "grounding_set P \<sigma>" shows "\<exists>D. D \<in> P \<and> (mset_cl (C', \<sigma>), mset_ecl (D, \<sigma>)) \<in> mult (mult trm_ord)"
assms: shows "Measurable.pred (stream_space (count_space UNIV)) P"
simps_2: fixes f :: "'fsym"   and Tl :: "'fsym trm list" shows "wt (Fn f Tl) = (wtFsym f \<and> list_all wt Tl \<and> arOf f = map tpOf Tl)"
Chn_vcomp: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "seq \<nu> \<mu>" shows "Chn (\<nu> \<bullet> \<mu>) = Chn \<nu> \<cdot> Chn \<mu>"
sample_coprime_e_gt_1: fixes e :: "nat"   and N :: "nat" assumes "e \<in> set_spmf (sample_coprime N)" shows "1 < e"
coef_sym_gt_0: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and f :: "'b"   and i :: "nat" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "0 < coef_sym f i"
dist_fs_zero2: fixes z1 :: "complex_homo"   and z2 :: "complex_homo" assumes "dist_fs z1 z2 = 0" shows "z1 = z2"
take_update_last: fixes n :: "nat"   and list :: "'b list"   and x :: "'b" assumes "n < length list" shows "(take (Suc n) list)[n := x] = take n list @ [x]"
jozsa_transform_times_\<psi>\<^sub>1_is_\<psi>\<^sub>2: shows "U\<^sub>f * Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))) = Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))"
nat_pow_0: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" shows "x [^]\<^bsub>G\<^esub> 0 = \<one>\<^bsub>G\<^esub>"
diff_eq_eq: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.group_add minus plus zero uminus" shows "(minus a b = c) = (a = plus c b)"
LIMSEQ_iff: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" shows "X \<longlonglongrightarrow> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r)"
supplement2_Jacobi': fixes p :: "int" assumes "odd p"   and "1 < p" shows "Jacobi 2 p = (if p mod 8 = 1 \<or> p mod 8 = 7 then 1 else - 1)"
is_weak_ranking_def: fixes xs :: "'b set list" shows "is_weak_ranking xs = ({} \<notin> set xs \<and> (\<forall>i j. i < length xs \<and> j < length xs \<and> i \<noteq> j \<longrightarrow> xs ! i \<inter> xs ! j = {}))"
passivate: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and t :: "trm"   and T :: "Syntax.type"   and x :: "nat" assumes "\<Gamma> , \<Delta> \<turnstile>\<^sub>T t : T"   and "\<Delta> x = T" shows "\<Gamma> , \<Delta> \<turnstile>\<^sub>C <x>t"
lift_opts_ind: fixes list :: "(cl \<times> char list \<times> ty) list"   and P :: "cld list"   and ctx :: "ctx" assumes "\<And>x. x \<in> set list \<Longrightarrow> case x of (cl, var, ty) \<Rightarrow> find_type_f P ctx cl = Some ty" shows "lift_opts (map (case_vd (\<lambda>clk vark. find_type_f P ctx clk) \<circ> (\<lambda>(cl, var, ty). vd_def cl var)) list) = Some (map (\<lambda>(cl, var, ty). ty) list)"
root_info_def: fixes p :: "int poly" shows "root_info p = (if degree p = 1 then let x = Rat.Fract (- coeff p 0) (coeff p 1) in Root_Info (\<lambda>l r. if l \<le> x \<and> x \<le> r then 1 else 0) (\<lambda>b. if x \<le> b then 1 else 0) else let rp = of_int_poly p; ps = sturm_rat rp in Root_Info (\<lambda>a b. sign_changes_rat ps a - sign_changes_rat ps b) (\<lambda>a. sign_changes_neg_number_rootat ps - sign_changes_rat ps a))"