hcju/isabelleps huggingface dataset:
qc_wf_relto_iff: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" assumes "R O S \<subseteq> (R \<union> S)\<^sup>* O R" shows "wf (S\<^sup>* O R O S\<^sup>*) = wf R"
RETURN_rule: fixes \<Phi> :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<Phi> x" shows "RETURN x \<le> SPEC \<Phi>"
AxLines: fixes m :: "Body"   and k :: "Body"   and x :: "'a Point"   and p :: "'a Point"   and q :: "'a Point" assumes "IOb m"   and "IOb k"   and "collinear x p q" shows "collinear (wvt k m x) (wvt k m p) (wvt k m q)"
at_neg1: fixes c :: "cars"   and \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. (\<forall>v'. v = c > v' \<longrightarrow> \<not> \<phi> ts v') \<longrightarrow> \<not> (\<forall>v'. v = c > v' \<longrightarrow> \<phi> ts v')"
preserves_inv: fixes f :: "'a" assumes "local.iso f" shows "src (local.inv f) = local.inv (src f)"
min_distI2: fixes v :: "nat"   and v' :: "nat"   and Q :: "nat \<Rightarrow> bool" assumes "connected v v'"   and "\<And>d. \<lbrakk>dist v d v'; \<And>d'. dist v d' v' \<Longrightarrow> d \<le> d'\<rbrakk> \<Longrightarrow> Q d" shows "Q (min_dist v v')"
wf_digraphI_app_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "wf_digraph (app_iso hom G)"
primitive_part_idemp: fixes f :: "'a poly" shows "primitive_part (primitive_part f) = primitive_part f"
g_BC: shows "g \<in> Hom\<^bsub>set_cat U\<^esub> B C"
set_bd_1: fixes x :: "('l, 'k, 'm, 'n) F" shows "|F'set3 x| \<le>o F'bd"
order_trans: fixes x :: "nat \<Rightarrow> nat"   and y :: "nat \<Rightarrow> nat"   and z :: "nat \<Rightarrow> nat" assumes "x \<lesssim> y"   and "y \<lesssim> z" shows "x \<lesssim> z"
induct: fixes x1 :: "'b list"   and x2 :: "'b list"   and P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "x1 \<approx>2 x2"   and "\<And>a b xs. P (a # b # xs) (b # a # xs)"   and "P [] []"   and "\<And>xs ys. \<lbrakk>xs \<approx>2 ys; P xs ys\<rbrakk> \<Longrightarrow> P ys xs"   and "\<And>a xs. P (a # a # xs) (a # xs)"   and "\<And>xs ys a. \<lbrakk>xs \<approx>2 ys; P xs ys\<rbrakk> \<Longrightarrow> P (a # xs) (a # ys)"   and "\<And>xs1 xs2 xs3. \<lbrakk>xs1 \<approx>2 xs2; P xs1 xs2; xs2 \<approx>2 xs3; P xs2 xs3\<rbrakk> \<Longrightarrow> P xs1 xs3" shows "P x1 x2"
at_infty_ereal_eq_at_top: shows "at \<infinity> = filtermap ereal at_top"
simps_4: fixes t :: "nat"   and s\<^sub>1 :: "stmt"   and s\<^sub>2 :: "stmt" shows "valid_sops_stmt t (Seq s\<^sub>1 s\<^sub>2) = (valid_sops_stmt t s\<^sub>1 \<and> valid_sops_stmt t s\<^sub>2)"
drop_take_inv: fixes n :: "nat"   and u :: "'b list"   and w :: "'b list" assumes "n \<le> \<^bold>|u\<^bold>|" shows "drop n (take n u \<cdot> w) = w"
Inf_M_telescoping: fixes k' :: "nat"   and k :: "nat" assumes "k' \<le> k" shows "{\<Sqinter> (M k)..} \<subseteq> {\<Sqinter> (M k')..}"
from_nat_not_eq: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Rep :: "'c \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'c"   and a :: "nat"   and b :: "'c" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "a \<noteq> mod_type.to_nat Rep b"   and "a < CARD('c)" shows "mod_type.from_nat Abs a \<noteq> b"
induct: fixes x :: "hf"   and P :: "hf \<Rightarrow> bool" assumes "x \<in> Special_ax"   and "\<And>v x ax y sx. \<lbrakk>AbstForm v 0 x ax; SubstForm v y x sx; Form x; is_Var v; Term y\<rbrakk> \<Longrightarrow> P (q_Imp sx (q_Ex ax))" shows "P x"
PO_l2_inv6': shows "reach l2 \<subseteq> l2_inv6'"
fmupd_fmsubset_restrict_set: fixes s :: "('a, bool) fmap"   and v :: "'a"   and X :: "'a set"   and x :: "bool" assumes "s \<in> {s. fmdom' s = insert v X}"   and "fmlookup s v = Some x" shows "s = fmupd v x (fmrestrict_set X s)"
HPhase1or2ReadThen_Inv2b: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "Inv2b s"   and "HPhase1or2ReadThen s s' p d q" shows "Inv2b s'"
elims: fixes x :: "nat"   and xa :: "real mpoly"   and xb :: "real mpoly"   and xc :: "atom fm"   and xd :: "nat"   and y :: "atom fm" assumes "linear_substitution_fm_helper x xa xb xc xd = y" obtains var :: "nat" and b :: "real mpoly" and c :: "real mpoly" and F :: "atom fm" and z :: "nat" where "x = var" and "xa = b" and "xb = c" and "xc = F" and "xd = z" and "y = liftmap (\<lambda>x A. fm.Atom (linear_substitution (var + x) (liftPoly 0 x b) (liftPoly 0 x c) A)) F z"
well_shaped_update_rightmost: fixes t :: "'b tree"   and f :: "'b \<Rightarrow> 'b" assumes "well_shaped t" shows "well_shaped (update_rightmost f t)"
mult_Mp_2: fixes f :: "int poly"   and g :: "int poly" shows "f * Mp g =m f * g"
ve: shows "ve \<in> smaps_over (vars p \<times> UNIV) (proc_poss p)"
fmlas_def: shows "fmlas = sinterleave atoms (smerge (smap snd (siterate extendLevel (atoms, depth1))))"
wf_J_prog_wf_J_prog': fixes P :: "'a J_prog" assumes "wf_J_prog P" shows "wf_J_prog' P"
Cons_3: shows "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"
mset_permute_list: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'c list" assumes "f permutes {..<length xs}" shows "mset (permute_list f xs) = mset xs"
0: shows "const_res_subseq k s 0 \<in> carrier Zp"
Bonk_Schramm_extension_unfolded_wo_props_3: shows "\<forall>a b. (a, middle a b) \<in> Bonk_Schramm_extension_unfolded_wo"
extended_monoidal_category_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b" shows "extended_monoidal_category C T \<alpha> \<iota> \<equiv> (monoidal_category C T \<alpha> \<iota> \<and> inverse_transformation (product_category.comp C (product_category.comp C C)) C (binary_endofunctor.ToTC C T) (binary_endofunctor.ToCT C T) \<alpha> \<and> natural_isomorphism C C (\<lambda>f. T (monoidal_category.unity C \<iota>, f)) (identity_functor.map C) (\<lambda>f. if partial_magma.arr C f then C f (monoidal_category.lunit C T \<alpha> \<iota> (partial_magma.dom C f)) else partial_magma.null C)) \<and> inverse_transformation C C (\<lambda>f. T (monoidal_category.unity C \<iota>, f)) (identity_functor.map C) (\<lambda>f. if partial_magma.arr C f then C f (monoidal_category.lunit C T \<alpha> \<iota> (partial_magma.dom C f)) else partial_magma.null C) \<and> natural_isomorphism C C (\<lambda>f. T (f, monoidal_category.unity C \<iota>)) (identity_functor.map C) (\<lambda>f. if partial_magma.arr C f then C f (monoidal_category.runit C T \<alpha> \<iota> (partial_magma.dom C f)) else partial_magma.null C) \<and> inverse_transformation C C (\<lambda>f. T (f, monoidal_category.unity C \<iota>)) (identity_functor.map C) (\<lambda>f. if partial_magma.arr C f then C f (monoidal_category.runit C T \<alpha> \<iota> (partial_magma.dom C f)) else partial_magma.null C)"
measure_completion: fixes S :: "'c set"   and M :: "'c measure" assumes "S \<in> sets M" shows "Sigma_Algebra.measure (completion M) S = Sigma_Algebra.measure M S"
T_invT: fixes h :: "'n" assumes "h \<in> N" shows "T (invT h) = h"
ket_zero_is_state: shows "state 1 |Deutsch.zero\<rangle>"
supteq_size: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" assumes "s \<unrhd> t" shows "size t \<le> size s"
not_in_interior_convex_hull_3_3: fixes c :: "complex"   and a :: "complex"   and b :: "complex" shows "c \<notin> interior (convex hull {a, b, c})"
diff_freshAbs_vsubstAbs: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and y :: "'var"   and y1 :: "'var"   and ys :: "'varSort" assumes "goodAbs A"   and "y \<noteq> y1" shows "freshAbs ys y (A $[y1 // y]_ys)"
supp_Act: fixes \<alpha> :: "'a" shows "supp (S_action.Act \<alpha>) = supp \<alpha>"
i_shrink_last_i_drop_mult: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "f \<Up> (n * k) \<div>\<^sub>l k = f \<div>\<^sub>l k \<Up> n"
n: shows "n \<in>\<^sub>\<circ> \<omega>"
elimVar_eq: fixes xs :: "real list"   and var :: "nat"   and p :: "real mpoly"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" assumes "length xs = var"   and "Eq p \<in> set L"   and "MPoly_Type.degree p var = 1 \<or> MPoly_Type.degree p var = 2" shows "(\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)) = (\<exists>x. eval (elimVar var L F (Eq p)) (xs @ x # \<Gamma>)) \<or> (\<forall>x. aEval (Eq p) (xs @ x # \<Gamma>))"
Zleast: fixes A :: "int set"   and n :: "int" assumes "A \<noteq> {}"   and "A \<subseteq> Zset"   and "LB A n" shows "Zleast A \<in> A \<and> (\<forall>x\<in>A. Zleast A \<le> x)"
Sigma_empty2: fixes A :: "'c set" shows "A \<times> {} = {}"
subst_cls_list_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and Cs :: "'b literal multiset list"   and \<sigma> :: "'c" shows "substitution_ops.subst_cls_list subst_atm Cs \<sigma> = map (\<lambda>A. substitution_ops.subst_cls subst_atm A \<sigma>) Cs"
has_derivative_subset: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set"   and t :: "'c set" assumes "(f has_derivative f') (at x within s)"   and "t \<subseteq> s" shows "(f has_derivative f') (at x within t)"
snd_quot_of_fract_nonzero: fixes x :: "'b fract" shows "denominator x \<noteq> (0::'b)"
set_intros_1: fixes x11 :: "'a"   and x12 :: "bool" shows "x11 \<in> set1_message <x11, inM x12>"
w_values_greater: fixes vs0 :: "'a \<times> addr_loc \<Rightarrow> 'a val set"   and P :: "'c prog"   and obs :: "('a, 'b) obs_event action list" shows "vs0 \<le> w_values P vs0 obs"
additive_delta_split: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and x :: "'b" assumes "Transformers.additive t"   and "healthy t"   and "sound P" shows "t P x = (\<Sum>y\<in>UNIV. P y * t \<guillemotleft> \<lambda>z. z = y \<guillemotright> x)"
B: shows "sinks_aux I D U xs \<subseteq> sinks_aux I D V xs"
simps_4: fixes f :: "char list \<Rightarrow> aexp \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "bexp \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'a"   and b :: "bexp"   and P1 :: "'a"   and C1 :: "'a acom"   and P2 :: "'a"   and C2 :: "'a acom"   and Q :: "'a" shows "Step f g S (IF b THEN {P1} C1 ELSE {P2} C2 {Q}) = IF b THEN {g b S} Step f g P1 C1 ELSE {g (bexp.Not b) S} Step f g P2 C2 {post C1 \<squnion> post C2}"
hom_induced_id: fixes c :: "(((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int) set"   and p :: "int"   and X :: "'b topology"   and S :: "'b set" assumes "c \<in> carrier (relative_homology_group p X S)" shows "hom_induced p X S X S id c = c"
vid_on_mono: fixes A :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> B" shows "vid_on A \<subseteq>\<^sub>\<circ> vid_on B"
reset_set: fixes cs :: "'a list"   and u :: "'a \<Rightarrow> 'b"   and d :: "'b"   and ts :: "'b list" assumes "\<forall>c\<in>set cs. u c = d" shows "[cs\<rightarrow>d]set_clocks cs ts u = u"
density_marg_dens2_eq: fixes x :: "nat"   and y :: "nat"   and \<rho> :: "nat \<Rightarrow> val" assumes "x \<in> V"   and "y \<in> V"   and "x \<noteq> y"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "density (stock_measure (PRODUCT (\<Gamma> x) (\<Gamma> y))) (marg_dens2 \<Y> x y \<rho>) = distr (dens_ctxt_measure \<Y> \<rho>) (stock_measure (PRODUCT (\<Gamma> x) (\<Gamma> y))) (\<lambda>\<sigma>. <|\<sigma> x, \<sigma> y|>)"
G1i: fixes i :: "nat" assumes "i < length (map gs2.gso [0..<m])" shows "map gs2.gso [0..<m] ! i \<in> Rn"
preserves_comp: fixes g :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr \<times> 'a2"   and f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr \<times> 'a2" assumes "A2_BxA2.seq g f" shows "E.map (g \<cdot>\<^sub>[\<^sub>A\<^sub>2\<^sub>,\<^sub>B\<^sub>]\<^sub>x\<^sub>A\<^sub>2 f) = E.map g \<cdot>\<^sub>B E.map f"
ex_x_axis_poincare_distance_negative: fixes d :: "real" assumes "0 \<le> d" shows "\<exists>z. is_real z \<and> Re z \<le> 0 \<and> - 1 < Re z \<and> of_complex z \<in> unit_disc \<and> of_complex z \<in> circline_set x_axis \<and> poincare_distance 0\<^sub>h (of_complex z) = d"
KBasic_4: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) \<^bold>\<equiv> \<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>& \<^bold>\<box>(\<psi> \<^bold>\<rightarrow> \<phi>) in v]"
exec_move_FAss1: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and F :: "String.literal"   and D :: "String.literal"   and e' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (e\<bullet>F{D} := e') h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
set_iterator_product_correct: fixes it_a :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and S_a :: "'b set"   and it_b :: "'b \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and S_b :: "'b \<Rightarrow> 'c set" assumes "set_iterator it_a S_a"   and "\<And>a. a \<in> S_a \<Longrightarrow> set_iterator (it_b a) (S_b a)" shows "set_iterator (set_iterator_product it_a it_b) (Sigma S_a S_b)"
int_iffD1: fixes F :: "'b \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" assumes "\<turnstile> F = G" shows "\<turnstile> F \<longrightarrow> G"
bounded_iff: fixes A :: "'d set"   and x :: "'d" assumes "finite A"   and "A \<noteq> {}" shows "(Max A \<le> x) = (\<forall>a\<in>A. a \<le> x)"
fv_strand_subst: fixes P :: "('c, 'd) strand_step \<Rightarrow> 'd set"   and ac :: "poscheckvariant"   and S :: "('c, 'd) strand_step list"   and \<delta> :: "'d \<Rightarrow> ('c, 'd) Term.term" assumes "P = fv\<^sub>s\<^sub>t\<^sub>p \<or> P = fv\<^sub>r\<^sub>c\<^sub>v \<or> P = fv\<^sub>s\<^sub>n\<^sub>d \<or> P = fv\<^sub>e\<^sub>q ac \<or> P = fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q"   and "bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` \<Union> (set (map P S))) = \<Union> (set (map P (S \<cdot>\<^sub>s\<^sub>t \<delta>)))"
residual_r_pure_induction: fixes x :: "'a" shows "(x \<rightarrow> x)\<^sup>\<star> \<le> x \<rightarrow> x"
ex_actions_ok1_conv_ex_actions_ok2: fixes ta1 :: "('a, 't, 'x1, 'm1, 'b, 'c) thread_action"   and ta2 :: "('a, 't, 'x2, 'm2, 'b, 'c) thread_action"   and t :: "'t" assumes "ta1 \<sim>m ta2" shows "(\<exists>s1. r1.actions_ok s1 t ta1) = (\<exists>s2. r2.actions_ok s2 t ta2)"
add_star_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and one :: "'c"   and star :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.left_kleene_algebra plus times less_eq less one star" shows "plus x (times (times y (star y)) x) = times (star y) x"
Cons_2: shows "folding_g.gallery (C # (E # Es) @ [D])"
weakBisimTransitive: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<approx> Q"   and "\<Psi> \<rhd> Q \<approx> R" shows "\<Psi> \<rhd> P \<approx> R"
that: fixes B :: "'a set" assumes "B \<subseteq> S"   and "pairwise orthogonal B"   and "span B = S" shows "thesis"
valid_Tree\<^sub>\<alpha>_eqvt: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and P :: "'b"   and t :: "('e, 'c, 'd) Tree\<^sub>\<alpha>"   and p :: "perm" assumes "nominal_ts satisfies transition"   and "nominal_ts.valid_Tree\<^sub>\<alpha> satisfies transition P t" shows "nominal_ts.valid_Tree\<^sub>\<alpha> satisfies transition (p \<bullet> P) (p \<bullet> t)"
cmp_lex_alt: fixes cmp :: "'a \<Rightarrow> 'b \<Rightarrow> comp_res"   and l :: "'a list"   and m :: "'b list" shows "cmp_lex cmp l m = cmp_lex' cmp l m"
elims_3: fixes x :: "'d"   and xa :: "'e"   and xb :: "('f, 'g, 'd, 'e, 'h) qAbs" assumes "\<not> qAFreshAbs x xa xb" obtains xs :: "'d" and x :: "'e" and ys :: "'d" and y :: "'e" and X :: "('f, 'g, 'd, 'e, 'h) qTerm" where "xa__ = xs" and "xa = x" and "xb = qAbs ys y X" and "\<not> ((xs \<noteq> ys \<or> x \<noteq> y) \<and> qAFresh xs x X)"
hb_irrefl: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and m1 :: "'c"   and m2 :: "'c" assumes "causal_network history msg_id"   and "network.hb history m1 m2" shows "m1 \<noteq> m2"
eval_\<MM>: shows "\<MM> = (\<lambda>x. \<Sum>p\<leftarrow>map fst (filter snd (List.enumerate 2 (sieve 1 (replicate (nat \<lfloor>x\<rfloor> - 1) True)))). ln (real p) / real p)"
length_interact: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xs :: "'b list list"   and ys :: "'b list list" assumes "class.monoid_add plus zero" shows "length (interact xs ys) = sum_list (map length xs) + sum_list (map length ys)"
simps_1: fixes \<eta> :: "nat"   and uu :: "unit" shows "vkey_oracle \<eta> None uu = key_gen \<eta> \<bind> (\<lambda>(vkey, sigkey). return_spmf (vkey, Some (vkey, sigkey, [])))"
gcd_proj1_iff: fixes m :: "'a"   and n :: "'a" shows "(gcd m n = normalize m) = (m dvd n)"
pps_to_list_sorted_wrt: fixes S :: "'t set" shows "sorted_wrt ord_term_strict_conv (pps_to_list S)"
joint_strategy_def: fixes \<sigma>\<^sub>e :: "'d list \<Rightarrow> 'c"   and \<sigma>\<^sub>o :: "'d list \<Rightarrow> 'c"   and p :: "'d list" shows "joint_strategy \<sigma>\<^sub>e \<sigma>\<^sub>o p = (if even (length p) then \<sigma>\<^sub>e p else \<sigma>\<^sub>o p)"
1: fixes \<mu> :: "'a arrow_of_spans_data" assumes "V.arr \<mu>" shows "V.arr (src \<mu>)"
atMost_Suc: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "prod g {..Suc n} = prod g {..n} * g (Suc n)"
sound_modes_invariant: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and cms :: "('a \<times> (Mode \<Rightarrow> 'b set)) list"   and mem :: "'b \<Rightarrow> 'c"   and cms' :: "('a \<times> (Mode \<Rightarrow> 'b set)) list"   and mem' :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'"   and "sifum_security.sound_mode_use eval (cms, mem)"   and "((cms, mem), cms', mem') \<in> sifum_security.meval eval" shows "sifum_security.sound_mode_use eval (cms', mem')"
quot_Zero: shows "\<guillemotleft>Zero\<guillemotright> = Zero"
IH: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "deg R p \<le> na__" shows "deg R (p [^]\<^bsub>P\<^esub> m) = m * deg R p"
transpose_transpose: fixes A :: "(('a, 'b) vec, 'c) vec" shows "Finite_Cartesian_Product.transpose (Finite_Cartesian_Product.transpose A) = A"
open_imp_subset: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and U :: "'c set" assumes "topological_space S is_open"   and "is_open U" shows "U \<subseteq> S"
spectrum_imp_cxt_quotient_ring: fixes \<pp> :: "('a \<times> 'a) set set" assumes "\<pp> \<in> local.local.spectrum" shows "Comm_Ring.quotient_ring (local.carrier_quotient_ring\<setminus>\<pp>) local.carrier_quotient_ring local.add_rel local.mult_rel local.zero_rel local.one_rel"
select_convs_2: fixes NTDom :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NTCod :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NatTransMap :: "'g \<Rightarrow> 'j"   and more :: "'m" shows "NTCod \<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more\<rparr> = NTCod"
ideD_1: fixes a :: "'a" assumes "B.ide a" shows "B.arr a"
simps_5: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "is_\<mu>LTL (\<phi> and\<^sub>n \<psi>) = (is_\<mu>LTL \<phi> \<and> is_\<mu>LTL \<psi>)"
wfE_min': fixes R :: "('d \<times> 'd) set"   and Q :: "'d set" assumes "wf R"   and "Q \<noteq> {}" obtains z :: "'d" where "z \<in> Q" and "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
even_set_encode_iff: fixes A :: "nat set" assumes "finite A" shows "even (set_encode A) = (0 \<notin> A)"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "distinct (x # xs) = (x \<notin> set xs \<and> distinct xs)"
operative_integralI: fixes f :: "'b \<Rightarrow> 'a" shows "operative (lift_option (+)) (Some (0::'a)) (\<lambda>i. if f integrable_on i then Some (integral i f) else None)"
restr_rtrancl: fixes m :: "'b \<Rightarrow> bool"   and l :: "'b"   and R :: "('b \<times> 'b) set" assumes "m l" shows "(R | m)\<^sup>* `` {l} = {l}"
map_add_assoc: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option"   and m3 :: "'a \<Rightarrow> 'b option" shows "Map.map_add m1 (Map.map_add m2 m3) = Map.map_add (Map.map_add m1 m2) m3"
deformation_retract_relative_homology_group_isomorphisms: fixes X :: "'a topology"   and Y :: "'a topology"   and r :: "'a \<Rightarrow> 'a"   and V :: "'a set"   and U :: "'a set"   and p :: "int" assumes "retraction_maps X Y r id"   and "V \<subseteq> U"   and "r ` U \<subseteq> V"   and "homotopic_with (\<lambda>h. h ` U \<subseteq> U) X X r id" shows "group_isomorphisms (relative_homology_group p X U) (relative_homology_group p Y V) (hom_induced p X U Y V r) (hom_induced p Y V X U id)"
inner_prod_smult_left_right: fixes u :: "complex vec"   and n :: "nat"   and v :: "complex vec"   and b :: "complex"   and a :: "complex" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n" shows "inner_prod (a \<cdot>\<^sub>v u) (b \<cdot>\<^sub>v v) = conjugate a * b * inner_prod u v"
has_ge_has_not: fixes n :: "nat"   and S :: "'b set"   and n' :: "nat" assumes "\<not> has n S"   and "n \<le> n'" shows "\<not> has n' S"
eventually_rev_mp: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter"   and Q :: "'b \<Rightarrow> bool" assumes "eventually P F"   and "\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x" shows "eventually Q F"
abs_idempotent: fixes a :: "'a" shows "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
smult_diag: fixes x :: "'b"   and y :: "'b" shows "x *\<^sub>S diag y = diag (x * y)"
lincomb_closed: fixes S :: "'c set"   and a :: "'c \<Rightarrow> 'a" assumes "S \<subseteq> carrier V"   and "a \<in> S \<rightarrow> carrier K" shows "lincomb a S \<in> carrier V"
distinct_mapI: fixes f :: "'e \<Rightarrow> 'd"   and l :: "'e list" assumes "distinct (map f l)" shows "distinct l"
prems_2: shows "all_chains (no_call_to c) \<Gamma> (Rule m' (Call c') # rrs)"
matrix_vector_mult_in_columnspace: fixes A :: "((real, 'b) vec, 'a) vec"   and x :: "(real, 'b) vec" shows "A *v x \<in> span (columns A)"
sub1_lt_0_iff: fixes n :: "num" shows "(Code_Numeral.sub n num.One < 0) = False"
elims_2: fixes x :: "'at set"   and xa :: "'at Literal set set" assumes "x \<Turnstile> xa" obtains I :: "'at set" and S :: "'at Literal set set" where "x = I" and "xa = S" and "\<forall>C. C \<in> S \<longrightarrow> I \<Turnstile> C"
clock_knowledge_def: fixes envObs :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and cec :: "'c odlist" shows "clock_knowledge envObs cec \<equiv> \<lambda>a s. ODList.filter (\<lambda>s'. envObs a s = envObs a s') cec"
IH2: assumes "p2 \<in> subterms (\<Gamma> pn)" shows "sterms \<Gamma> p2 \<subseteq> cterms \<Gamma>"
LT30: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and N :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "|~ P \<and> N \<longrightarrow> \<circle>P \<or> \<circle>Q" shows "|~ N \<longrightarrow> (P \<longrightarrow> \<circle>P) \<or> \<diamond>Q"
eq_GSpan_single: fixes v :: "'c" assumes "v \<in> V"   and "v \<noteq> (0::'c)" shows "V = GSpan [v]"
False: shows "\<not> (countable_preimages B Y i = {} \<or> countable_preimages B Y j = {})"
s0g_secrecy_def: shows "s0g_secrecy \<equiv> {s. kn s \<subseteq> az s \<union> lkr s}"