hcju/isabelleps huggingface dataset:
Afix_lookup_fresh: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and v :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "EdomArityAnalysis Aexp"   and "atom v \<sharp> \<Gamma>" shows "(ArityAnalysis.Afix Aexp \<Gamma>\<cdot>ae) v = ae v"
SFAssNone: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "\<nexists>b t. P \<turnstile> C has F,b:t in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NoSuchFieldError,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"
i_Exec_Stream_Acc_Output_causal: fixes n :: "nat"   and input1 :: "nat \<Rightarrow> 'd message_af"   and input2 :: "nat \<Rightarrow> 'd message_af"   and k :: "nat"   and output_fun :: "'f \<Rightarrow> 'e message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and c :: "'f" assumes "input1 \<Down> n = input2 \<Down> n" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input1 c \<Down> n = i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input2 c \<Down> n"
division_ofD_4: fixes s :: "'a set set"   and i :: "'a set"   and K :: "'a set" assumes "s division_of i"   and "K \<in> s" shows "\<exists>a b. K = cbox a b"
det_four_block_mat_lower_right_id: fixes A :: "'b mat"   and m :: "nat"   and B :: "'b mat"   and n :: "nat"   and C :: "'b mat"   and D :: "'b mat" assumes "A \<in> carrier_mat m m"   and "B = 0\<^sub>m m (n - m)"   and "C = 0\<^sub>m (n - m) m"   and "D = 1\<^sub>m (n - m)"   and "m < n" shows "Determinant.det (four_block_mat A B C D) = Determinant.det A"
set_apply_ocrit: fixes crit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and data :: "nat \<times> 'g"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps' :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "set (apply_ocrit crit data hs ps' ps) = {(p, q) |p q. (p, q) \<in> set ps \<and> \<not> crit data hs ps' p q}"
convol_expand_snd: fixes f :: "'a \<Rightarrow> 'b \<times> 'c"   and g :: "'a \<Rightarrow> 'b" assumes "fst \<circ> f = g" shows "\<langle>g, snd \<circ> f\<rangle> = f"
dom_glb_eq: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(ans_d x \<le> ans_d y \<cdot> ans_d z) = (ans_d x \<le> ans_d y \<and> ans_d x \<le> ans_d z)"
pos_le_minus_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 c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less_eq a (uminus (divide b c)) = less_eq (times a c) (uminus b)"
d: shows "abc_notfinal (abc_steps_l (0, args) A stpa) A \<and> abc_final (abc_steps_l (0, args) A (Suc stpa)) A"
is_RGraph: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" assumes "NFlow c s t f" shows "RGraph c s t (residualGraph c f)"
d: shows "Esigma (a * m) = a * m + a * n"
S_transition_eqvt: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and P :: "'a"   and \<alpha>\<^sub>SP' :: "(('c, 'b) S_action, 'a) residual"   and p :: "perm" assumes "nominal_ts satisfies transition"   and "nominal_ts.S_transition satisfies transition P \<alpha>\<^sub>SP'" shows "nominal_ts.S_transition satisfies transition (p \<bullet> P) (p \<bullet> \<alpha>\<^sub>SP')"
indRelEQp_indRelEQ_eq: fixes Enc :: "'a \<Rightarrow> 'b" shows "encoding.indRelEQp Enc = (\<lambda>x xa. (x, xa) \<in> encoding.indRelEQ Enc)"
simps: fixes a1 :: "Core.term"   and a2 :: "Core.term" shows "(a1 \<rightarrow>\<^sub>\<beta> a2) = ((\<exists>T s t. a1 = Abs T s $ t \<and> a2 = subst_bv2 s 0 t) \<or> (\<exists>s t u. a1 = s $ u \<and> a2 = t $ u \<and> s \<rightarrow>\<^sub>\<beta> t) \<or> (\<exists>s t u. a1 = u $ s \<and> a2 = u $ t \<and> s \<rightarrow>\<^sub>\<beta> t) \<or> (\<exists>s t T. a1 = Abs T s \<and> a2 = Abs T t \<and> s \<rightarrow>\<^sub>\<beta> t))"
min_def: fixes xs :: "('c, 'd) bintree option list" shows "Binomial_Queue.min xs = fold least (map (map_option priority) xs) None"
cmp_def: fixes g :: "'f \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'f" shows "cmp g f = (\<lambda>x. g (f x))"
TensorDiag_preserves_Diag_2: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag t"   and "Diag u" shows "Dom (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) = Dom t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Dom u"
ys_1: shows "start_remove xs k [] = as @ k # ys"
linorder_linear: fixes x :: "'b"   and y :: "'b" shows "x \<le> y \<or> y \<le> x"
mon_iff_hom_one: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and f :: "'a \<Rightarrow> 'c" assumes "Group.group G"   and "Group.group H" shows "(f \<in> mon G H) = (f \<in> hom G H \<and> (\<forall>x. x \<in> carrier G \<and> f x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>))"
Taylor_down: fixes n :: "nat"   and diff :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and c :: "real" assumes "0 < n"   and "diff 0 = f"   and "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> b \<longrightarrow> (diff m has_real_derivative diff (Suc m) t) (at t)"   and "a < c"   and "c \<le> b" shows "\<exists>t>a. t < c \<and> f a = (\<Sum>m<n. diff m c / fact m * (a - c) ^ m) + diff n t / fact n * (a - c) ^ n"
A: shows "1 / ereal (- epsilonG TYPE('a)) * ereal (- epsilonG TYPE('a)) = 1"
analytic_on_neg: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "f analytic_on S" shows "(\<lambda>z. - f z) analytic_on S"
assms: shows "\<forall>x. x |\<in>| A \<longrightarrow> f x = g x"
semiring_norm_72: fixes m :: "num"   and n :: "num" shows "(num.Bit0 m \<le> num.Bit1 n) = (m \<le> n)"
ht_hash_dom_map_of_disj: fixes l :: "('a \<times> 'b) list list"   and i :: "nat"   and j :: "nat" assumes "ht_hash l"   and "i < length l"   and "j < length l"   and "i \<noteq> j" shows "dom (map_of (l ! i)) \<inter> dom (map_of (l ! j)) = {}"
filtermap_bot_iff: fixes f :: "'b \<Rightarrow> 'a"   and F :: "'b filter" shows "(filtermap f F = bot) = (F = bot)"
span_minimal: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and T :: "'b set" assumes "module scale"   and "S \<subseteq> T"   and "module.subspace scale T" shows "module.span scale S \<subseteq> T"
cones_map_is_composition: fixes g :: "'c"   and a' :: "'c"   and a :: "'c"   and D :: "'j \<Rightarrow> 'c"   and \<chi> :: "'j \<Rightarrow> 'c" assumes "\<guillemotleft>g : a' \<rightarrow>\<^sub>C a\<guillemotright>"   and "Limit.cone (\<cdot>\<^sub>J) (\<cdot>) D a \<chi>" shows "J_C.MkArr (constant_functor.map (\<cdot>\<^sub>J) (\<cdot>) a') D ((\<lambda>\<chi>\<in>{\<chi>. Limit.cone (\<cdot>\<^sub>J) (\<cdot>) D (C.cod g) \<chi>}. \<lambda>j. if J.arr j then \<chi> j \<cdot> g else C.null) \<chi>) = J_C.MkArr (constant_functor.map (\<cdot>\<^sub>J) (\<cdot>) a) D \<chi> \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] local.map g"
that: fixes U :: "'a set set"   and s' :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "is_zariski_open U"   and "\<pp> \<in> U"   and "s' \<in> \<O> U"   and "s = st.class_of U s'" shows "thesis"
null_eqI: fixes n :: "'a" assumes "\<And>f. n \<cdot> f = n \<and> f \<cdot> n = n" shows "n = null"
module_on_def: fixes U\<^sub>M :: "'d set"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" shows "module_on U\<^sub>M scale \<equiv> ((\<forall>x y a. x \<in> U\<^sub>M \<longrightarrow> y \<in> U\<^sub>M \<longrightarrow> scale a (x + y) = scale a x + scale a y) \<and> (\<forall>x a b. x \<in> U\<^sub>M \<longrightarrow> scale (a + b) x = scale a x + scale b x) \<and> (\<forall>x a b. x \<in> U\<^sub>M \<longrightarrow> scale a (scale b x) = scale (a * b) x)) \<and> ((\<forall>x. x \<in> U\<^sub>M \<longrightarrow> scale (1::'e) x = x) \<and> (\<forall>x y. x \<in> U\<^sub>M \<longrightarrow> y \<in> U\<^sub>M \<longrightarrow> x + y \<in> U\<^sub>M)) \<and> (0::'d) \<in> U\<^sub>M \<and> (\<forall>x a. x \<in> U\<^sub>M \<longrightarrow> scale a x \<in> U\<^sub>M)"
q: shows "q = p - (lookup p (lt f) / lc f) \<cdot> f"
neq_beta2i_alpha2alpha5m: fixes q :: "'b"   and j :: "'b" assumes "(q, j) \<in> b\<inverse> \<union> d \<union> f \<union> ov\<inverse> \<union> m\<inverse>"   and "(q, j) \<in> ov \<union> s \<union> m \<union> f\<inverse> \<union> d\<inverse> \<union> e \<union> s\<inverse>" shows "False"
assms_6: fixes x :: "'a" assumes "x \<in> T" shows "q (p x) = q x"
\<PP>_altdef: shows "\<PP> = paths_le (length Doc) (\<Q> (length Doc))"
assms: fixes c :: "'a" shows "(a, c) \<notin> R"
subgroupE_3: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set"   and a :: "'e" assumes "Group.group G"   and "H \<le> G"   and "a \<in> H" shows "inv\<^bsub>G\<^esub> a \<in> H"
trm_prm_simp_3: fixes \<pi> :: "'b prm"   and A :: "'b trm"   and B :: "'b trm" shows "\<pi> \<cdot> App A B = App (\<pi> \<cdot> A) (\<pi> \<cdot> B)"
f2: shows "trans {(p, pa). (mset_cl p, mset_cl pa) \<in> mult (mult trm_ord)}"
SUP_upper: fixes i :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e \<times> 'f" assumes "i \<in> A" shows "less_eq_prod (f i) (Sup_prod (f ` A))"
mono_ndet_T: fixes P :: "'a process"   and P' :: "'a process"   and S :: "'a process"   and S' :: "'a process" assumes "P \<sqsubseteq>\<^sub>T P'"   and "S \<sqsubseteq>\<^sub>T S'" shows "(P |-| S) \<sqsubseteq>\<^sub>T (P' |-| S')"
maxsimpchain_rev: fixes X :: "'b set set"   and xs :: "'b set list" assumes "SimplicialComplex X"   and "SimplicialComplex.maxsimpchain X xs" shows "SimplicialComplex.maxsimpchain X (rev xs)"
lookup_trace_def: fixes k :: "'a" shows "lookup_trace k = State (\<lambda>(log, m). case run_state (lookup k) m of (None, m) \<Rightarrow> (None, (''Missed'', k) # log, m) | (Some v, m) \<Rightarrow> (Some v, (''Found'', k) # log, m))"
set_sel_2: fixes a :: "('g, 'h, 'i) generat" assumes "\<not> is_Pure a" shows "generat.output a \<in> generat_outs a"
RedNewFail: fixes h :: "heap"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and l :: "sfields"   and sh :: "sheap" assumes "new_Addr h = None"   and "is_class P C" shows "P \<turnstile> \<langle>new C,(h, l, sh),True\<rangle> \<rightarrow> \<langle>THROW OutOfMemory,(h, l, sh),False\<rangle>"
attach_converter_of_resource_conv_parallel_resource: fixes res :: "('y, 'aa) resource"   and res' :: "('z, 'ab) resource" shows "converter_of_resource res |\<^sub>\<propto> 1\<^sub>C \<rhd> res' = res \<parallel> res'"
simps_2: fixes \<gamma> :: "'a constr"   and \<Gamma> :: "'a constr list" shows "\<lbrakk>\<lbrakk> \<gamma> # \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk> \<gamma> \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
is_induced_map_iff: fixes w :: "'a" shows "h.is_induced_map w = (ide w \<and> (\<exists>\<nu> \<theta>. \<guillemotleft>\<theta> : tab\<^sub>0 h \<star> w \<rightarrow> tab\<^sub>0 h\<guillemotright> \<and> \<guillemotleft>\<nu> : tab\<^sub>1 h \<rightarrow> tab\<^sub>1 h \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> (h \<star> tab\<^sub>0 h) \<cdot> h.tab = (h \<star> \<theta>) \<cdot> \<a>[h, tab\<^sub>0 h, w] \<cdot> (h.tab \<star> w) \<cdot> \<nu>))"
elims_3: fixes x :: "'b trie"   and xa :: "'b list" assumes "\<not> isin x xa" obtains b :: "bool" and m :: "'b \<Rightarrow> 'b trie option" where "x = Nd b__ m__" and "xa = []" and "\<not> b__"   | b :: "bool" and m :: "'b \<Rightarrow> 'b trie option" and k :: "'b" and xs :: "'b list" where "x = Nd b m" and "xa = k # xs" and "\<not> (case m k of None \<Rightarrow> False | Some t \<Rightarrow> isin t xs)"
map_option_eq_Some: fixes f :: "'g \<Rightarrow> 'f"   and xo :: "'g option"   and y :: "'f" shows "(map_option f xo = \<lfloor>y\<rfloor>) = (\<exists>z. xo = \<lfloor>z\<rfloor> \<and> f z = y)"
f1: shows "((r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s')) \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s'') = ((s' \<otimes> r \<oplus> s \<otimes> r') \<otimes> r'' |\<^bsub>rel\<^esub> s \<otimes> s' \<otimes> s'')"
binomial_eq_0: fixes n :: "nat"   and k :: "nat" assumes "n < k" shows "n choose k = 0"
LIMSEQ_inverse_real_of_nat_add: fixes r :: "real" shows "(\<lambda>n. r + inverse (real (Suc n))) \<longlonglongrightarrow> r"
smult_distrib_right: fixes r :: "('b, 'a) aezfun"   and s :: "('b, 'a) aezfun"   and m :: "'c" assumes "r \<in> FG"   and "s \<in> FG"   and "m \<in> V" shows "(r + s) \<cdot> m = r \<cdot> m + s \<cdot> m"
Tokens_def: fixes k :: "nat"   and T :: "(symbol \<times> character list) set"   and I :: "item set" shows "Tokens k T I = Sel T (TokensAt k I)"
cod_dom: fixes f :: "(bool, bool list) concrete_category.arr" assumes "arr f" shows "cod (local.dom f) = local.dom f"
vector_space_pair_axioms: shows "vector_space_pair (*a) (*b)"
S11: fixes x :: "'a" shows "((1::'a) + x)\<^sup>\<star> = x\<^sup>\<star>"
simps: fixes cs :: "QDelta ns_constraint list"   and asi :: "('a, ('a \<times> QDelta atom) list) mapping"   and tv :: "(nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping"   and ui :: "'a list"   and s :: "('a, QDelta) state" shows "solution_simplex (Simplex_State (cs, (asi, tv, ui), s)) = \<langle>from_ns (tv (\<V> s)) cs\<rangle>"
tauActTauChain: fixes P :: "pi"   and P' :: "pi" assumes "P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'" shows "(P, P') \<in> {(P, P') |P P'. P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'}\<^sup>*"
span_superset: fixes S :: "'b set" shows "S \<subseteq> span S"
nn_integral_dens_ctxt_measure: fixes \<rho> :: "nat \<Rightarrow> val"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "f \<in> borel_measurable (state_measure (V \<union> V') \<Gamma>)" shows "integral\<^sup>N (dens_ctxt_measure \<Y> \<rho>) f = \<integral>\<^sup>+ x. \<delta> (merge V V' (x, \<rho>)) * f (merge V V' (x, \<rho>)) \<partial>state_measure V \<Gamma>"
\<rho>_ide_simp: fixes a :: "'a" assumes "ide a" shows "MC.\<rho> a = MC.runit a"
borel_measurableI_ge: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" assumes "\<And>y. {x \<in> space M. y \<le> f x} \<in> sets M" shows "f \<in> borel_measurable M"
bound_def: fixes z :: "'c"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c" shows "bound z n f \<equiv> \<forall>m>n. f m = z"
hom_components_homogeneous: fixes q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "q \<in> hom_components p" shows "homogeneous q"
length_filter_replace2: fixes x :: "'a"   and xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and ys :: "'a list" assumes "x \<in> set xs"   and "P x" shows "|filter P (replace x ys xs)| = |filter P xs| + |filter P ys| - 1"
dependent_zero: fixes A :: "('b, 'c) vec set" assumes "0 \<in> A" shows "vec.dependent A"
simps_6: fixes \<phi> :: "'a fm" shows "depth (Neg \<phi>) = depth \<phi> + 1"
in_open_segment_rotD: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> {a<--<b}" shows "(x - a) \<bullet> rev.rot (b - a) = 0 \<and> x \<bullet> (b - a) \<in> {a \<bullet> (b - a)<..<b \<bullet> (b - a)}"
that: fixes pi :: "('a \<times> 'a) list" assumes "(pi \<bullet> Xs) \<sharp>* c"   and "set pi \<subseteq> Xs \<times> pi \<bullet> Xs" shows "thesis"
reachable_adjI: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow> v" shows "u \<rightarrow>\<^sup>* v"
iter_tensor_of_gate_is_gate: fixes m :: "nat"   and A :: "complex Matrix.mat"   and n :: "nat" assumes "gate m A"   and "1 \<le> n" shows "gate (m * n) (A \<otimes>\<^bsup>n\<^esup>)"
leq: shows "l = g'c * (f b - f a) - f'c * (g b - g a)"
inf_ge_any: fixes x :: "ant" shows "x \<le> \<infinity>"
pell_valuation_solution_pos_nat: fixes D :: "nat"   and z :: "nat \<times> nat" assumes "pell D"   and "pell.solution D z" shows "0 < pell.pell_valuation D (case z of (x, y) \<Rightarrow> (int x, int y))"
that: fixes e' :: "nat \<Rightarrow> real" assumes "e' \<in> UNIV \<rightarrow> I"   and "pdevs_val e (pdevs_of_list (ccw.selsort 0 xs)) = pdevs_val e' (pdevs_of_list xs)" shows "thesis"
SOC_wrt_equivalence_iff_indRelTEQ_strong_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "equivalence TRel" shows "(((\<forall>S S'. S \<longmapsto>Source S' \<longrightarrow> (\<exists>T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel)) \<and> (\<forall>S T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<longrightarrow> (\<exists>S'. S \<longmapsto>Source S' \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel))) \<and> strong_reduction_simulation TRel Target \<and> (\<forall>P Q Q'. (P, Q) \<in> TRel \<and> Q \<longmapsto>Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Target P' \<and> (P', Q') \<in> TRel))) = ((\<forall>P Q P'. P \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q')) \<and> (\<forall>P Q Q'. P \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> P' \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q')))"
mapping_eqI: fixes m :: "('c, 'd) mapping"   and m' :: "('c, 'd) mapping" assumes "\<And>x. Mapping.lookup m x = Mapping.lookup m' x" shows "m = m'"
angelic_def: fixes Q :: "'c \<Rightarrow> 'd"   and p :: "'d" shows "{: Q :} p = {s. Q s \<sqinter> p \<noteq> \<bottom>}"
fixp_induct: fixes P :: "('a \<Rightarrow> 'a spmf) \<Rightarrow> bool" assumes "spmf.admissible P"   and "P (\<lambda>while. return_pmf None)"   and "\<And>s. P s \<Longrightarrow> P (\<lambda>sa. if guard sa then body sa \<bind> s else return_spmf sa)" shows "P local.while"
tms_any_less: fixes k :: "nat"   and ap :: "abc_inst list"   and a :: "action"   and b :: "nat" assumes "k < length ap"   and "(a, b) \<in> set (tms_of ap ! k)" shows "b \<le> start_of (layout_of ap) (length ap)"
ineq_model_subst: fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and X :: "'a list"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" assumes "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> set X = {}"   and "ineq_model (\<delta> \<circ>\<^sub>s \<theta>) X F" shows "ineq_model \<theta> X (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>)"
\<R>\<^sub>3_mem_eq: fixes c\<^sub>1 :: "(Eg1.var, aexp, bexp) Stmt"   and mds :: "Mode \<Rightarrow> Eg1.var set"   and mem\<^sub>1 :: "Eg1.var \<Rightarrow> nat"   and \<Gamma>' :: "Eg1.var \<Rightarrow> bexp set option"   and \<S>' :: "Eg1.var set \<times> Eg1.var set"   and P' :: "bexp set"   and c\<^sub>2 :: "(Eg1.var, aexp, bexp) Stmt"   and mem\<^sub>2 :: "Eg1.var \<Rightarrow> nat" assumes "A.\<R>\<^sub>3_aux \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>\<^sub>A \<Gamma>' \<S>' P' \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>\<^sub>A" shows "A.low_mds_eq mds mem\<^sub>1 mem\<^sub>2"
simps_2: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "splice (x # xs) ys = x # splice ys xs"
tendsto_arctan: fixes f :: "'a \<Rightarrow> real"   and x :: "real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. arctan (f x)) \<longlongrightarrow> arctan x) F"
hom_minus: fixes x :: "int poly"   and y :: "int poly" shows "of_int_poly (x - y) = of_int_poly x - of_int_poly y"
A_authenticates_B: fixes K :: "nat"   and Ta :: "nat"   and evs :: "event list"   and A :: "agent"   and Tk :: "nat"   and B :: "agent"   and X :: "msg" assumes "Crypt K (Number Ta) \<in> parts (knows Spy evs)"   and "Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace> \<in> parts (knows Spy evs)"   and "Key K \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerb_gets" shows "Says B A (Crypt K (Number Ta)) \<in> set evs"
usubstappf_and_usubstappp_antimon_1: fixes V :: "variable set"   and U :: "variable set"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and \<phi> :: "fml" assumes "V \<subseteq> U"   and "usubstappf \<sigma> U \<phi> \<noteq> undeff" shows "usubstappf \<sigma> U \<phi> = usubstappf \<sigma> V \<phi>"
bridge_proper_dia: fixes p :: "('c, 'b) fm"   and v :: "nat"   and v' :: "nat"   and xs :: "(nat \<times> nat) set"   and k :: "'b"   and j :: "'b" assumes "\<nexists>a. p = Nom a" shows "bridge k j xs v v' (\<^bold>\<diamond> p) = (\<^bold>\<diamond> p)"
simps_2: fixes p :: "real \<times> real \<times> real" shows "substNegInfinityUni (LessUni p) = alternateNegInfinity' p"
dis_elim: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "dis \<phi> \<psi> \<pi>l" obtains "\<phi> \<pi>l"   | "\<psi> \<pi>l"
sup_commute: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
row_length_mat: fixes m1 :: "'a list list"   and m2 :: "'a list list" shows "row_length (m1 \<otimes> m2) = row_length m1 * row_length m2"
emb_prj: fixes x :: "udom" shows "EMB('a)\<cdot>(PRJ('a)\<cdot>x) = cast\<cdot>DEFL('a)\<cdot>x"
simps: fixes a :: "'a \<times> exp \<times> stack_elem list" shows "boring_step a = (\<exists>e \<Gamma> x S. a = (\<Gamma>, e, Upd x # S) \<and> isVal e)"
assms_4: fixes x :: "real" assumes "x \<in> {a..b}" shows "g x \<in> s"
assms_1: shows "r \<cdot> l \<le> l \<cdot> r"
of_nat_mpoly_eq: fixes n :: "nat" shows "of_nat n = Const (of_nat n)"
primeI: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero x"   and "normalize x = x" shows "normalization_semidom.prime zero times one normalize x"