hcju/isabelleps huggingface dataset:
return_time_def: fixes P :: "'a set"   and x :: "'a" shows "return_time P x = (if returns_to P x then SOME t. 0 < t \<and> t \<in> existence_ivl0 x \<and> flow0 x t \<in> P \<and> (\<forall>s\<in>{0<..<t}. flow0 x s \<notin> P) else 0)"
member_filter: fixes x :: "'c"   and P :: "'c \<Rightarrow> bool"   and A :: "'c set" shows "(x \<in> Set.filter P A) = (x \<in> A \<and> P x)"
IH_2: fixes X :: "'c set" assumes "reduced t2 X"   and "t2 \<noteq> Trueif"   and "t2 \<noteq> Falseif" shows "\<exists>s1 s2. val_ifex t2 s1 \<noteq> val_ifex t2 s2"
max_ge_y: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.non_strict_order less_eq" shows "less_eq y (ord.max less_eq x y)"
eval_equality: fixes i :: "char list \<Rightarrow> tv"   and p :: "fm"   and q :: "fm" shows "eval i (Eql' p q) = (if eval i p = eval i q then Det True else if eval i p = Det True then eval i q else if eval i q = Det True then eval i p else if eval i p = Det False then eval i (Neg' q) else if eval i q = Det False then eval i (Neg' p) else Det False)"
Ext_subclseq_Base: shows "tprg\<turnstile>Ext\<preceq>\<^sub>C Base"
sep_conj_sep_true_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and h :: "'c" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_conj plus sep_disj P Q h" shows "sep_algebra.sep_conj plus sep_disj (\<lambda>s. True) Q h"
akra_bazzi_measure_decreases: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and x :: "real"   and i :: "nat" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "x\<^sub>1 \<le> x"   and "i < k" shows "akra_bazzi_measure (bs ! i * x + (hs ! i) x) < akra_bazzi_measure x"
det_zero_low_rank: fixes A :: "'a mat" assumes "A \<in> carrier_mat n n"   and "det A = (0::'a)" shows "rank A < n"
three: fixes xs :: "('b com \<times> 'b set \<times> ('b \<times> 'b) set \<times> ('b \<times> 'b) set \<times> 'b set) list"   and rely :: "('b \<times> 'b) set"   and pre :: "'b set"   and clist :: "('b com option \<times> 'b) list list"   and x :: "('b com option list \<times> 'b) list"   and s :: "'b"   and i :: "nat"   and j :: "nat" assumes "xs \<noteq> []"   and "\<And>i. i < length xs \<Longrightarrow> rely \<union> (\<Union>j\<in>{j. j < length xs \<and> j \<noteq> i}. Guar (xs ! j)) \<subseteq> Rely (xs ! i)"   and "pre \<subseteq> (\<Inter>i\<in>{i. i < length xs}. Pre (xs ! i))"   and "\<And>i. i < length xs \<Longrightarrow> \<Turnstile> Com (xs ! i) sat [Pre (xs ! i), Rely (xs ! i), Guar (xs ! i), Post (xs ! i)]"   and "length xs = length clist"   and "x \<in> par_cp (ParallelCom xs) s"   and "x \<in> par_assum (pre, rely)"   and "\<And>i. i < length clist \<Longrightarrow> clist ! i \<in> cp (Some (Com (xs ! i))) s"   and "x \<propto> clist"   and "i < length clist \<and> Suc j < length x"   and "clist ! i ! j -e\<rightarrow> clist ! i ! Suc j" shows "(snd (clist ! i ! j), snd (clist ! i ! Suc j)) \<in> rely \<union> (\<Union>j\<in>{j. j < length xs \<and> j \<noteq> i}. Guar (xs ! j))"
wf_interp_for_formula_FEXISTS: fixes \<Sigma> :: "'b list"   and I :: "(nat + nat set) list"   and \<phi> :: "'b formula"   and w :: "'b list" assumes "formula \<Sigma>"   and "formula.wf_formula \<Sigma> (length I) (FEXISTS \<phi>)"   and "w \<noteq> []" shows "formula.wf_interp_for_formula \<Sigma> (w, I) (FEXISTS \<phi>) = (\<forall>P\<subseteq>{0..length w - 1}. formula.wf_interp_for_formula \<Sigma> (w, Inr P # I) \<phi>)"
bt_dfs_comp_2: fixes f :: "'f \<Rightarrow> 'c"   and g :: "('d, 'e) bintree \<Rightarrow> 'f"   and ts :: "('d, 'e) bintree list" shows "bts_dfs (f \<circ> g) ts = map f (bts_dfs g ts)"
disjnt_iff: fixes A :: "'a set"   and B :: "'a set" shows "disjnt A B = (\<forall>x. \<not> (x \<in> A \<and> x \<in> B))"
out: shows "a \<in> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
listrel1_mono: fixes r :: "('g \<times> 'g) set"   and s :: "('g \<times> 'g) set" assumes "r \<subseteq> s" shows "listrel1 r \<subseteq> listrel1 s"
AAA: fixes r :: "'d rexp"   and r' :: "'d rexp"   and s :: "'d rexp"   and s' :: "'d rexp" assumes "(~)\<^sup>*\<^sup>* r r'"   and "(~)\<^sup>*\<^sup>* s s'" shows "(~)\<^sup>*\<^sup>* (Alt r s) (Alt r' s')"
FOREACHc_rule: fixes S :: "'a set"   and I :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma>0 :: "'b"   and c :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b nres"   and P :: "'b \<Rightarrow> bool" assumes "finite S"   and "I S \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> it; it \<subseteq> S; I it \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (it - {x}))"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>it \<sigma>. \<lbrakk>it \<noteq> {}; it \<subseteq> S; I it \<sigma>; \<not> c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "FOREACH\<^sub>C S c f \<sigma>0 \<le> SPEC P"
trancl_rtrancl_absorb: fixes R :: "('d \<times> 'd) set" shows "(R\<^sup>+)\<^sup>* = R\<^sup>*"
monad_catch_optionT': assumes "monad Monad_Overloading.return Monad_Overloading.bind" shows "monad_catch Monad_Overloading.return Monad_Overloading.bind fail catch"
sol_lemma: fixes ODE :: "('sf, 'sz) ODE"   and t :: "real"   and I :: "('sf, 'sc, 'sz) interp"   and x :: "'sz"   and y :: "'sz"   and \<phi> :: "('sf, 'sc, 'sz) formula"   and sol :: "real \<Rightarrow> (real, 'sz) vec"   and b :: "(real, 'sz) vec" assumes "ORadmit ODE"   and "0 \<le> t"   and "\<And>\<nu>. (\<nu> \<in> fml_sem I (FUrename x y \<phi>)) = (Radj x y \<nu> \<in> fml_sem I \<phi>)"   and "(sol solves_ode (\<lambda>a. ODE_sem I (OUrename x y ODE))) {0..t} {xaa. mk_v I (OUrename x y ODE) (sol 0, b) xaa \<in> fml_sem I (FUrename x y \<phi>)}" shows "((\<lambda>t. RSadj x y (sol t)) solves_ode (\<lambda>a. ODE_sem I ODE)) {0..t} {xaa. mk_v I ODE (RSadj x y (sol 0), RSadj x y b) xaa \<in> fml_sem I \<phi>}"
times_float_def: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "a * b = fmul To_nearest a b"
f_shrink_take_mult: fixes n :: "nat"   and k :: "nat"   and xs :: "'d message_af list" shows "xs \<down> (n * k) \<div> k = xs \<div> k \<down> n"
rep_eq: fixes x :: "'a sq_mtx" shows "($$) x = ($) (to_vec x)"
disjSD2aux: fixes a :: "('b set set, 'c) Combinators"   and b :: "('b set set, 'd) Combinators" assumes "disjSD_2 a b"   and "\<not> member DenyAll a"   and "\<not> member DenyAll b" shows "disjSD_2 (DenyAllFromTo (first_srcNet a) (first_destNet a) \<oplus> DenyAllFromTo (first_destNet a) (first_srcNet a) \<oplus> a) b"
eFresh_simp2: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and X :: "'c"   and ys :: "'varSort"   and y :: "'var" assumes "\<forall>s. \<not> igWls MOD s X" shows "eFresh MOD ys y (OK X)"
ide_cod: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.ide C (partial_magma.cod C f)"
uminus_key_compare: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "invert_order (key_compare ko x y) = key_compare ko y x"
the_\<theta>_simps_4: shows "local.dom w.the_\<theta> = s\<^sub>0 \<star> w"
\<pi>_def: shows "\<pi> \<equiv> M_lookup \<pi>i"
replace_primitive_matchexpr_preserves_not_has_disc: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and disc2 :: "'b \<Rightarrow> bool"   and replace_f :: "'c negation_type \<Rightarrow> 'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<not> has_disc disc2 m"   and "has_disc disc m \<Longrightarrow> \<forall>pts. \<not> has_disc disc2 (replace_f pts)" shows "\<not> has_disc disc2 (replace_primitive_matchexpr (disc, sel) replace_f m)"
fls_regpart_shift_conv_fps_shift: fixes m :: "int"   and f :: "'a fls" assumes "0 \<le> m" shows "fls_regpart (fls_shift m f) = fps_shift (nat m) (fls_regpart f)"
prefixes_append: fixes xs :: "'c list"   and ys :: "'c list" shows "prefixes (xs @ ys) = prefixes xs @ map ((@) xs) (tl (prefixes ys))"
that: fixes w :: "'a" assumes "w \<in> fv t"   and "v \<in> fv (\<delta> w)" shows "thesis"
f21_def: shows "f21 = snd (split_face f v w ws)"
UN_I: fixes a :: "'f"   and A :: "'f set"   and b :: "'g"   and B :: "'f \<Rightarrow> 'g set" assumes "a \<in> A"   and "b \<in> B a" shows "b \<in> \<Union> (B ` A)"
laplace_expansion_column: fixes A :: "'b mat"   and n :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat n n"   and "j < n" shows "Determinant.det A = (\<Sum>i<n. A $$ (i, j) * Determinant.cofactor A i j)"
restrict_empty: fixes P :: "('a \<times> 'b) set" shows "P || {} = {}"
move_matrix_row_mult: fixes A :: "'b matrix"   and B :: "'b matrix"   and j :: "int" shows "move_matrix (A * B) j 0 = move_matrix A j 0 * B"
BIGTHETA_CONST_def: fixes c :: "'b"   and A :: "('a \<Rightarrow> 'b) set" shows "BIGTHETA_CONST c A = set_mult \<Theta>(\<lambda>_. c) A"
interval_subset: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A" shows "interval r a b \<subseteq> A"
append_self_conv: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs @ ys = xs) = (ys = [])"
finite_code: fixes A :: "'a set" assumes "class.finite TYPE('a)" shows "finite A = True"
integrable_iff_bounded: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" shows "integrable M f = (f \<in> borel_measurable M \<and> \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M < \<infinity>)"
Dom_Diagonalize_rep: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "monoidal_language.Dom C (monoidal_language.Diagonalize C (free_monoidal_category.rep C f)) = monoidal_language.Diagonalize C (free_monoidal_category.rep C (partial_magma.dom (free_monoidal_category.comp C) f))"
restrict_top_alt: fixes n :: "'b ifex"   and var :: "'b"   and val :: "bool" shows "restrict_top n var val = (case n of IF v t e \<Rightarrow> if v = var then if val then t else e else IF v t e | _ \<Rightarrow> n)"
assms_3: fixes v :: "'b" shows "((k, v) \<in> set (entries t1)) = ((k, v) \<in> set (entries t2))"
conSim: shows "weak_reduction_contrasimulation (indRelRTPO TRel) (STCal Source Target)"
f_def: shows "f = (\<lambda>t. poly q t / poly p t)"
replicate_0: fixes x :: "'a" shows "x \<up> 0 = []"
prAbs_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'a) \<Rightarrow> 'a"   and gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'a) \<Rightarrow> ('varSort, 'a) sAbs"   and SEM :: "('b, 'c, 'd, 'sort, 'e, 'a, 'f) semDom_scheme" shows "prAbs g gA SEM \<equiv> \<forall>xs s x X val. isInBar (xs, s) \<and> wls s X \<and> sWlsVal SEM val \<longrightarrow> gA (Abs xs x X) val = sAbs xs (\<lambda>sX. if sWls SEM (asSort xs) sX then g X (val (x := sX)_xs) else sDummy SEM s)"
IH: fixes err :: "v error_result"   and s :: "'a state" assumes "is_cupcake_all_env env"   and "list_all is_cupcake_exp es"   and "sequence_result ys = Rerr err" shows "\<exists>s'. Big_Step_Unclocked_Single.evaluate_list (Big_Step_Unclocked_Single.evaluate env) s es (s', Rerr err)"
inangle_one_side: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" assumes "\<not> Col A B C"   and "\<not> Col A B P"   and "\<not> Col A B Q"   and "P InAngle A B C"   and "Q InAngle A B C" shows "A B OS P Q"
alternating_list_in_lists: fixes a :: "'b"   and A :: "'b set"   and b :: "'b"   and n :: "nat" assumes "a \<in> A"   and "b \<in> A" shows "alternating_list n a b \<in> lists A"
assms_4: shows "(v, w, v') \<in> G.E"
f_def: shows "f = (\<lambda>n. ln (2 * pi * real n) / 2 + real n * ln (real n) - real n)"
b_non_zero: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "proper_design point_set block_collection" shows "int (size block_collection) \<noteq> 0"
subset_iff: fixes A :: "'g set"   and B :: "'g set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
subst_clause_def: fixes \<Phi> :: "'b ltln fset"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "subst_clause \<Phi> m = \<Otimes>\<^sub>m {min_dnf (subst \<phi> m) |\<phi>. \<phi> \<in> fset \<Phi>}"
weakBisimp_weakBisim_eq: shows "weakBisimp = (\<approx>)"
integral_norm_bound_integral: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> real" assumes "integrable M f"   and "integrable M g"   and "\<And>x. x \<in> space M \<Longrightarrow> norm (f x) \<le> g x" shows "norm (integral\<^sup>L M f) \<le> integral\<^sup>L M g"
vsv_is_VLambda: shows "VLambda (\<D>\<^sub>\<circ> r) (app r) = r"
infinity: fixes undiago :: "'a \<Rightarrow> nat"   and diago :: "nat \<Rightarrow> 'a"   and tree :: "'a set" assumes "\<forall>n. undiago (diago n) = n"   and "\<forall>n. diago n \<in> tree" shows "infinite tree"
oc_projection_exist: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs')" shows "map of_int_hom.vec_hom fs' ! i - gs2.gso i \<in> gs.span (gs2.gso ` {0..<i})"
lam_pr1_eq: shows "\<Lambda>_pr1 = \<Lambda>_pr1'"
XH_largest_def: shows "XH_largest = {y. Xd y \<in> ds \<and> y \<in> Field (Pd (Xd y)) \<and> (\<forall>x\<in>dX X (Xd y). (x, y) \<in> Pd (Xd y))}"
conjugate_subinf: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f" shows "conjugate f (x \<sqinter> y) \<le> conjugate f x \<sqinter> conjugate f y"
reachable_def: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" shows "greachable A w V \<equiv> \<lambda>p. {x. transition_system.reachablep (\<lambda>x _. x) (\<lambda>u (k, p). w !! k \<in> alphabet A \<and> u \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V) p x}"
subset_iff': fixes A :: "'c set"   and B :: "'c set" shows "(A \<subseteq> B) = (\<forall>t\<in>A. t \<in> B)"
range_prod: fixes f :: "'c \<Rightarrow> 'a \<times> 'b" shows "range f \<subseteq> range (fst \<circ> f) \<times> range (snd \<circ> f)"
assms_2: shows "\<forall>x. P x \<longrightarrow> (\<forall>i. Q i \<longrightarrow> 0 \<le> x i \<and> x i \<le> 1)"
WTrtSeq: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e\<^sub>1 :: "expr"   and T\<^sub>1 :: "ty"   and e\<^sub>2 :: "expr"   and T\<^sub>2 :: "ty" assumes "P,E,h \<turnstile> e\<^sub>1 : T\<^sub>1"   and "P,E,h \<turnstile> e\<^sub>2 : T\<^sub>2" shows "P,E,h \<turnstile> e\<^sub>1;; e\<^sub>2 : T\<^sub>2"
assms_1: shows "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta>"
regular_dense_top: fixes x :: "'a" assumes "regular x"   and "dense x" shows "x = top"
pr_Iso_same_no_nodes: fixes \<phi> :: "'a \<Rightarrow> 'b"   and Fs\<^sub>1 :: "'a list set"   and Fs\<^sub>2 :: "'b list set" assumes "is_pr_Iso \<phi> Fs\<^sub>1 Fs\<^sub>2"   and "finite Fs\<^sub>1" shows "card (\<Union> (set ` Fs\<^sub>1)) = card (\<Union> (set ` Fs\<^sub>2))"
hom_add_pow_nat: fixes x :: "int \<times> int"   and n :: "nat" assumes "x \<in> carrier R" shows "\<phi> ([n] \<cdot>\<^bsub>R\<^esub> x) = [n] \<cdot>\<^bsub>S\<^esub> \<phi> x"
distinct_1: fixes x11 :: "'c"   and x12 :: "'d"   and x21 :: "'e"   and x22 :: "'a \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm option"   and x23 :: "'b \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs option" shows "qVar x11 x12 \<noteq> qOp x21 x22 x23"
sum_mult_ring_multiplicationTr: fixes A :: "'a set"   and B :: "'a set"   and r :: "'a"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "A \<subseteq> carrier R"   and "ideal R B"   and "r \<in> carrier R" shows "(\<forall>j\<le>n. f j \<in> set_mult R A B) \<longrightarrow> r \<cdot>\<^sub>r \<Sigma>\<^sub>e R f n \<in> sum_mult R A B"
Pareto_strictI: fixes x :: "'b"   and y :: "'b"   and i :: "'a" assumes "\<And>i. i \<in> dom \<Longrightarrow> R i x y"   and "i \<in> dom"   and "x \<prec>[R i] y" shows "x \<prec>[Pareto R] y"
assms_2: fixes u :: "'b" assumes "u \<in> Basis" shows "((\<lambda>x. g x \<bullet> u) has_vector_derivative g' \<bullet> u) (at x within S)"
hyps: assumes "channel cid = None" shows "msgs (s init t n) cid = []"
edge'_simps_5: shows "edge' vertex.SINK vertex.SINK = False"
valid_weak_action_modality: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and P :: "'c"   and \<alpha> :: "'b"   and x :: "('a, 'd, 'b) formula" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "nominal_ts.valid satisfies transition P (indexed_weak_nominal_ts.weak_action_modality \<tau> \<alpha> x) = (\<exists>\<alpha>'' x' P'. Act \<alpha> x = Act \<alpha>'' x' \<and> weak_nominal_ts.weak_transition transition \<tau> P \<alpha>'' P' \<and> nominal_ts.valid satisfies transition P' x')"
cosh_dist_perp_divide: fixes l :: "proj2_line"   and m :: "proj2_line"   and a :: "proj2"   and b :: "proj2"   and c :: "proj2" assumes "M_perp l m"   and "a \<in> hyp2"   and "b \<in> hyp2"   and "c \<in> hyp2"   and "proj2_incident a l"   and "proj2_incident b l"   and "proj2_incident b m"   and "proj2_incident c m" shows "cosh_dist b c = cosh_dist a c / cosh_dist b a"
InitConf_def: fixes A :: "('a, 'b, 'c) hierauto" shows "InitConf A \<equiv> (HAInitStates A \<times> HAInitStates A \<inter> ChiRel A)\<^sup>* `` {HAInitState A}"
elims_1: fixes x :: "value"   and xa :: "value"   and y :: "bool" assumes "(x < xa) = y" obtains n :: "int" and s :: "String.literal" where "x = Num n__" and "xa = Str s__" and "y"   | s :: "String.literal" and n :: "int" where "x = Str s" and "xa = Num n" and "\<not> y"   | s1 :: "String.literal" and s2 :: "String.literal" where "x = Str s1" and "xa = Str s2" and "y = (s1 < s2)"   | n1 :: "int" and n2 :: "int" where "x = Num n1" and "xa = Num n2" and "y = (n1 < n2)"
select_convs_2: fixes prism_match :: "'d \<Rightarrow> 'c option"   and prism_build :: "'c \<Rightarrow> 'd"   and more :: "'e" shows "build\<^bsub>\<lparr>prism_match = prism_match, prism_build = prism_build, \<dots> = more\<rparr>\<^esub> = prism_build"
NSCauchy_Cauchy: fixes X :: "nat \<Rightarrow> 'b" assumes "NSCauchy X" shows "Cauchy X"
Sum1_2: fixes P :: "pi"   and u :: "name"   and P'' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P'" shows "P \<oplus> Q \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P'"
SigmaD2: fixes a :: "'d"   and b :: "'e"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" assumes "(a, b) \<in> Sigma A B" shows "b \<in> B a"
expectation_gpv_1_le_inline: fixes gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s"   and f :: "'b \<Rightarrow> ennreal" assumes "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; I s\<rbrakk> \<Longrightarrow> pfinite_gpv \<I>' (callee s x)"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s"   and "\<And>x. f x \<le> 1" shows "expectation_gpv 1 \<I> f gpv \<le> expectation_gpv 1 \<I>' (\<lambda>(x, s). f x) (inline callee gpv s)"
simps: fixes \<Gamma> :: "(Vars.var \<times> 'a) list"   and S :: "stack_elem list" shows "heap_upds_ok (\<Gamma>, S) = (domA \<Gamma> \<inter> upds S = {} \<and> distinct (upds_list S))"
arev_dom: fixes a :: "'b" shows "(a \<in> arcs H) = (rev_H' a \<noteq> a)"
linear_span_image: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and S :: "('a, 'b) vec set" assumes "Vector_Spaces.linear (*s) (*s) f" shows "vec.span (f ` S) = f ` vec.span S"
map_add_dom_app_simps_1: fixes m :: "'c"   and l2 :: "'c \<Rightarrow> 'd option"   and l1 :: "'c \<Rightarrow> 'd option" assumes "m \<in> dom l2" shows "(l1 ++ l2) m = l2 m"
eval_AddNum_correct: fixes xs :: "dynamic list" assumes "length xs = 2" shows "eval_AddNum xs = eval_Add xs"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) L local.map \<ll>.map"
cryptinverts_hilf1: fixes p :: "nat"   and m :: "nat"   and k :: "nat" assumes "prime p" shows "m * m ^ (k * pred p) mod p = m mod p"
mbisim_final_thread_preserve1: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" assumes "s1 \<approx>m s2"   and "r1.final_thread s1 t" shows "r2.final_thread s2 t"
list_of_dnelist: fixes x :: "'a dnelist" shows "list_of_dnelist x \<in> {xs. distinct xs \<and> xs \<noteq> []}"
uniqify_mset: fixes q :: "('c, 'd) SkewBinomialTree list" assumes "queue_invar q" shows "queue_to_multiset q = queue_to_multiset (uniqify q)"
secure_def: fixes c :: "IMP" shows "secure c = (\<forall>s t ss tt.  s \<approx> t  \<longrightarrow> s , c \<Down> ss  \<longrightarrow> t , c \<Down> tt  \<longrightarrow>  ss \<approx> tt )"
mat_vec_as_mat_mat_mult: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'a vec" assumes "A \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc" shows "A *\<^sub>v v = col (A * mat_of_cols nc [v]) 0"
simps: fixes p :: "nat"   and r :: "nat"   and a :: "nat" shows "newleft p r a = (if a = 0 \<or> a = 1 then newleft0 [p, r] else if a = 2 then newleft2 [p, r] else if a = 3 then newleft3 [p, r] else p)"
valid_mbind'_mt: fixes \<sigma> :: "'d"   and f :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) option"   and P :: "'e list \<Rightarrow> bool" shows "(\<sigma> \<Turnstile> _bind_SE s (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p [] f) (unit\<^sub>S\<^sub>E (P s))) = P []"
M_times_1: fixes m :: "int"   and x :: "int"   and y :: "int" shows "poly_mod.M m (poly_mod.M m x * y) = poly_mod.M m (x * y)"
wt_\<delta>\<^sub>h_imp_\<delta>\<^sub>h_eq_\<epsilon>\<^sub>h: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and s :: "('s, 'v) tm" assumes "legal_zpassign A"   and "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s)) = zhmset_of \<delta>\<^sub>h" shows "\<delta>\<^sub>h = \<epsilon>\<^sub>h"
completeI: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y\<rbrakk> \<Longrightarrow> x \<^bsub>r\<^esub>\<preceq> y \<or> y \<^bsub>r\<^esub>\<preceq> x" shows "complete A r"
pp_dist_inf: fixes x :: "'a"   and y :: "'a" shows "- - (x \<sqinter> y) = - - x \<sqinter> - - y"
ordering_axioms: shows "ordering (\<^bold>\<le>*) (\<^bold><*)"
inf_continuous_gfp: fixes F :: "'c \<Rightarrow> 'c" assumes "inf_continuous F" shows "gfp F = (\<Sqinter>i. (F ^^ i) \<top>)"
suma2__conga: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a"   and G' :: "'a"   and H' :: "'a"   and I' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G' H' I'" shows "Tarski_neutral_dimensionless.CongA Bet Cong G H I G' H' I'"
guaranteesI: fixes F :: "'b program"   and X :: "'b program set"   and Y :: "'b program set" assumes "\<And>G. \<lbrakk>F ok G; F \<squnion> G \<in> X\<rbrakk> \<Longrightarrow> F \<squnion> G \<in> Y" shows "F \<in> X guarantees Y"
swing_pos_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "swing_pos w1 w2" shows "kauff_mat w1 = kauff_mat w2"