hcju/isabelleps huggingface dataset:
isVal_stops: fixes e :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "stack_elem list"   and T :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and z :: "exp" assumes "isVal e"   and "(\<Gamma>, e, S) \<Rightarrow>\<^sup>b\<^sup>*\<^bsub>T\<^esub> (\<Delta>, z, S)" shows "T = []"
simps_2: shows "to_real_Gromov_completion \<infinity> = abs_Gromov_completion real"
substT_suc: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and zer :: "'b"   and suc :: "'b \<Rightarrow> 'b"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tms :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t1 :: "'b"   and t :: "'b"   and x :: "'a" assumes "Syntax_Arith_aux var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"   and "t1 \<in> trm"   and "t \<in> trm"   and "x \<in> var" shows "substT (suc t1) t x = suc (substT t1 t x)"
assign_global_def: fixes upd :: "('c \<Rightarrow> 'c) \<Rightarrow> 'b control_state_scheme \<Rightarrow> 'b control_state_scheme"   and rhs :: "'b control_state_scheme \<Rightarrow> 'c" shows "assign_global upd rhs = assign (\<lambda>\<sigma>. upd (\<lambda>_. rhs \<sigma>) \<sigma>)"
incid_inter_left: fixes l :: "Lines"   and m :: "Lines" shows "incid (Pascal_Property.inter l m) l"
permsI: fixes f :: "'b \<Rightarrow> 'b" assumes "bij f"   and "\<forall>\<^sub>\<infinity>x. f x = x" shows "f \<in> perms"
nat_pow_closed: fixes a :: "'a"   and n :: "nat" assumes "a \<in> H" shows "a [^] n \<in> H"
right_angle_to_compass: fixes p :: "proj2"   and a :: "proj2"   and q :: "proj2" assumes "right_angle p a q" shows "\<exists>J. is_K2_isometry J \<and> apply_cltn2 p J = east \<and> apply_cltn2 a J = K2_centre \<and> apply_cltn2 q J = north"
card_bind: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd set" assumes "finite S"   and "\<forall>X\<in>S. finite (f X)"   and "disjoint_family_on f S" shows "card (S \<bind> f) = (\<Sum>x\<in>S. card (f x))"
bigo_mult5: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>x. f x \<noteq> (0::'b)" shows "O(f * g) \<subseteq> f *o O(g)"
induct: fixes P :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> ('c, 'b) FingerTreeStruc \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "('c, 'b) FingerTreeStruc" assumes "\<And>p i. P p i Empty"   and "\<And>p i ea. P p i (Single ea)"   and "\<And>p i uu_ pr m sf. (\<And>x xa. \<lbrakk>x = i + gmd pr; xa = x + gmft m; \<not> p x; p xa\<rbrakk> \<Longrightarrow> P p x m) \<Longrightarrow> P p i (Deep uu_ pr m sf)" shows "P a0 a1 a2"
one_le_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq one (numeral.numeral one plus n)"
euclid_diff: fixes x :: "nat"   and y :: "nat" shows "rdom \<lceil>\<lambda>s. s ''x'' = x \<and> s ''y'' = y \<and> zero_class.zero < x \<and> zero_class.zero < y\<rceil> \<subseteq> wp (WHILE (\<lambda>s. s ''x'' \<noteq> s ''y'') INV (\<lambda>s. gcd (s ''x'') (s ''y'') = gcd x y) DO (IF (\<lambda>s. s ''y'' < s ''x'') THEN (''x'' ::= (\<lambda>s. s ''x'' - s ''y'')) ELSE (''y'' ::= (\<lambda>s. s ''y'' - s ''x'')) FI) OD) \<lceil>\<lambda>s. s ''x'' = gcd x y\<rceil>"
not_Some_eq: fixes x :: "'f option" shows "(\<forall>y. x \<noteq> Some y) = (x = None)"
cnf_of_positive_transition_frame_axiom: fixes \<Pi> :: "'b strips_problem"   and k :: "nat"   and v :: "'b" shows "cnf (encode_positive_transition_frame_axiom \<Pi> k v) = {{(State k (index (\<Pi>\<^sub>\<V>) v))\<^sup>+, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<inverse>} \<union> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (add_effects_of op)}}"
not_prefix_induct: fixes ps :: "'c list"   and ls :: "'c list"   and P :: "'c list \<Rightarrow> 'c list \<Rightarrow> bool" assumes "\<not> prefix ps ls"   and "\<And>x xs. P (x # xs) []"   and "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>x xs y ys. \<lbrakk>x = y; \<not> prefix xs ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P ps ls"
intro: fixes lookup_st :: "'c \<Rightarrow> (heap, 'd option) state"   and update_st :: "'c \<Rightarrow> 'd \<Rightarrow> (heap, unit) state"   and P :: "heap \<Rightarrow> bool"   and Q :: "heap \<Rightarrow> bool"   and lookup :: "'c \<Rightarrow> 'd option Heap"   and update :: "'c \<Rightarrow> 'd \<Rightarrow> unit Heap" assumes "dp_consistency lookup_st update_st P"   and "dp_heap_axioms Q lookup lookup_st update update_st" shows "dp_heap P Q lookup lookup_st update update_st"
valid_not_I: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<upsilon> x" shows "\<tau> \<Turnstile> \<upsilon> not x"
rank_mono: fixes a :: "V"   and b :: "V" assumes "a \<subseteq>\<^sub>\<circ> b" shows "rank a \<subseteq>\<^sub>\<circ> rank b"
summarize_aformsE: fixes e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) list"   and d :: "nat"   and p :: "nat"   and C :: "(real \<times> real pdevs) list \<Rightarrow> nat \<Rightarrow> real list \<Rightarrow> bool" assumes "e \<in> funcset UNIV {- 1..1}"   and "degree_aforms X \<le> d" obtains e' :: "nat \<Rightarrow> real" where "aform_vals e X = aform_vals e' (summarize_aforms p C d X)" and "\<And>i. i < d \<Longrightarrow> e i = e' i" and "e' \<in> funcset UNIV {- 1..1}"
web_flow_iff: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "countable_bipartite_web \<Gamma>" shows "web_flow \<Gamma> f = current \<Gamma> f"
Says_imp_spies: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says A B X \<in> set evs" shows "X \<in> knows Spy evs"
head_isnpolyh_Suc: fixes p :: "'b poly"   and n :: "nat" assumes "isnpolyh p (Suc n)" shows "head p = p"
bounded_linear_vec: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>i. bounded_linear (f i)" shows "bounded_linear (\<lambda>x. \<chi>i. f i x)"
le_Ssegment_Ssegment: fixes D :: "'c Order"   and a :: "'c"   and b :: "'c" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq>\<^bsub>D\<^esub> b = (Ssegment (SIod D (Ssegment D b)) a = Ssegment D a)"
com_det: fixes c :: "com"   and s :: "state"   and t :: "state"   and u :: "state" assumes "<c,s> -c-> t"   and "<c,s> -c-> u" shows "u = t"
equivalent_objects_symmetric: fixes a :: "'d"   and b :: "'d" assumes "D.equivalent_objects a b" shows "D.equivalent_objects b a"
abs_eq: fixes xa :: "'d"   and x :: "'d" shows "bdoubleton xa x = Abs_bset {xa, x}"
Nmlize_Inv: fixes t :: "'c local.term" assumes "Can t" shows "\<^bold>\<lfloor>Inv t\<^bold>\<rfloor> = Inv \<^bold>\<lfloor>t\<^bold>\<rfloor>"
cycle_free_diag_intro: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs" shows "cycle_free M n"
prems_1: fixes M :: "'a" assumes "\<Psi> \<otimes> \<one> \<turnstile> the (subject \<alpha>) \<leftrightarrow> M"   and "B \<sharp>* M" shows "thesis"
list_nth_append0: fixes i :: "nat"   and x :: "'a list"   and z :: "'a list" assumes "i < length x" shows "x ! i = (x @ z) ! i"
rows_iarray_def: fixes A :: "'d iarray iarray" shows "rows_iarray A = {row_iarray i A |i. i \<in> {..<nrows_iarray A}}"
at_within_domain: fixes x :: "'c"   and c :: "('c, 'd) chart" assumes "x \<in> domain c" shows "at x within domain c = at x"
ideal_1: fixes a :: "'a"   and b :: "'a" assumes "a \<in> R"   and "b \<in> I" shows "a \<cdot> b \<in> I"
strictified_bicategory_axioms: shows "strictified_bicategory (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"
bind_spmf_cong: fixes p :: "'d spmf"   and q :: "'d spmf"   and f :: "'d \<Rightarrow> 'e spmf"   and g :: "'d \<Rightarrow> 'e spmf" assumes "p = q"   and "\<And>x. x \<in> set_spmf q \<Longrightarrow> f x = g x" shows "p \<bind> f = q \<bind> g"
ide_right: shows "ide (tab\<^sub>0 f)\<^sup>*"
mtype_mbody: fixes CT :: "nat \<Rightarrow> classDef option"   and m :: "nat"   and C :: "nat"   and Cs :: "nat list"   and C0 :: "nat" assumes "mtype(CT,m,C) = Cs \<rightarrow> C0" shows "\<exists>xs e. mbody(CT,m,C) = xs . e \<and> length xs = length Cs"
MkI_eq_1_iff: fixes n :: "int" shows "(MkI\<cdot>n = 1) = (n = 1)"
vector_diff_chain_within: fixes f :: "real \<Rightarrow> real"   and f' :: "real"   and x :: "real"   and s :: "real set"   and g :: "real \<Rightarrow> 'a"   and g' :: "'a" assumes "(f has_vector_derivative f') (at x within s)"   and "(g has_vector_derivative g') (at (f x) within f ` s)" shows "(g \<circ> f has_vector_derivative f' *\<^sub>R g') (at x within s)"
UN_derivations_eq_derivable: shows "\<Union> (Collect Q.derivation) = Collect Q.derivable"
v: shows "pseudo_divmod_main_list lc (hd (rev r) ## map ((*) lc) q) (rev (rev (tl (minus_poly_rev_list (map ((*) lc) (rev r)) (map ((*) (hd (rev r))) (rev d)))))) (rev d) n = (q', rev r')"
\<pi>_nonneg: fixes x :: "real" shows "0 \<le> primes_pi x"
preal_add_commute: fixes x :: "preal"   and y :: "preal" shows "x + y = y + x"
frequently_mono_filter: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and F' :: "'a filter" assumes "frequently P F"   and "F \<le> F'" shows "frequently P F'"
emb_trans: fixes t :: "('c, 'd) tm"   and u :: "('c, 'd) tm"   and s :: "('c, 'd) tm" assumes "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b u"   and "u \<unrhd>\<^sub>e\<^sub>m\<^sub>b s" shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
base_simps_2: shows "hseq r\<^sub>0\<^sup>* s\<^sub>1"
inverse_unique: fixes f :: "'c"   and g :: "'c" assumes "C.inverse_arrows f g" shows "C.inv f = g"
proj_def: fixes r :: "('c \<times> 'b) set"   and x :: "'c" shows "Equiv_Relations.proj r x = r `` {x}"
normal_Int_subgroup: fixes H :: "'a set"   and N :: "'a set" assumes "subgroup H G"   and "N \<lhd> G" shows "N \<inter> H \<lhd> G\<lparr>carrier := H\<rparr>"
union_inter: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "finite B" shows "sum g (A \<union> B) + sum g (A \<inter> B) = sum g A + sum g B"
abs_eq: fixes x :: "'d list" shows "pdevs_of_list x = Abs_pdev (\<lambda>i. if i < length x then x ! i else (0::'d))"
Y_injective_on_homs: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and a :: "'a"   and a' :: "'a"   and f :: "'a"   and f' :: "'a" assumes "yoneda_lemma_for_hom C S \<phi> a a'"   and "partial_magma.in_hom C f a a'"   and "partial_magma.in_hom C f' a a'"   and "curried_functor'.map (dual_category.comp C) C S (hom_functor.map C S \<phi>) f = curried_functor'.map (dual_category.comp C) C S (hom_functor.map C S \<phi>) f'" shows "f = f'"
IH: fixes x :: "('fun, 'atom) Term.term \<times> nat" assumes "\<Gamma> (Var x) \<in> set U"   and "\<exists>y\<in>fv\<^sub>s\<^sub>e\<^sub>t (set M). \<Gamma> (Var x) = \<Gamma> (Var y)"   and "Fun f T \<sqsubseteq> \<Gamma> (Var x)" shows "\<exists>g U. Fun g U \<in> set M \<and> \<Gamma> (Fun g U) = Fun f T \<and> Ball (set U) is_Var \<and> distinct U"
snd_apsnd: fixes f :: "'d \<Rightarrow> 'b"   and x :: "'c \<times> 'd" shows "snd (apsnd f x) = f (snd x)"
cy_def: fixes x :: "'a"   and y :: "'a" shows "cy x y = x + y * ((1::'a) - x\<^sup>2 - y\<^sup>2)"
steps_impl_exec: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and s :: "('d, 'f) xstate"   and t :: "('d, 'f) xstate" assumes "\<Gamma>\<turnstile> ([c], [], s) \<rightarrow>\<^sup>* ([], [], t)" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
lookup_filter: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('b, 'a) RBT.rbt"   and k :: "'b" shows "RBT.lookup (RBT.filter P t) k = (case RBT.lookup t k of None \<Rightarrow> None | Some v \<Rightarrow> if P k v then Some v else None)"
inf_absorb1: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x \<cdot> y = x"
secure_mult_asymp_def: fixes q :: "nat \<Rightarrow> nat" shows "secure_mult_asymp q \<equiv> \<forall>n. secure_mult (q n)"
prems_3: fixes C :: "'a set" assumes "C \<in> \<M>" shows "aff_dim C \<le> int n"
pow_add_list: fixes x :: "'b list"   and a :: "nat"   and b :: "nat" shows "x \<^sup>@ (a + b) = x \<^sup>@ a \<cdot> x \<^sup>@ b"
eq: shows "ideal F = ideal ((\<lambda>x. monomial (1::'b) (monomial 1 x) - monomial (a x) 0) ` X)"
color_paint_Black: fixes t :: "('b \<times> color) tree" shows "color (paint Black t) = Black"
IH_1: assumes "send\<langle>t\<rangle> \<in> set A" shows "send\<langle>t \<cdot> \<theta>\<rangle> \<in> set (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
dyn_standard_control_dependence_def_variant: fixes n :: "'node"   and n' :: "'node"   and as :: "'edge list" shows "n controls\<^sub>s n' via as = (n -as\<rightarrow>* n' \<and> n \<noteq> n' \<and> \<not> n' postdominates n \<and> n' \<notin> set (sourcenodes as) \<and> (\<forall>n''\<in>set (targetnodes as). n' postdominates n''))"
one_reorient: fixes x :: "'b" shows "((1::'b) = x) = (x = (1::'b))"
dehomogenize_def: fixes x :: "'b" shows "dehomogenize x = poly_subst (dehomo_subst x)"
sigma_finite: fixes I :: "'i set" assumes "finite I" shows "sigma_finite_measure (Pi\<^sub>M I M)"
pres_eq_iff_svb: fixes K :: "('a \<times> 'b) set" shows "(((=), (=)) \<in> K \<rightarrow> K \<rightarrow> bool_rel) = (single_valued K \<and> single_valued (K\<inverse>))"
nth_restrict: fixes i :: "nat"   and A :: "nat set"   and v :: "'a option list" assumes "i \<in> A"   and "i < length v" shows "restrict A v ! i = v ! i"
assms_1: shows "t \<sqsubseteq> s \<cdot> \<theta>"
simps_2: fixes x :: "'b"   and hs :: "'b heap list" shows "is_root' (Hp x hs) = no_Emptys hs"
simps_2: fixes a :: "'b"   and x :: "'b list" shows "mset (a # x) = a, mset x"
ignores_old_vars_conj: fixes R :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list \<Rightarrow> bool"   and P :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list \<Rightarrow> bool"   and Q :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list \<Rightarrow> bool" assumes "\<And>x. R x = (P x \<and> Q x)"   and "ignores_old_vars P"   and "ignores_old_vars Q" shows "ignores_old_vars R"
v11_veqpollE: fixes A :: "V"   and B :: "V" assumes "A \<approx>\<^sub>\<circ> B" obtains f :: "V" where "v11 f" and "\<D>\<^sub>\<circ> f = A" and "\<R>\<^sub>\<circ> f = B"
R_def: shows "R \<equiv> \<lambda>N. \<Sum>n. real_of_int (B * b (n + N + 1)) / real_of_int (prod a {N..n + N + 1})"
fair_implies_Liminf_saturated: fixes Bot :: "'b set"   and Inf :: "'b inference set"   and entails :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Red_I :: "'b set \<Rightarrow> 'b inference set"   and Red_F :: "'b set \<Rightarrow> 'b set"   and Ns :: "'b set llist" assumes "calculus Bot Inf entails Red_I Red_F"   and "chain (calculus.derive Red_F) Ns"   and "calculus.fair Inf Red_I Ns" shows "calculus.saturated Inf Red_I (Liminf_llist Ns)"
closest_supchamber_closest: fixes C :: "'a set"   and F :: "'a set"   and D :: "'a set" assumes "domain.chamber C"   and "order.greater_eq C F" shows "order.greater_eq (domain.chamber_distance C D) (domain.chamber_distance (domain.closest_supchamber F D) D)"
axioms_1: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "VectorSpaceEnd smult V T" shows "VectorSpaceHom smult V smult T"
heq0: fixes t :: "real" assumes "t \<in> {0..1}" shows "h (t, 0) = g1 0"
r_badblock_last: fixes n :: "nat"   and v :: "nat" shows "e_nth (the (eval r_badblock [n, v])) n = 1 - v"
that: fixes env\<^sub>1 :: "(name, 'a) fmap" assumes "matchs ps ts\<^sub>1 = Some env\<^sub>1"   and "P_env env\<^sub>1 env\<^sub>2" shows "thesis"
assms_2: shows "t \<in> ball z (Re z / 2)"
flat_interpretation: fixes b :: "'b" shows "partial_function_definitions (flat_ord b) (flat_lub b)"
step_r_complete: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and l :: "'d"   and u :: "'e \<Rightarrow> real"   and l' :: "'d"   and u' :: "'e \<Rightarrow> real"   and \<R> :: "('e \<Rightarrow> real) set set"   and X :: "'e set"   and k :: "'e \<Rightarrow> nat" assumes "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>"   and "\<R> = {region X I r |I r. valid_region X k I r}"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "\<forall>x\<in>X. 0 \<le> u x" shows "\<exists>R'. A,\<R> \<turnstile> \<langle>l, [u]\<^sub>\<R>\<rangle> \<leadsto> \<langle>l', R'\<rangle> \<and> u' \<in> R' \<and> R' \<in> \<R>"
distinct_42: fixes x71 :: "nat"   and x72 :: "msg"   and x61 :: "msg"   and x62 :: "msg" shows "Crypt x71 x72 \<noteq> \<lbrace>x61, x62\<rbrace>"
set_of_plus: fixes A :: "'a interval"   and B :: "'a interval" shows "set_of (A + B) = set_of A + set_of B"
map_add_def: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option" shows "m2 \<Oplus> m1 = (\<lambda>x. case m2 x of \<bottom> \<Rightarrow> m1 x | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>y\<rfloor>)"
has_bochner_integral_null_eq: fixes N :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and i :: "'b" assumes "N \<in> null_sets lebesgue" shows "has_bochner_integral (lebesgue_on N) f i = (i = (0::'b))"
qFreshAll_igFreshAll_qInitAll: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model"   and ys :: "'d"   and y :: "'a"   and qX :: "('b, 'c, 'd, 'a, 'f) qTerm"   and qA :: "('b, 'c, 'd, 'a, 'f) qAbs" assumes "FixSyn TYPE('a) Delta"   and "igFreshClsSTR MOD" shows "(qFresh ys y qX \<longrightarrow> igFresh MOD ys y (FixSyn.qInit MOD qX)) \<and> (qFreshAbs ys y qA \<longrightarrow> igFreshAbs MOD ys y (FixSyn.qInitAbs MOD qA))"
inf_sup_aci_1: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.lattice inf less_eq less sup" shows "inf x y = inf y x"
simps_9: fixes f :: "'c \<Rightarrow> 'b"   and x1 :: "'c" shows "map_literal f (Pos x1) = Pos (f x1)"
prec_st_inpt: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" shows "prec_st (inpt_st r) r = \<box> lft_pred_st (inpt_st r)"
invertible_mult_JNF: fixes A :: "'b mat"   and n :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "invertible_mat A"   and "invertible_mat B" shows "invertible_mat (A * B)"
hoare_trans_UN: fixes P :: "'a set"   and R :: "'b \<Rightarrow> ('a \<times> 'a) set"   and Q :: "'a set" shows "{P} \<Union> (range R) {> Q} = (\<forall>x. {P} R x {> Q})"
restrict0: fixes T :: "'c \<Rightarrow> 'd"   and G :: "'c set" assumes "UGroupHom T"   and "Group G" shows "GroupHom G (restrict0 T G)"
L_infinity_1: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" shows "eNorm (\<LL> \<infinity> M) f = (if f \<in> borel_measurable M then e2ennreal (esssup M (\<lambda>x. ereal \<bar>f x\<bar>)) else \<infinity>)"
set_stake_snth: fixes x :: "'a"   and n :: "nat"   and xs :: "'a stream" shows "(x \<in> set (stake n xs)) = (\<exists>i<n. xs !! i = x)"
rel_mode_rel: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and x :: "Eg1.var"   and SomeMode :: "Mode"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = Skip@[x -=\<^sub>m SomeMode]"   and "c\<^sub>C = Skip@[Eg2_var\<^sub>C_of_Eg1 x -=\<^sub>m SomeMode]"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "\<forall>v\<^sub>C. v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C AsmNoReadOrWrite"   and "\<forall>mds\<^sub>A' mem\<^sub>A' mem\<^sub>A'' mds\<^sub>C'' mem\<^sub>C' mem\<^sub>C''. mds\<^sub>A' = mds\<^sub>A_of mds\<^sub>C'' \<and> mem\<^sub>A' = mem\<^sub>A_of mem\<^sub>C' \<and> mem\<^sub>A'' = mem\<^sub>A_of mem\<^sub>C'' \<and> eval_abv\<^sub>A \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A''\<rangle>\<^sub>A \<langle>Stop, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A \<and> eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C''\<rangle>\<^sub>C \<langle>Stop, mds\<^sub>C'', mem\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<langle>Stop, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A, \<langle>Stop, mds\<^sub>C'', mem\<^sub>C'\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2" shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
simps_1: fixes l :: "'a \<Rightarrow> 'a"   and r :: "'a \<Rightarrow> 'a"   and s :: "'a" shows "root (tree_iterate l r s) = s"
QTrans: shows "Q \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> Q'"
shiftl_over_and_dist: fixes a :: "'b word"   and b :: "'b word"   and c :: "nat" shows "a AND b << c = (a << c) AND (b << c)"
i_def: shows "i \<in> A \<and> Sup {(f i)\<^sup>2 |i. i \<in> A} = (f i)\<^sup>2"
mset_insert: fixes x :: "'b"   and t :: "('b \<times> nat) tree" shows "mset_tree (insert x t) = mset_tree t + {#x#}"
prism_diff_intro: fixes X :: "'a \<Longrightarrow>\<^sub>\<triangle> 'c"   and Y :: "'b \<Longrightarrow>\<^sub>\<triangle> 'c" assumes "\<And>s\<^sub>1 s\<^sub>2. build\<^bsub>X\<^esub> s\<^sub>1 = build\<^bsub>Y\<^esub> s\<^sub>2 \<Longrightarrow> False" shows "X \<nabla> Y"
bot_synlr_def: shows "bot \<equiv> mksynlr ({\<bottom>} \<times> UNIV)"
induct: fixes P :: "dbtm \<Rightarrow> bool"   and dbtm :: "dbtm" assumes "P DBZero"   and "\<And>x. P (DBVar x)"   and "\<And>x. P (DBInd x)"   and "\<And>x1 x2a. \<lbrakk>P x1; P x2a\<rbrakk> \<Longrightarrow> P (DBEats x1 x2a)" shows "P dbtm"
Rf_subs_Rf_diff_Rf: fixes N :: "'a literal multiset set" shows "Rf N \<subseteq> Rf (N - Rf N)"
tuple_simps_3: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "commutative_square f g h k" shows "cod \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = f \<down>\<down> g"
B4: shows "B4 \<in> carrier_mat (Suc m) (Suc m)"
select_convs_3: fixes Field1 :: "nat message_af"   and Field2 :: "int message_af"   and Field3 :: "int message_af"   and more :: "'b" shows "Field3 \<lparr>Field1 = Field1, Field2 = Field2, Field3 = Field3, \<dots> = more\<rparr> = Field3"
Map_ide_1: fixes a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.ide a" shows "J_C.Map a = J_C.Dom a"
associative: fixes M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "Group_Theory.monoid M composition unit"   and "a \<in> M"   and "b \<in> M"   and "c \<in> M" shows "composition (composition a b) c = composition a (composition b c)"
ydrin: shows "y \<in> set (fst (TSdet init h (as @ x # bs @ [x]) (Suc (length as + length bs))))"
id: fixes i :: "nat"   and j :: "nat" assumes "i < nc"   and "j < nr" shows "m1 ! i ! j = m2 ! i ! j"