hcju/isabelleps huggingface dataset:
simp_in_frac: fixes r :: "'a"   and s :: "'a"   and s' :: "'a" assumes "(r, s) \<in> carrier rel"   and "s' \<in> S" shows "(r |\<^bsub>rel\<^esub> s) = (s' \<otimes> r |\<^bsub>rel\<^esub> s' \<otimes> s)"
merge_less_sets_hd: fixes as :: "nat list list"   and bs :: "nat list list"   and us :: "nat list list"   and vs :: "nat list list" assumes "merge as bs us vs"   and "strict_sorted (concat as)"   and "strict_sorted (concat bs)"   and "bs \<in> lists (- {[]})" shows "list.set (hd us) \<lless> list.set (concat vs)"
vfsequence_vproduct: fixes n :: "V"   and xs :: "V"   and A :: "V \<Rightarrow> V" assumes "n \<in>\<^sub>\<circ> \<omega>"   and "xs \<in>\<^sub>\<circ> vproduct n A" shows "vfsequence xs"
WT_callee_full: fixes callee :: "'b \<Rightarrow> ('c \<times> 'd) spmf" shows "\<I>_full \<turnstile>c callee \<surd>"
that: fixes ys :: "'a list"   and x :: "'a"   and zs :: "'a list" assumes "xs = ys @ x # zs"   and "P x" shows "thesis"
field_cont_on_typeI_region_cont_on_edges: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and F :: "real \<times> real \<Rightarrow> 'a"   and k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "typeI_twoCube twoC"   and "continuous_on (cubeImage twoC) F"   and "(k, \<gamma>) \<in> boundary twoC" shows "continuous_on (\<gamma> ` {0..1}) F"
butlast_snoc: fixes xs :: "'b list"   and x :: "'b" shows "butlast (xs @ [x]) = xs"
covering_space_monodromy: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and g1 :: "real \<Rightarrow> 'd"   and g2 :: "real \<Rightarrow> 'd"   and h1 :: "real \<Rightarrow> 'c"   and h2 :: "real \<Rightarrow> 'c" assumes "covering_space C p S"   and "path g1"   and "path_image g1 \<subseteq> S"   and "path g2"   and "path_image g2 \<subseteq> S"   and "homotopic_paths S g1 g2"   and "path h1"   and "path_image h1 \<subseteq> C"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p (h1 t) = g1 t"   and "path h2"   and "path_image h2 \<subseteq> C"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p (h2 t) = g2 t"   and "pathstart h1 = pathstart h2" shows "pathfinish h1 = pathfinish h2"
rel_interior_convex_cone_aux: fixes S :: "'a set"   and c :: "real"   and x :: "'a" assumes "convex S" shows "((c, x) \<in> rel_interior (cone hull {1} \<times> S)) = (0 < c \<and> x \<in> (*\<^sub>R) c ` rel_interior S)"
iso_def: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "iso r r' f \<equiv> embed r r' f \<and> bij_betw f (Field r) (Field r')"
e: shows "extreme {p \<in> A. f p \<sim> p \<or> f p = p} (\<lambda>x y. y \<sqsubseteq> x) p"
when_simps_2: fixes zero :: "'a"   and a :: "'a" shows "zero.when zero a False = zero"
border_per_root: fixes x :: "'b list"   and w :: "'b list" assumes "x \<le>b w" shows "w \<le>p w\<^sup><\<inverse>x  \<cdot> w"
count_mset_set_3: fixes x :: "'k"   and A :: "'k set" assumes "x \<notin> A" shows "count (mset_set A) x = 0"
UP_a_closed: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "p \<oplus>\<^bsub>UP R\<^esub> q \<in> carrier (UP R)"
CFG_Construct_linorder_def: fixes \<alpha>e :: "'f \<Rightarrow> ('g \<times> 'h \<times> 'g) set"   and \<alpha>n :: "'f \<Rightarrow> 'g list"   and invar :: "'f \<Rightarrow> bool"   and inEdges' :: "'f \<Rightarrow> 'g \<Rightarrow> ('g \<times> 'h) list"   and Entry :: "'f \<Rightarrow> 'g"   and defs :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set"   and "uses" :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set" shows "CFG_Construct_linorder \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<equiv> CFG_Construct_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses"
triangle_permutes_4: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.kinematic_triangle \<E> \<P> a b c" shows "MinkowskiPrimitive.kinematic_triangle \<E> \<P> c a b"
v: shows "b * p + e * a * q = (p\<^sup>2 + N * q\<^sup>2) * v"
decr_grading_p_monomial: fixes d :: "'c \<Rightarrow> nat"   and n :: "nat"   and v :: "'c"   and c :: "'d" shows "Quasi_PM_Power_Products.punit.decr_grading_p d n (monomial c v) = monomial c (Quasi_PM_Power_Products.punit.decr_grading_term d n v)"
the_default_def: shows "the_default \<equiv> \<lambda>uu uua. rec_option (\<lambda>x. x) (\<lambda>x _. x) uua uu"
Uniq_def: fixes P :: "'c \<Rightarrow> bool" shows "Uniq P \<equiv> \<forall>x y. P x \<longrightarrow> P y \<longrightarrow> y = x"
simps: shows "subst_bv (Fv x \<tau>') (bind_fv (x, \<tau>') s) = s"
f_def: shows "f \<equiv> \<lambda>i. if a = i then 1 else 0"
weakPar2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and R :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P : R \<rhd> Q \<Longrightarrow>\<alpha> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "bn \<alpha> \<sharp>* P"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* R" shows "\<Psi> : P \<parallel> R \<rhd> P \<parallel> Q \<Longrightarrow>\<alpha> \<prec> P \<parallel> Q'"
residues_prime_def: fixes p :: "nat" shows "residues_prime p \<equiv> prime p"
map_id: fixes t :: "('a, 'b, 'c) var0" shows "map_var0 id id t = t"
supr_def: fixes A :: "'a set" shows "supr A \<equiv> minim (local.Above A)"
\<epsilon>_def: shows "\<phi>\<psi>.\<epsilon> \<equiv> transformation_by_components.map (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) \<eta>\<epsilon>.FG.map \<phi>\<psi>.\<epsilon>o"
dgrad_p_set_le_def: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and G :: "('t \<Rightarrow>\<^sub>0 'e) set" shows "dgrad_p_set_le d F G = dgrad_set_le d (pp_of_term ` Keys F) (pp_of_term ` Keys G)"
isTable_def: fixes f :: "'a \<Rightarrow> 'b"   and vs :: "'a list"   and t :: "('a \<times> 'b) list" shows "isTable f vs t \<equiv> \<forall>p. p \<in> set t \<longrightarrow> snd p = f (fst p) \<and> fst p \<in> set vs"
oghoare_Catch: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and p1 :: "('e, 'd, 'f) com"   and p2 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P (Catch p1 p2) Q, A" shows "\<exists>P\<^sub>1 P\<^sub>2 P' Q' A'. P = AnnComp P\<^sub>1 P\<^sub>2 \<and> \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P\<^sub>1 p1 Q', P' \<and> P' \<subseteq> pre P\<^sub>2 \<and> \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P\<^sub>2 p2 Q', A' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A"
le_prod_encode_1: fixes a :: "nat"   and b :: "nat" shows "a \<le> prod_encode (a, b)"
card_Diff_subset_Int: fixes A :: "'c set"   and B :: "'c set" assumes "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
steps: shows "(a, b) \<in> (R \<union> S)\<^sup>*"
actions_change_prefix: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and n :: "enat"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "a \<in> actions E"   and "ltake n E [\<approx>] ltake n E'"   and "enat a < n" shows "a \<in> actions E'"
i_Exec_Comp_Stream_Acc_Output__Init__eq_Msg_iUntil_start_event_conv: fixes k :: "nat"   and m :: "'a message_af"   and event :: "nat \<Rightarrow> bool"   and t0 :: "nat"   and t :: "nat"   and s :: "nat \<Rightarrow> 'a message_af"   and output_fun :: "'b \<Rightarrow> 'a message_af"   and trans_fun :: "'c message_af \<Rightarrow> 'b \<Rightarrow> 'b"   and input :: "nat \<Rightarrow> 'c message_af"   and c :: "'b" assumes "0 < k"   and "m \<noteq> NoMsg"   and "\<And>t. event t = ((t + k - Suc 0) mod k = 0)"   and "t0 = Suc (t * k)"   and "s = output_fun \<circ> i_Exec_Comp_Stream_Init trans_fun (input \<odot> k) c" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m) = (s t0 = m \<and> (\<circle> t' t0 [0\<dots>]. s t1 = NoMsg. t1 \<U> t2 [0\<dots>] \<oplus> t'. event t2) \<or> (\<circle> t' t0 [0\<dots>]. \<not> event t1. t1 \<U> t2 [0\<dots>] \<oplus> t'. s t2 = m \<and> \<not> event t2 \<and> (\<circle> t'' t2 [0\<dots>]. s t3 = NoMsg. t3 \<U> t4 [0\<dots>] \<oplus> t''. event t4)))"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) \<lparr>Chn = fgh.chine_assoc', Dom = Cod (f \<star> g \<star> h), Cod = Dom ((f \<star> g) \<star> h)\<rparr>"
simps: fixes \<delta>\<^sub>L :: "int"   and p\<^sub>0\<^sub>L :: "int \<times> int"   and p\<^sub>1\<^sub>L :: "int \<times> int"   and \<delta>\<^sub>R :: "int"   and p\<^sub>0\<^sub>R :: "int \<times> int"   and p\<^sub>1\<^sub>R :: "int \<times> int"   and l :: "int"   and ps :: "(int \<times> int) list" shows "combine_code (\<delta>\<^sub>L, p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (\<delta>\<^sub>R, p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps = (let (\<delta>, c\<^sub>0, c\<^sub>1) = if \<delta>\<^sub>L < \<delta>\<^sub>R then (\<delta>\<^sub>L, p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) else (\<delta>\<^sub>R, p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) in Let (filter (\<lambda>p. (fst p - l)\<^sup>2 < \<delta>) ps) (find_closest_pair_code (\<delta>, c\<^sub>0, c\<^sub>1)))"
SubstFormP_Zero: fixes i :: "name"   and A :: "fm" shows "{} \<turnstile> SubstFormP \<guillemotleft>Var i\<guillemotright> Zero \<guillemotleft>A\<guillemotright> \<guillemotleft>A(i::=Zero)\<guillemotright>"
lt_ubound_def: fixes c :: "'a"   and b :: "'a option" shows "c <\<^sub>u\<^sub>b b = \<lhd>\<^sub>u\<^sub>b (<) c b"
assms: fixes r :: "char list"   and s :: "char list" shows "\<forall>x\<in>set xs. showsx x (r @ s) = showsx x r @ s"
hoare_triple_effect: fixes P :: "assn"   and c :: "'d Heap"   and Q :: "'d \<Rightarrow> assn"   and h :: "heap"   and as :: "nat set" assumes "<P> c <Q>"   and "(h, as) \<Turnstile> P" shows "\<exists>h' r. effect c h h' r \<and> (h', new_addrs h as h') \<Turnstile> Q r"
insert_Diff1: fixes x :: "'a"   and B :: "'a set"   and A :: "'a set" assumes "x \<in> B" shows "insert x A - B = A - B"
fdia_add1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows " |x\<rangle> (y + z) =  |x\<rangle> y +  |x\<rangle> z"
\<sigma>l: shows "labels \<sigma> = [hd (labels \<sigma>)]"
higher_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'t" shows "higher p t = except p {s. s \<preceq>\<^sub>t t}"
convergent_mult_const_iff: fixes c :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "convergent (\<lambda>n. c * f n) = convergent f"
fold_fun_left_comm: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c set"   and x :: "'b"   and z :: "'c set" assumes "finite A" shows "f x \<union> Finite_Set.fold (\<lambda>\<phi>. (\<union>) (f \<phi>)) z A = Finite_Set.fold (\<lambda>\<phi>. (\<union>) (f \<phi>)) (f x \<union> z) A"
inter_rel_br: fixes a :: "'a \<Rightarrow> 'd set"   and I :: "'a \<Rightarrow> bool"   and b :: "'c \<Rightarrow> 'd set"   and J :: "'c \<Rightarrow> bool" shows "\<langle>br a I, br b J\<rangle>Refine_Intersection.inter_rel = br (\<lambda>(x, y). a x \<inter> b y) (\<lambda>x. I (fst x) \<and> J (snd x))"
notStuck_CatchD2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e"   and s' :: "'e" assumes "\<Gamma>\<turnstile> \<langle>Catch c1 c2,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"   and "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> Abrupt s'" shows "\<Gamma>\<turnstile> \<langle>c2,Normal s'\<rangle> \<Rightarrow>\<notin>{Stuck}"
ae: shows "almost_everywhere (T x) (alw (\<lambda>\<omega>. HLD Y \<omega> \<longrightarrow> ev (HLD X) \<omega>))"
elims: fixes x :: "'a tree"   and y :: "nat set" assumes "braun_indices x = y" obtains "x = \<langle>\<rangle>" and "y = {}"   | l :: "'a tree" and uu_ :: "'a" and r :: "'a tree" where "x = \<langle>l, uu_, r\<rangle>" and "y = {1} \<union> (*) 2 ` braun_indices l \<union> Suc ` (*) 2 ` braun_indices r"
nres_order_simps_6: fixes X :: "'b nres set" shows "(Sup X = FAIL) = (FAIL \<in> X)"
lookup_except_when: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and S :: "'c set" shows "lookup (except p S) = (\<lambda>t. lookup p t when t \<notin> S)"
maxsplit_lemma: fixes P :: "'a list \<Rightarrow> bool"   and res :: "'a list \<times> 'a list"   and ps :: "'a list"   and qs :: "'a list"   and xs :: "'a list"   and ys :: "'a list" shows "(maxsplit P res ps qs = (xs, ys)) = (if \<exists>us. prefix us qs \<and> P (ps @ us) then xs @ ys = ps @ qs \<and> is_maxpref P xs (ps @ qs) else (xs, ys) = res)"
unit_simps_2: fixes a :: "'b" assumes "B.obj a" shows "src\<^sub>C (H.unit a) = H.map\<^sub>0 a"
iMOD_imp_iMODb: fixes x :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "x \<in> [ r, mod m ]"   and "x \<le> r + m * c" shows "x \<in> [ r, mod m, c ]"
d: shows "k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1)"
getBackjumpLevel_def: fixes state :: "State" shows "getBackjumpLevel state \<equiv> if getC state = [opposite (getCl state)] then 0 else elementLevel (getCll state) (getM state)"
simps_2: fixes fp :: "Syntax.var"   and l :: "Syntax.var list"   and y :: "Syntax.var" shows "map_fixed fp l (Var y) = (if y \<in> set l then Var y else Var fp)"
comm_groupE_4: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
r_prefixes_def: shows "r_prefixes \<equiv> Pr 1 r_singleton_encode r_next"
mset_update: fixes i :: "nat"   and ls :: "'a list"   and v :: "'a" assumes "i < length ls" shows "mset (ls[i := v]) = add_mset v (mset ls - {#ls ! i#})"
lower_pi_float_interval: fixes prec :: "nat" shows "lower (pi_float_interval prec) = lb_pi prec"
splitAt_simp1: fixes ram :: "'a"   and a :: "'a list"   and b :: "'a list" assumes "ram \<notin> set a"   and "ram \<notin> set b" shows "fst (splitAt ram (a @ ram # b)) = a"
int_of_integer_of_int: fixes k :: "int" shows "int_of_integer (ring_1_class.of_int k) = k"
assms_1: shows "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> m"
distinct_disc_20: fixes "term" :: "'b local.term" assumes "is_Tensor term" shows "term \<noteq> \<^bold>\<I>"
GF_advice_af: fixes w :: "'b set list"   and w' :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and X :: "'b ltln set" assumes "(w \<frown> w') \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<nu>" shows "w' \<Turnstile>\<^sub>n af \<phi> w[X]\<^sub>\<nu>"
sd_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "sd_prod x y = (fst x \<cdot> fst y, snd x \<squnion> \<alpha> (fst x) (snd y))"
p_\<alpha>_def: shows "p_\<alpha> \<equiv> map seg [0..<length B]"
lsl_inf_closed_var: fixes x :: "'b"   and y :: "'b" shows "\<nu> (\<nu> x \<sqinter> \<nu> y) = \<nu> x \<sqinter> \<nu> y"
SUCC_SUBS: fixes x :: "tm"   and y :: "tm"   and H :: "fm set" shows "insert (SUCC x SUBS SUCC y) H \<turnstile> x SUBS y"
induct: fixes P :: "nat \<Rightarrow> nat ifex \<Rightarrow> bdd \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat ifex"   and a2 :: "bdd" assumes "\<And>bdd. P 0 Falseif bdd"   and "\<And>bdd. P (Suc 0) Trueif bdd"   and "\<And>n v t e bdd. \<lbrakk>\<And>x y xa ya. \<lbrakk>(x, y) = pm_pth (dpm bdd) n; (xa, ya) = y\<rbrakk> \<Longrightarrow> P xa t bdd; \<And>x y xa ya. \<lbrakk>(x, y) = pm_pth (dpm bdd) n; (xa, ya) = y\<rbrakk> \<Longrightarrow> P ya e bdd\<rbrakk> \<Longrightarrow> P (Suc (Suc n)) (IF v t e) bdd"   and "\<And>va uw_. P (Suc (Suc va)) Trueif uw_"   and "\<And>va uw_. P (Suc (Suc va)) Falseif uw_"   and "\<And>v uw_. P (Suc v) Falseif uw_"   and "\<And>va vb vc uw_. P (Suc 0) (IF va vb vc) uw_"   and "\<And>uw_. P 0 Trueif uw_"   and "\<And>va uw_. P (Suc (Suc va)) Trueif uw_"   and "\<And>v va vb uw_. P 0 (IF v va vb) uw_"   and "\<And>v va vb uw_. P (Suc 0) (IF v va vb) uw_" shows "P a0 a1 a2"
cases: fixes x :: "('b \<times> color) tree" obtains "x = \<langle>\<rangle>"   | l :: "('b \<times> color) tree" and xa :: "'b" and c :: "color" and r :: "('b \<times> color) tree" where "x = \<langle>l, (xa, c), r\<rangle>"
is_natural_2: fixes f :: "'j" assumes "J.arr f" shows "A.map (J.cod f) \<cdot> A.map f = A.map f"
x_2: fixes n :: "nat" shows "dist (x n) (u n) \<le> f (x n)"
iso_lunit: fixes f :: "'d" assumes "D.ide f" shows "D.iso \<l>\<^sub>D[f]"
mono_Mprefix2: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process"   and Q :: "'b \<Rightarrow> 'b process" assumes "\<forall>x\<in>A. P x \<sqsubseteq> Q x" shows "\<forall>s. s \<notin> D (Mprefix A P) \<longrightarrow> Ra (Mprefix A P) s = Ra (Mprefix A Q) s"
rel_gpv_Done: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'d"   and y :: "'e" shows "rel_gpv A C (Generative_Probabilistic_Value.Done x) (Generative_Probabilistic_Value.Done y) = A x y"
monic_Prod: shows "monic (\<Prod>(a, i)\<in>as. a ^ i)"
mtf2_mono: fixes A :: "'a list"   and q :: "'a"   and x :: "'a"   and n :: "nat" assumes "distinct A"   and "q \<in> set A"   and "q < x in A" shows "q < x in mtf2 n q A"
fps_conv_radius_add: fixes f :: "'a fps"   and g :: "'a fps" shows "min (fps_conv_radius f) (fps_conv_radius g) \<le> fps_conv_radius (f + g)"
execute_def: fixes h :: "'n"   and p :: "('n, 'l, 'm) prog" shows "h \<turnstile> p = the_prog p h"
conj_qf: fixes p :: "fm"   and q :: "fm" assumes "qfree p"   and "qfree q" shows "qfree (Parametric_Ferrante_Rackoff.conj p q)"
prj_simps_7: shows "C.cod \<mu>\<nu>\<pi>.Prj\<^sub>1\<^sub>1 = \<mu>.apex"
fun_upd_def: fixes f :: "'g \<Rightarrow> 'h"   and a :: "'g"   and b :: "'h" shows "f(a := b) = (\<lambda>x. if x = a then b else f x)"
cases: fixes a1 :: "'a set"   and a2 :: "'a \<Rightarrow> nat"   and a3 :: "'a \<Rightarrow> Regions.intv"   and a4 :: "('a \<times> 'a) set" assumes "Regions.valid_region a1 a2 a3 a4" obtains X\<^sub>0 :: "'a set" and X :: "'a set" and I :: "'a \<Rightarrow> Regions.intv" and r :: "('a \<times> 'a) set" and k :: "'a \<Rightarrow> nat" where "a1 = X" and "a2 = k" and "a3 = I" and "a4 = r" and "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Regions.intv.Intv d}" and "refl_on X\<^sub>0 r" and "trans r" and "total_on X\<^sub>0 r" and "\<forall>x\<in>X. Regions.valid_intv (k x) (I x)"
boundedI: fixes b :: "'b multiset"   and a :: "'b multiset"   and c :: "'b multiset" assumes "b \<subseteq># a"   and "c \<subseteq># a" shows "b \<union># c \<subseteq># a"
subset_subseq: fixes X :: "'b set"   and ys :: "'b list" shows "(X \<subseteq> set ys) = (\<exists>xs. X = set xs \<and> subseq xs ys)"
subset_basis_in_imp_indep_in: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and X :: "'b set"   and Y :: "'b set" assumes "indep_system carrier indep"   and "\<E> \<subseteq> carrier"   and "indep_system.basis_in indep \<E> X"   and "Y \<subseteq> X" shows "indep_system.indep_in indep \<E> Y"
interaction_bound_inline1'_aux: fixes gpv :: "('b, 'call, 'ret) gpv"   and p :: "enat"   and s :: "'s" assumes "interaction_bound consider' gpv \<le> p" shows "set_spmf (local.inline1' gpv s) \<subseteq> {Inr (out', out, c', rpv) |out' out c' rpv. if consider' out' then (\<forall>input. (if consider out then eSuc (interaction_bound consider (c' input)) else interaction_bound consider (c' input)) \<le> q) \<and> (\<forall>x. eSuc (interaction_bound consider' (rpv x)) \<le> p) else \<not> consider out \<and> (\<forall>input. interaction_bound consider (c' input) = 0) \<and> (\<forall>x. interaction_bound consider' (rpv x) \<le> p)} \<union> range Inl"
frontier_Un_subset: fixes S :: "'a set"   and T :: "'a set" shows "frontier (S \<union> T) \<subseteq> frontier S \<union> frontier T"
Init_def: fixes s :: "state" shows "Init s = (range (inpt s) \<subseteq> Inputs \<and> outpt s = (\<lambda>p. NotAnInput) \<and> disk s = (\<lambda>d p. InitDB) \<and> phase s = (\<lambda>p. 0) \<and> dblock s = (\<lambda>p. InitDB) \<and> disksWritten s = (\<lambda>p. {}) \<and> blocksRead s = (\<lambda>p d. {}))"
is_cupcake_all_envE: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and env :: "v sem_env" assumes "cakeml_static_env static_cenv"   and "cakeml_static_env.is_cupcake_all_env static_cenv env" obtains v :: "(char list \<times> v) list" and c :: "(char list, char list, nat \<times> tid_or_exn) namespace" where "env = make_sem_env (Bind v []) c" and "c = static_cenv" and "list_all (cakeml_static_env.is_cupcake_value static_cenv \<circ> snd) v"
stat_trpl_snd: fixes l :: "nat"   and st :: "nat"   and r :: "nat" shows "stat (trpl l st r) = st"
pseudonatural_transformation_whisker_left_axioms: shows "pseudonatural_transformation_whisker_left (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1"
abs_eq: fixes x :: "'a \<times> 'a" assumes "fractrel x x" shows "quot_of_fract (abs_fract x) = normalize_quot x"
assms_1: shows "xs @ [Broadcast (r, Rem ix e)] prefix of j"
column_iarray_def: fixes k :: "nat"   and A :: "'b iarray iarray" shows "column_iarray k A = IArray.of_fun (\<lambda>m. A !! m !! k) (IArray.length A)"
1: shows "\<forall>rs. wf s0 rs \<longrightarrow> real (T_on A s0 rs) \<le> c * real (T_opt s0 rs) + b"
det_nfa_accepts: fixes w :: "bool list list" assumes "list_all (is_alph n) w" shows "dfa_accepts (det_nfa A) w = nfa_accepts A w"
leftderives1_implies_leftderives: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.leftderives1 \<NN> \<TT> \<RR> a b" shows "CFG.leftderives \<NN> \<TT> \<RR> a b"
FAssRed2: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and v :: "val"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>Val v\<bullet>F{D} := e,s,b\<rangle> \<rightarrow> \<langle>Val v\<bullet>F{D} := e',s',b'\<rangle>"
regressively_finite_acyclic: fixes x :: "'a" assumes "regressively_finite x" shows "is_acyclic x"
median_3_def: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "median_3 a b c = (if a \<le> b then if b \<le> c then b else max a c else if c \<le> b then b else min a c)"
LID\<^sub>S_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option" shows "LID\<^sub>S \<sigma> \<equiv> dom \<sigma> \<union> \<Union> (LID\<^sub>V ` ran \<sigma>)"
plus_set_def: fixes A :: "'a set"   and B :: "'a set" shows "A + B = A \<union> B"
fSum_eqTr: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "(\<forall>j\<in>nset n (n + m). f j \<in> carrier A \<and> g j \<in> carrier A \<and> f j = g j) \<longrightarrow> \<Sigma>\<^sub>f A f n (n + m) = \<Sigma>\<^sub>f A g n (n + m)"
mulex_on_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "mulex_on P A = (restrict_to (mulex1 P) (multisets A))\<^sup>+\<^sup>+"
distinct_79: fixes x51 :: "'a set"   and x52 :: "('a, 'b, 'c) com"   and x53 :: "('a, 'b, 'c) com"   and x111 :: "('a, 'b, 'c) com"   and x112 :: "('a, 'b, 'c) com" shows "Cond x51 x52 x53 \<noteq> Catch x111 x112"
Node: shows "t = \<langle>l, (a_, b_), r\<rangle>"
source: fixes S :: "'procS"   and TRel :: "('procT \<times> 'procT) set" shows "SourceTerm S \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> SourceTerm S"
info_2: fixes y :: "rat" assumes "info_2 x = Inl y" shows "real_of_2 x = real_of_rat y"