hcju/isabelleps huggingface dataset:
set_add_pairs: fixes ps1 :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd"   and gs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "\<And>xs ys. set (comb xs ys) = set xs \<union> set ys"   and "ps1 = apply_ncrit ncrit data gs bs hs (apply_icrit icrit data gs bs hs (np gs bs hs data))" shows "set (add_pairs gs bs ps hs data) = {(p, q) |p q. (False, p, q) \<in> set ps1 \<or> (p, q) \<in> set ps \<and> \<not> ocrit data hs ps1 p q}"
nat_pow_distrib: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat"   and n :: "nat" assumes "x \<in> carrier S"   and "y \<in> carrier S" shows "(x \<otimes>\<^bsub>S\<^esub> y) [^]\<^bsub>S\<^esub> n = x [^]\<^bsub>S\<^esub> n \<otimes>\<^bsub>S\<^esub> y [^]\<^bsub>S\<^esub> n"
card1: shows "n1 = card {i. i < m1 \<and> sel1 i}"
vNM_utility_implies_independence: fixes outcomes :: "'b set"   and \<R> :: "('b pmf \<times> 'b pmf) set" assumes "finite outcomes"   and "outcomes \<noteq> {}"   and "\<exists>u. ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)" shows "independent_vnm (\<P> outcomes) \<R>"
s_1: fixes i :: "nat" shows "simple_function (N \<Otimes>\<^sub>M M) (s i)"
PAR_IMAGE_def: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f nres"   and X :: "'e set" shows "PAR_IMAGE P f X = list_spec X \<bind> (\<lambda>xs. nres_of_nress (\<lambda>(x, y). P x y) (Parallel.map (\<lambda>x. f x \<bind> (\<lambda>y. RETURN (x, y))) xs) \<bind> (\<lambda>fxs. RETURN (set fxs)))"
alphaAbs_qAbs_iff_ex_qAFresh: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
add_arc_def: fixes a :: "'b" shows "add_arc a = \<lparr>verts = verts G \<union> {tail G a, head G a}, arcs = insert a (arcs G), tail = tail G, head = head G\<rparr>"
rep_preserves_seq: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b local.term set"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) (free_monoidal_category.comp C g f)" shows "monoidal_language.Arr C (free_monoidal_category.rep C g) \<and> monoidal_language.Arr C (free_monoidal_category.rep C f) \<and> monoidal_language.Dom C (free_monoidal_category.rep C g) = monoidal_language.Cod C (free_monoidal_category.rep C f)"
u_def: shows "u = (\<lambda>t. pp f t + pp g t)"
algorithmic_path_type_unicity: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty"   and u :: "trm"   and T' :: "ty" assumes "\<Gamma> \<turnstile> s \<leftrightarrow> t : T"   and "\<Gamma> \<turnstile> s \<leftrightarrow> u : T'" shows "T = T'"
cast_document_ptr_not_node_ptr_2: fixes node_ptr :: "(_) node_ptr"   and document_ptr :: "(_) document_ptr" shows "cast node_ptr \<noteq> cast document_ptr"
DisjointPathsPlusOne_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list" shows "DisjointPathsPlusOne G v0 v1 paths P_new \<equiv> DisjointPaths G v0 v1 paths \<and> DisjointPathsPlusOne_axioms G v0 v1 paths P_new"
thetaWhile_def: shows "thetaWhile \<equiv> {(while tst do c, while tst do d) |tst c d. compatTst tst \<and> c \<approx>s d} \<union> {(c1 ;; while tst do c, d1 ;; while tst do d) |tst c1 d1 c d. compatTst tst \<and> c1 \<approx>s d1 \<and> c \<approx>s d}"
cases: fixes a :: "('b \<times> 'b) set" assumes "asym a" obtains R :: "('b \<times> 'b) set" where "a = R" and "\<And>a b. (a, b) \<in> R \<Longrightarrow> (b, a) \<notin> R"
315: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> y) \<sqinter> x = x"
closure_mono: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T" shows "closure S \<subseteq> closure T"
C12: shows "\<E> f\<^sub>1 \<noteq> (\<E> f\<^sub>2)\<inverse>"
prepend: fixes xs :: "'a list"   and ys :: "'a list"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "(xs, ys) \<in> genPrefix r"   and "(x, y) \<in> r" shows "(x # xs, y # ys) \<in> genPrefix r"
long_power: fixes r :: "'b list"   and x :: "'c list" assumes "r \<noteq> \<epsilon>" shows "\<^bold>|x\<^bold>| \<le> \<^bold>|r \<^sup>@ \<^bold>|x\<^bold>|\<^bold>|"
condensation_acyclic: fixes g :: "'g"   and P :: "'val set" shows "acyclic (condensation_edges g P)"
additive_def: fixes f :: "'b \<Rightarrow> 'c" shows "Modules.additive f \<equiv> \<forall>x y. f (x + y) = f x + f y"
inject_7: fixes x71 :: "nat"   and x72 :: "msg"   and y71 :: "nat"   and y72 :: "msg" shows "(Crypt x71 x72 = Crypt y71 y72) = (x71 = y71 \<and> x72 = y72)"
rep_eq: fixes x :: "ennreal"   and xa :: "ennreal" shows "enn2ereal (x - xa) = max 0 (enn2ereal x - enn2ereal xa)"
continuous_on_diff_ennreal: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "continuous_on A f"   and "continuous_on A g"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> top"   and "\<And>x. x \<in> A \<Longrightarrow> g x \<noteq> top" shows "continuous_on A (\<lambda>z. f z - g z)"
atm_of_Max_lit: fixes C :: "'a literal multiset" assumes "C \<noteq> {#}" shows "atm_of (Max_mset C) = Max (atms_of C)"
init: fixes \<xi> :: "'a"   and p :: "'b" assumes "(\<xi>, p) \<in> init A" shows "P (\<xi>, p)"
pr_pequiv_trans: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and c :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "a \<sqsubseteq> b"   and "b \<simeq> c" shows "a \<sqsubseteq> c"
openin_diff: fixes U :: "'b topology"   and S :: "'b set"   and T :: "'b set" assumes "openin U S"   and "closedin U T" shows "openin U (S - T)"
sinh_def: fixes x :: "'a" shows "sinh x = (exp x - exp (- x)) /\<^sub>R 2"
successive_dest_head: fixes xs :: "'b list"   and w :: "'b"   and x :: "'b"   and ys :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "xs = w # x # ys"   and "successive P xs" shows "successive P (x # xs) \<and> \<not> P w \<or> successive P xs \<and> \<not> P x"
added_ids_Deliver_Add_diff_collapse: fixes e :: "'b"   and e' :: "'b"   and i :: "'c"   and j :: "'c" assumes "e \<noteq> e'" shows "added_ids [Deliver (i, Add j e)] e' = []"
prv_eqv_neg_prv_fls: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "prv (eqv \<phi> (neg \<phi>))" shows "prv fls"
gb_schema_aux_domI2: fixes data :: "nat \<times> nat \<times> 'd"   and args :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list \<times> ((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "struct_spec sel ap ab compl" shows "gb_schema_aux_dom (data, args)"
hatom_of_nat_nat_of_hatom: fixes l :: "char list \<times> hterm list" shows "hatom_of_nat (nat_of_hatom l) = l"
dsg: shows "matrix_sum d g n \<in> carrier_mat d d"
HT'set_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and \<Theta> :: "(((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> 'b) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> com) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool)) set" shows "HT'set \<pi> \<Theta> \<equiv> \<forall>(n, P, c, Q)\<in>\<Theta>. HT' \<pi> P c Q"
fm_measurable: fixes n :: "nat"   and N :: "nat set set"   and M :: "'e measure" assumes "Utn ` J n \<in> N" shows "fm n \<in> Pi\<^sub>M (J n) (\<lambda>_. M) \<rightarrow>\<^sub>M Pi\<^sub>F N (\<lambda>_. M)"
assms_1: fixes var :: "Vars.var" shows "P1 (Var var)"
no_smallest_element_skippedD: fixes carrier :: "'c set"   and indep :: "'c set \<Rightarrow> bool"   and weight :: "'c \<Rightarrow> 'd"   and E :: "'c set"   and F :: "'c set"   and e :: "'c"   and g :: "'d" assumes "weighted_matroid carrier indep"   and "weighted_matroid.no_smallest_element_skipped carrier indep weight E F"   and "e \<in> carrier - E"   and "weight e < g"   and "indep (insert e (weighted_matroid.limi weight F g))" shows "e \<in> weighted_matroid.limi weight F g"
admissible_leI: fixes lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and luba :: "'d set \<Rightarrow> 'd"   and orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "class.ccpo lub ord (mk_less ord)"   and "mcont luba orda lub ord f"   and "mcont luba orda lub ord g" shows "ccpo.admissible luba orda (\<lambda>x. ord (f x) (g x))"
is_trail_path: fixes v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" shows "is_trail v ps v' = (is_path v ps v' \<and> distinct ps \<and> set ps \<inter> set (rev_path ps) = {})"
submap_imp_state_succ_submap_a: fixes s1 :: "('a, 'b) fmap"   and s2 :: "('a, 'b) fmap"   and s3 :: "('a, 'b) fmap" assumes "s1 \<subseteq>\<^sub>f s2"   and "s2 \<subseteq>\<^sub>f s3" shows "s1 \<subseteq>\<^sub>f s3"
reflection: shows "rel_weakly_reflects_barbs (indRelRTPO TRel) (STCalWB SWB TWB)"
hyps_8: shows "l\<^sub>2' = [this \<mapsto> Addr a, pns [\<mapsto>] vs]"
set_zmset_empty: shows "set_zmset {#}\<^sub>z = {}"
Un_empty_right: fixes A :: "'c set" shows "brouwer.comp_inf.vector A"
prems: shows "v \<in> fv\<^sub>s\<^sub>t (x # S)"
simps_2: fixes x :: "nat"   and xs :: "nat list" shows "tuple_encode 0 (x # xs) = x"
runit_naturality: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "\<mu> \<cdot> PB.runit (local.dom \<mu>) = PB.runit (cod \<mu>) \<cdot> (\<mu> \<star> src \<mu>)"
exhaust: fixes y :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" obtains x1 :: "'a" and x2 :: "'b" and x3 :: "nat" and x4 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" where "y = SkewBinomialHeapStruc.Node x1 x2 x3 x4"
irreducibleD2: fixes C :: "'s set"   and a :: "'s"   and b :: "'s" assumes "C \<in> UNIV // communicating"   and "a \<in> C"   and "(a, b) \<in> communicating" shows "b \<in> C"
retract_of_refl: fixes S :: "'c set" shows "S retract_of S"
alw_discrCf_indis: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and \<omega> :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream" assumes "enabled cf \<omega>"   and "discrCf cf" shows "alw (holds (\<lambda>cf'. snd cf' \<approx> snd cf)) (cf ## \<omega>)"
Sum2: fixes Q :: "pi"   and Rs :: "Late_Semantics.residual"   and P :: "pi" assumes "Q \<longmapsto>\<^sub>l Rs" shows "P \<oplus> Q \<longmapsto>\<^sub>l Rs"
sets_borel: shows "sets borel = sigma_sets UNIV {S. open S}"
simps_4: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list"   and xs :: "nat list" shows "eval_wellf (Cn n f gs) xs = (if \<forall>g\<in>set gs. eval_wellf g xs \<down> then eval_wellf f (map (\<lambda>g. the (eval_wellf g xs)) gs) else None)"
simps_1: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "real" shows "aEvalUni (EqUni (a, b, c)) x = (a * x\<^sup>2 + b * x + c = 0)"
seq_type: fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and P :: "bexp set"   and c\<^sub>1 :: "(Eg1.var, aexp, bexp) Stmt"   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 \<Gamma>'' :: "Eg1.var \<Rightarrow> bexp set option"   and \<S>'' :: "Eg1.var set \<times> Eg1.var set"   and P'' :: "bexp set" assumes "has_type \<Gamma> \<S> P c\<^sub>1 \<Gamma>' \<S>' P'"   and "has_type \<Gamma>' \<S>' P' c\<^sub>2 \<Gamma>'' \<S>'' P''" shows "has_type \<Gamma> \<S> P (c\<^sub>1 ;; c\<^sub>2) \<Gamma>'' \<S>'' P''"
iMOD_subset_imp_divisor_mod_0: fixes r' :: "nat"   and m' :: "nat"   and r :: "nat"   and m :: "nat" assumes "[ r', mod m' ] \<subseteq> [ r, mod m ]" shows "m' mod m = 0"
in_set_of_interval_of: fixes x :: "'b"   and y :: "'b" shows "(x \<in>\<^sub>i interval_of y) = (x = y)"
l2_inv2I: fixes x :: "l2_state" assumes "\<And>R. in_progress (progress x R) xsk \<Longrightarrow> \<exists>X N. sklvl1.guessed_frame R xsk = Some (Exp X (NonceF (R $ N)))" shows "x \<in> l2_inv2"
norm_conv_dist: fixes x :: "'c" shows "norm x = dist x (0::'c)"
op_map_update_def: fixes k :: "'g"   and v :: "'h"   and m :: "'g \<Rightarrow> 'h option" shows "op_map_update k v m \<equiv> m(k \<mapsto> v)"
has_domain_iff_arr: fixes f :: "'a" shows "(C.domains f \<noteq> {}) = C.arr f"
iMOD_inter: fixes r :: "nat"   and r' :: "nat"   and m :: "nat" assumes "r \<le> r'"   and "r mod m = r' mod m" shows "[ r, mod m ] \<inter> [ r', mod m ] = [ r', mod m ]"
lo_power: fixes x :: "nat"   and r :: "nat" assumes "Suc 0 < x" shows "lo (x ^ r) x = r"
comp_arr_inv': fixes f :: "'a \<times> 'a" assumes "VV.iso f" shows "VV.comp f (VV.inv f) = VV.cod f"
refl: fixes a :: "'b::order" shows "(a::'b::order) \<le> a"
atLeastLessThan_rev_at_least_Suc_atMost: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" shows "sum g {n..<m} = (\<Sum>i = Suc n..m. g (m + n - i))"
min_elems_no: fixes x :: "'b list"   and A :: "'b list set"   and t :: "'b list" assumes "x \<in> min_elems A"   and "t \<in> A"   and "t \<le> x" shows "x = t"
B_def: shows "B = (\<lambda>b. insert (n - 1) {x \<in> {..<n - 1}. \<not> b x})"
w: fixes n :: "nat" assumes "N \<le> n" shows "AE x in M. norm (s (X n) x) \<le> w x"
updS_def: fixes s :: "'b bdd_scheme"   and x :: "nat \<times> nat \<times> nat"   and r :: "nat" shows "updS s x r = dcl_update (\<lambda>m. m(x \<mapsto> r)) s"
hyps_4: shows "\<zero>\<^bsub>L\<^esub> = indexed_const \<zero>"
g_subdist: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "is_conjugation f g" shows "g (x \<cdot> y) \<le> g x"
sup_eq_bot_iff: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "(sup x y = bot) = (x = bot \<and> y = bot)"
approx_slp_outer_def: fixes p :: "nat"   and n :: "nat"   and slp :: "floatarith list"   and XS :: "(real \<times> real pdevs) list" shows "approx_slp_outer p n slp XS = (let d = degree_aforms XS; XSe = map (\<lambda>x. (x, 0)) XS in approx_slp p slp XSe \<bind> (\<lambda>rs. let rs' = take n rs; d' = max d (degree_aforms_err rs') in Some (aforms_err_to_aforms d' rs')))"
assms_1: fixes \<xi> :: "'a" shows "S \<xi> \<xi>"
Un_Int_eq_3: fixes S :: "'f set"   and T :: "'f set" shows "S \<inter> (S \<union> T) = S"
mapCollectE: fixes x :: "'e"   and f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and m :: "'f \<Rightarrow> 'g option" assumes "x \<in> mapCollect f m" obtains k :: "'f" and v :: "'g" where "m k = Some v" and "x = f k v"
X2': shows "flowsto (X2 - trap) T CX (\<Union> F)"
dichotomous_Lazard: fixes x :: "'b"   and n :: "nat"   and y :: "'b" assumes "to_fract x ^ n / to_fract y ^ (n - 1) \<in> range to_fract" shows "to_fract (dichotomous_Lazard x y n) = to_fract x ^ n / to_fract y ^ (n - 1)"
simps_2: fixes s :: "g_state" shows "g_out s Count = Some s"
add_divide_eq_if_simps_1: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and z :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "plus a (divide b z) = (if z = zero then a else divide (plus (times a z) b) z)"
mult_cancel_right1: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and c :: "'k"   and b :: "'k" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(c = times b c) = (c = zero \<or> b = one)"
sraux: fixes c :: "nat"   and n :: "nat"   and ts :: "'a \<Rightarrow> thread"   and ci :: "nat"   and ni :: "nat"   and tsi :: "'a \<Rightarrow> thread" assumes "sim_rel1 (c, n, ts) (ci, ni, tsi)" shows "ci = c mod N \<and> ni = n mod (k * N)"
comp_apply: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "(f \<circ> g) x = f (g x)"
irreducibleI': fixes a :: "'b" assumes "a \<noteq> (0::'b)"   and "\<not> a dvd (1::'b)"   and "\<And>b. b dvd a \<Longrightarrow> a dvd b \<or> b dvd (1::'b)" shows "irreducible a"
inject: fixes x :: "'d \<Rightarrow> 'e"   and ya :: "'d \<Rightarrow> 'e" shows "(StateT x = StateT ya) = (x = ya)"
l5: fixes x :: "int set"   and y :: "int set" assumes "x \<in> local.S2"   and "y \<in> local.S2"   and "x \<noteq> y" shows "x \<inter> y = {}"
simps_2: fixes l :: "brick"   and ls :: "brick list" shows "blockmat (l # ls) = brickmat l \<otimes> blockmat ls"
scale_left_distrib: fixes a :: "'d"   and b :: "'d"   and x :: "'d" shows "(a + b) * x = a * x + b * x"
eq_iff: fixes a :: "'t"   and b :: "'t" shows "(a = b) = (le_of_nat_term_order cmp_term a b \<and> le_of_nat_term_order cmp_term b a)"
config_n_bv: fixes s0 :: "'c list"   and qs :: "'c list" shows "map_pmf (fst \<circ> snd) (config'_rand BIT (fst BIT s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs) = bv (length s0)"
ltl_prop_implies_equiv: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<phi> \<sim>\<^sub>P \<psi> = (\<phi> \<longrightarrow>\<^sub>P \<psi> \<and> \<psi> \<longrightarrow>\<^sub>P \<phi>)"
pure_tree_simps_1: fixes x :: "'b" shows "root (pure x) = x"
infinite_ascending: fixes A :: "nat set" assumes "infinite A" obtains f :: "nat \<Rightarrow> nat" where "range f = A" and "\<forall>i. f i < f (Suc i)"
c_types_agree_not_bot_exists: fixes ts :: "checker_type" assumes "ts \<noteq> Bot" shows "\<exists>ts_c. c_types_agree ts ts_c"
n_prod_ideal: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set" assumes "\<forall>k\<le>n. ideal R (J k)" shows "ideal R (i\<Pi>\<^bsub>R,n\<^esub> J)"
hyps: shows "Q \<longmapsto>\<^sub>e \<alpha> \<prec> Q'"
typeII_edges_are_valid_paths: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "typeII_twoCube twoC"   and "(k, \<gamma>) \<in> boundary twoC" shows "valid_path \<gamma>"
ent_refl_true: fixes A :: "assn" shows "A \<Longrightarrow>\<^sub>A A * true"
ln_root: fixes n :: "nat"   and b :: "real" assumes "0 < n"   and "0 < b" shows "ln (root n b) = ln b / real n"
map_comp: fixes g1 :: "'d \<Rightarrow> 'a"   and g2 :: "'e \<Rightarrow> 'b"   and f1 :: "'f \<Rightarrow> 'd"   and f2 :: "'g \<Rightarrow> 'e"   and v :: "('f, 'g, 'c) gpv" shows "map_gpv g1 g2 (map_gpv f1 f2 v) = map_gpv (g1 \<circ> f1) (g2 \<circ> f2) v"
verts_del_arc: fixes G :: "('c, 'd) pre_digraph"   and a :: "'d" shows "verts (pre_digraph.del_arc G a) = verts G"
max_factor_degree: fixes ws :: "'b poly list"   and vs :: "'b poly list" assumes "mset ws \<subseteq># mset vs"   and "length ws + length ws \<le> length vs" shows "degree (prod_list ws) \<le> max_factor_degree (map degree vs)"
case_2: fixes f1 :: "'d"   and f2 :: "nat \<Rightarrow> 'd"   and x2 :: "nat" shows "(case Suc x2 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f2 x2"
assms: shows "f permutes (x \<rightleftharpoons>\<^sub>F y) ` S"
sum_unfold_sum_mset: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "sum f A = \<Sum>\<^sub># (image_mset f (mset_set A))"
prime_factorization_lcm_factorial: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)"   and "b \<noteq> (0::'a)" shows "prime_factorization (lcm_factorial a b) = prime_factorization a \<union># prime_factorization b"
simps: fixes F :: "(('b, 'a) Term.term \<times> ('c, 'a) Term.term) list" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F = \<Union> (fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r ` set F)"
sup_continuous_def: fixes F :: "'c \<Rightarrow> 'd" shows "sup_continuous F = (\<forall>M. incseq M \<longrightarrow> F (\<Squnion> range M) = (\<Squnion>i. F (M i)))"
implicit_span_with: shows "span.with (0::'b) (+) (*s) = local.span"
assms: shows "\<langle>x\<rangle> \<Turnstile>\<^sub>c index_geq_n i n"
generated_ringI_Basic: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> sets M" shows "A \<in> sets.generated_ring M"
prog_instrs_append: fixes xs :: "'b memref list"   and ys :: "'b memref list" shows "prog_instrs (xs @ ys) = prog_instrs xs @ prog_instrs ys"
conj_chaos_left: fixes conj :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and chaos :: "'b::refinement_lattice"   and a :: "'b::refinement_lattice" assumes "conjunction (conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (chaos::'b::refinement_lattice)" shows "(conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (chaos::'b::refinement_lattice) (a::'b::refinement_lattice) = a"
assms_5: fixes x :: "'b" assumes "x \<in> A" shows "f x \<le>\<^sub>o\<^sub>w g x"
distinct_36: fixes x81 :: "nat"   and x82 :: "msg"   and x3 :: "nat" shows "Crypt x81 x82 \<noteq> Nonce x3"
card_1_singletonE: fixes A :: "'d set" assumes "card A = 1" obtains x :: "'d" where "A = {x}"
odd_iff_mod_2_eq_one: fixes a :: "'a" shows "odd a = (a mod (2::'a) = (1::'a))"
star_comm: fixes P :: "bool"   and Q :: "bool"   and H :: "nat \<Rightarrow> nat option" shows "P ** Q = Q ** P"
sum_gp_strict: fixes x :: "'b"   and n :: "nat" shows "sum ((^) x) {..<n} = (if x = (1::'b) then of_nat n else ((1::'b) - x ^ n) / ((1::'b) - x))"
Suc_0_mod_eq: fixes n :: "nat" shows "Suc 0 mod n = of_bool (n \<noteq> Suc 0)"
sum_zero_power: fixes c :: "nat \<Rightarrow> 'a"   and A :: "nat set" shows "(\<Sum>i\<in>A. c i * (0::'a) ^ i) = (if finite A \<and> 0 \<in> A then c 0 else (0::'a))"
eq: shows "f \<equiv> C (set.fixp_fun (\<lambda>f. U (F (C f))))"
extremum_uniqueI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and a :: "'b" assumes "ordering_top less_eq less top"   and "less_eq top a" shows "a = top"
numgcd_pos: fixes t :: "Ferrack.num" shows "0 \<le> numgcd t"
push_in_let_conv_2: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'c" shows "Let x (RETURN \<circ> f) = RETURN (Let x f)"
a2: shows "Suc (fMax ((\<lambda>x. x - Suc 0) |`| (P |-| {|0|}))) \<noteq> fMax P"
simps_2: fixes uv :: "const" shows "Any \<squnion> uv = Any"
select_convs_1: fixes carrier :: "'a set"   and eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and more :: "'b" shows "(\<sqsubseteq>\<^bsub>\<lparr>carrier = carrier, eq = eq, le = le, \<dots> = more\<rparr>\<^esub>) = le"
enc_length: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and w' :: "'a list"   and I' :: "(nat + nat set) list" assumes "enc (w, I) = enc (w', I')" shows "length I = length I'"
continuous_on_iff: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on s f = (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
r''_2: shows "action_obs E r' \<approx> action_obs E' r'"