hcju/isabelleps huggingface dataset:
swap_swap_id: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "(x \<rightleftharpoons>\<^sub>F y) ((x \<rightleftharpoons>\<^sub>F y) z) = z"
mset_hp_empty: fixes hp :: "'b hp" shows "mset_hp hp \<noteq> {#}"
prod_lens_sublens_cong: fixes X\<^sub>1 :: "'d \<Longrightarrow> 'e"   and X\<^sub>2 :: "'f \<Longrightarrow> 'e"   and Y\<^sub>1 :: "'g \<Longrightarrow> 'h"   and Y\<^sub>2 :: "'i \<Longrightarrow> 'h" assumes "X\<^sub>1 \<subseteq>\<^sub>L X\<^sub>2"   and "Y\<^sub>1 \<subseteq>\<^sub>L Y\<^sub>2" shows "X\<^sub>1 \<times>\<^sub>L Y\<^sub>1 \<subseteq>\<^sub>L X\<^sub>2 \<times>\<^sub>L Y\<^sub>2"
left_idem: fixes a :: "'b word"   and b :: "'b word" shows "signed.max a (signed.max a b) = signed.max a b"
exhaust: fixes y :: "'b literal" obtains x1 :: "'b" where "y = Pos x1"   | x2 :: "'b" where "y = Neg x2"
r: shows "\<forall>n. cmod (s n) \<le> r"
circ_circ_sup: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "(mone \<oplus> x)\<^sup>\<odot> = x\<^sup>\<odot>\<^sup>\<odot>"
star_square: fixes x :: "'a" shows "(x \<cdot> x)\<^sup>\<star> \<le> x\<^sup>\<star>"
has_real_derivative_pos_inc_left: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and x :: "real"   and S :: "real set" assumes "(f has_real_derivative l) (at x within S)"   and "0 < l" shows "\<exists>d>0. \<forall>h>0. x - h \<in> S \<longrightarrow> h < d \<longrightarrow> f (x - h) < f x"
sel_1: fixes x21 :: "('a, 'b) rbt"   and x22 :: "color"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt" shows "lsub (Node x21 x22 x23 x24 x25) = x21"
bind_return_pmf: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'c pmf" shows "return_pmf x \<bind> f = f x"
bernoulli_conv_num_denom: fixes n :: "nat" shows "bernoulli n = real_of_int (bernoulli_num n) / real (bernoulli_denom n)"
a_def: shows "a \<equiv> \<lambda>u. v $ (THE i. u = unit_vec n i)"
simps: fixes x :: "nat"   and t :: "Formula.trm"   and y :: "event_data option list" shows "eval_assignment (x, t) y = y[x := Some (meval_trm t y)]"
wls_induct_fresh: fixes P :: "('index, 'bindex, 'varSort, 'var, 'opSym, 'sort) paramS"   and phi :: "'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> bool"   and phiAbs :: "'varSort \<times> 'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> bool"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "wlsPar P"   and "\<And>xs x. phi (asSort xs) (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>wlsInp delta inp; wlsBinp delta binp; liftAll2 phi (arOf delta) inp; liftAll2 (\<lambda>(us, s). phiAbs (us, s)) (barOf delta) binp\<rbrakk> \<Longrightarrow> phi (stOf delta) (Op delta inp binp)"   and "\<And>s xs x X. \<lbrakk>isInBar (xs, s); wls s X; x \<notin> varsOfS P xs; \<And>s' Y. Y \<in> termsOfS P s' \<Longrightarrow> fresh xs x Y; \<And>us s' A. A \<in> absOfS P (us, s') \<Longrightarrow> freshAbs xs x A; \<And>rho. rho \<in> envsOfS P \<Longrightarrow> freshEnv xs x rho; phi s X\<rbrakk> \<Longrightarrow> phiAbs (xs, s) (Abs xs x X)" shows "(wls s X \<longrightarrow> phi s X) \<and> (wlsAbs (xs, s') A \<longrightarrow> phiAbs (xs, s') A)"
freshFreeDerivative_1: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and y :: "name" assumes "P \<longmapsto> \<alpha> \<prec> P'"   and "y \<sharp> P" shows "y \<sharp> \<alpha>"
second_countable_topology_def: fixes "open" :: "'c set \<Rightarrow> bool" shows "class.second_countable_topology open \<equiv> class.topological_space open \<and> class.second_countable_topology_axioms open"
next_noteq: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<circle>(F \<noteq> G) = (\<circle>F \<noteq> \<circle>G)"
SUP_bot_conv_2: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and B :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(bot = Sup (B ` A)) = (\<forall>x\<in>A. B x = bot)"
untilI: fixes n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and t :: "nat \<Rightarrow> cnf"   and \<gamma>' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<exists>n''\<ge>n. \<gamma> t n'' \<and> (\<forall>n'\<ge>n. n' < n'' \<longrightarrow> \<gamma>' t n')" shows "(\<gamma>' \<UU>\<^sub>c \<gamma>) t n"
finite_UNIV_nat_def: shows "finite_UNIV = Phantom(nat) False"
prj0_in_hom: fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and b :: "'a" assumes "cospan f g"   and "a = f \<down>\<down> g"   and "b = local.dom g" shows "\<guillemotleft>\<p>\<^sub>0[f, g] : a \<rightarrow> b\<guillemotright>"
mrconcNone: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and x :: "'b"   and p :: "'a list"   and b :: "'a" assumes "applied_rule_rev C x p = \<bottom>" shows "applied_rule_rev C x (b # p) = applied_rule_rev C x [b]"
advantage_def: fixes \<A> :: "'a \<Rightarrow> nat spmf" shows "local.dis_log.advantage \<A> \<equiv> spmf (local.dis_log.dis_log \<A>) True"
comparable_cases: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "connex A less_eq"   and "x \<in> A"   and "y \<in> A" obtains "less_eq x y"   | "less_eq y x"
length_transpose: fixes xs :: "'g list list" shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
simps: fixes A :: "rat mat" shows "matrix_to_lpolies A = map vec_to_lpoly (rows A)"
success_shiftr_f: fixes h :: "heap"   and i :: "nat ref"   and a :: "'b array" assumes "Ref.get h i < Array.length h a" shows "success (shiftr_f a i) h"
map_poly_map_poly: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and p :: "'c poly" assumes "f (0::'b) = (0::'a)" shows "map_poly f (map_poly g p) = map_poly (f \<circ> g) p"
deriv_rec_correct: fixes f :: "floatarith"   and xs :: "real list"   and t :: "real"   and n :: "nat"   and S :: "real set" assumes "isDERIV 0 f (xs[0 := t])"   and "0 < length xs" shows "((\<lambda>x. interpret_floatarith (deriv_rec f n) (xs[0 := x])) has_real_derivative interpret_floatarith (deriv_rec f (Suc n)) (xs[0 := t])) (at t within S)"
sel_1: fixes x1 :: "int"   and x2 :: "int" shows "ReZ (Gauss_Int x1 x2) = x1"
simps_2: fixes U :: "complex mat"   and \<rho> :: "complex mat" shows "denote (Utrans U) \<rho> = U * \<rho> * adjoint U"
iso_is_arr: fixes f :: "'a \<times> 'a" assumes "VV.iso f" shows "VV.arr f"
finite_subgroup_card_neq_0: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.group G"   and "subgroup H G"   and "finite H" shows "card H \<noteq> 0"
subseq_Cons2_neq: fixes x :: "'d"   and xs :: "'d list"   and y :: "'d"   and ys :: "'d list" assumes "subseq (x # xs) (y # ys)"   and "x \<noteq> y" shows "subseq (x # xs) ys"
nextElem_notin: fixes a :: "'a"   and as :: "'a list"   and c :: "'a" assumes "a \<notin> set as" shows "nextElem as c a = c"
until_def: fixes p :: "'a"   and q :: "'a" shows "p until q = (SUP n. Inf ((!!) p ` {i. i < n}) \<sqinter> q !! n)"
fv_snd_rcv_strand: fixes S :: "('d, 'c) strand_step list" shows "fv\<^sub>s\<^sub>t S = \<Union> (set (map fv\<^sub>s\<^sub>n\<^sub>d S)) \<union> \<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v S)) \<union> \<Union> (set (map (fv\<^sub>e\<^sub>q assign) S)) \<union> \<Union> (set (map (fv\<^sub>e\<^sub>q check) S)) \<union> \<Union> (set (map fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q S))"
h_add: shows "\<forall>p q. h (p + q) = h p + h q"
2: shows "ran f \<subseteq> {0..2 * card (nodes A)}"
map_disc_iff: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d llist" shows "(lmap f a = LNil) = (a = LNil)"
inv_DirProd: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and g :: "'c"   and h :: "'e" assumes "Group.group G"   and "Group.group H"   and "g \<in> carrier G"   and "h \<in> carrier H" shows "inv\<^bsub>G \<times>\<times> H\<^esub> (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"
l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf DocumentClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs"
coeff_0_reflect_poly_0_iff: fixes p :: "'b poly" shows "(coeff (reflect_poly p) 0 = (0::'b)) = (p = 0)"
cont2cont_fun: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and y :: "'c" assumes "cont f" shows "cont (\<lambda>x. f x y)"
image_paired_Times: fixes f :: "'i \<Rightarrow> 'g"   and g :: "'j \<Rightarrow> 'h"   and A :: "'i set"   and B :: "'j set" shows "(\<lambda>(x, y). (f x, g y)) ` (A \<times> B) = f ` A \<times> g ` B"
seqI': fixes f :: "'c"   and a :: "'c"   and b :: "'c"   and g :: "'c"   and c :: "'c" assumes "J'.in_hom f a b"   and "J'.in_hom g b c" shows "J'.seq g f"
mat_of_rows_carrier_2: fixes n :: "nat"   and vs :: "'a vec list" shows "dim_row (mat_of_rows n vs) = length vs"
inext_nth_imirror_iprev_nth_conv: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "I \<noteq> {}" shows "imirror I \<rightarrow> n = mirror_elem (I \<leftarrow> n) I"
assms_4: shows "\<forall>(x, K)\<in>p. \<exists>c d. c < d \<and> K = {c..d}"
transC_mayDiverge: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "('a, 'c) com"   and s :: "'b"   and c' :: "('a, 'c) com"   and s' :: "'b" assumes "PL.transC tval aval (c, s) (c', s')"   and "PL.mayDiverge tval aval c' s'" shows "PL.mayDiverge tval aval c s"
closure_idem_weak: fixes A :: "'a set" shows "closure_of (closure_of A) {.=} closure_of A"
neg_strictly_subsumes_variants: fixes E :: "'a literal multiset"   and D :: "'a literal multiset"   and D' :: "'a literal multiset" assumes "\<not> strictly_subsumes E D"   and "variants D D'" shows "\<not> strictly_subsumes E D'"
CONS_surj: fixes \<AA> :: "interp" assumes "0 < Length \<AA>" shows "\<exists>x \<BB>. \<AA> = CONS x \<BB> \<and> #\<^sub>V \<BB> = #\<^sub>V \<AA> \<and> size_atom x = #\<^sub>V \<AA>"
bounded_linear_scaleR_left: fixes x :: "'a" shows "bounded_linear (\<lambda>r. r *\<^sub>R x)"
scaleRat_left_distrib: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleRat :: "rat \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "rat"   and b :: "rat"   and x :: "'b" assumes "class.rational_vector plus zero minus uminus scaleRat" shows "scaleRat (a + b) x = plus (scaleRat a x) (scaleRat b x)"
iso_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "local.iso \<a>[a, b, c]"
unitary_def: fixes U :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "unitary U = (U* o\<^sub>C\<^sub>L U = id_cblinfun \<and> U o\<^sub>C\<^sub>L U* = id_cblinfun)"
rep_eq: fixes x :: "Float.float" shows "is_float_pos x = (0 < real_of_float x)"
ideD_1: fixes a :: "nat" assumes "J.ide a" shows "J.arr a"
i_th_strict2: fixes s :: "'a Stream" shows "s !! \<bottom> = \<bottom>"
P3: shows "Bet A Y Z \<and> Cong Y Z Y P"
reachablep_reachable_eq: fixes w :: "'b stream"   and A :: "('b, 'a) nba"   and V :: "(nat \<times> 'a) set"   and p :: "nat \<times> 'a" shows "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 = (\<lambda>x. x \<in> greachable A w V p)"
ce_intersect: fixes A :: "nat set"   and B :: "nat set" assumes "A \<in> ce_sets"   and "B \<in> ce_sets" shows "A \<inter> B \<in> ce_sets"
insert_mapping_def: shows "insert_mapping \<equiv> \<lambda>uu. case uu of (k, v) \<Rightarrow> \<lambda>m. case Mapping.lookup m k of None \<Rightarrow> Mapping.update k {v} m | Some V \<Rightarrow> Mapping.update k (insert v V) m"
funion_fempty: fixes A :: "'b fset"   and B :: "'b fset" shows "(A |\<union>| B = {||}) = (A = {||} \<and> B = {||})"
prems_3: shows "all_unshared (x # sb) \<inter> dom \<S>' - dom \<S> = {}"
ineq: shows "\<forall>i\<in>net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2). \<sigma>' i = \<sigma> i"
delta_min: fixes qd :: "(QDelta \<times> QDelta) list" assumes "\<forall>qd1 qd2. (qd1, qd2) \<in> set qd \<longrightarrow> qd1 \<le> qd2" shows "\<forall>\<epsilon>. 0 < \<epsilon> \<and> \<epsilon> \<le> \<delta>_min qd \<longrightarrow> (\<forall>qd1 qd2. (qd1, qd2) \<in> set qd \<longrightarrow> val qd1 \<epsilon> \<le> val qd2 \<epsilon>)"
drop_all_conc: fixes a :: "'a list"   and b :: "'a list" shows "drop (length a) (a @ b) = b"
q: shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. q (fst y) (snd y))"
scale_zero_right: fixes a :: "'b" shows "a * (0::'b) = (0::'b)"
AAcc_\<tau>execI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and i :: "'addr expr1" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (e\<lfloor>i\<rceil>) h s s'"
assms_2: shows "\<I> \<Turnstile>\<^sub>c \<langle>S,\<theta>\<rangle>"
HEndPhase2_Inv2c: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2c s"   and "HEndPhase2 s s' p"   and "Inv2a s" shows "Inv2c s'"
assms_2: fixes r :: "nat" assumes "CHOnextConfig A r (rho r) (HOs r) (coords (Suc r)) (rho (Suc r))" shows "P r"
ntrancl_on_Suc: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and F :: "'c set"   and n :: "nat"   and c :: "'c" assumes "(a, b) \<in> r"   and "ntrancl_onp F r n b c"   and "a \<in> F" shows "ntrancl_onp F r (Suc n) a c"
riD: shows "P \<turnstile> \<langle>RI (D,throw a) ; Cs' \<leftarrow> e\<^sub>0,(h', l', sh'(C \<mapsto> (sfs, Error)))\<rangle> \<Rightarrow> \<langle>Val v',s\<^sub>1\<rangle>"
amult_pos: fixes b :: "int"   and x :: "ant" assumes "0 < b"   and "0 \<le> x" shows "x \<le> b *\<^sub>a x"
pell_mul_trivial_left: fixes z :: "'b \<times> 'b" shows "pell_mul (1::'b, 0::'b) z = z"
strict_mono_cancel_eq: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "OrdinalInduct.strict_mono f" shows "(f x = f y) = (x = y)"
pi: shows "pi \<in> set (factors_of_int_poly (poly_add p (poly_uminus p)))"
decseq_def: fixes X :: "nat \<Rightarrow> 'c" shows "decseq X = (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m)"
has_bochner_integral_sum: fixes I :: "'c set"   and M :: "'d measure"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and x :: "'c \<Rightarrow> 'e" assumes "\<And>i. i \<in> I \<Longrightarrow> has_bochner_integral M (f i) (x i)" shows "has_bochner_integral M (\<lambda>x. \<Sum>i\<in>I. f i x) (sum x I)"
ls1: shows "ls = take i ls @ ls ! i # drop (Suc i) ls"
sorted_Cons_le: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c" assumes "ASSUMPTION (sorted (x # xs))"   and "y \<le> x" shows "sorted (y # xs)"
principal_eq_bot_iff: fixes X :: "'c set" shows "(principal X = bot) = (X = {})"
foldrS_Stream: fixes s :: "'b"   and f :: "'d \<rightarrow> 'c \<rightarrow> 'c"   and z :: "'c"   and h :: "'b \<rightarrow> ('d, 'b) Step" assumes "s \<noteq> \<bottom>" shows "foldrS\<cdot>f\<cdot>z\<cdot>(Stream\<cdot>h\<cdot>s) = (case h\<cdot>s of Done \<Rightarrow> z | Skip\<cdot>s' \<Rightarrow> foldrS\<cdot>f\<cdot>z\<cdot>(Stream\<cdot>h\<cdot>s') | Yield\<cdot>x\<cdot>s' \<Rightarrow> f\<cdot>x\<cdot>(foldrS\<cdot>f\<cdot>z\<cdot>(Stream\<cdot>h\<cdot>s')))"
conservative_power: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "conservative M T" shows "conservative M (T ^^ n)"
P3: shows "Bet B C C0 \<and> Cong C C0 E F"
rel_reflp: fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "reflp R1"   and "reflp R2" shows "reflp (rel_prod R1 R2)"
Node: shows "h = \<langle>hs1, x, hs\<rangle>"
step_atomicity: shows "\<forall>s a. current (kstep s a) = current s"
assms_2: shows "k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1"
real_cond_exp_cong: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "AEeq M f g"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "AEeq M (real_cond_exp M F f) (real_cond_exp M F g)"
distinct_7: fixes x2 :: "'d"   and x3 :: "'e" shows "Abrupt x2 \<noteq> Fault x3"
3: fixes x :: "real" assumes "- (pi / 2) < x"   and "x * 2 < pi" shows "isCont (\<lambda>x. inverse (1 + x\<^sup>2)) (tan x)"
correctCompositionDiffLevelsA5: shows "correctCompositionDiffLevels sA5"
assms_1: shows "expectation_gpv2 \<equiv> expectation_gpv 1 \<I>'"
deleteMin_xlate: fixes q :: "('c, 'd) BsSkewBinomialTree list" assumes "q \<noteq> []" shows "map bsmapt (deleteMin q) = SkewBinomialHeapStruc.deleteMin (map bsmapt q)"
pfun_comp_assoc: fixes f :: "('c, 'b) pfun"   and g :: "('d, 'c) pfun"   and h :: "('a, 'd) pfun" shows "f \<circ>\<^sub>p (g \<circ>\<^sub>p h) = f \<circ>\<^sub>p g \<circ>\<^sub>p h"
OclNot_def: fixes X :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "not X \<equiv> \<lambda>\<tau>. case X \<tau> of None \<Rightarrow> None | \<lfloor>None\<rfloor> \<Rightarrow> \<lfloor>None\<rfloor> | \<lfloor>\<lfloor>x\<rfloor>\<rfloor> \<Rightarrow> \<lfloor>\<lfloor>\<not> x\<rfloor>\<rfloor>"
prems_1: shows "subpath g v1 (e' # es) v2"
SUP_empty: fixes f :: "'e \<Rightarrow> 'c \<times> 'd" shows "Sup_prod (f ` {}) = bot_prod"
union_disjoint: fixes A :: "'k set"   and B :: "'k set"   and g :: "'k \<Rightarrow> 'l" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "sum g (A \<union> B) = sum g A + sum g B"
P3_b: fixes x :: "'a" shows "(x r\<rightarrow> (1::'a)) l\<rightarrow> (1::'a) = (1::'a)"
trans_IfFalse: fixes tst :: "'test"   and s :: "'state"   and c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "\<not> tval tst s" shows "(if tst then c1 else c2, s) \<rightarrow>c (c2, s)"
m_ivl_widen: fixes y :: "ivl"   and x :: "ivl" assumes "\<not> y \<le> x" shows "m_ivl (x \<nabla> y) < m_ivl x"
infs_snth: fixes P :: "'b \<Rightarrow> bool"   and w :: "'b stream" shows "infs P w = (\<forall>n. \<exists>k\<ge>n. P (w !! k))"
sorted_list_of_set_empty: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_set less_eq {} = []"
gt_a0_ge_aN: fixes x :: "ant"   and N :: "nat" assumes "0 < x"   and "N \<noteq> 0" shows "ant (int N) \<le> int N *\<^sub>a x"
mult_adjugate_det: fixes A :: "('a, 'b) sq_matrix" shows "A * Square_Matrix.adjugate A = diag (Square_Matrix.det A)"
echelon_form_of_invertible: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A :: "(('d, 'f) vec, 'e) vec" assumes "is_bezout_ext bezout" shows "\<exists>P. invertible P \<and> P ** A = echelon_form_of A bezout \<and> echelon_form (echelon_form_of A bezout)"
combine_options_def: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e option"   and y :: "'e option" shows "combine_options f x y = (case x of None \<Rightarrow> y | Some x \<Rightarrow> case y of None \<Rightarrow> Some x | Some y \<Rightarrow> Some (f x y))"
list_match_lel_lel: fixes c1 :: "'a list"   and qs :: "'a"   and c2 :: "'a list"   and c1' :: "'a list"   and qs' :: "'a"   and c2' :: "'a list" assumes "c1 @ qs # c2 = c1' @ qs' # c2'" obtains c21' :: "'a list" where "c1 = c1' @ qs' # c21'" and "c2' = c21' @ qs # c2"   | "c1' = c1" and "qs' = qs" and "c2' = c2"   | c21 :: "'a list" where "c1' = c1 @ qs # c21" and "c2 = c21 @ qs' # c2'"
simps: fixes a1 :: "nat"   and a2 :: "('c, 'd) bintree option list" shows "is_binqueue a1 a2 = ((\<exists>l. a1 = l \<and> a2 = []) \<or> (\<exists>l xs. a1 = l \<and> a2 = None # xs \<and> is_binqueue (Suc l) xs) \<or> (\<exists>l xs t. a1 = l \<and> a2 = Some t # xs \<and> is_binqueue (Suc l) xs \<and> is_bintree_list l (children t) \<and> is_heap_list (priority t) (children t)))"
distinct_29: fixes x51 :: "'b rexp"   and x52 :: "'b rexp"   and x6 :: "'b rexp" shows "Times x51 x52 \<noteq> Star x6"
pr_in_hom_2: fixes a0 :: "'a"   and a1 :: "'a" assumes "ide a0"   and "ide a1" shows "\<guillemotleft>\<p>\<^sub>1[a0, a1] : a0 \<otimes> a1 \<rightarrow> a1\<guillemotright>"
Nil_eq_concat_conv: fixes xss :: "'b list list" shows "(\<epsilon> = concat xss) = (\<forall>xs\<in>set xss. xs = \<epsilon>)"
g_bigtheta: shows "g \<in> \<Theta>(\<lambda>x. real x powr p * ln (real x) powr - 1)"
intro: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "inj_on \<alpha> S" shows "injective_map_axioms \<alpha> S"
max2_iff: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "wo_rel r"   and "a \<in> Field r"   and "b \<in> Field r" shows "((wo_rel.max2 r a b, c) \<in> r) = ((a, c) \<in> r \<and> (b, c) \<in> r)"
cogen: fixes c :: "'c"   and cs :: "'c list"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and w :: "'d" assumes "c \<in> set cs"   and "P c w" shows "cogen P cs w"
get_disconnected_nodes_new_shadow_root: fixes h :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr' :: "(_) document_ptr" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "r \<in> get_disconnected_nodes_locs ptr'" shows "r h h'"
UN_extend_simps_5: fixes A :: "'b set"   and B :: "'c \<Rightarrow> 'b set"   and C :: "'c set" shows "A \<inter> \<Union> (B ` C) = (\<Union>x\<in>C. A \<inter> B x)"
uw\<theta>\<omega>\<nu>_axioms_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c"   and \<omega> :: "'c"   and \<nu> :: "'c" shows "uw\<theta>\<omega>\<nu>_axioms V H \<a> r \<rho> f g u w \<theta> \<omega> \<nu> \<equiv> partial_magma.ide V w \<and> partial_magma.in_hom V \<theta> (H f w) u \<and> partial_magma.in_hom V \<nu> (partial_magma.dom V \<omega>) (H g w) \<and> category.iso V \<nu> \<and> V (H r \<theta>) (V (\<a> r f w) (V (H \<rho> w) \<nu>)) = \<omega>"
UPairSingleton: fixes u :: "ZF"   and v :: "ZF" shows "Upair u v = HOLZF.union (Singleton u) (Singleton v)"
rename_stable: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and A :: "'c set" assumes "bij h" shows "(rename h F \<in> stable (h ` A)) = (F \<in> stable A)"
vwalk_to_vpath_vpath: fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph" assumes "vwalk p G" shows "vpath (vwalk_to_vpath p) G"
S7: shows "x + y \<le> c_sum (c_pair x y)"
fls_zero_nth: fixes n :: "int" shows "fls_nth 0 n = (0::'a)"
Gets_imp_knows_agents: fixes A :: "agent"   and X :: "msg"   and evs :: "event list" assumes "A \<noteq> Spy"   and "Gets A X \<in> set evs" shows "X \<in> knows A evs"