hcju/isabelleps huggingface dataset:
distinct_3: fixes x1 :: "nat"   and x3 :: "val option" shows "VNat x1 \<noteq> Abs x3"
rel_set_OO: fixes R :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_set R OO rel_set S = rel_set (R OO S)"
conc_def: fixes w :: "'b list"   and x :: "nat \<Rightarrow> 'b" shows "w \<frown> x \<equiv> \<lambda>n. if n < length w then w ! n else x (n - length w)"
Lp_le_1_1: fixes p :: "real"   and M :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "0 < p"   and "p \<le> 1" shows "eNorm (\<LL> (ennreal p) M) f = (if f \<in> borel_measurable M \<and> integrable M (\<lambda>x. \<bar>f x\<bar> powr p) then ennreal ((LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p)) else \<infinity>)"
factorize_rat_poly_basic_def: fixes p :: "rat poly" shows "factorize_rat_poly_basic p = factorize_rat_poly_main 1 [] [p]"
dom_one: shows "am_d (1::'a) = (1::'a)"
l3_inv3_init: shows "init l3 \<subseteq> l3_inv3"
DERIV_neg_imp_decreasing: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "a < b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. (f has_real_derivative y) (at x) \<and> y < 0" shows "f b < f a"
distinct_disc_5: fixes event :: "('a, 'b, 'c) event" assumes "isTrans event" shows "\<not> isSnapshot event"
axiom5_F: fixes f1 :: "'h \<Rightarrow> 'l"   and f2 :: "'i \<Rightarrow> 'k"   and f3 :: "'j \<Rightarrow> 'm" shows "F1set1 \<circ> Fmap f1 f2 f3 = (`) f2 \<circ> F1set1"
eval_poly_with_sum: fixes v :: "linear_poly"   and X :: "nat \<Rightarrow> rat" shows "v \<lbrace> X \<rbrace> = (\<Sum>x\<in>vars v. Abstract_Linear_Poly.coeff v x * X x)"
rtrancl_0: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and x :: "'t \<Rightarrow>\<^sub>0 'c" assumes "(red F)\<^sup>*\<^sup>* 0 x" shows "x = 0"
g: fixes i :: "nat" shows "flow \<Delta> (f_i i)"
cong_def: fixes b :: "'b"   and c :: "'b"   and a :: "'b" shows "[b = c] (mod a) = (b mod a = c mod a)"
exhaust: fixes y :: "trilean" obtains "y = true"   | "y = false"   | "y = invalid"
elims: fixes x :: "'a op\<^sub>q"   and xa :: "'a list \<times> 'a list"   and y :: "'a list \<times> 'a list" assumes "nxt_q2 x xa = y" obtains a :: "'a" and xs :: "'a list" and ys :: "'a list" where "x = Enq a" and "xa = (xs__, ys__)" and "y = balance (a # xs__, ys__)"   | xs :: "'a list" and ys :: "'a list" where "x = Deq" and "xa = (xs, ys)" and "y = balance (xs, tl ys)"
poly_nate_def: fixes bs :: "'b list"   and p :: "'b poly" shows "poly_nate bs p = map (Ipoly bs \<circ> decrpoly) (coefficients (polynate p))"
elims: fixes x :: "tv"   and y :: "char list" assumes "string_tv x = y" obtains "x = Det True" and "y = ''*''"   | "x = Det False" and "y = ''o''"   | n :: "nat" where "x = Indet n" and "y = string_of_nat n"
ev_imp_shift: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream" assumes "ev \<phi> xs" shows "\<exists>xl xs2. xs = xl @- xs2 \<and> \<phi> xs2"
comp_null_1: fixes f :: "'a \<times> 'a" shows "VV.comp VV.null f = VV.null"
list_emb_length: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "list_emb P xs ys" shows "length xs \<le> length ys"
takeWhile_append2: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and ys :: "'a list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x" shows "takeWhile P (xs @ ys) = xs @ takeWhile P ys"
h7: assumes "Av < 0"   and "0 < Bv" shows "(Av + Bv * sqrt Cv \<le> 0) = (Av \<le> 0 \<and> Bv\<^sup>2 * Cv \<le> Av\<^sup>2 \<or> Bv \<le> 0 \<and> Av\<^sup>2 \<le> Bv\<^sup>2 * Cv)"
cross7_simps_1: fixes a :: "(real, 7) vec"   and b :: "(real, 7) vec" shows "a \<times>\<^sub>7 b \<equiv> vector [a $ 2 * b $ 4 - a $ 4 * b $ 2 + a $ 3 * b $ 7 - a $ 7 * b $ 3 + a $ 5 * b $ 6 - a $ 6 * b $ 5, a $ 3 * b $ 5 - a $ 5 * b $ 3 + a $ 4 * b $ 1 - a $ 1 * b $ 4 + a $ 6 * b $ 7 - a $ 7 * b $ 6, a $ 4 * b $ 6 - a $ 6 * b $ 4 + a $ 5 * b $ 2 - a $ 2 * b $ 5 + a $ 7 * b $ 1 - a $ 1 * b $ 7, a $ 5 * b $ 7 - a $ 7 * b $ 5 + a $ 6 * b $ 3 - a $ 3 * b $ 6 + a $ 1 * b $ 2 - a $ 2 * b $ 1, a $ 6 * b $ 1 - a $ 1 * b $ 6 + a $ 7 * b $ 4 - a $ 4 * b $ 7 + a $ 2 * b $ 3 - a $ 3 * b $ 2, a $ 7 * b $ 2 - a $ 2 * b $ 7 + a $ 1 * b $ 5 - a $ 5 * b $ 1 + a $ 3 * b $ 4 - a $ 4 * b $ 3, a $ 1 * b $ 3 - a $ 3 * b $ 1 + a $ 2 * b $ 6 - a $ 6 * b $ 2 + a $ 4 * b $ 5 - a $ 5 * b $ 4]"
M_def: shows "M \<equiv> {y. \<exists>x\<in>set xs. P x y}"
T_simp: fixes f :: "'a"   and g :: "'a" assumes "arr f"   and "arr g" shows "MC.tensor f g = f \<otimes> g"
clear_locals_def: fixes \<xi> :: "state" shows "clear_locals \<xi> = \<xi>\<lparr>msg := SOME x. True, data := SOME x. True, dests := SOME x. True, rreqid := SOME x. True, dip := SOME x. True, oip := SOME x. True, hops := SOME x. True, dsn := SOME x. True, dsk := SOME x. True, osn := SOME x. True, sip := SOME x. x \<noteq> ip \<xi>\<rparr>"
y_def: shows "y = Octo 1 3 0 0 0 1 0 0"
bipartite_E: fixes x :: "'v"   and y :: "'v" assumes "edge \<Gamma>' x y" shows "x \<in> A \<Gamma>' \<and> y \<in> B \<Gamma>'"
assms_3: shows "\<gamma> (lnth (\<pi>\<^bsub>cid\<^esub>inf_llist t @\<^sub>l inf_llist t')) (\<^bsub>cid\<^esub>\<down>\<^bsub>t\<^esub>n)"
new_document_new_ptr: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_document_ptr :: "(_) document_ptr" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr" shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr|}"
cases: fixes K :: "'a \<Rightarrow> 'a pmf"   and a1 :: "'a"   and a2 :: "'a stream" assumes "MC_syntax.enabled K a1 a2" obtains \<omega> :: "'a stream" and s :: "'a" where "a1 = s" and "a2 = \<omega>" and "MC_syntax.enabled K (shd \<omega>) (stl \<omega>)" and "shd \<omega> \<in> set_pmf (K s)"
wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t_split_2: fixes S :: "('fun, 'var) extstrand_step list"   and S' :: "('fun, 'var) extstrand_step list" shows "wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t (S @ S') = wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t S \<union> wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t S'"
has_subprob_density_imp_has_density: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_subprob_density M N f" shows "has_density M N f"
exhaust: fixes y :: "agent" obtains "y = Server"   | x2 :: "nat" where "y = Friend x2"   | "y = Spy"
col_cop_orth__orth_at: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and U :: "'p"   and V :: "'p"   and X :: "'p" assumes "A B C Orth U V"   and "Coplanar A B C X"   and "Col U V X" shows "X OrthAt A B C U V"
equivclp_least: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "r \<le> s"   and "equivp s" shows "equivclp r \<le> s"
set_less_aux_Compl_fusion_def: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and proper_interval :: "'c option \<Rightarrow> 'c option \<Rightarrow> bool"   and g1 :: "('c, 'd) generator"   and g2 :: "('c, 'e) generator"   and ao :: "'c option"   and s1 :: "'d"   and s2 :: "'e" shows "proper_intrvl.set_less_aux_Compl_fusion less proper_interval g1 g2 ao s1 s2 = proper_intrvl.set_less_aux_Compl less proper_interval ao (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
base_in_hom_1: shows "\<guillemotleft>r : src r \<rightarrow> trg r\<guillemotright>"
induct: fixes x :: "nat \<times> nat list \<times> 'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and t :: "nat \<times> nat list \<times> 'c list"   and P :: "nat \<times> nat list \<times> 'c list \<Rightarrow> bool" assumes "x \<in> gcsort_set index key p t"   and "P t"   and "\<And>u ns xs. \<lbrakk>(u, ns, xs) \<in> gcsort_set index key p t; P (u, ns, xs)\<rbrakk> \<Longrightarrow> P (gcsort_round index key p ns xs)" shows "P x"
semiring_norm_71: fixes m :: "num"   and n :: "num" shows "(num.Bit0 m \<le> num.Bit0 n) = (m \<le> n)"
rel_gpv''_mono': fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C' :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and gpv :: "('a, 'c, 'e) gpv"   and gpv' :: "('b, 'd, 'f) gpv" assumes "\<And>x y. A x y \<longrightarrow> A' x y"   and "\<And>x y. C x y \<longrightarrow> C' x y"   and "\<And>x y. R' x y \<longrightarrow> R x y" shows "rel_gpv'' A C R gpv gpv' \<longrightarrow> rel_gpv'' A' C' R' gpv gpv'"
sum_list_insort: fixes d :: "'b"   and xs :: "'b list" shows "sum_list (insort d xs) = d + sum_list xs"
prems_3: shows "P \<turnstile> (S # Ss) [\<le>] Ts"
flush_all_until_volatile_wirte_mem_independent: fixes a :: "nat"   and ls :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and m' :: "nat \<Rightarrow> nat"   and m :: "nat \<Rightarrow> nat" assumes "a \<in> (\<Union>(uu_, uu_, uu_, sb, uu_, uu_, uu_)\<in>set ls. outstanding_refs is_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb))" shows "flush_all_until_volatile_write ls m' a = flush_all_until_volatile_write ls m a"
lens_override_id: fixes S\<^sub>1 :: "'a"   and S\<^sub>2 :: "'a" shows "S\<^sub>1 \<triangleleft>\<^bsub>1\<^sub>L\<^esub> S\<^sub>2 = S\<^sub>2"
in_closure_iff_infdist_zero: fixes A :: "'c set"   and x :: "'c" assumes "A \<noteq> {}" shows "(x \<in> closure A) = (infdist x A = 0)"
Inf_inner_def: fixes X :: "'c set"   and y :: "'c" shows "Inf_inner X y = SPEC (\<lambda>r. \<forall>x\<in>X. r \<le> x \<bullet> y)"
sdrop_stl: fixes n :: "nat"   and s :: "'c stream" shows "sdrop n (stl s) = stl (sdrop n s)"
image_eqI: fixes b :: "'e"   and f :: "'f \<Rightarrow> 'e"   and x :: "'f"   and A :: "'f set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
max_absorb1: fixes y :: "'a"   and x :: "'a" assumes "y \<in> U"   and "x \<in> U"   and "y \<le>\<^sub>o\<^sub>w x" shows "local.max x y = x"
lad_on_substitutes_on_irc_on: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" assumes "f_range_on A f"   and "substitutes_on A f"   and "lad_on A f" shows "irc_on A f"
memoryless_on_cfg_onI: fixes f :: "'s \<Rightarrow> 's pmf"   and s :: "'s" assumes "\<And>s. f s \<in> K s" shows "memoryless_on f s \<in> cfg_on s"
fmd_nullI: fixes P :: "cld list"   and ty :: "ty"   and meth :: "char list" assumes "find_path_ty P ty None" shows "find_meth_def P ty meth None"
ide_char': fixes a :: "'s" shows "ide a = (arr a \<and> (local.dom a = a \<or> cod a = a))"
simps_4: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list" shows "encode (Cn n f gs) = quad_encode 3 n (encode f) (Partial_Recursive.list_encode (map encode gs))"
Sup_closed_coclop: fixes X :: "'b set" assumes "Sup_closed_set X" shows "coclop (\<lambda>y. \<Squnion> {x \<in> X. x \<le> y})"
subgroup_generated_restrict: fixes G :: "('e, 'f) monoid_scheme"   and S :: "'e set" shows "subgroup_generated G (carrier G \<inter> S) = subgroup_generated G S"
norm_rep_ARR: fixes f :: "'c local.term set" assumes "ARR f" shows "\<^bold>\<parallel>rep f\<^bold>\<parallel> = rep f"
wf_graph_union_edges: fixes V :: "'a set"   and E :: "('a \<times> 'a) set"   and E' :: "('a \<times> 'a) set" assumes "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "wf_graph \<lparr>nodes = V, edges = E'\<rparr>" shows "wf_graph \<lparr>nodes = V, edges = E \<union> E'\<rparr>"
nn_integral_has_integral_lebesgue: fixes f :: "'a \<Rightarrow> real"   and I :: "real"   and \<Omega> :: "'a set" assumes "\<And>x. 0 \<le> f x"   and "(f has_integral I) \<Omega>" shows "\<integral>\<^sup>+ x. ennreal (indicat_real \<Omega> x * f x) \<partial>lborel = ennreal I"
map_def: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b Predicate.pred" shows "Predicate.map f P = Predicate.bind P (Predicate.single \<circ> f)"
rtrancl_idemp: fixes r :: "('a \<times> 'a) set" shows "(r\<^sup>*)\<^sup>* = r\<^sup>*"
Pair_eqD1: fixes x :: "'c"   and y :: "'d"   and x' :: "'c"   and y' :: "'d" assumes "(x, y) = (x', y')" shows "x = x'"
U_1: shows "Z = \<Union> {Z \<inter> R |R. R \<in> U}"
sor: shows "(\<sigma>, s) \<in> oreachable (opnet onp p\<^sub>1) (otherwith S (net_tree_ips p\<^sub>1) (oarrivemsg I)) (other U (net_tree_ips p\<^sub>1))"
sp_append: fixes g :: "('c, 'd) rgraph_scheme"   and v1 :: "'c"   and es1 :: "'c edge list"   and es2 :: "'c edge list"   and v2 :: "'c" shows "Graph.subpath g v1 (es1 @ es2) v2 = (\<exists>v. Graph.subpath g v1 es1 v \<and> Graph.subpath g v es2 v2)"
top_left_mult_increasing: fixes x :: "'a" shows "x \<le> top \<sqinter> x"
linear_add: fixes f :: "('d, 'e) vec \<Rightarrow> ('d, 'f) vec"   and b1 :: "('d, 'e) vec"   and b2 :: "('d, 'e) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f (b1 + b2) = f b1 + f b2"
parallel_append: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list"   and d :: "'b list" assumes "a \<parallel> b" shows "a @ c \<parallel> b @ d"
R: fixes c :: "'b set" assumes "c \<subseteq> carrier\<^bsub>T\<^esub>"   and "c closed\<^bsub>T\<^esub>" shows "f -` c closed"
independent_imp_finite: fixes B :: "'a set" assumes "independent B" shows "finite B"
bind_mzero: fixes k :: "'c \<rightarrow> 'a\<cdot>'b" shows "mzero \<bind> k = mzero"
path_pd_pd0: fixes \<pi> :: "nat \<Rightarrow> 'n"   and l :: "nat"   and n :: "'n" assumes "is_path \<pi>"   and "\<pi> l pd\<rightarrow> n"   and "n pd\<rightarrow> \<pi> 0" obtains k :: "nat" where "k \<le> l" and "\<pi> k = n"
trmc_conj_overlap: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A" shows "(K \<div>\<^bsub>\<gamma>\<^sub>T\<^sub>R\<^esub> \<phi>) \<inter> (K \<div>\<^bsub>\<gamma>\<^sub>T\<^sub>R\<^esub> \<psi>) \<subseteq> K \<div>\<^bsub>\<gamma>\<^sub>T\<^sub>R\<^esub> (\<phi> .\<and>. \<psi>)"
Abs_poly_inverse: fixes y :: "nat \<Rightarrow> 'a" assumes "y \<in> {f. \<forall>\<^sub>\<infinity>n. f n = (0::'a)}" shows "poly.coeff (Abs_poly y) = y"
SAT_restrict_current: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "current \<Gamma> g" shows "SAT (quotient_web \<Gamma> f) (g \<upharpoonleft> \<Gamma> / f) = RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<union> (SAT \<Gamma> g - A \<Gamma>)"
distrib_imp1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<And>x y z. x \<sqinter> y \<squnion> z = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" shows "x \<squnion> (y \<sqinter> z) = x \<squnion> y \<sqinter> x \<squnion> z"
gets_def: fixes f :: "'d \<Rightarrow> 'e" shows "gets f \<equiv> do s \<leftarrow> get; return (f s) od"
Local: fixes f :: "state \<Rightarrow> state"   and c :: "com"   and g :: "state \<Rightarrow> state \<Rightarrow> state"   and cs :: "com list"   and s :: "state" shows "((LOCAL f; c; g) # cs, s) \<rightarrow> (c # Do (\<lambda>t. {g s t}) # cs, f s)"
rel_nabla_bin_def: fixes X :: "('a \<times> 'a) set"   and Q :: "('a \<times> 'a) set" shows "rel_nabla_bin X Q = \<Union> {P. P \<subseteq> relfdia X P \<union> rdom Q}"
reachable_def: fixes E :: "('a \<times> 'a) set"   and src :: "'a"   and tgt :: "'a" shows "reachable E src tgt = ((src, tgt) \<in> E\<^sup>*)"
PPs_closed_adds: fixes s :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "s \<in> .[X]"   and "t adds s" shows "t \<in> .[X]"
2: shows "avl \<langle>l, (a, n), r\<rangle>"
mprod_expR_def: fixes R :: "('c, 'd) Ring_scheme"   and e :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "mprod_expR R e f n = e\<Pi>\<^bsub>R,n\<^esub> (\<lambda>j. f j^\<^bsup>R e j\<^esup>)"
distr_distr: fixes g :: "'b \<Rightarrow> 'c"   and N :: "'b measure"   and L :: "'c measure"   and f :: "'d \<Rightarrow> 'b"   and M :: "'d measure" assumes "g \<in> N \<rightarrow>\<^sub>M L"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "distr (distr M N f) L g = distr M L (g \<circ> f)"
simps_2: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and uu :: "'a" shows "sorted_by cmp [uu] = True"
bigtheta_const_ln: fixes a :: "real" assumes "0 < a" shows "(\<lambda>x. ln (a * x)) \<in> \<Theta>(ln)"
set_integral_add_1: fixes M :: "'c measure"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "set_integrable M A f"   and "set_integrable M A g" shows "set_integrable M A (\<lambda>x. f x + g x)"
weak_preservation_of_barbs_and_closures_3: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "rel_weakly_preserves_barbs Rel CWB" shows "rel_weakly_preserves_barbs (Rel\<^sup>*) CWB"
DIAndCutP12_def: shows "DIAndCutP12 = (([[$\<alpha> vid1]]Pc pid1 \<rightarrow> Pc pid2 \<rightarrow> (Pc pid1 && Pc pid2)) \<rightarrow> ([[$\<alpha> vid1]]Pc pid1) \<rightarrow> [[$\<alpha> vid1]]Pc pid2 \<rightarrow> (Pc pid1 && Pc pid2))"
cases: fixes a :: "'b"   and r :: "('c \<times> 'b) set" assumes "a \<in> Range r" obtains aa :: "'c" and b :: "'b" where "a = b" and "(aa, b) \<in> r"
AtCondExpr1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and a1 :: "('e, 'd, 'f) ann"   and c1 :: "('e, 'd, 'f) com"   and a :: "'e set \<times> ('e, 'd, 'f) com"   and r :: "'e set"   and a2 :: "('e, 'd, 'f) ann"   and b :: "'e set"   and c2 :: "('e, 'd, 'f) com" assumes "atomicsR \<Gamma> \<Theta> a1 c1 a" shows "atomicsR \<Gamma> \<Theta> (AnnBin r a1 a2) (Cond b c1 c2) a"
subgroup_conjugation_is_inj_aux: fixes g :: "'a"   and H1 :: "'a set"   and H2 :: "'a set" assumes "g \<in> carrier G"   and "H1 \<subseteq> carrier G"   and "H2 \<subseteq> carrier G"   and "g <# H1 #> inv g = g <# H2 #> inv g" shows "H1 \<subseteq> H2"
rephrase1: shows "\<Union> (\<G>_F_L_q q ` NL) = no_labels.\<G>_Fset_q q (fst ` NL)"
G_def: shows "G \<equiv> some_gcd.listgcd ys"
applicableLearn_def: fixes state :: "State" shows "applicableLearn state \<equiv> \<exists>state'. appliedLearn state state'"
elims: fixes x :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list \<times> ('a, 'b) rose_tree\<^sub>m list) list \<times> ('a, 'b) rose_tree\<^sub>m"   and y :: "(('a \<times> ('a, 'b) rose_tree\<^sub>m list \<times> ('a, 'b) rose_tree\<^sub>m list) list \<times> ('a, 'b) rose_tree\<^sub>m) list" assumes "zipper_children\<^sub>m x = y" obtains p :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list \<times> ('a, 'b) rose_tree\<^sub>m list) list" and a :: "'a" and ts :: "('a, 'b) rose_tree\<^sub>m list" where "x = (p, Tree\<^sub>m (Unblinded (a, ts)))" and "y = map (\<lambda>(l, t, r). ((a, l, r) # p, t)) (splits ts)"   | v :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list \<times> ('a, 'b) rose_tree\<^sub>m list) list" and vc :: "('b \<times> 'b rose_tree\<^sub>h list) blindable\<^sub>h" where "x = (v, Tree\<^sub>m (Blinded vc))" and "y = []"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b"   and acc_modi :: "acc_modi" shows "P (case acc_modi of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = ((acc_modi = Private \<longrightarrow> P f1) \<and> (acc_modi = Package \<longrightarrow> P f2) \<and> (acc_modi = Protected \<longrightarrow> P f3) \<and> (acc_modi = Public \<longrightarrow> P f4))"
document_ptr_casts_commute2: fixes document_ptr :: "(_) document_ptr" shows "cast (cast document_ptr) = Some document_ptr"
sqrt_sum_squares_half_less: fixes x :: "real"   and u :: "real"   and y :: "real" assumes "x < u / 2"   and "y < u / 2"   and "0 \<le> x"   and "0 \<le> y" shows "sqrt (x\<^sup>2 + y\<^sup>2) < u"
clamp_def: fixes a :: "'b"   and b :: "'b"   and x :: "'b" shows "clamp a b x = (if \<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i then \<Sum>i\<in>Basis. (if x \<bullet> i < a \<bullet> i then a \<bullet> i else if x \<bullet> i \<le> b \<bullet> i then x \<bullet> i else b \<bullet> i) *\<^sub>R i else a)"
Up: shows "Up \<equiv> \<lambda>(s\<^sub>1, s\<^sub>2) e. if e \<in> Tr\<^sub>1 s\<^sub>1 then (Up\<^sub>1 s\<^sub>1 e, s\<^sub>2) else if e \<in> Tr\<^sub>2 s\<^sub>2 then (s\<^sub>1, Up\<^sub>2 s\<^sub>2 e) else (s\<^sub>1, s\<^sub>2)"
is_sup_def: fixes x :: "'b"   and y :: "'b"   and sup :: "'b" shows "is_sup x y sup = (x \<sqsubseteq> sup \<and> y \<sqsubseteq> sup \<and> (\<forall>z. x \<sqsubseteq> z \<and> y \<sqsubseteq> z \<longrightarrow> sup \<sqsubseteq> z))"
inner_mult_right: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and a :: "'b"   and m :: "real"   and b :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "a \<bullet> (of_real m * b) = m * (a \<bullet> b)"
orbit_equiv: fixes b :: "'b"   and f :: "'b perm"   and a :: "'b" assumes "b \<in> orbit f a" shows "orbit f b = orbit f a"
efficient_power_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "efficient_power f \<equiv> \<forall>x z. f x (f x z) = f (f x x) z"
rtranclpD: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "R\<^sup>*\<^sup>* a b" shows "a = b \<or> a \<noteq> b \<and> R\<^sup>+\<^sup>+ a b"
atMost_Suc: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "sum g {..Suc n} = sum g {..n} + g (Suc n)"
Node_in_trees_iff: fixes l :: "'d tree"   and v :: "'d"   and r :: "'d tree"   and S :: "'d set" shows "(\<langle>l, v, r\<rangle> \<in> trees S) = (l \<in> trees S \<and> v \<in> S \<and> r \<in> trees S)"
assms_2: shows "\<forall>x y z. fst (g x y) \<longrightarrow> snd (g y z) \<longrightarrow> fst (g x z)"
Ana\<^sub>f_assm2_alt: fixes f :: "'fun"   and K :: "(('fun, 'atom, 'sets) prot_fun, nat) Term.term list"   and M :: "nat list"   and i :: "nat" assumes "Ana\<^sub>f f = (K, M)"   and "i \<in> fv\<^sub>s\<^sub>e\<^sub>t (set K) \<union> set M" shows "i < arity\<^sub>f f"
extreme_unique: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set"   and x :: "'c"   and y :: "'c" assumes "antisymmetric A less_eq"   and "X \<subseteq> A"   and "extreme X less_eq x" shows "extreme X less_eq y = (x = y)"
flat_Stream: fixes xs :: "'c list"   and ws :: "'c list stream" assumes "xs \<noteq> []" shows "flat (xs ## ws) = xs @- flat ws"
eventually_filtermap: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and F :: "'b filter" shows "eventually P (filtermap f F) = (\<forall>\<^sub>F x in F. P (f x))"
Rep_zet_inverse: fixes x :: "'b zet" shows "Abs_zet (Rep_zet x) = x"
LAss: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and l' :: "char list \<Rightarrow> val option"   and V :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h, l)\<rangle>"   and "l' = l(V \<mapsto> v)" shows "P \<turnstile> \<langle>V:=e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,(h, l')\<rangle>"