hcju/isabelleps huggingface dataset:
assms_2: shows "freshAbs zs z (A $[y1 // y]_ys)"
distinct_95: fixes x7 :: "rat"   and x101 :: "nat"   and x102 :: "msg" shows "Pri_AgrK x7 \<noteq> Crypt x101 x102"
module_hom_scale_self: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'a" assumes "module scale" shows "module_hom scale scale (scale c)"
trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "order.greater_eq b a"   and "order.greater_eq c b" shows "order.greater_eq c a"
assms_1: shows "clauseFalse (removeAll literal clause1) valuation"
wf_iff_no_infinite_down_chain: fixes r :: "('g \<times> 'g) set" shows "wf r = (\<nexists>f. \<forall>i. (f (Suc i), f i) \<in> r)"
subst_var: fixes \<sigma> :: "'f \<Rightarrow> 'g"   and x :: "'e \<Longrightarrow> 'g" shows "\<sigma> \<dagger> x = \<langle>\<sigma>\<rangle>\<^sub>s x"
None_eq_map_option_iff: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d option" shows "(\<bottom> = map_option f x) = (x = \<bottom>)"
code: fixes xyax :: "x_y_ax" shows "F xyax = (if xyax \<noteq> ??.Process.x_y_ax.x then Action (if xyax = ??.Process.x_y_ax.y then ''b'' else ''a'') (F ??.Process.x_y_ax.x) else Choice (F ??.Process.x_y_ax.ax) (F ??.Process.x_y_ax.y))"
parPres: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<approx> Q" shows "P \<parallel> R \<approx> Q \<parallel> R"
d_reaches_unique_cases: fixes M :: "('e, 'f, 'd) FSM"   and xs :: "'e list" shows "{q. d_reaches M (initial M) xs q} = {} \<or> (\<exists>q2. {q. d_reaches M (initial M) xs q} = {q2})"
diameter_bounded_bound: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "bounded S"   and "x \<in> S"   and "y \<in> S" shows "dist x y \<le> diameter S"
linear_cf_def: fixes r :: "('d \<times> 'd) set"   and linear_cf :: "'d set \<Rightarrow> 'd set"   and X :: "'d set" assumes "linear_cf r linear_cf" shows "linear_cf X \<equiv> set_option (MaxR.MaxR_opt r X)"
mem_simps_6: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" shows "(c \<in> A - B) = (c \<in> A \<and> c \<notin> B)"
edc_fst: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and e :: "expr"   and f :: "cexpr"   and t :: "pdf_type"   and t' :: "pdf_type" assumes "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c e \<Rightarrow> f"   and "\<Gamma> \<turnstile> e : PRODUCT t t'" shows "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c Fst $$ e \<Rightarrow> \<integral>\<^sub>c (map_vars (case_nat 0 (\<lambda>x. x + 2)) f \<circ>\<^sub>c <CVar 1, CVar 0>\<^sub>c) \<partial>t'"
def_to_eq: fixes P :: "'b"   and Q :: "'b" assumes "P = Q" shows "P \<equiv> Q"
span_explicit': fixes b :: "'b set" shows "m1.span b = {\<Sum>v | f v \<noteq> (0::'a). f v *a v |f. finite {v. f v \<noteq> (0::'a)} \<and> (\<forall>v. f v \<noteq> (0::'a) \<longrightarrow> v \<in> b)}"
pp_isotone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "- - x \<le> - - y"
obj_src: fixes \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.arr \<mu>" shows "S.obj (S.src \<mu>)"
that: fixes N :: "nat" assumes "\<And>m n. N \<le> m \<Longrightarrow> norm (sum f {m..n}) < e" shows "thesis"
monom_mult_in_pmdl: fixes b :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and B :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and c :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "b \<in> B" shows "local.punit.monom_mult c t b \<in> local.punit.pmdl B"
reads_bind_pure: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and h' :: "'a"   and S :: "('a \<Rightarrow> 'a \<Rightarrow> bool) set"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog" assumes "pure f h"   and "pure f h'"   and "reads S f h h'"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> reads S (g x) h h'" shows "reads S (Heap_Error_Monad.bind f g) h h'"
"3_1": assumes "x = y"   and "\<And>y. \<lbrakk>y \<in> set ys; P y\<rbrakk> \<Longrightarrow> False" shows "filter P (merge_wrt rel xs ys) = filter P xs"
axioms_2: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and altc :: "'f cset \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e" assumes "monad_altc3 return bind altc" shows "three TYPE('f)"
tauChainCasesSym: fixes P :: "ccs"   and P' :: "ccs"   and Prop :: "ccs \<Rightarrow> bool" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "Prop P"   and "\<And>P''. \<lbrakk>P \<longmapsto>\<tau> \<prec> P''; P'' \<Longrightarrow>\<^sub>\<tau> P'\<rbrakk> \<Longrightarrow> Prop P'" shows "Prop P'"
keys_Mapping: fixes xs :: "('c \<times> 'd) list" shows "Mapping.keys (AList_Mapping.Mapping xs) = set (map fst xs)"
wt_subst_trm': fixes \<sigma> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma>"   and "\<Gamma> s = \<Gamma> t" shows "\<Gamma> (s \<cdot> \<sigma>) = \<Gamma> (t \<cdot> \<sigma>)"
continuous_on_norm: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. norm (f x))"
inverse_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" shows "inverse_aform_err p X = map_aform_err (inverse_aform p) p X"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "gcd (gcd a b) c = gcd a (gcd b c)"
andalso_eq_bottom_iff: fixes x :: "tr"   and y :: "tr" shows "((x andalso y) = \<bottom>) = (x = \<bottom> \<or> x = TT \<and> y = \<bottom>)"
ideI: fixes a :: "'a" assumes "right a a"   and "ide a" shows "Right_a.ide a"
EqButGH_def: fixes p :: "pvar"   and p' :: "pvar" shows "EqButGH p p' \<equiv> Scon {EqOnUC p p' u c |u c. (u, c) \<in> (UNIV - GH) \<times> UNIV}"
eq_neutr_iff: fixes a :: "nat"   and b :: "nat" shows "(max a b = 0) = (a = 0 \<and> b = 0)"
terminal_path_end_is_terminal: fixes v :: "'vertex"   and v' :: "'vertex"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" assumes "terminal_path v v' pth" shows "terminal_vertex v'"
obtains_obtainable: fixes s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and e :: "((nat \<times> nat) \<times> transition) fset"   and t :: "(String.literal \<times> value list) list" assumes "obtains s r e 0 <> t" shows "obtainable s r e"
vconst_on_iff: fixes a :: "V"   and c :: "V"   and A :: "V" shows "(\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vconst_on A c) = (a \<in>\<^sub>\<circ> A)"
distinct_3: fixes x1 :: "'e" shows "prot_atom.Atom x1 \<noteq> SetType"
clock_commonKnowledge_relation_image: fixes s :: "'s"   and Y :: "'s odlist"   and as :: "'a list" assumes "s \<in> toSet Y" shows "toSet (clock_commonKnowledge envObs Y as s) = (\<Union> (relations (clock_repMC (toSet Y)) ` set as))\<^sup>+ `` {s}"
simps_1: fixes at :: "('a, 'b) atm" shows "nvL (Pos at) = nvA at"
left: shows "(ix p d - 1) * 2 ^ (lv p' d - lv p d) \<le> ix p' d"
fermat_witness_def: shows "local.fermat_witness = [a ^ (n - 1) \<noteq> 1] (mod n)"
Let_def: fixes s :: "'c"   and f :: "'c \<Rightarrow> 'b" shows "HOL.Let s f \<equiv> f s"
total_mult_closed: fixes x :: "'a"   and y :: "'a" assumes "times_top_class.total x"   and "times_top_class.total y" shows "times_top_class.total (x * y)"
STA_sub': fixes A :: "'b set"   and n :: "nat"   and branch :: "(('d, 'b) fm list \<times> 'b) list"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>f i A. \<lbrakk>finite A; i \<notin> A\<rbrakk> \<Longrightarrow> \<exists>j. j \<notin> f ` A"   and "finite A"   and "A, n \<turnstile> branch" shows "f ` A \<turnstile> sub_branch f branch"
a: shows "a \<in> {1..p - 1} \<and> {1..p - 1} = {a ^ i mod p |i. i \<in> UNIV}"
sum_intro: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "(\<forall>a\<in>A. \<exists>b\<in>B. a \<le> b) \<longrightarrow> \<Sum> A \<le> \<Sum> B"
next_initial_ps2_edge: fixes G :: "diagram"   and V :: "node fset"   and \<Lambda> :: "node \<Rightarrow> assertion_gadget"   and E :: "(node fset \<times> command_gadget \<times> node fset) list"   and G' :: "diagram"   and V' :: "node fset"   and E' :: "(node fset \<times> command_gadget \<times> node fset) list"   and e :: "node fset \<times> command_gadget \<times> node fset"   and S :: "node fset" assumes "G = Graph V \<Lambda> E"   and "G' = Graph V' \<Lambda> E'"   and "V' = V |-| fst3 e"   and "E' = removeAll e E"   and "e \<in> set E"   and "fst3 e |\<subseteq>| S"   and "S |\<subseteq>| initials G"   and "wf_dia G" shows "initial_ps2 (S |-| fst3 e) G' = initial_ps2 S G \<ominus> fst3 e ++\<^sub>f [ thd3 e |=> Top ]"
Stable_eq_stable: fixes F :: "'b program"   and A :: "'b set" shows "(F \<in> Stable A) = (F \<in> stable (reachable F \<inter> A))"
Zero_not_eq_fromArr: fixes Arr :: "'a set" shows "quiver.Zero \<notin> quiver.fromArr Arr ` Arr"
pvalid_def: fixes p :: "(symbol \<times> character list) list"   and x :: "item" shows "pvalid p x = (\<exists>u \<gamma>. wellformed_tokens p \<and> wellformed_item x \<and> u \<le> length p \<and> charslength p = item_end x \<and> charslength (take u p) = item_origin x \<and> is_derivation (terminals (take u p) @ [item_nonterminal x] @ \<gamma>) \<and> derives (item_\<alpha> x) (terminals (drop u p)))"
Int_atLeastAtMostR2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.linorder less_eq less" shows "ord.atLeast less_eq a \<inter> ord.atLeastAtMost less_eq c d = ord.atLeastAtMost less_eq (ord.max less_eq a c) d"
dependent_zero: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set" assumes "module scale"   and "(0::'b) \<in> A" shows "module.dependent scale A"
W_respects_labels: fixes B :: "'b set"   and \<phi> :: "'a \<Rightarrow> 'b"   and w :: "'a permutation"   and v :: "'a" assumes "label_wrt B \<phi>"   and "w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>"   and "v \<in> \<Union> X" shows "\<phi> (w \<rightarrow> v) = \<phi> v"
tendsto_compose_eventually: fixes g :: "'b \<Rightarrow> 'c"   and m :: "'c"   and l :: "'b"   and f :: "'d \<Rightarrow> 'b"   and F :: "'d filter" assumes "g \<midarrow>l\<rightarrow> m"   and "(f \<longlongrightarrow> l) F"   and "\<forall>\<^sub>F x in F. f x \<noteq> l" shows "((\<lambda>x. g (f x)) \<longlongrightarrow> m) F"
assms: shows "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> (\<A> @ \<A>')"
simps: fixes a :: "real"   and b :: "real"   and c :: "real" shows "convertDerivativeUni (a, b, c) = OrUni (AtomUni (LessUni (a, b, c))) (AndUni (AtomUni (EqUni (a, b, c))) (OrUni (AtomUni (LessUni (0, 2 * a, b))) (AndUni (AtomUni (EqUni (0, 2 * a, b))) (AtomUni (LessUni (0, 0, 2 * a))))))"
PowI: fixes A :: "'g set"   and B :: "'g set" assumes "A \<subseteq> B" shows "A \<in> Pow B"
space_expr_sem: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and w :: "expr"   and t :: "pdf_type"   and \<sigma> :: "nat \<Rightarrow> val" assumes "\<Gamma> \<turnstile> w : t"   and "\<forall>x\<in>free_vars w. val_type (\<sigma> x) = \<Gamma> x" shows "space (expr_sem \<sigma> w) = type_universe t"
image_vimage_eq: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'a set" shows "f ` f -` A = A \<inter> range f"
simps_2: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; send\<langle>t\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>c' = (\<lambda>\<I>. M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c t \<cdot> \<I> \<and> \<lbrakk>M; S\<rbrakk>\<^sub>c' \<I>)"
that: assumes "get h' r = v"   and "present h' r"   and "\<not> present h r" shows "thesis"
a_get_M_def: fixes ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> 'u" shows "get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter = Heap_Error_Monad.bind get_heap (\<lambda>h. case get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h of None \<Rightarrow> error SegmentationFault | Some res \<Rightarrow> return (getter res))"
ctr: fixes f :: "('c \<Rightarrow> 'b) tree"   and x :: "'c tree" shows "f \<diamondop> x = Node (root f (root x)) (left f \<diamondop> left x) (right f \<diamondop> right x)"
top_Assertion_def: shows "\<top> = abs_wpt (1::'a)"
coprimeI_primes: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a) \<or> b \<noteq> (0::'a)"   and "\<And>p. \<lbrakk>prime p; p dvd a; p dvd b\<rbrakk> \<Longrightarrow> False" shows "algebraic_semidom_class.coprime a b"
LAss_\<tau>execI: 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 V :: "nat" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (V:=e) h s s'"
y_axis_gauge_integrable: fixes x :: "'b" shows "(\<lambda>y. f (y, x)) integrable_on UNIV"
constr_rep_2: fixes x2 :: "tyvrs"   and x1 :: "ty" shows "Rep_binding (TVarB x2 x1) = TVarB_Rep x2 x1"
elims_2: fixes x :: "('b rexp set \<times> 'b rexp set) list \<times> ('b rexp set \<times> 'b rexp set) list" assumes "test x" obtains ws :: "('b rexp set \<times> 'b rexp set) list" and ps :: "('b rexp set \<times> 'b rexp set) list" where "x = (ws, ps)" and "case ws of [] \<Rightarrow> False | (R, S) # x \<Rightarrow> Nullable R = Nullable S"
Stirling_same: fixes n :: "nat" shows "Stirling n n = 1"
Sum_any_when_dependent_prod_right: fixes g :: "'d \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'e" shows "(\<Sum>(a, b). g a when b = h a) = Sum_any g"
assms_1: shows "(r, s) \<in> carrier R \<times> S"
condensation_finite: fixes g :: "'g"   and P :: "'val set" shows "finite (condensation_edges g P)"
v_TS_defs_3: fixes Quorum :: "process set set" assumes "quorum_process Quorum" shows "quorum_process.v_trans Quorum = (\<Union>r v_f. \<Union> (range (quorum_process.v_round Quorum r v_f)))\<^sup>="
intro: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and i :: "real \<times> real"   and j :: "real \<times> real" assumes "typeI_twoCube twoC"   and "valid_two_cube twoC"   and "analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> i) j" shows "green_typeI_cube_axioms i j twoC F"
openin_topology_generated_by_iff: fixes S :: "'c set set"   and s :: "'c set" shows "openin (topology_generated_by S) s = generate_topology_on S s"
f_def: fixes u :: "nat" shows "f u = (\<Sum>x\<in>{i. i < u}. g1 x u)"
assms_4: shows "w' \<in> worlds (gen_model M' w)"
"2_4": fixes m :: "C" assumes "m \<in> hbase b"   and "evalC b n = evalC b m" shows "n = m"
part_emb: fixes ps :: "char list \<Rightarrow> int option"   and q :: "char list \<Rightarrow> int" assumes "dom ps = UNIV" shows "part (Partial_Evaluation.emb ps q) = ps"
restrict_def: fixes S :: "'a set"   and s :: "'a \<Rightarrow> 'b" shows "restrict S s = (\<lambda>x. if x \<in> S then Some (s x) else None)"
eq_snd_iff: fixes b :: "'g"   and p :: "'h \<times> 'g" shows "(b = snd p) = (\<exists>a. p = (a, b))"
R_O_Id: fixes R :: "('a \<times> 'b) set" shows "R O Id = R"
component_in_hom: fixes j :: "'j" assumes "J.arr j" shows "\<guillemotleft>\<chi>' j : x' \<rightarrow>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] D (J.cod j)\<guillemotright>"
cil_subset_il': fixes w :: "'d list"   and w1 :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and w2 :: "'d list" assumes "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2" shows "w \<in> w1 \<otimes> w2"
cases: fixes a1 :: "'b prog"   and a2 :: "ty"   and a3 :: "bop"   and a4 :: "ty"   and a5 :: "ty" assumes "a1 \<turnstile> a2\<guillemotleft>a3\<guillemotright>a4 :: a5" obtains P :: "'b prog" and T1 :: "ty" and T2 :: "ty" where "a1 = Pa__" and "a2 = T1a__" and "a3 = Eq" and "a4 = T2a__" and "a5 = Boolean" and "Pa__ \<turnstile> T1a__ \<le> T2a__ \<or> Pa__ \<turnstile> T2a__ \<le> T1a__"   | P :: "'b prog" and T1 :: "ty" and T2 :: "ty" where "a1 = Pb__" and "a2 = T1" and "a3 = NotEq" and "a4 = T2" and "a5 = Boolean" and "Pb__ \<turnstile> T1 \<le> T2 \<or> Pb__ \<turnstile> T2 \<le> T1"   | P :: "'b prog" where "a1 = Pc__" and "a2 = Integer" and "a3 = LessThan" and "a4 = Integer" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Pd__" and "a2 = Integer" and "a3 = LessOrEqual" and "a4 = Integer" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Pe__" and "a2 = Integer" and "a3 = GreaterThan" and "a4 = Integer" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Pf__" and "a2 = Integer" and "a3 = GreaterOrEqual" and "a4 = Integer" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Pg__" and "a2 = Integer" and "a3 = Add" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Ph__" and "a2 = Integer" and "a3 = Subtract" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pi__" and "a2 = Integer" and "a3 = Mult" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pj__" and "a2 = Integer" and "a3 = Div" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pk__" and "a2 = Integer" and "a3 = Mod" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pl__" and "a2 = Boolean" and "a3 = BinAnd" and "a4 = Boolean" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Pm__" and "a2 = Integer" and "a3 = BinAnd" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pn__" and "a2 = Boolean" and "a3 = BinOr" and "a4 = Boolean" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Po__" and "a2 = Integer" and "a3 = BinOr" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pp__" and "a2 = Boolean" and "a3 = BinXor" and "a4 = Boolean" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Pq__" and "a2 = Integer" and "a3 = BinXor" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pr__" and "a2 = Integer" and "a3 = ShiftLeft" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Ps__" and "a2 = Integer" and "a3 = ShiftRightZeros" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = P" and "a2 = Integer" and "a3 = ShiftRightSigned" and "a4 = Integer" and "a5 = Integer"
Dg2_is_gamma1_pathimg: shows "path_image gamma1 = {p. \<exists>x. x \<in> cbox a b \<and> p = (x, g2 x)}"
length: shows "\<forall>e\<in>A. llength e = 2 * N"
takeWhile_empty: shows "takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb = []"
weight_finite: fixes \<Gamma> :: "('k, 'l) web_scheme"   and x :: "'k" assumes "countable_bipartite_web \<Gamma>" shows "weight \<Gamma> x \<noteq> \<top>"
cut_less_def: fixes I :: "'a set"   and t :: "'a" shows "I \<down>< t \<equiv> {x \<in> I. x < t}"
box_eq_empty_2: fixes a :: "'a"   and b :: "'a" shows "(cbox a b = {}) = (\<exists>i\<in>Basis. b \<bullet> i < a \<bullet> i)"
fls_nonzeroI: fixes f :: "'a fls"   and n :: "int" assumes "fls_nth f n \<noteq> (0::'a)" shows "f \<noteq> 0"
\<oo>_Times: fixes r :: "'a language"   and s :: "'a language" shows "\<oo> (Times r s) = (\<oo> r \<and> \<oo> s)"
simps_5: fixes x1 :: "'g" shows "NewExit \<noteq> Node x1"
lift_def: fixes f :: "'g \<Rightarrow> 'd \<Rightarrow> (('e + 'f) \<times> 'd) \<times> bool"   and v :: "'e + 'g" shows "lift f v \<equiv> case v of Inl e \<Rightarrow> throwError e | Inr v' \<Rightarrow> f v'"
ltl_simps_2: fixes x21 :: "'b"   and x22 :: "'b llist" shows "ltl (LCons x21 x22) = x22"
EntryPath_snoc: fixes g :: "'g"   and ns :: "'node list"   and m :: "'node" assumes "EntryPath g ns"   and "shortestPath g m = Suc (shortestPath g (last ns))" shows "EntryPath g (ns @ [m])"
Ln_eq_zero_iff: fixes x :: "complex" assumes "x \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "(Ln x = 0) = (x = 1)"
lexord_app_right_linorder: fixes u :: "'a list"   and w :: "'a list"   and v :: "'a list"   and z :: "'a list" assumes "u <lex w"   and "\<^bold>|w\<^bold>| \<le> \<^bold>|u\<^bold>|" shows "u \<cdot> v <lex w \<cdot> z"
spec_refine: fixes P :: "'a upred"   and R :: "'a upred"   and Q :: "'a upred" assumes "Q \<sqsubseteq> (P \<and> R)" shows "(P \<Rightarrow> Q) \<sqsubseteq> R"
ospec_alt: fixes m :: "'d option"   and P :: "'d \<Rightarrow> bool" shows "ospec m P = (case m of None \<Rightarrow> False | Some x \<Rightarrow> P x)"
as_y: shows "y \<in> at_least_as_good x carrier relation - as_good_as x carrier relation"
map_tpOf_getTvars: fixes \<sigma>l :: "'tp list" shows "map TE.tpOf (TE.getTvars \<sigma>l) = \<sigma>l"
Method_i_i_i_o_o_o_oE: fixes x :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and xa :: "char list"   and xb :: "char list"   and xc :: "ty list"   and xd :: "ty"   and xe :: "'b"   and xf :: "char list" assumes "pred.eval (Method_i_i_i_o_o_o_o x xa xb) (xc, xd, xe, xf)" obtains "x \<turnstile> xa sees xb: xc\<rightarrow>xd = xe in xf"
diff_less_divisor_imp_sub_mod_eq: fixes x :: "nat"   and y :: "nat"   and m :: "nat" assumes "x \<le> y"   and "y - x < m" shows "x = y - (y - x) mod m"
atLeastatMost_empty_iff2: fixes a :: "'e word"   and b :: "'e word" shows "({} = signed.atLeastAtMost a b) = (\<not> a \<le>s b)"
induct: fixes n :: "nat"   and P :: "nat \<Rightarrow> 'b mat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'b mat" assumes "\<And>j A. (\<And>x xa. \<lbrakk>\<not> n \<le> j; x = A $$ (j, j); xa = (case ??.Jordan_Normal_Form_Existence.lookup_ev x j A of None \<Rightarrow> A | Some i \<Rightarrow> ??.Jordan_Normal_Form_Existence.swap_cols_rows_block (Suc i) j A)\<rbrakk> \<Longrightarrow> P (Suc j) xa) \<Longrightarrow> P j A" shows "P a0 a1"
X: shows "random_variable (Pi\<^sub>M I M') (\<lambda>x. \<lambda>i\<in>I. X i x)"
Un_eq_UN: fixes A :: "'b set"   and B :: "'b set" shows "A \<union> B = (\<Union>b. if b then A else B)"
is_map_def: fixes M :: "('d \<times> 'e) set" shows "is_map M = (\<forall>(a\<^sub>1, b\<^sub>1)\<in>M. \<forall>(a\<^sub>2, b\<^sub>2)\<in>M. a\<^sub>1 = a\<^sub>2 \<longrightarrow> b\<^sub>1 = b\<^sub>2)"
distinct_13: fixes x81 :: "'b ltln"   and x82 :: "'b ltln" shows "true\<^sub>n \<noteq> x81 U\<^sub>n x82"
not_less: fixes x :: "'b"   and y :: "'b" shows "(\<not> x < y) = (y \<le> x)"
not_empty_eq_Ici_eq_empty: fixes b :: "'b"   and l :: "'b" shows "{} \<noteq> flat_ord.atLeast b l"
ends_in_segment_2: fixes b :: "'c"   and a :: "'c" shows "b \<in> {a--b}"
order_witness_unfold: fixes n :: "nat"   and f :: "'a perm"   and a :: "'a" assumes "0 < n"   and "(f ^ n) \<langle>$\<rangle> a = a" shows "order f a = card ((\<lambda>m. (f ^ m) \<langle>$\<rangle> a) ` {0..<n})"
domain_eq: shows "Domainp (pcr_program (=)) = (\<lambda>x. x \<simeq> x \<and> well_def x \<and> well_def x)"
circle_cube_is_type_I: assumes "0 < d" shows "typeI_twoCube circle_cube"
simps_1: fixes x1 :: "'c"   and x2 :: "'c stream"   and y1 :: "'c"   and y2 :: "'c stream" shows "(x1 ## x2 = y1 ## y2) = (x1 = y1 \<and> x2 = y2)"
Compr_image_eq: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and P :: "'c \<Rightarrow> bool" shows "{x \<in> f ` A. P x} = f ` {x \<in> A. P (f x)}"
While: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c :: "com" assumes "\<turnstile> {\<lambda>s. P s \<and> bval b s} c {P}" shows "\<turnstile> {P} WHILE b DO c {\<lambda>s. P s \<and> \<not> bval b s}"
simps: fixes a1 :: "('Var, 'AExp, 'BExp) Stmt"   and a2 :: "'Var \<Rightarrow> 'Val"   and a3 :: "('Var, 'AExp, 'BExp) Stmt"   and a4 :: "'Var \<Rightarrow> 'Val" shows "eval\<^sub>w_simple_abv (a1, a2) (a3, a4) = ((\<exists>x e mem. a1 = x \<leftarrow> e \<and> a2 = mem \<and> a3 = Stop \<and> a4 = mem(x := ev\<^sub>A mem e)) \<or> (\<exists>mem. a1 = Skip \<and> a2 = mem \<and> a3 = Stop \<and> a4 = mem) \<or> (\<exists>c mem. a1 = Stop ;; c \<and> a2 = mem \<and> a3 = c \<and> a4 = mem) \<or> (\<exists>mem b t e. a1 = Stmt.If b t e \<and> a2 = mem \<and> a3 = t \<and> a4 = mem \<and> ev\<^sub>B mem b) \<or> (\<exists>mem b t e. a1 = Stmt.If b t e \<and> a2 = mem \<and> a3 = e \<and> a4 = mem \<and> \<not> ev\<^sub>B mem b) \<or> (\<exists>b c mem. a1 = While b c \<and> a2 = mem \<and> a3 = Stmt.If b (c ;; While b c) Stop \<and> a4 = mem))"