hcju/isabelleps huggingface dataset:
simps: fixes a1 :: "'a list"   and a2 :: "'a list multiset"   and a3 :: "('b, 'c) label list"   and a4 :: "'a list"   and a5 :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" shows "(((a1, a2), a3, a4, a5) \<in> ntrs fg) = (\<exists>u r ce p u' w v ce'. a1 = u # r \<and> a2 = ce \<and> a3 = LCall p # w \<and> a4 = v # u' # r \<and> a5 = ce' \<and> ((u # r, ce), LCall p, entry fg p # u' # r, ce) \<in> trss fg \<and> (([entry fg p], ce), w, [v], ce') \<in> trcl (trss fg))"
finite_nyinitcls: fixes G :: "prog"   and s :: "state" shows "finite (nyinitcls G s)"
dtor1_diff: fixes x :: "'d JF1"   and y :: "'d JF1" shows "(dtor1 x = dtor1 y) = (x = y)"
simps_1: fixes x :: "'a"   and y :: "'a" shows "(extT.Val x \<le> extT.Val y) = (x \<le> y)"
emeasure_eq_measure: fixes A :: "('i \<Rightarrow> 'a) set" shows "emeasure (Pi\<^sub>M I M) A = ennreal (P.prob A)"
bst_delete: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t :: "('b \<times> 'c) tree"   and x :: "'b" assumes "Set2_Join join inv"   and "Tree2.bst t" shows "Tree2.bst (Set2_Join.delete join x t)"
f15: shows "fmask2 - (fmask1 - c1) = c1"
reach\<^sub>t_def: fixes \<Sigma> :: "'b set"   and \<delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>0 :: "'a" shows "reach\<^sub>t \<Sigma> \<delta> q\<^sub>0 = {run\<^sub>t \<delta> q\<^sub>0 w n |w n. range w \<subseteq> \<Sigma>}"
meet_greatest: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "x \<sqsubseteq> z" shows "x \<sqsubseteq> y \<sqinter> z"
atLeastAtMost_reindex: fixes h :: "nat \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat"   and g :: "'c \<Rightarrow> 'd" assumes "bij_betw h {m..n} {h m..h n}" shows "sum g {h m..h n} = sum (g \<circ> h) {m..n}"
hom_induced_default: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and c :: "(((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int) set"   and p :: "int" assumes "\<not> (continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and> c \<in> carrier (relative_homology_group p X S))" shows "hom_induced p X S Y T f c = \<one>\<^bsub>relative_homology_group p Y T\<^esub>"
assms_2: shows "V = (\<Union>(A, X)\<in>Q. A)"
remove_hd: fixes z :: "'c"   and xs :: "'c list" shows "remove1 z (z # xs) = xs"
euclidean_size_ratfps_def: fixes x :: "'b ratfps" shows "euclidean_size x = (if x = 0 then 0 else 2 ^ ratfps_subdegree x)"
P: shows "v \<noteq> (0::'a) \<and> v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \<and> v * det A4 = det B4 \<and> B3 \<in> carrier_mat (Suc m) n \<and> B4 \<in> carrier_mat (Suc m) (Suc m) \<and> (\<forall>i<m. B4 $$ (i, m) = (0::'a))"
Let_def: fixes s :: "'e"   and f :: "'e \<Rightarrow> 'd" shows "Let s f \<equiv> f s"
inv_into_image_cancel: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and S :: "'b set" assumes "inj_on f A"   and "S \<subseteq> A" shows "inv_into A f ` f ` S = S"
of_int_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and k :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (numeral k) = numeral.numeral one plus k"
errMOD_igSubstClsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igWlsAllDisj MOD"   and "igConsIPresIGWls MOD"   and "igWlsAbsIsInBar MOD"   and "igSubstAllIPresIGWlsAll MOD"   and "igSubstCls MOD" shows "igSubstClsSTR (errMOD MOD)"
minus_add_distrib: fixes a :: "'b"   and b :: "'b" shows "- (a + b) = - a + - b"
reindex_cong: fixes l :: "'d \<Rightarrow> 'e"   and B :: "'d set"   and A :: "'e set"   and g :: "'e \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'a" assumes "inj_on l B"   and "A = l ` B"   and "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" shows "F.F g A = F.F h B"
dedekind: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * y \<sqinter> z \<le> (x \<sqinter> z * y\<^sup>T) * (y \<sqinter> (x[[z]]))"
update_convs_4: fixes disconnected_nodes' :: "(_) node_ptr list \<Rightarrow> (_) node_ptr list"   and nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p" shows "disconnected_nodes_update disconnected_nodes' \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes' disconnected_nodes, \<dots> = more\<rparr>"
responses_\<I>_uniform: fixes A :: "'e set"   and B :: "'d set"   and x :: "'e" shows "responses_\<I> (\<I>_uniform A B) x = (if x \<in> A then B else {})"
s2: shows "Partial_Cost_Model.config'_rand BIT (type0 init x y) a = type1 init x y"
numeral_unfold_funpow: fixes k :: "num" shows "numeral k = ((+) (1::'a) ^^ numeral k) (0::'a)"
C_merge_adj': fixes xss :: "'b list list" shows "C_merge_adj xss \<le> length (concat xss)"
inf_bres: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> y = \<Sqinter> {z. y \<le> x \<rightarrow> z}"
isShortestPath_min_dist_def: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "Graph.isShortestPath c u p v = (Graph.isPath c u p v \<and> length p = Graph.min_dist c u v)"
induced_map_unique: fixes r\<^sub>0 :: "'a"   and \<omega> :: "'a"   and w :: "'a"   and w' :: "'a" assumes "\<rho>.is_induced_by_cell w r\<^sub>0 \<omega>"   and "\<rho>.is_induced_by_cell w' r\<^sub>0 \<omega>" shows "w \<cong> w'"
A_1: shows "\<forall>i j. i \<le> n \<longrightarrow> j \<le> n \<longrightarrow> M i j \<preceq> M' i j"
mem_cong_ax_holds: fixes e :: "(name, hf) finfun" shows "eval_fm e mem_cong_ax"
"2_5": shows "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'"
PS\<^sub>L_def: shows "PS\<^sub>L = ps\<^sub>L - {p\<^sub>0}"
span_scale: fixes x :: "'a \<Rightarrow>\<^sub>0 'b"   and S :: "('a \<Rightarrow>\<^sub>0 'b) set"   and c :: "'b" assumes "x \<in> phull S" shows "c \<cdot> x \<in> phull S"
lessThan_nat_numeral: fixes k :: "num" shows "{..<numeral k} = insert (pred_numeral k) {..<pred_numeral k}"
denote_while_n_iter_dim: fixes \<rho> :: "complex mat"   and m :: "nat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and M0 :: "complex mat"   and n :: "nat" assumes "\<rho> \<in> carrier_mat m m"   and "partial_density_operator \<rho>"   and "M1 \<in> carrier_mat m m"   and "adjoint M1 * M1 \<le>\<^sub>L 1\<^sub>m m"   and "\<And>\<rho>. \<lbrakk>\<rho> \<in> carrier_mat m m; partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> DS \<rho> \<in> carrier_mat m m \<and> partial_density_operator (DS \<rho>)" shows "denote_while_n_iter M0 M1 DS n \<rho> \<in> carrier_mat m m \<and> partial_density_operator (denote_while_n_iter M0 M1 DS n \<rho>)"
asp_z_Z: fixes z :: "int"   and x :: "int" shows "z *\<^sub>a ant x \<in> Z\<^sub>\<infinity>"
has_density_emeasure: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal"   and X :: "'c set" assumes "has_density M N f"   and "X \<in> sets M" shows "emeasure M X = set_nn_integral N X f"
limsup_subseq_lim: fixes u :: "nat \<Rightarrow> 'a" shows "\<exists>r. strict_mono r \<and> (u \<circ> r) \<longlonglongrightarrow> limsup u"
comp_fun_commute_const: fixes f :: "'c \<Rightarrow> 'c" shows "comp_fun_commute (\<lambda>_. f)"
post_mult_testNk_neg: fixes k :: "nat" assumes "\<not> f k" shows "post * testN k = 0\<^sub>m N N"
phi_diff: shows "(*) (1 / pi) \<circ> (\<lambda>x. if - 1 \<le> x \<and> x \<le> 1 then arccos x else if x < - 1 then - x + pi else 1 - x) \<circ> (\<lambda>t. 2 * x_coord (1 - t)) piecewise_C1_differentiable_on {0..1}"
quot_equality: fixes a :: "'d"   and b :: "'d" shows "(\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>) = (a \<sim> b)"
tendsto_norm_zero_iff: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> (0::'b)) F"
sharing_consis_nth_update: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and i :: "nat"   and \<O> :: "nat set"   and sb :: "'b memref list"   and p :: "'b"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<R> :: "nat \<Rightarrow> bool option" assumes "sharing_consis \<S> ts"   and "i < length ts"   and "sharing_consistent \<S> \<O> sb" shows "sharing_consis \<S> (ts[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
I: shows "Q \<Longrightarrow>\<^sub>t Q'"
simps_49: fixes x17 :: "nat"   and x11 :: "floatarith"   and x12 :: "floatarith" shows "Var x17 \<noteq> Add x11 x12"
intros_2: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and m :: "'Var ModeUpd" assumes "is_final c" shows "is_final (c@[m])"
exec_strip: shows "\<Gamma>\<turnstile> \<langle>strip_guards F c,s\<rangle> \<Rightarrow> t"
fsubset_refl: fixes A :: "'b fset" shows "A |\<subseteq>| A"
combine_set: fixes c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and p\<^sub>0\<^sub>L :: "int \<times> int"   and p\<^sub>1\<^sub>L :: "int \<times> int"   and p\<^sub>0\<^sub>R :: "int \<times> int"   and p\<^sub>1\<^sub>R :: "int \<times> int"   and l :: "int"   and ps :: "(int \<times> int) list" assumes "(c\<^sub>0, c\<^sub>1) = combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps" shows "c\<^sub>0 \<in> set ps \<and> c\<^sub>1 \<in> set ps \<or> c\<^sub>0 = p\<^sub>0\<^sub>L \<and> c\<^sub>1 = p\<^sub>1\<^sub>L \<or> c\<^sub>0 = p\<^sub>0\<^sub>R \<and> c\<^sub>1 = p\<^sub>1\<^sub>R"
context_equiv_entailment: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and P :: "'BExp set"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and P' :: "'BExp set" assumes "\<Gamma> =:\<^sub>P \<Gamma>'"   and "P' \<turnstile> P" shows "\<Gamma> =:\<^sub>P' \<Gamma>'"
dup_simps_1: fixes f :: "'a" assumes "arr f" shows "arr (dup f)"
agent_ii_abs_def: fixes a :: "'a"   and t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace" shows "agent_ii_abs a t \<equiv> {tLast t' |t'. t' \<in> SPR.jkbpC \<and> spr_jview a t' = spr_jview a t}"
span_scale: fixes x :: "'b"   and S :: "'b set"   and c :: "'b" assumes "x \<in> ideal S" shows "c * x \<in> ideal S"
sym: fixes s :: "'a"   and t :: "'a" assumes "s = t" shows "t = s"
arc_connected_trans: fixes g :: "real \<Rightarrow> 'b"   and h :: "real \<Rightarrow> 'b" assumes "arc g"   and "arc h"   and "pathfinish g = pathstart h"   and "pathstart g \<noteq> pathfinish h" obtains i :: "real \<Rightarrow> 'b" where "arc i" and "path_image i \<subseteq> path_image g \<union> path_image h" and "pathstart i = pathstart g" and "pathfinish i = pathfinish h"
insert_ident: fixes x :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "x \<notin> A"   and "x \<notin> B" shows "(insert x A = insert x B) = (A = B)"
hom_induced_empty_hom: fixes p :: "int"   and X :: "'b topology"   and X' :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "hom_induced p X {} X' {} f \<in> hom (homology_group p X) (homology_group p X')"
simps_2: fixes f :: "'b \<Rightarrow> bool"   and as :: "'b negation_type list"   and ass :: "'b negation_type list list" shows "dnf_to_bool f (as # ass) = (cnf_to_bool f as \<or> dnf_to_bool f ass)"
snd3_def: shows "snd3 \<equiv> fst \<circ> snd"
ex_y: shows "\<forall>x. x \<in># X \<longrightarrow> (\<exists>y. y \<in># Y \<and> R x y)"
read_dcache2_mod_state: fixes s :: "'b sparc_state"   and addr :: "word32"   and new_undef :: "bool"   and new_traps :: "Trap set"   and new_state_var :: "sparc_state_var"   and new_dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and new_user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32" shows "fst (case read_data_cache s addr of None \<Rightarrow> (None, s) | Some w \<Rightarrow> (Some w, s)) = fst (case read_data_cache (s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>) addr of None \<Rightarrow> (None, s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>) | Some w \<Rightarrow> (Some w, s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>))"
inv_f_f: fixes x :: "'a" shows "inv (inv (inv hom)) (inv (inv hom) x) = x"
y_def: shows "y = c_assoc_value (pr_gr upb1) (c_pair (c_snd (c_snd n)) x)"
Span_subgroup_props_4: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list"   and v :: "'a" assumes "ring R"   and "subfield K R"   and "set Us \<subseteq> carrier R"   and "v \<in> ring.Span R K Us" shows "\<ominus>\<^bsub>R\<^esub> v \<in> ring.Span R K Us"
ex_point_lg: fixes l :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p" assumes "QCong l" shows "\<exists>B. l A B"
stermsl_sterms: fixes q :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp" assumes "q \<in> stermsl p"   and "not_call q"   and "wellformed \<Gamma>" shows "q \<in> sterms \<Gamma> p"
assms_3: shows "m \<noteq> m' \<and> dominates g m m'"
fmrel_on_fsubset: fixes S :: "'d fset"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'d \<rightharpoonup>\<^sub>f 'e"   and n :: "'d \<rightharpoonup>\<^sub>f 'f"   and T :: "'d fset" assumes "fmrel_on_fset S R m n"   and "T |\<subseteq>| S" shows "fmrel_on_fset T R m n"
vfun_le: fixes t1 :: "(val \<times> val) fset"   and t2 :: "(val \<times> val) fset" assumes "fset t1 \<subseteq> fset t2" shows "VFun t1 \<sqsubseteq> VFun t2"
prems_10: shows "InvariantQCharacterization (getConflictFlag state') (getQ state') (getF state') (getM state')"
thr: shows "- row i A \<in> vec.span {row j A |j. j \<noteq> i}"
assms_3: shows "\<not> m cd\<^bsup>\<pi>\<^esup>\<rightarrow> n"
arr_char: fixes f :: "'a" shows "S.arr f = (C.arr f \<and> Ide (C.dom f) \<and> Ide (C.cod f))"
image_inverse_permutations: fixes S :: "'a set" shows "{inv p |p. p permutes S} = {p. p permutes S}"
DSTEP_append: fixes s :: "('c + 'd) list"   and t :: "('c + 'd) list"   and G :: "('c \<times> ('c, 'd) RHS) list"   and L :: "('c + 'd) list"   and R :: "('c + 'd) list" assumes "s -G\<rightarrow> t" shows "L \<cdot> s \<cdot> R -G\<rightarrow> L \<cdot> t \<cdot> R"
zero_hom_0_axioms: shows "zero_hom_0 (inv (inv hom))"
lastMarked_def: fixes t :: "('a \<times> bool) list" shows "lastMarked t = last (markedElements t)"
th0: shows "{(k, m). k + m \<le> n} = (\<Union>s\<in>{0..n}. \<Union>i\<in>{0..s}. {(i, s - i)})"
inv_is_inverse: fixes f :: "'b" assumes "\<phi>.B.iso f" shows "\<phi>.B.inverse_arrows f (\<phi>.B.inv f)"
less_eq_finite_3_def: fixes x :: "finite_3"   and y :: "finite_3" shows "(x \<le> y) = (x = y \<or> x < y)"
VectorSpaceHom_axioms: shows "VectorSpaceHom (\<cdot>) V (\<cdot>) T"
recursive: shows "G,A \<union> {{P} Methd-\<succ> {Q} | ms}|\<Turnstile>\<Colon>{{P} body G-\<succ> {Q} | ms}"
assms: shows "prime (p\<^sup>2 + N * q\<^sup>2) \<and> p\<^sup>2 + N * q\<^sup>2 dvd a\<^sup>2 + N * b\<^sup>2"
AsSeqExpr2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and Q :: "'e set"   and A :: "'e set"   and a2 :: "('e, 'd, 'f) ann"   and c2 :: "('e, 'd, 'f) com"   and a :: "'e set"   and a1 :: "('e, 'd, 'f) ann"   and c1 :: "('e, 'd, 'f) com" assumes "assertionsR \<Gamma> \<Theta> Q A a2 c2 a" shows "assertionsR \<Gamma> \<Theta> Q A (AnnComp a1 a2) (Seq c1 c2) a"
sigma_set_union_count: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and X :: "'c set"   and Y :: "'c set set" assumes "\<forall>y\<in>A. B y \<in> sigma_sets X Y"   and "countable A" shows "\<Union> (B ` A) \<in> sigma_sets X Y"
gamma_deriv_at_within_gen: fixes a :: "real"   and b :: "real"   and x :: "real"   and s :: "real set"   and \<gamma> :: "real \<Rightarrow> 'a" assumes "a < b"   and "x \<in> s"   and "s \<subseteq> {a..b}"   and "\<forall>x\<in>s. \<gamma> differentiable at x" shows "vector_derivative \<gamma> (at x within {a..b}) = vector_derivative \<gamma> (at x)"
P: shows "\<forall>P. eventually P net \<longrightarrow> y \<le> Sup (f ` Collect P)"
pred_entailment_def: fixes P :: "bexp set"   and P' :: "bexp set" shows "pred_entailment P P' \<equiv> \<forall>mem. pred P mem \<longrightarrow> pred P' mem"
assms_1: shows "a \<rightarrow>\<^sup>+ b"
comp_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "B.VxV.comp g f = (if B.seq (fst g) (fst f) \<and> B.seq (snd g) (snd f) then (fst g \<cdot> fst f, snd g \<cdot> snd f) else (B.null, B.null))"
widens_refl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and xs :: "ty list" shows "P \<turnstile> xs [\<le>] xs"
IH: shows "degree (p \<circ>\<^sub>p q) \<le> degree p * degree q"
arith_extra_simps_18: fixes a :: "'d" shows "(0::'d) * a = (0::'d)"
fmrestrict_set_null: fixes m :: "('c, 'd) fmap" shows "fmrestrict_set {} m = {$$}"
Cons_3: shows "\<forall>a\<in>set (zip (s # ss) (t # ts)). case a of (s, t) \<Rightarrow> s = t \<or> fst (f s t) \<or> fst (f t s)"
power_one_right: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.monoid_mult one times" shows "power.power one times a 1 = a"
wf_gt: shows "wfP (\<lambda>y x. y \<in> A \<and> x \<in> A \<and> gt y x)"
inf: fixes g :: "nat \<Rightarrow> V" shows "small (range g)"
obj_ty_widenD: fixes G :: "prog"   and obj :: "obj"   and t :: "ref_ty" assumes "G\<turnstile>obj_ty obj\<preceq>RefT t" shows "(\<exists>C. tag obj = CInst C) \<or> (\<exists>T k. tag obj = Arr T k)"
assms_2: shows "\<forall>(P, Q)\<in>RelB. CondB (P, Q)"
cases: fixes x :: "('c + 'd) + 'e" obtains a :: "'c" where "x = Inl (Inl a)"   | b :: "'d" where "x = Inl (Inr b)"   | c :: "'e" where "x = Inr c"
Gromov_product_le_dist_2: fixes e :: "'a"   and x :: "'a"   and y :: "'a" shows "Gromov_product_at e x y \<le> dist e y"
isof_app: fixes c1 :: "Const"   and t1 :: "Ty"   and t2 :: "Ty"   and c2 :: "Const" assumes "c1 isof t1 -> t2"   and "c2 isof t1" shows "c_app c1 c2 isof t2"
Gets_imp_analz_Spy: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> yahalom" shows "X \<in> analz (knows Spy evs)"
preR: shows "pre_adjust \<langle>l, (ab, lv), AA_Map.delete x r\<rangle>"
coinductive_eq_I: fixes a :: "'b llist"   and b :: "'b llist" assumes "llength a = llength b \<and> (\<forall>n. enat n < llength a \<longrightarrow> a $ n = b $ n)" shows "a = b"
left_unique_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "left_unique R = (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
disc_eq_case_2: fixes option :: "\<langle>'a\<rangle>\<^sub>\<bottom>" shows "(option \<noteq> \<bottom>) = (case option of \<bottom> \<Rightarrow> False | \<lfloor>uu_\<rfloor> \<Rightarrow> True)"
simps_10: fixes v :: "'f" shows "generalize_instr (IOp v) = IOp v"
run_stl: fixes xs :: "'a stream" assumes "run xs" shows "run (stl xs)"
evaluate_list_total: fixes env :: "v sem_env"   and s :: "'b state"   and es :: "exp0 list" shows "\<exists>s' r. evaluate_list True env s es (s', r)"
obs_consistent_conj1: fixes R :: "('e \<times> 'f) set"   and pi :: "'g \<Rightarrow> 'h"   and Sa :: "('e, 'h) spec"   and Sc :: "('f, 'g) spec"   and R' :: "('e \<times> 'f) set" assumes "obs_consistent R pi Sa Sc" shows "obs_consistent (R \<inter> R') pi Sa Sc"
Diff_Diff_Int: fixes A :: "'k set"   and B :: "'k set" shows "A - (A - B) = A \<inter> B"
v11_vconverse_app_in_vdomain: fixes y :: "V" assumes "y \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_uminus" shows "vreal_uminus\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vreal_uminus"
iso_in_hom_def: fixes f :: "'c"   and a :: "'c"   and b :: "'c" shows "C.iso_in_hom f a b \<equiv> \<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright> \<and> C.iso f"
cod_dom: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.VV.cod (B.VV.dom f) = B.VV.dom f"
zero_eq_lcm_iff: fixes a :: "'a"   and b :: "'a" shows "((0::'a) = lcm a b) = (a = (0::'a) \<or> b = (0::'a))"
Domain_converse: fixes r :: "('e \<times> 'd) set" shows "Domain (r\<inverse>) = Range r"
take_add: fixes i :: "nat"   and j :: "nat"   and xs :: "'b list" shows "xs \<down> (i + j) = xs \<down> i @ xs \<up> i \<down> j"
Abs_freelist_append: fixes xs :: "'b list"   and ys :: "'b list" shows "Abs_freelist (xs @ ys) = Abs_freelist xs + Abs_freelist ys"
distinct_2: fixes lam1' :: "lam"   and lam2' :: "lam"   and name :: "name" shows "lam1' \<degree> lam2' \<noteq> Var name"
one_reducible_to_refl: fixes A :: "nat set" shows "one_reducible_to A A"
filtrationI: fixes M :: "'c measure"   and F :: "'b \<Rightarrow> 'c measure" assumes "\<forall>t. subalgebra M (F t)"   and "\<forall>s t. s \<le> t \<longrightarrow> subalgebra (F t) (F s)" shows "Filtration.filtration M F"
args_to_set_alt: fixes gs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "args_to_set (gs, bs, ps) = fst ` set gs \<union> fst ` set bs \<union> fst ` fst ` set ps \<union> fst ` snd ` set ps"
ennreal_mult': fixes a :: "real"   and b :: "real" assumes "0 \<le> a" shows "ennreal (a * b) = ennreal a * ennreal b"
p_unfold: shows "p = pq * xx ^ min op oq * (pp div pq * xx ^ (op - min op oq))"
sums_zeta: assumes "1 < Re s" shows "(\<lambda>n. of_nat (Suc n) powr - s) sums zeta s"
Rep_run_cases: fixes y :: "nat \<Rightarrow> clock \<Rightarrow> bool \<times> 'b tag_const" assumes "y \<in> {\<rho>. \<forall>c. mono (\<lambda>n. time (\<rho> n c))}" obtains x :: "'b run" where "y = Rep_run x"
dpd3: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and domain_op :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.domain_pre_dioid_one plus times one less_eq less domain_op" shows "less_eq (domain_op x) one"
hyp: shows "lincomb f (set ws) = lincomb_list (\<lambda>i. f (ws ! i)) ws"
set_sorted_list_of_set: fixes less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and A :: "'x set" assumes "class.linorder less_eq less"   and "finite A" shows "set (linorder.sorted_list_of_set less_eq A) = A"
before_in_def: fixes x :: "'c"   and y :: "'c"   and xs :: "'c list" shows "x < y in xs = (index xs x < index xs y \<and> y \<in> set xs)"
inv_delete: fixes t :: "('a \<times> 'b) tree"   and x :: "'a" assumes "inv t" shows "inv (delete x t)"
simps: fixes a1 :: "'b"   and a2 :: "'b"   and Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" shows "a1 \<R>\<mapsto><Rel,Cal> a2 = ((\<exists>P Q. a1 = P \<and> a2 = Q \<and> (P, Q) \<in> Rel) \<or> (\<exists>P P'. a1 = P' \<and> a2 = P \<and> P \<longmapsto>Cal* P') \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> P \<R>\<mapsto><Rel,Cal> Q \<and> Q \<R>\<mapsto><Rel,Cal> R))"
im_sheaf_morphisms_def: fixes U :: "'c set"   and V :: "'c set" shows "im_sheaf_morphisms U V \<equiv> \<rho> (f \<^sup>\<inverse> S U) (f \<^sup>\<inverse> S V)"