hcju/isabelleps huggingface dataset:
ideal_prod_subTr: fixes I :: "'a set"   and J :: "'a set"   and C :: "'a set" assumes "ideal R I"   and "ideal R J"   and "ideal R C"   and "\<forall>i\<in>I. \<forall>j\<in>J. i \<cdot>\<^sub>r j \<in> C" shows "I \<diamondsuit>\<^sub>r J \<subseteq> C"
pm_set_def: fixes i :: "'e \<Rightarrow> 'e"   and A :: "'e set" shows "pm_set i A = A \<union> minus_set i A"
set_times_mono3: fixes a :: "'a"   and C :: "'a set"   and D :: "'a set" assumes "a \<in> C" shows "a *o D \<subseteq> C * D"
vec_upd_def: fixes s :: "('b, 'c) vec"   and i :: "'c"   and a :: "'b" shows "HS_VC_Spartan.vec_upd s i a = (\<chi>j. ((($) s)(i := a)) j)"
urel_integer_eq: fixes pp :: "integer"   and x :: "integer"   and y :: "int"   and x' :: "integer"   and y' :: "int" assumes "p = int_of_integer pp"   and "urel_integer x y"   and "urel_integer x' y'" shows "(x = x') = (y = y')"
reflp_not_less: shows "reflp (\<lambda>a b. cmp a b \<noteq> Less)"
funion_fminus: fixes A :: "'b fset"   and B :: "'b fset"   and C :: "'b fset" shows "A |\<union>| B |-| C = A |-| C |\<union>| (B |-| C)"
zero_less_abs_iff: fixes abs :: "'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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less zero (abs a) = (a \<noteq> zero)"
finite_range_constant_imp_connected: fixes S :: "'a set" assumes "\<And>f. \<lbrakk>continuous_on S f; finite (f ` S)\<rbrakk> \<Longrightarrow> f constant_on S" shows "connected S"
elims_3: fixes x :: "primfun"   and xa :: "primfun" assumes "\<not> x < xa" obtains x :: "nat" and y :: "nat" where "xa__ = LnChain x" and "xa = LnChain y" and "\<not> x < y"
fps_unit_factor_fls_regpart: fixes f :: "'a fls" assumes "0 \<le> fls_subdegree f" shows "unit_factor (fls_regpart f) = fls_base_factor_to_fps f"
initial_stateD: fixes sys :: "('g, 'h, 'i, 'j, 'k, 'l) pre_system_scheme"   and s :: "'i \<Rightarrow> ('g, 'h, 'j, 'k) com list \<times> 'h option \<times> 'k" assumes "initial_state sys s" shows "AT \<lparr>GST = s, HST = []\<rparr> = atC \<circ> PGMs sys \<and> INIT sys \<lparr>GST = s, HST = []\<rparr>\<down> \<and> (\<forall>p l. \<not> taken p l \<lparr>GST = s, HST = []\<rparr>)"
mset_subset_diff_self: fixes c :: "'a"   and B :: "'a multiset" assumes "c \<in># B" shows "B - {#c#} \<subset># B"
leg_simps_1: shows "C.arr HH\<nu>\<pi>\<rho>.leg0"
left: fixes x :: "'a"   and t :: "'a tree" assumes "x \<in> set_tree (left t)"   and "P x (left t)" shows "P x t"
no_marker_left_in_final_state: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and cid :: "nat" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final" shows "Marker \<notin> set (msgs final cid)"
Recv_ev: shows "ev = Recv i' (occurs_on ev) r u u' m"
set_intros_2: fixes y :: "'c"   and x2 :: "'c stream"   and x1 :: "'c" assumes "y \<in> sset x2" shows "y \<in> sset (x1 ## x2)"
left_null_space_def: fixes A :: "(('d, 'f) vec, 'e) vec" shows "left_null_space A = {x. x v* A = 0}"
simps_11: fixes sh :: "sheap"   and e :: "'a exp"   and M :: "char list"   and es :: "'a exp list" shows "iconf sh (e\<bullet>M(es)) = (case val_of e of None \<Rightarrow> iconf sh e \<and> \<not> sub_RIs es | \<lfloor>v\<rfloor> \<Rightarrow> iconfs sh es)"
right_neutral: fixes a :: "'b" shows "a OR (0::'b) = a"
lookup_combine: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m1 :: "('c, 'b) mapping"   and m2 :: "('c, 'b) mapping"   and x :: "'c" shows "lookup (combine f m1 m2) x = combine_options f (lookup m1 x) (lookup m2 x)"
totalFormulaImpliesTotalClause: fixes clause :: "Literal list"   and formula :: "Literal list list"   and valuation :: "Literal list" assumes "clause el formula"   and "CNF.total valuation (vars formula)" shows "CNF.total valuation (vars clause)"
induct: fixes x1 :: "('fun, 'var) Term.term set"   and x2 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and x3 :: "('fun, 'var) extstrand_step list"   and P :: "('fun, 'var) Term.term set \<Rightarrow> ('var \<Rightarrow> ('fun, 'var) Term.term) \<Rightarrow> ('fun, 'var) extstrand_step list \<Rightarrow> bool" assumes "sem\<^sub>e\<^sub>s\<^sub>t_c x1 x2 x3"   and "\<And>M\<^sub>0 \<I>. P M\<^sub>0 \<I> []"   and "\<And>M\<^sub>0 \<I> S t. \<lbrakk>sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S; P M\<^sub>0 \<I> S; ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c t \<cdot> \<I>\<rbrakk> \<Longrightarrow> P M\<^sub>0 \<I> (S @ [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)])"   and "\<And>M\<^sub>0 \<I> S t. \<lbrakk>sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S; P M\<^sub>0 \<I> S\<rbrakk> \<Longrightarrow> P M\<^sub>0 \<I> (S @ [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)])"   and "\<And>M\<^sub>0 \<I> S t t' a. \<lbrakk>sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S; P M\<^sub>0 \<I> S; Unifier \<I> t t'\<rbrakk> \<Longrightarrow> P M\<^sub>0 \<I> (S @ [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"   and "\<And>M\<^sub>0 \<I> S X F. \<lbrakk>sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S; P M\<^sub>0 \<I> S; ineq_model \<I> X F\<rbrakk> \<Longrightarrow> P M\<^sub>0 \<I> (S @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])"   and "\<And>M\<^sub>0 \<I> S t K M. \<lbrakk>sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S; P M\<^sub>0 \<I> S; ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c t \<cdot> \<I>; Ana t = (K, M); \<And>k. k \<in> set K \<Longrightarrow> ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c k \<cdot> \<I>\<rbrakk> \<Longrightarrow> P M\<^sub>0 \<I> (S @ [Decomp t])" shows "P x1 x2 x3"
new_Addr_def: fixes h :: "heap" shows "new_Addr h \<equiv> if \<exists>a. h a = None then \<lfloor>LEAST a. h a = None\<rfloor> else None"
totalorder_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "class.totalorder less_eq less \<equiv> total_ordering UNIV less_eq less"
eval_ternary_simps_3_1: fixes x :: "ternaryformula"   and T :: "ternaryvalue" shows "(eval_ternary_And (ternary_ternary_eval x) T = TernaryTrue) = (ternary_ternary_eval x = TernaryTrue \<and> T = TernaryTrue)"
cSUP_upper: fixes x :: "'k"   and A :: "'k set"   and f :: "'k \<Rightarrow> 'l multiset" assumes "x \<in> A"   and "subset_mset.bdd_above (f ` A)" shows "f x \<subseteq># Sup (f ` A)"
Dags_subdags: fixes t :: "dag"   and nodes :: "ref set"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and st :: "dag" assumes "t \<in> Dags nodes low high"   and "st \<le> t"   and "st \<noteq> Tip" shows "st \<in> Dags nodes low high"
le_left_mono: fixes x :: "'a"   and y :: "'a"   and a :: "'a" assumes "x \<le> y" shows "y \<le> a \<longrightarrow> x \<le> a"
assoc: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "a \<sqinter> b \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
strip_all_body_unchanged_iff_strip_all_single_body_unchanged: fixes B :: "Core.term" shows "(strip_all_body B = B) = (strip_all_single_body B = B)"
rep_eq: fixes x :: "rat" shows "map_prod int_of_integer int_of_integer (quotient_of' x) = quotient_of x"
indRelRTp_indRelRT_eq: fixes TRel :: "('procT \<times> 'procT) set" shows "indRelRTp (\<lambda>x xa. (x, xa) \<in> TRel) = (\<lambda>x. indRelRTinfix x TRel)"
push_forward_is_derivation: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a"   and x :: "'b \<Rightarrow> real"   and y :: "'b \<Rightarrow> real" assumes "\<And>x y. \<lbrakk>x \<in> src.diff_fun_space; y \<in> src.diff_fun_space\<rbrakk> \<Longrightarrow> X (x * y) = x p * X y + y p * X x"   and "x \<in> dest.diff_fun_space"   and "y \<in> dest.diff_fun_space"   and "p \<in> src.carrier" shows "push_forward X (x * y) = x (f p) * push_forward X y + y (f p) * push_forward X x"
finite_blocks: fixes b :: "'a set" assumes "b \<in># str_del_point_blocks x" shows "finite b"
acost_def: fixes f :: "'op"   and ss :: "'s list" shows "acost f ss = real (cost f ss) + \<Phi> (exec f ss) - sum_list (map \<Phi> ss)"
DirProds_group_iff: fixes G :: "'c \<Rightarrow> ('d, 'e) monoid_scheme"   and I :: "'c set" shows "Group.group (DirProds G I) = (\<forall>i\<in>I. Group.group (G i))"
ODEo_undef: fixes x :: "char"   and \<theta> :: "trm option" shows "(ODEo x \<theta> = undefg) = (\<theta> = undeft)"
drop_bit_take_bit: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit m (take_bit n a) = take_bit (n - m) (drop_bit m a)"
subsumed_by_one: shows "\<forall>v\<in>subsumees subs. \<exists>!v'. (v, v') \<in> subs"
X5_ALL: fixes P :: "X5 set \<Rightarrow> bool" shows "(\<forall>X''. P X'') = (\<forall>X''\<in>set ` set (subseqs [X5.Xd1, X5.Xd1', X5.Xd2, X5.Xd2', Xd2'']). P X'')"
emb_Liminf_llist_mono_aux: fixes Xs :: "'c set llist"   and Ys :: "'c set llist"   and i :: "nat"   and x :: "'c" assumes "emb Xs Ys"   and "\<not> lfinite Xs"   and "\<not> lfinite Ys"   and "\<forall>j\<ge>i. x \<in> lnth Ys j" shows "\<forall>j\<ge>i. x \<in> lnth Xs j"
prem: fixes x :: "'b"   and y :: "'c"   and x' :: "'b"   and y' :: "'c" assumes "h (x, y) = h (x', y')" shows "x = x'"
map_vals_of_spec: fixes es :: "'c exp list"   and vs :: "val list" assumes "map_vals_of es = \<lfloor>vs\<rfloor>" shows "es = map Val vs"
preimage_of_inter: fixes f :: "'c \<Rightarrow> 'd"   and V :: "'d set"   and V' :: "'d set"   and X :: "'c set" shows "f \<^sup>\<inverse> X (V \<inter> V') = f \<^sup>\<inverse> X V \<inter> f \<^sup>\<inverse> X V'"
max_less_iff_conj: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.linorder less_eq less" shows "less (ord.max less_eq x y) z = (less x z \<and> less y z)"
silent_moves_intra_path_obs: fixes m' :: "'node"   and m :: "'node"   and S :: "'node SDG_node set"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and msx' :: "'node list" assumes "m' \<in> obs_intra m \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "length s = length (m # msx')"   and "Ball (set msx') return_node" obtains as' :: "'edge list" where "S,slice_kind S \<turnstile> (m # msx',s) =as'\<Rightarrow>\<^sub>\<tau> (m' # msx',s)"
eq: shows "monomial (lookup (p + monom_mult c t b) v) v = monomial (lookup p v) v + monomial (lookup (monom_mult c t b) v) v"
successive_snd_last: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b"   and y :: "'b" assumes "successive P (xs @ [x, y])"   and "P y" shows "\<not> P x"
scaleR_left_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and a :: "real" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "less_eq x y"   and "0 \<le> a" shows "less_eq (scaleR a x) (scaleR a y)"
consr: fixes xs :: "'a list"   and ys :: "'b list"   and x :: "'a"   and y :: "'b" assumes "list_all2 Q xs ys"   and "Q x y"   and "P xs ys" shows "P (x # xs) (y # ys)"
wt_elim_cases_12: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and va :: "Term.var"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (Acc va)\<Colon>T" obtains T :: "ty" where "Ta__ = Inl T" and "E,dt\<Turnstile>va\<Colon>=T"
split_paired_all: fixes P :: "'e \<times> 'f \<Rightarrow> prop" shows "(\<And>x. PROP P x) \<equiv> (\<And>a b. PROP P (a, b))"
inv_inv: fixes dir :: "dir" shows "Grid_Point.inv (Grid_Point.inv dir) = dir"
simps_3: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "less_eq_list (x # xs) (y # ys) = (x < y \<or> x = y \<and> less_eq_list xs ys)"
set_mult_assoc: fixes M :: "'a set"   and H :: "'a set"   and K :: "'a set" assumes "M \<subseteq> carrier G"   and "H \<subseteq> carrier G"   and "K \<subseteq> carrier G" shows "M <#>\<^bsub>G\<^esub> H <#>\<^bsub>G\<^esub> K = M <#>\<^bsub>G\<^esub> (H <#>\<^bsub>G\<^esub> K)"
map_cong: fixes x :: "'c stream"   and ya :: "'c stream"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "x = ya"   and "\<And>z. z \<in> sset ya \<Longrightarrow> f z = g z" shows "smap f x = smap g ya"
CAS: fixes D\<^sub>a :: "nat set"   and \<theta> :: "nat \<Rightarrow> nat option"   and D\<^sub>c :: "nat set"   and t :: "nat"   and s\<^sub>e :: "expr"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and a :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and c :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and A :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and L :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and R :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and W :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set" assumes "D\<^sub>a \<subseteq> dom \<theta>"   and "D\<^sub>c \<subseteq> dom \<theta>"   and "eval_expr t s\<^sub>e = (D, f)" shows "\<theta>\<turnstile> (CAS (Tmp (D\<^sub>a, a)) (Tmp (D\<^sub>c, c)) s\<^sub>e A L R W, t) \<rightarrow>\<^sub>p ((Skip, Suc (t + used_tmps s\<^sub>e)), issue_expr t s\<^sub>e @ [RMW (a \<theta>) (t + used_tmps s\<^sub>e) (D, f) (\<lambda>\<theta>. the (\<theta> (t + used_tmps s\<^sub>e)) = c \<theta>) (\<lambda>v\<^sub>1 v\<^sub>2. v\<^sub>1) (A \<theta>) (L \<theta>) (R \<theta>) (W \<theta>)])"
Q_def: shows "Q \<equiv> {q. not_blocked p R q}"
zeroA_minus: fixes a :: "'e"   and i :: "'e \<Rightarrow> 'e"   and z :: "'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z1 :: "'e" assumes "ipp_cond1 {a} i"   and "zeroA z i f {a} z1" shows "zeroA z i f {\<^sub>i- a} z1"
ag_inv_unique: fixes A :: "('c, 'd) aGroup_scheme"   and x :: "'c"   and y :: "'c" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "x \<plusminus>\<^bsub>A\<^esub> y = \<zero>\<^bsub>A\<^esub>" shows "y = -\<^sub>a\<^bsub>A\<^esub> x"
proj_finite2: fixes t :: "cnf llist"   and n :: "nat"   and c :: "'id" assumes "\<forall>n'. enat n' \<le> llength t \<longrightarrow> n \<le> n' \<longrightarrow> \<not> \<parallel>c\<parallel>\<^bsub>lnth t n'\<^esub>" shows "lfinite (\<pi>\<^bsub>c\<^esub>t)"
tconfD: fixes P :: "'a prog"   and h :: "'heap"   and t :: "'thread_id" assumes "P,h \<turnstile> t \<surd>t" shows "\<exists>C. typeof_addr h (thread_id2addr t) = \<lfloor>Class_type C\<rfloor> \<and> P \<turnstile> C \<preceq>\<^sup>* Thread"
powsum_shift1: fixes x :: "'a"   and m :: "nat"   and n :: "nat" shows "x \<cdot> x\<^bsub>m\<^esub>\<^bsup>n\<^esup> = x\<^bsub>m + 1\<^esub>\<^bsup>n\<^esup>"
\<nu>LTL_def: shows "\<nu>LTL = {\<phi>. is_\<nu>LTL \<phi>}"
Cons: fixes xs :: "'c list"   and ys :: "'c list"   and z :: "'c" assumes "xs <~~> ys" shows "z # xs <~~> z # ys"
SCallInit: fixes P :: "(char list list \<times> char list exp) prog"   and ps :: "char list exp list"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and vs :: "val list"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "sfields"   and sh\<^sub>1 :: "sheap"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and v' :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and l\<^sub>2' :: "sfields"   and e' :: "char list exp"   and h\<^sub>3 :: "heap"   and l\<^sub>3 :: "sfields"   and sh\<^sub>3 :: "sheap" assumes "P \<turnstile> \<langle>ps,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = (pns, body) in D"   and "\<nexists>sfs. sh\<^sub>1 D = \<lfloor>(sfs, Done)\<rfloor>"   and "M \<noteq> clinit"   and "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle> \<Rightarrow> \<langle>Val v',(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "length vs = length pns"   and "l\<^sub>2' = [pns [\<mapsto>] vs]"   and "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2', sh\<^sub>2)\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, l\<^sub>3, sh\<^sub>3)\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, l\<^sub>2, sh\<^sub>3)\<rangle>"
comm_ring_1_ow_axioms: shows "comm_ring_1_ow U (*\<^sub>o\<^sub>w) 1\<^sub>o\<^sub>w (+\<^sub>o\<^sub>w) 0\<^sub>o\<^sub>w (-\<^sub>o\<^sub>w) uminus"
sel_3: fixes x1 :: "('c, 'd) hd" shows "fun (Hd x1) = Hd x1"
take_Suc_Cons: fixes n :: "nat"   and x :: "'d"   and xs :: "'d list" shows "take (Suc n) (x # xs) = x # take n xs"
limp_trace_def: fixes Z :: "('a \<times> ('b \<times> 'a) list) set"   and Y :: "('a \<times> ('b \<times> 'a) list) set" shows "limp_trace Z Y = \<Union> {X. t_prod X Y \<subseteq> Z}"
hyps: fixes p :: "dir list"   and q :: "dir list"   and t :: "('a, 'b) tm" assumes "length p + length q < length pa__ + length qa__"   and "take (length q) p \<noteq> q"   and "take (length p) q \<noteq> p" shows "position_of t (p @ [d1]) = position_of (emb_step_at q d2 t) (p @ [d1])"
S_imp_NS: fixes s :: "('f, 'a) Term.term"   and t :: "('f, 'a) Term.term" assumes "S s t" shows "NS s t"
MGT_implies_complete: fixes c :: "com"   and P :: "state \<Rightarrow> state \<Rightarrow> bool"   and Q :: "state \<Rightarrow> state \<Rightarrow> bool" assumes "{} \<tturnstile> {MGT c}"   and "\<Turnstile> {P} c {Q}" shows "{} \<turnstile> {P} c {Q}"
monom_eq_1: shows "Polynomial.monom (1::'a) 0 = 1"
subdomain_axioms_def: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme" shows "subdomain_axioms H R \<equiv> \<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub> \<and> (\<forall>h1 h2. h1 \<in> H \<longrightarrow> h2 \<in> H \<longrightarrow> h1 \<otimes>\<^bsub>R\<^esub> h2 = \<zero>\<^bsub>R\<^esub> \<longrightarrow> h1 = \<zero>\<^bsub>R\<^esub> \<or> h2 = \<zero>\<^bsub>R\<^esub>)"
plus_complex_interval_tendsto: fixes A :: "nat \<Rightarrow> complex_interval"   and a :: "complex"   and B :: "nat \<Rightarrow> complex_interval"   and b :: "complex" assumes "A \<longlonglongrightarrow>\<^sub>c a"   and "B \<longlonglongrightarrow>\<^sub>c b" shows "(\<lambda>i. A i + B i) \<longlonglongrightarrow>\<^sub>c a + b"
one_im_sheaf_def: shows "im0.one_im_sheaf \<equiv> \<lambda>V. 0::'e"
f_additive: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "is_conjugation f g" shows "f (x + y) = f x + f y"
circle_top_edge_def: shows "circle_top_edge = (- 1, \<lambda>t. (x_coord t * d, d * circle_y (x_coord t)))"
dom_eq_SomeD1: fixes m :: "'b \<Rightarrow> 'c option"   and n :: "'b \<Rightarrow> 'd option"   and x :: "'b"   and y :: "'c" assumes "dom m = dom n"   and "m x = Some y" shows "n x \<noteq> None"
det_eq_0_rank: fixes A :: "((real, 'a) vec, 'a) vec" shows "(det A = 0) = (rank A < CARD('a))"
lossless_exec_gpv: fixes gpv :: "('c, 'a, 'b) gpv"   and callee :: "'d \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'd) spmf"   and s :: "'d" assumes "lossless_gpv \<I>_full gpv"   and "\<And>s out. \<lbrakk>out \<in> outs_\<I> \<I>_full; True\<rbrakk> \<Longrightarrow> lossless_spmf (callee s out)"   and "\<I>_full \<turnstile>g gpv \<surd>"   and "True" shows "lossless_spmf (exec_gpv callee gpv s)"
star_n_diff: fixes X :: "nat \<Rightarrow> 'c"   and Y :: "nat \<Rightarrow> 'c" shows "star_n X - star_n Y = star_n (\<lambda>n. X n - Y n)"
conj_mono_left: fixes c\<^sub>0 :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "(c\<^sub>0::'b::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'b::refinement_lattice)" shows "(c\<^sub>0::'b::refinement_lattice) \<iinter> (d::'b::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'b::refinement_lattice) \<iinter> d"
zero_U_single_not_in_FIN: shows "{0\<^sup>\<infinity>} \<union> U_single \<notin> FIN"
homeomorphic_map_path_component_of: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "homeomorphic_map X Y f"   and "x \<in> topspace X" shows "path_component_of_set Y (f x) = f ` path_component_of_set X x"
Gen_Shleg_def: fixes k :: "nat"   and x :: "real" shows "local.Gen_Shleg k x = poly (of_int_poly (local.gen_shleg_poly k)) x"
e_type_const1: fixes e :: "e"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list" assumes "is_const e" shows "\<exists>t. \<S>\<bullet>\<C> \<turnstile> [e] : ts _> ts @ [t]"
unconst_const: fixes name :: "name" shows "unconst (const name) = Some name"
sum_mono: fixes K :: "'d set"   and f :: "'d \<Rightarrow> 'e multiset"   and g :: "'d \<Rightarrow> 'e multiset" assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<subseteq># g i" shows "comm_monoid_add.sum (+) {#} f K \<subseteq># comm_monoid_add.sum (+) {#} g K"
set_category_given_img_axioms: shows "set_category_given_img (\<cdot>\<acute>) S'.some_img \<AA>"
case_2: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and x21 :: "nat"   and x22 :: "nat" shows "(case Eq x21 x22 of Less x xa \<Rightarrow> f1 x xa | Eq x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
degree_prod_sum_eq: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b poly" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0" shows "degree (prod f A) = (\<Sum>x\<in>A. degree (f x))"
g1': shows "nextnat (s |\<^sub>s n) = 0 \<or> nextnat (s |\<^sub>s n) = 1"
False: shows "\<not> (u = v \<or> u = w \<or> v = w)"
imp_conj_lemma: fixes P :: "bool"   and Q :: "bool"   and R :: "bool" assumes "P \<Longrightarrow> Q \<longrightarrow> R" shows "P \<and> Q \<longrightarrow> R"
coalesce_eq_OK_ub_D: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and xs :: "'b list"   and n :: "nat"   and ys :: "'b list"   and zs :: "'b list"   and us :: "'b list" assumes "semilat (err A, Err.le r, lift2 f)"   and "xs \<in> list n A"   and "ys \<in> list n A"   and "coalesce (xs [+f] ys) = OK zs \<and> xs [\<sqsubseteq>\<^bsub>r\<^esub>] us \<and> ys [\<sqsubseteq>\<^bsub>r\<^esub>] us \<and> us \<in> list n A" shows "zs [\<sqsubseteq>\<^bsub>r\<^esub>] us"
fdisjnt_alt_def: fixes M :: "'b fset"   and N :: "'b fset" shows "fdisjnt M N = (M |\<inter>| N = {||})"
that: fixes l :: "'b list"   and l' :: "'b list" assumes "ls = l @ l'"   and "map f l = fl"   and "map f l' = fl'" shows "thesis"
multiset_inter_assoc: fixes A :: "'k multiset"   and B :: "'k multiset"   and C :: "'k multiset" shows "A \<inter># (B \<inter># C) = A \<inter># B \<inter># C"