hcju/isabelleps huggingface dataset:
well_base\<^sub>h_mono_hmset: fixes base :: "nat"   and M :: "hmultiset"   and N :: "hmultiset" assumes "2 \<le> base"   and "well_base\<^sub>h base M"   and "hmsetmset N \<subseteq># hmsetmset M" shows "well_base\<^sub>h base N"
gdd_index_zero_iff: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and index :: "int"   and ps :: "'b set" assumes "GDD point_set block_collection groups index"   and "ps \<subseteq> point_set"   and "card ps = 2" shows "(block_collection index ps = 0) = (size (filter_mset ((\<subseteq>) ps) (mset_set groups)) = 1)"
locals_upd_gobj: fixes r :: "oref"   and n :: "vn"   and v :: "val"   and s :: "st" shows "locals (upd_gobj r n v s) = locals s"
image_compactin: fixes X :: "'a topology"   and S :: "'a set"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "compactin X S"   and "continuous_map X Y f" shows "compactin Y (f ` S)"
NC2_not_CardSecret: fixes EKj :: "nat"   and K :: "nat"   and p :: "nat"   and D :: "agent"   and N :: "nat"   and evs :: "event list"   and EKi :: "nat"   and K' :: "nat"   and p' :: "nat" assumes "Crypt EKj \<lbrace>Key K, Pan p, Hash \<lbrace>Agent D, Nonce N\<rbrace>\<rbrace> \<in> parts (knows Spy evs)"   and "Key K \<notin> analz (knows Spy evs)"   and "Nonce N \<notin> analz (knows Spy evs)"   and "evs \<in> set_cr" shows "Crypt EKi \<lbrace>Key K', Pan p', Nonce N\<rbrace> \<notin> parts (knows Spy evs)"
lbutlast_snoc: fixes xs :: "'b llist"   and A :: "'b set"   and x :: "'b" assumes "xs \<in> A\<^sup>\<star>" shows "lbutlast (xs @@ x ## LNil) = xs"
oparright: fixes t :: "'c"   and a :: "'d seq_action"   and t' :: "'c"   and T :: "('c \<times> 'd seq_action \<times> 'c) set"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'e"   and s :: "'f"   and S :: "(((nat \<Rightarrow> 'e) \<times> 'f) \<times> 'd seq_action \<times> (nat \<Rightarrow> 'e) \<times> 'f) set" assumes "(t, a, t') \<in> T"   and "\<And>m. a \<noteq> send m"   and "\<sigma>' i = \<sigma> i" shows "((\<sigma>, s, t), a, \<sigma>', s, t') \<in> oparp_sos i S T"
mcont_cont: fixes luba :: "'f set \<Rightarrow> 'f"   and orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lubb :: "'g set \<Rightarrow> 'g"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g" assumes "mcont luba orda lubb ordb f" shows "cont luba orda lubb ordb f"
rel_spmf_return_spmf1: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and p :: "'c spmf" shows "rel_spmf R (return_spmf x) p = (lossless_spmf p \<and> (\<forall>a\<in>set_spmf p. R x a))"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>src\<^sub>B f : src\<^sub>B a \<Rightarrow>\<^sub>B src\<^sub>B b\<guillemotright>"
nxt_diverg: fixes s :: "nat \<Rightarrow> nat option"   and b :: "nat" assumes "total_cons s"   and "eval (total_cons.r_search s) [b] \<up>" shows "eval (total_cons.r_next s) [b] \<up>"
refl_on_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "refl_on A r = (r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. x \<succeq>[r] x))"
p_prob: shows "\<forall>\<^sup>\<infinity>n. 0 < p n \<and> p n < 1"
iter_state_def: fixes f :: "'a \<Rightarrow> ('d, 'e) state" shows "iter_state f \<equiv> wfrec {(nxt x, x) |x. cnt x} (\<lambda>rec x. if cnt x then f x \<bind> (\<lambda>_. rec (nxt x)) else State_Monad.return ())"
Done: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" assumes "g s = Done" shows "s \<in> productive_on g"
openin_Int_derived_set_of_eq: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin X S" shows "S \<inter> X derived_set_of T = S \<inter> X derived_set_of (S \<inter> T)"
step_Abrupt_end: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c\<^sub>1' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and x :: "'b" assumes "\<Gamma>\<turnstile> (c\<^sub>1, s) \<rightarrow> (c\<^sub>1', s')"   and "s' = Abrupt x" shows "s = Abrupt x"
gga_to_list_loc_def: fixes \<alpha> :: "'d \<Rightarrow> ('e, 'f) graph"   and invar :: "'d \<Rightarrow> bool"   and nodes_list_it :: "'d \<Rightarrow> ('e list \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'e list \<Rightarrow> 'e list) \<Rightarrow> 'e list \<Rightarrow> 'e list"   and edges_list_it :: "'d \<Rightarrow> (('e \<times> 'f \<times> 'e) list \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<times> 'e \<Rightarrow> ('e \<times> 'f \<times> 'e) list \<Rightarrow> ('e \<times> 'f \<times> 'e) list) \<Rightarrow> ('e \<times> 'f \<times> 'e) list \<Rightarrow> ('e \<times> 'f \<times> 'e) list" shows "gga_to_list_loc \<alpha> invar nodes_list_it edges_list_it \<equiv> graph \<alpha> invar \<and> graph_nodes_it \<alpha> invar nodes_list_it \<and> graph_edges_it \<alpha> invar edges_list_it"
ex_ivl_eq: fixes s :: "'a" assumes "s \<in> S" shows "ex_ivl s = T"
Mp_irreducible\<^sub>d_m: fixes m :: "int"   and f :: "int poly" shows "poly_mod.irreducible\<^sub>d_m m (poly_mod.Mp m f) = poly_mod.irreducible\<^sub>d_m m f"
pre_between_before: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "pre_between vs ram1 ram2" shows "before vs ram1 ram2 \<or> before vs ram2 ram1"
cases: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "finite_chains ord" obtains "\<And>Y. Complete_Partial_Order.chain ord Y \<Longrightarrow> finite Y"
the_inv_lconjby: fixes x :: "'a" shows "the_inv (lconjby x) = rconjby x"
invertible_det: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "invertible_mat A = (det A \<noteq> (0::'b))"
ts''_def_2: shows "\<C> \<turnstile> [e] : ts'' _> ts'"
init_inv: shows "\<forall>s. initial_state \<lparr>PGMs = gc_coms, INIT = gc_system_init, FAIR = \<lambda>s. True\<rparr> s \<longrightarrow> I \<lparr>GST = s, HST = []\<rparr>"
drinks_1_rejects_trace: fixes e :: "String.literal \<times> value list"   and r :: "nat \<Rightarrow>f value option"   and es :: "(String.literal \<times> value list) list" assumes "e \<noteq> (STR ''vend'', [])"   and "\<nexists>i. e = (STR ''coin'', [i])" shows "\<not> recognises_execution drinks 1 r (e # es)"
Spy_know_RPwd_iff: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> certified_mail" shows "(Key (RPwd A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
distinct_129: shows "GreaterThan \<noteq> BinAnd"
c_tr_singleton: fixes step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s :: "'g"   and x :: "'e" shows "c_tr step out s [x] = [(x, out s x)]"
symm: shows "universal (\<lambda>x y. x r y \<longrightarrow> y r x)"
dvd_cong_not_eq_mod: fixes xa :: "'b"   and c :: "'b"   and xb :: "'b" assumes "xa mod c \<noteq> xb"   and "c dvd xa mod c - xb"   and "c \<noteq> (0::'b)" shows "xb mod c \<noteq> xb"
reflexive_linorder_matrix_2: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "\<forall>i. f (i, i) = top" shows "matrix_idempotent_semiring.reflexive f"
assms_3: fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> \<F> \<and> T \<in> \<F>" shows "S \<subseteq> T \<or> T \<subseteq> S"
divide_less_eq: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and b :: "'e"   and c :: "'e"   and a :: "'e" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide b c) a = (if less zero c then less b (times a c) else if less c zero then less (times a c) b else less zero a)"
weakBisimSubstClosed: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "\<Psi> \<rhd> P \<approx>\<^sub>s Q" shows "p \<bullet> \<Psi> \<rhd> p \<bullet> P \<approx>\<^sub>s p \<bullet> Q"
d\<mu>_def: fixes i :: "nat"   and j :: "nat" shows "d\<mu> i j = int_of_rat (rat_of_int (d fs (Suc j)) * gs.\<mu> i j)"
ring_distribs_2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a + b) \<cdot> c = a \<cdot> c + b \<cdot> c"
simps_4: fixes r :: "'b rexp"   and s :: "'b rexp" shows "lang (Plus r s) = lang r \<union> lang s"
cSUP_upper2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and x :: "'c"   and u :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_above less_eq (f ` A)"   and "x \<in> A"   and "less_eq u (f x)" shows "less_eq u (Sup (f ` A))"
vbrelation_vconst_on: fixes A :: "V"   and c :: "V" shows "vbrelation (vconst_on A c)"
exec_to_exec_mark_guards: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate"   and f :: "'f" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "\<not> isFault t" shows "\<Gamma>\<turnstile> \<langle>mark_guards f c,s\<rangle> \<Rightarrow> t"
assms_2: shows "theta\<inverse> \<subseteq> ZOretr (theta\<inverse> \<union> ZObis)"
A_7: shows "v \<notin> \<lbrace>GE x c\<rbrace>"
known_ptr_defs: fixes ptr :: "(_) object_ptr" shows "ShadowRootClass.known_ptr ptr = (DocumentClass.known_ptr ptr \<or> is_shadow_root_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)"
dist_not_less_zero: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.metric_space dist uniformity open" shows "\<not> dist x y < 0"
x_3: shows "[x\<^sup>2 + 1 = 0] (mod p)"
restrict_id: fixes e :: "'a ifex"   and v :: "'a"   and v' :: "'a"   and val :: "bool" assumes "ifex_ordered e"   and "ifex_top_var e = Some v"   and "v' < v" shows "restrict e v' val = e"
mask_eq_sum_exp_nat: fixes n :: "nat" shows "2 ^ n - Suc 0 = sum ((^) 2) {q. q < n}"
fminus_finter2: fixes A :: "'a fset"   and C :: "'a fset"   and B :: "'a fset" shows "A |\<inter>| C |-| (B |\<inter>| C) = A |\<inter>| C |-| B"
Kp_def: shows "Kp \<equiv> \<lambda>(a, b). pair_pmf (K1 a) (K2 b)"
cSUP_singleton: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "\<Squnion> (f ` {x}) = f x"
midpoint_eq_endpoint_1: fixes a :: "'c"   and b :: "'c" shows "(midpoint a b = a) = (a = b)"
simps_1: fixes f :: "'a fps"   and a :: "'a" shows "fps_left_inverse_constructor f a 0 = a"
UINF_ind_const: fixes P :: "('a, 'b) uexpr" shows "(\<Sqinter> i \<bullet> P) = P"
that: fixes c :: "'a" assumes "c \<in> closed_segment a b"   and "d \<bullet> c = k"   and "\<And>z. z \<in> closed_segment a c \<Longrightarrow> d \<bullet> z \<le> k"   and "\<And>z. z \<in> closed_segment c b \<Longrightarrow> k \<le> d \<bullet> z" shows "thesis"
row_transpose: fixes j :: "nat"   and A :: "'b mat" assumes "j < dim_col A" shows "row A\<^sup>T j = col A j"
l_sort_out_def: fixes X :: "'b list \<times> 'b list \<times> 'b list" shows "l_sort_out X \<equiv> snd (snd X)"
lprefix_ltake: fixes u :: "'a llist"   and v :: "'a llist" assumes "u \<le> v" shows "u = ltake (llength u) v"
sup_ge1: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_sup sup less_eq less" shows "less_eq x (sup x y)"
obtainable_if_unreachable: fixes s :: "nat"   and e :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat \<Rightarrow>f value option" assumes "\<not> reachable s e" shows "\<not> obtainable s r e"
elims_2: fixes x :: "'a Vector"   and xa :: "'a Vector" assumes "x \<bottom> xa" obtains u :: "'a Vector" and v :: "'a Vector" where "x = u" and "xa = v" and "(u dot v) = (0::'a)"
\<mu>_non_zero: fixes M :: "'a measure"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "prob_space_with_indicators M I A" shows "0 < prob_space_with_indicators.\<mu> M I A"
assms_1: shows "(\<sigma>, p) \<in> oreachable A S U"
assms: shows "A' \<equiv> mult_row (interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n)) i ((1::'a) / interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) $ i $ j)"
measure_preserving_def: fixes M :: "'c measure"   and N :: "'d measure" shows "measure_preserving M N = {f \<in> M \<rightarrow>\<^sub>M N. \<forall>A\<in>sets N. emeasure M (f -` A \<inter> space M) = emeasure N A}"
ag_pOp_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier A"   and "y \<in> carrier A"   and "z \<in> carrier A" shows "x \<plusminus> y \<plusminus> z = x \<plusminus> (y \<plusminus> z)"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>B.\<ll> f : B.L a \<rightarrow>\<^sub>B B.map b\<guillemotright>"
sum_bounded_below: fixes A :: "'c set"   and K :: "'d"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>i. i \<in> A \<Longrightarrow> K \<le> f i" shows "of_nat (card A) * K \<le> sum f A"
word_less_cases: fixes x :: "'a word"   and y :: "'a word" assumes "x < y" shows "x = y - 1 \<or> x < y - 1"
length_stream_dec: fixes n :: "nat"   and FO :: "nat set"   and x :: "('a \<times> bool list) stream" shows "length (stream_dec n FO x) = n"
rep_eq: fixes x :: "'b interval"   and xa :: "'b interval" shows "bounds_of_interval (max_interval x xa) = (case bounds_of_interval x of (l1, u1) \<Rightarrow> \<lambda>(l2, u2). (max l1 l2, max u1 u2)) (bounds_of_interval xa)"
reduced_bdt_checkers_def: fixes ifex_of :: "'d \<Rightarrow> 'e ifex"   and val :: "'d \<Rightarrow> ('e \<Rightarrow> bool) \<Rightarrow> bool" shows "reduced_bdt_checkers ifex_of val \<equiv> (\<forall>b s. val_ifex (ifex_of b) s = val b s) \<and> (\<forall>b. reduced (ifex_of b) {})"
in_in_arcs_conv: fixes e :: "'a"   and G :: "('b, 'a) pre_digraph"   and v :: "'b" shows "(e \<in> in_arcs G v) = (e \<in> arcs G \<and> head G e = v)"
abc_comp_first_step_eq_pre: fixes s :: "nat"   and A :: "abc_inst list"   and lm :: "nat list"   and B :: "abc_inst list" assumes "s < length A" shows "abc_step_l (s, lm) (abc_fetch s (A [+] B)) = abc_step_l (s, lm) (abc_fetch s A)"
contour_integral_local_primitive_any: fixes g :: "real \<Rightarrow> complex"   and a :: "real"   and b :: "real"   and S :: "complex set"   and f :: "complex \<Rightarrow> complex" assumes "g piecewise_differentiable_on {a..b}"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> (\<forall>y. cmod (y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within S))"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> S" shows "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}"
build_rec_env_def: fixes funs :: "(char list \<times> char list \<times> exp0) list"   and cl_env :: "v sem_env"   and add_to_env :: "(char list, char list, v) namespace" shows "build_rec_env funs cl_env add_to_env = foldr (\<lambda>x. case x of (f, x, e) \<Rightarrow> nsBind f (Recclosure cl_env funs f)) funs add_to_env"
altern: shows "i = pmin \<or> i \<in> kmin f PR"
vertex_2_int_def: shows "vertex_2 \<equiv> 2"
one_stalk_def: fixes V :: "'a set" shows "one_stalk V \<equiv> \<lfloor>V, \<one>\<^bsub>V\<^esub>\<rfloor>"
vintersection_def: fixes A :: "V"   and B :: "V" shows "A \<inter>\<^sub>\<circ> B = set {x. x \<in>\<^sub>\<circ> A \<and> x \<in>\<^sub>\<circ> B}"
filter_insort_triv: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and x :: "'d"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" assumes "class.linorder less_eq less"   and "\<not> P x" shows "filter P (linorder.insort_key less_eq f x xs) = filter P xs"
inject_2: fixes x21 :: "nat"   and x22 :: "pol"   and y21 :: "nat"   and y22 :: "pol" shows "(Pinj x21 x22 = Pinj y21 y22) = (x21 = y21 \<and> x22 = y22)"
dvd_lcmI2: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a dvd c" shows "a dvd lcm b c"
deterministic: fixes cs :: "com \<times> (char list \<Rightarrow> int)"   and cs' :: "com \<times> (char list \<Rightarrow> int)"   and cs'' :: "com \<times> (char list \<Rightarrow> int)" assumes "cs \<rightarrow> cs'"   and "cs \<rightarrow> cs''" shows "cs'' = cs'"
times: shows "(R ===> R ===> R) (arith_ops_record.times ops) (*)"
initial_impl_def: fixes v0 :: "'v"   and I :: "'v \<Rightarrow> node_state option" shows "initial_impl v0 I \<equiv> GS_initial_impl I v0 (E `` {v0})"
token_fails_def: fixes x :: "nat" shows "token_fails x = (\<exists>n. sink (token_run x n) \<and> token_run x n \<notin> F)"
ring_axioms: shows "ring R' (+') (\<cdot>') \<zero>' \<one>'"
normNum_nilpotent: fixes x :: "int \<times> int" shows "normNum (normNum x) = normNum x"
sup_continuous_def: fixes F :: "'d \<Rightarrow> 'e" shows "sup_continuous F = (\<forall>M. incseq M \<longrightarrow> F (\<Squnion> range M) = (\<Squnion>i. F (M i)))"
pair_map_pmf2: fixes A :: "'e pmf"   and f :: "'g \<Rightarrow> 'f"   and B :: "'g pmf" shows "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
preserves_comp: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "V.seq g f" shows "src (g \<bullet> f) = src g \<bullet> src f"
effect_updateI: fixes h' :: "heap"   and r :: "'b ref"   and v :: "'b"   and h :: "heap"   and x :: "unit" assumes "h' = Ref.set r v h" shows "effect (r := v) h h' x"
lip: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.lip_order less_eq less"   and "less_eq x y" shows "\<forall>v w. less_eq x v \<and> less_eq v y \<and> less_eq x w \<and> less_eq w y \<longrightarrow> less_eq v w \<or> less_eq w v"
pseudo_proj_True_prefix: fixes n :: "nat"   and m :: "nat"   and x :: "bool stream"   and y :: "bool stream" assumes "n \<le> m"   and "pseudo_proj_True m x = pseudo_proj_True m y" shows "pseudo_proj_True n x = pseudo_proj_True n y"
stateful_strand_step_fv_subset_cases_2: fixes t :: "('e, 'f) Term.term"   and S :: "('e, 'f) stateful_strand_step list" assumes "receive\<langle>t\<rangle> \<in> set S" shows "fv t \<subseteq> fv\<^sub>s\<^sub>s\<^sub>t S"
left_alt: fixes xs :: "int list"   and j :: "nat" shows "left_spec xs j = (if has_left xs j then Some (the_left xs j) else None)"
case_prod_conv: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'f"   and a :: "'g"   and b :: "'h" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
shadow_root_eq2_h2: fixes ptr' :: "(_) element_ptr" shows "|h2 \<turnstile> get_shadow_root ptr'|\<^sub>r = |h3 \<turnstile> get_shadow_root ptr'|\<^sub>r"
WT_gpv_ContD: fixes \<Gamma> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and out :: "'a"   and c :: "('c, 'a, 'b) rpv"   and input :: "'b" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<Gamma> out" shows "\<Gamma> \<turnstile>g c input \<surd>"
mon_w_empty: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme" shows "mon_w fg [] = {}"
simps_2: fixes n :: "'a"   and y :: "'a"   and ys :: "'a list" shows "local.times n (y # ys) = (n * y) # local.times n ys"
zero_closed: shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
case_2: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and f3 :: "'d \<Rightarrow> 'a"   and x2 :: "'c" shows "(case In2 x2 of In1 x \<Rightarrow> f1 x | In2 x \<Rightarrow> f2 x | In3 x \<Rightarrow> f3 x) = f2 x2"
proper_implies_part: fixes x :: "i"   and y :: "i" assumes "PP x y" shows "P x y"
path_connected_continuous_image: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on S f"   and "path_connected S" shows "path_connected (f ` S)"
lhd_LCons_ltl: fixes llist :: "'a llist" assumes "\<not> lnull llist" shows "LCons (lhd llist) (ltl llist) = llist"
M_perp_to_compass: fixes l :: "proj2_line"   and m :: "proj2_line"   and a :: "proj2"   and b :: "proj2" assumes "M_perp l m"   and "a \<in> hyp2"   and "proj2_incident a l"   and "b \<in> hyp2"   and "proj2_incident b m" shows "\<exists>J. is_K2_isometry J \<and> apply_cltn2_line equator J = l \<and> apply_cltn2_line meridian J = m"
check_instr'_stk_offer: shows "ci_stk_offer check_instr'"
composite_imp_factors_nat: fixes m :: "nat" assumes "1 < m"   and "\<not> prime m" shows "\<exists>n k. m = n * k \<and> 1 < n \<and> n < m \<and> 1 < k \<and> k < m"
target: fixes T1 :: "'a"   and T2 :: "'a"   and TRel :: "('a \<times> 'a) set"   and Enc :: "'b \<Rightarrow> 'a" assumes "(T1, T2) \<in> TRel" shows "(TargetTerm T1, TargetTerm T2) \<in> encoding.indRelTEQ Enc TRel"
dvd_mult2: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a b" shows "dvd.dvd times a (times b c)"
mod_ring_def: fixes n :: "int"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a" shows "mod_ring n Rep Abs \<equiv> mod_type n Rep Abs"
E_def: shows "E = Pascal_Property.inter (line A C) (line R Q)"
invar_abs': fixes s :: "'C" assumes "s \<in> wa_invar WAC" shows "\<alpha> s \<in> wa_invar WAA"
in_carrier_def: fixes R :: "('c, 'd) ring_scheme"   and xs :: "'c list" assumes "cring R" shows "cring.in_carrier R xs = (\<forall>x\<in>set xs. x \<in> carrier R)"
statC_prop: shows "\<forall>statC. statT = ClassT statC \<longrightarrow> is_class G statC"
valid_prefix_E: fixes pf :: "'b prefix_match" assumes "valid_prefix pf" shows "pfxm_mask pf && pfxm_prefix pf = 0"
inter_union_distrib_left: fixes A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" shows "A \<inter># B + C = (A + C) \<inter># (B + C)"
remdups_wrt_rev_notin: fixes v :: "'a"   and vs :: "'a list"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "v \<in> set vs" shows "v \<notin> f ` set (remdups_wrt_rev f xs vs)"
simps: fixes a1 :: "'p"   and a2 :: "'p" shows "rel_12_23 a1 a2 = (a1 = \<^bold>1 \<and> a2 = \<^bold>2 \<or> a1 = \<^bold>2 \<and> a2 = \<^bold>3)"
p2r_conj_hom: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "p2r P \<inter> p2r Q = p2r (P \<sqinter> Q)"
bindE: fixes p :: "'c + 'e"   and f :: "'e \<Rightarrow> 'c + 'd"   and x :: "'d" assumes "p \<bind> f = Inr x" obtains y :: "'e" where "p = Inr y" and "f y = Inr x"
assms: shows "run_one_step d i (s, vs, ves, $Store x201 x202 x203 x204) = (s', vs', res)"
tm_wf_shift: fixes y :: "nat"   and tp :: "(action \<times> nat) list"   and off :: "nat" assumes "list_all (\<lambda>(acn, st). st \<le> y) tp" shows "list_all (\<lambda>(acn, st). st \<le> y + off) (shift tp off)"
iso_class_elems_isomorphic: fixes F :: "'a set"   and f :: "'a"   and f' :: "'a" assumes "is_iso_class F"   and "f \<in> F"   and "f' \<in> F" shows "f \<cong> f'"
assms: shows "(\<lambda>(x, y). f x y) abs_summable_on Sigma A B"
norm_matrix_le_op_norm: fixes x :: "('d, 'e) vec"   and A :: "(('d, 'e) vec, 'f) vec" assumes "\<parallel>x\<parallel> = 1" shows "\<parallel>A *v x\<parallel> \<le> \<parallel>A\<parallel>\<^sub>o\<^sub>p"
zhmset_of_enat_le_\<omega>\<^sub>z: fixes n :: "enat" shows "zhmset_of_enat n \<le> \<omega>\<^sub>z"
prems: shows "\<not> list_all (\<lambda>x. x = dir.Left) (a # p)"
exp_not_eq_zero: fixes x :: "'a" shows "exp x \<noteq> (0::'a)"
dom_comp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C (C g f)" shows "partial_magma.dom C (C g f) = partial_magma.dom C f"