hcju/isabelleps huggingface dataset:
wf_implies_termi_reach: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list"   and s :: "('d, 'f) xstate"   and c1 :: "('d, 'e, 'f) com list \<times> (('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list \<times> ('d, 'f) xstate"   and cs1 :: "('d, 'e, 'f) com list"   and css1 :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list"   and s1 :: "('d, 'f) xstate" assumes "wf {(c2, c1). \<Gamma>\<turnstile> (cs, css, s) \<rightarrow>\<^sup>* c1 \<and> \<Gamma>\<turnstile> c1 \<rightarrow> c2}"   and "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow>\<^sup>* c1"   and "c1 = (cs1, css1, s1)" shows "\<Gamma>\<turnstile>cs1,css1 \<Down> s1"
ndet_id: fixes P :: "'a process" shows "(P |-| P) = P"
of_int_eq_iff: 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 w :: "int"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus w = ring_1.of_int one plus zero minus z) = (w = z)"
bijI: fixes f :: "'d \<Rightarrow> 'e" assumes "inj f"   and "surj f" shows "bij f"
cong: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_rvar_incdec_eq :: "('b, 'c) Direction \<Rightarrow> ('b, 'c) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'b list + nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> ('b, 'c) state \<Rightarrow> ('b, 'c) state" shows "PivotUpdateMinVars.check' eq_idx_for_lvar min_rvar_incdec_eq pivot_and_update \<equiv> PivotUpdateMinVars.check' eq_idx_for_lvar min_rvar_incdec_eq pivot_and_update"
is_finite_weak_ranking_Cons_empty: fixes xs :: "'b set list" shows "\<not> is_finite_weak_ranking ({} # xs)"
return_partition_basics_3: fixes A :: "'a set" assumes "A \<in> sets MA" shows "(\<Union>n. A.return_partition A (n + 1)) = A.recurrent_subset A"
iT_Plus_iEx_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<diamond> t I \<oplus> k. P t) = (\<diamond> t I. P (t + k))"
inf_absorb2: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.semilattice_inf inf less_eq less"   and "less_eq y x" shows "inf x y = y"
circ_left_unfold: fixes x :: "'a" shows "(1::'a) \<squnion> x\<^sup>+ = x\<^sup>\<star>"
leading_monomial_tail: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "p = monomial (ordered_term.lc term_of_pair ord_term p) (ordered_term.lt term_of_pair ord_term p) + ordered_term.tail term_of_pair ord_term p"
trancl_image_by_rtrancl: fixes E :: "('b \<times> 'b) set"   and Vi :: "'b set" shows "E\<^sup>+ `` Vi \<union> Vi = E\<^sup>* `` Vi"
card_of_lists: fixes A :: "'a set" shows "|A| \<le>o |lists A|"
eq: shows "adjoint (tensor_P mat_O (1\<^sub>m K)) = tensor_P mat_O (1\<^sub>m K)"
Nml_HcompD_6: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml (t \<^bold>\<star> u)" shows "\<^bold>\<langle>src\<^sub>C (un_Prim t)\<^bold>\<rangle>\<^sub>0 = Trg u"
map_ds_map_SSup_inv: shows "map_ds \<circ> map_SSup = id"
LocalSecretsComposition_ine2_k: fixes k :: "Keys"   and PQ :: "specID"   and P :: "specID"   and Q :: "specID" assumes "kKS k \<in> LocalSecrets PQ"   and "subcomponents PQ = {P, Q}"   and "correctCompositionLoc PQ"   and "\<not> ine P (kE k)"   and "kKS k \<notin> LocalSecrets P"   and "kKS k \<notin> LocalSecrets Q" shows "ine Q (kE k)"
Pd'_d'_linear: shows "Linear_order (Pd' d')"
rtrancl_refl: fixes a :: "'x"   and r :: "('x \<times> 'x) set" shows "(a, a) \<in> r\<^sup>*"
distinct_1: fixes "term" :: "'a"   and psi :: "('a, 'b, 'c) psi"   and name' :: "name"   and input' :: "('a, 'b, 'c) input" shows "\<rparr>term.psi \<noteq> \<nu>name'input'"
radicalE: fixes f :: "'c"   and F :: "'c set" assumes "f \<in> \<surd>F" obtains m :: "nat" where "f ^ m \<in> F"
wf_lprod: fixes R :: "('b \<times> 'b) set" assumes "wf R" shows "wf (lprod R)"
mult_commute: fixes a :: "'c"   and b :: "'c" shows "a \<sqinter> b = b \<sqinter> a"
assms: shows "infer \<Gamma> (Fn x T A) = Some \<tau>"
analz_insert_freshCryptK: fixes evs :: "event list"   and K :: "nat"   and Seskey :: "nat"   and X :: "msg" assumes "evs \<in> orb"   and "Key K \<notin> analz (knows Spy evs)"   and "Seskey \<notin> range shrK" shows "(Crypt K X \<in> analz (insert (Key Seskey) (knows Spy evs))) = (Crypt K X \<in> analz (knows Spy evs))"
prems_3: shows "\<forall>aa. ListMem aa (a # as) \<longrightarrow> agree (snd aa) s"
set_mset_def: fixes M :: "'k multiset" shows "set_mset M = {x. 0 < count M x}"
StackMark_correct: shows "Hoare_dgr (R_a .. SetMarkInv) StackMark_a ((R_a .. SetMarkInv) \<sqinter> - grd (step StackMark_a))"
zero_dagger: shows "(0::'a)\<^sup>\<star> = (1::'a)"
assms_3: fixes args :: "term list"   and name :: "name" assumes "linears args"   and "\<And>arg. arg \<in> set args \<Longrightarrow> P arg" shows "P (name $$ args)"
exists_trace_for_any_i_j: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and i :: "nat"   and j :: "nat" assumes "\<exists>c'. trace c t c'"   and "i \<le> j" shows "trace (s c t i) (take (j - i) (drop i t)) (s c t j)"
P_def: shows "P = map_poly Re ((p - [:z:]) \<circ>\<^sub>p [:a, b - a:])"
joinable_def: fixes p :: "'a list"   and q :: "'a list" shows "joinable p q \<equiv> last p = hd q \<and> p \<noteq> [] \<and> q \<noteq> []"
D_ndet: fixes P :: "'a process"   and Q :: "'a process" shows "D (P |-| Q) = D P \<union> D Q"
inject_2: fixes x21 :: "char list"   and x22 :: "aexp"   and x23 :: "'a"   and y21 :: "char list"   and y22 :: "aexp"   and y23 :: "'a" shows "(x21 ::= x22 {x23} = y21 ::= y22 {y23}) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
OUTfromVCorrect1_data10: shows "OUTfromVCorrect1 data10"
Confidentiality_B_temporal: fixes B :: "agent"   and Tk :: "nat"   and A :: "agent"   and K :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace> \<in> parts (knows Spy evs)"   and "\<not> expiredK Tk evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos" shows "Key K \<notin> analz (knows Spy evs)"
is_span: shows "C.span f.dom.leg0 f.dom.leg1"
SourcesA81_L1: shows "Sources level1 sA81 = {sA6, sA71, sA81, sA91}"
assms_4: fixes \<sigma> :: "'\<sigma>" assumes "I {} \<sigma>" shows "P \<sigma>"
assms_2: shows "time (Rep_run \<rho> n K) = \<tau>"
wf_subset_mset_rel: shows "wf {(M, N). M \<subset># N}"
dvd_square_part_iff: fixes x :: "'b"   and n :: "'b" shows "(x dvd square_part n) = (x\<^sup>2 dvd n)"
segments_neq: assumes "segment G.face_cycle_succ rev.a' a' \<noteq> {} \<or> segment G.face_cycle_succ a' rev.a' \<noteq> {}" shows "segment G.face_cycle_succ a' rev.a' \<noteq> segment G.face_cycle_succ rev.a' a'"
lc_set_def: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "lc_set F = lc ` (F - {0})"
alpha_preserves_qGood1: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "X #= X'"   and "qGood X" shows "qGood X'"
g_def: shows "g \<equiv> Cn 1 r_enumdom [r_const i, recf.Id 1 0]"
InitDone\<^sub>1: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and P :: "nat exp prog"   and C' :: "char list"   and Cs :: "char list list"   and e :: "nat exp"   and h :: "heap"   and l :: "val list"   and e' :: "nat exp"   and s' :: "heap \<times> val list \<times> sheap" assumes "sh C = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile>\<^sub>1 \<langle>INIT C' (Cs,True) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
compact_space_topspace_empty: fixes X :: "'b topology" assumes "topspace X = {}" shows "compact_space X"
psubset_imp_subset: fixes A :: "'b set"   and B :: "'b set" assumes "order.greater B A" shows "order.greater_eq B A"
isObjV_def: fixes v :: "Value" shows "isObjV v = (case v of objV C a \<Rightarrow> True | _ \<Rightarrow> False)"
span_card_ge_dim: fixes B :: "'c set"   and V :: "'c set" assumes "B \<subseteq> V"   and "V \<subseteq> vs2.span B"   and "finite B" shows "vs2.dim V \<le> card B"
preprm_inv_def: fixes \<pi> :: "('b \<times> 'b) list" shows "preprm_inv \<pi> \<equiv> rev \<pi>"
IH: shows "norm ((\<Prod>n\<in>A. (1::'a) + f n) - (1::'a)) \<le> (\<Prod>n\<in>A. 1 + norm (f n)) - 1"
return_id: fixes f :: "('a, 'b, 'c) prog" shows "f \<bind> return = f"
proper_interval_simps_4: fixes x :: "'a"   and y :: "'a" shows "proper_interval (Some x) (Some y) = (\<exists>z>x. z < y)"
reg\<^sub>C_the_only_concrete_only_var: fixes v\<^sub>C :: "var\<^sub>C" assumes "v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1" shows "v\<^sub>C = reg\<^sub>C"
simps_5: fixes r :: "'a rexp"   and s :: "'a rexp" shows "lang (Times r s) = lang r \<cdot> lang s"
simps_2: fixes x :: "'b" shows "embed_dir (x, sort_dir.Right) = (x, Suc (Suc 0))"
topr: fixes x :: "'a" shows "- (x \<cdot> \<top>) \<cdot> \<top> = - (x \<cdot> \<top>)"
UP_simps_3: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "trg (UP \<mu>) = UP\<^sub>0 (trg\<^sub>B \<mu>)"
assms: fixes x :: "'a" shows "mf x \<equiv> RETURN (f x)"
id_oracle_def: fixes s :: "'e"   and x :: "'f" shows "id_oracle s x = Generative_Probabilistic_Value.Pause x (\<lambda>x. Generative_Probabilistic_Value.Done (x, s))"
yi_biggest: shows "\<forall>i\<in>set (map fst (xs @ [(oid, oper)] @ ys)). i < yi"
degree_unit_factor: fixes f :: "'b poly" shows "degree (unit_factor f) = 0"
idx_in_elem_inserted: fixes i :: "'c"   and e :: "'c \<times> 'd \<times> bool"   and n :: "'c option"   and xs :: "('c \<times> ('c, 'd) operation) event list" assumes "Deliver (i, Insert e n) \<in> set xs" shows "fst e \<in> set (indices xs)"
qpm_nat_inf_term_def: fixes to :: "(('a, 'b) pp \<times> nat) nat_term_order" shows "qpm_nat_inf_term to \<equiv> gd_nat_term (\<lambda>x. x) (\<lambda>x. x) to"
finfun_Diag_const2: fixes f :: "'a \<Rightarrow>f 'b"   and c :: "'c" shows "($f, K$ c$) = (\<lambda>b. (b, c)) \<circ>$ f"
assms_3: shows "snd (cs (S t (i + 1)) cid) \<noteq> NotStarted"
induct: fixes P :: "'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b up\<^sub>d \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> bool"   and a0 :: "'b tree234"   and a1 :: "'b"   and a2 :: "'b tree234"   and a3 :: "'b"   and a4 :: "'b up\<^sub>d"   and a5 :: "'b"   and a6 :: "'b tree234" assumes "\<And>t1 a t2 b t3 c t4. P t1 a t2 b (T\<^sub>d t3) c t4"   and "\<And>t1 a t2 b t3 c t4 d t5. P t1 a \<langle>t2, b, t3\<rangle> c (Up\<^sub>d t4) d t5"   and "\<And>t1 a t2 b t3 c t4 d t5 e t6. P t1 a \<langle>t2, b, t3, c, t4\<rangle> d (Up\<^sub>d t5) e t6"   and "\<And>t1 a t2 b t3 c t4 d t5 e t6 f t7. P t1 a \<langle>t2, b, t3, c, t4, d, t5\<rangle> e (Up\<^sub>d t6) f t7"   and "\<And>a b d v f g. P a b \<langle>\<rangle> d (Up\<^sub>d v) f g" shows "P a0 a1 a2 a3 a4 a5 a6"
Diff_eq_empty_iff: fixes A :: "'b set"   and B :: "'b set" shows "(A - B = {}) = order.greater_eq B A"
trelRS: shows "rel_reflects_binary_pred TRel (\<lambda>b. hasBarb b TWB)"
array_set_def: shows "array_set \<equiv> rec_array (\<lambda>A n a. Array (A[n := a]))"
no_zero_height_Exec_derivs1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and H :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and v :: "Val" assumes "(M, l, (os, S, H), 0, h, v) \<in> Exec" shows "False"
strict_trans: fixes b :: "'b word"   and a :: "'b word"   and c :: "'b word" assumes "b <s a"   and "c <s b" shows "c <s a"
all_components_smooth_one_pw_smooth_is_pw_smooth: fixes j :: "'a"   and f :: "real \<Rightarrow> 'a"   and s :: "real set" assumes "\<And>i. i \<in> Basis - {j} \<Longrightarrow> (\<lambda>x. f x \<bullet> i) C1_differentiable_on s"   and "(\<lambda>x. f x \<bullet> j) piecewise_C1_differentiable_on s" shows "f piecewise_C1_differentiable_on s"
wf_lderiv: fixes \<Sigma> :: "nat \<Rightarrow> 'a set"   and project :: "'a \<Rightarrow> 'a"   and embed :: "'a \<Rightarrow> 'a list"   and wf_atom :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and r :: "'b rexp"   and lookup :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "'a" assumes "Pi_Derivatives.embed \<Sigma> project embed"   and "alphabet.wf wf_atom n r" shows "alphabet.wf wf_atom n (embed.lderiv lookup embed w r)"
basic_cong_4: fixes p :: "int" assumes "2 < p" shows "[1 \<noteq> 0] (mod p)"
subst_pp_def: fixes f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and t :: "'e \<Rightarrow>\<^sub>0 nat" shows "subst_pp f t = (\<Prod>x\<in>keys t. f x ^ lookup t x)"
at_StatCh: fixes c :: "secprop set"   and A :: "agent"   and B :: "agent"   and M :: "atom list"   and H :: "chmsg set"   and At :: "atom" assumes "StatCh c A B (Msg M) \<in> H"   and "At \<in> set M" shows "At \<in> atoms H"
unat_p2: fixes n :: "nat" assumes "n < LENGTH('b)" shows "unat (2 ^ n) = 2 ^ n"
trancl_Int_subset: fixes r :: "('e \<times> 'e) set"   and s :: "('e \<times> 'e) set" assumes "r \<subseteq> s"   and "(r\<^sup>+ \<inter> s) O r \<subseteq> s" shows "r\<^sup>+ \<subseteq> s"
max_in_subcomplex: fixes Y :: "'a set set"   and y :: "'a set" assumes "order.greater_eq X Y \<and> SimplicialComplex Y"   and "y \<in> Y"   and "folding.chamber y" shows "SimplicialComplex.maxsimp Y y"
ring_iso_imp_img_ring: fixes R :: "('e, 'f) ring_scheme"   and h :: "'e \<Rightarrow> 'g"   and S :: "('g, 'h) ring_scheme" assumes "ring R"   and "h \<in> ring_iso R S" shows "ring (S\<lparr>zero := h \<zero>\<^bsub>R\<^esub>\<rparr>)"
in_tms: fixes i :: "nat"   and ap :: "abc_inst list" assumes "i < length (tm_of ap)" shows "\<exists>k<length ap. tm_of ap ! i \<in> set (tms_of ap ! k)"
OV_inv2_def: shows "OV_inv2 = {s. \<forall>r\<ge>v_state.next_round s. obsv_state.obs s r = Map.empty}"
CollectE: fixes a :: "'e"   and P :: "'e \<Rightarrow> bool" assumes "a \<in> {x. P x}" obtains "P a"
merge_dup_coeff: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset"   and xs :: "char list multiset"   and m :: "int"   and n :: "int" assumes "normalize_poly_p p q" shows "normalize_poly_p (add_mset (xs, m) (add_mset (xs, n) p)) (add_mset (xs, m + n) q)"
set_iterator_union_def: fixes it_a :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and it_b :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c" shows "set_iterator_union it_a it_b \<equiv> \<lambda>c f \<sigma>0. it_b c f (it_a c f \<sigma>0)"
assms: fixes x :: "'a" assumes "x \<in> A" shows "f x \<in> \<int>"
T_spaceM_stable_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat" assumes "qmpt M T"   and "x \<in> space M" shows "(T ^^ n) x \<in> space M"
MapsToMorDomCod_2: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e" assumes "Category C"   and "f \<approx>>\<^bsub>C\<^esub> g" shows "dom\<^bsub>C\<^esub> (f ;;\<^bsub>C\<^esub> g) = dom\<^bsub>C\<^esub> f"
NatTransP_def: fixes \<eta> :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans" shows "NatTransP \<eta> \<equiv> (Functor (NTDom \<eta>) \<and> Functor (NTCod \<eta>) \<and> NTCatDom \<eta> = CatDom (NTCod \<eta>)) \<and> NTCatCod \<eta> = CatCod (NTDom \<eta>) \<and> (\<forall>X. X \<in> obj\<^bsub>NTCatDom \<eta>\<^esub> \<longrightarrow> \<eta> $$ X maps\<^bsub>NTCatCod \<eta>\<^esub> NTDom \<eta> @@ X to NTCod \<eta> @@ X) \<and> (\<forall>f X Y. f maps\<^bsub>NTCatDom \<eta>\<^esub> X to Y \<longrightarrow> NTDom \<eta> ## f ;;\<^bsub>NTCatCod \<eta>\<^esub> (\<eta> $$ Y) = \<eta> $$ X ;;\<^bsub>NTCatCod \<eta>\<^esub> (NTCod \<eta> ## f))"
foo: shows "h' \<turnstile> map_M (to_tree_order \<circ> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r) (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr # disc_nodes_h3) \<rightarrow>\<^sub>r [cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr] # foo"
canonize_state_sim: fixes A :: "('b \<Rightarrow> bool) set"   and s :: "'b" shows "canonize A s ~A~ s"
assms: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "1 < degree p" shows "splitted p"
Cons_1: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and vtx2 :: "'a"   and vtx1 :: "'a" assumes "\<forall>x y. x \<le> f x y \<and> y \<le> f x y"   and "\<forall>x y. x \<le> g x y \<and> y \<le> g x y"   and "\<forall>vtx. ListMem vtx G \<longrightarrow> \<not> R vtx vtx"   and "R vtx2 vtx1"   and "ListMem vtx1 G"   and "top_sorted_abs R G" shows "f (w vtx2) (wlp R w g f vtx1 G) \<le> wlp R w g f vtx2 G"
less_eq_type_def: shows "(\<le>) \<equiv> (\<sqsubset>)\<^sup>*\<^sup>*"
fd_subdist: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows " |x\<rangle> (ad (ad y) \<cdot> ad (ad z)) \<le>  |x\<rangle> y \<cdot>  |x\<rangle> z"
hash_embedded_simp: fixes h :: "'b \<Rightarrow> 'a"   and e :: "'c \<Rightarrow> 'b"   and st :: "'c rose_tree" shows "hash_tree h (embed_source_tree e st) = hash_source_tree (h \<circ> e) st"
inverse_scaleC_distrib: fixes a :: "complex"   and x :: "'b" shows "inverse (a *\<^sub>C x) = inverse x /\<^sub>C a"
Q_invar_when_discharged1: fixes u :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'capacity"   and Q :: "nat list" assumes "QD_invar u f Q"   and "excess f u = (0::'capacity)" shows "Q_invar f Q"
V: shows "(\<V>, \<V>') \<in> \<langle>var_rel\<rangle>set_rel"
is_generalised_output_of_def: fixes t' :: "transition"   and t :: "transition"   and r :: "nat"   and p :: "nat" shows "is_generalised_output_of t' t r p = (t' = generalise_output t r p)"
Ord_succ: fixes k :: "hf" assumes "Ord k" shows "Ord (succ k)"
S3: shows "c_fold (c_fst u # c_unfold k (c_snd u)) = c_pair (c_fst u) (c_fold (c_unfold k (c_snd u)))"
dghm_id_def: fixes \<CC> :: "V" shows "dghm_id \<CC> = [vid_on (\<CC>\<lparr>Obj\<rparr>), vid_on (\<CC>\<lparr>Arr\<rparr>), \<CC>, \<CC>]\<^sub>\<circ>"
bounded_linear_axioms_def: fixes f :: "'d \<Rightarrow> 'e" shows "bounded_linear_axioms f \<equiv> \<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
pref_iffI: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" assumes "|~ F \<longrightarrow> G"   and "|~ G \<longrightarrow> F" shows "|~ F = G"
monomial_pmdl_field: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and c :: "'c" assumes "is_monomial_set B"   and "p \<in> pmdl B"   and "v \<in> keys p" shows "monomial c v \<in> pmdl B"
diameter_bounded_1: fixes S :: "'b set" assumes "bounded S" shows "\<forall>x\<in>S. \<forall>y\<in>S. dist x y \<le> diameter S"
order_subst2: fixes a :: "'e"   and b :: "'e"   and f :: "'e \<Rightarrow> 'f"   and c :: "'f" assumes "a \<le> b"   and "f b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a \<le> c"
sup_aci_1: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
ctx_compose: fixes E1 :: "ctx"   and E2 :: "ctx"   and t :: "lam" shows "E1 \<circ> E2\<lbrakk>t\<rbrakk> = E1\<lbrakk>E2\<lbrakk>t\<rbrakk>\<rbrakk>"
intros_2: fixes xs :: "'a list"   and i :: "nat"   and y :: "'a"   and x :: "'a" assumes "nth_el1 xs i y" shows "nth_el1 (x # xs) (Suc i) y"
norm_cos_plus1_le: fixes z :: "complex" shows "cmod (1 + cos z) \<le> 2 * exp (cmod z)"
case_optionE: fixes P :: "bool"   and Q :: "'e \<Rightarrow> bool"   and x :: "'e option" assumes "case x of None \<Rightarrow> P | Some y \<Rightarrow> Q y" obtains "x = None" and "P"   | y :: "'e" where "x = Some y" and "Q y"
aezfun_restrict_and_extend0: fixes A :: "'a set"   and x :: "('b, 'a) aezfun" shows "aezfun x \<down> A \<in> aezfun_set"
map_simp_ide: fixes a :: "'a \<times> 'a" assumes "VV.ide a" shows "\<Phi>.map a = CMP (fst a) (snd a)"
ord_inj: fixes a :: "'a" assumes "a \<in> carrier (mult_of R)" shows "inj_on (([^]) a) {0..G.ord a - 1}"
empty_tree_empty_V: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set" assumes "TreeDecomposition G T bag"   and "V\<^bsub>T\<^esub> = {}" shows "V\<^bsub>G\<^esub> = {}"
BB: assumes "t1 \<in> B n"   and "t2 \<in> B n" shows "thesis"
fresh_subst: fixes x :: "name"   and t :: "lam"   and u :: "lam"   and y :: "name" assumes "x \<sharp> t"   and "x \<sharp> u" shows "x \<sharp> t[y::=u]"
DiffD2: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set"   and P :: "bool" assumes "c \<in> A - B"   and "c \<in> B" shows "P"
prems_1: shows "(Catch a1 a2 \<inter>\<^sub>g c2) = Some c"
HEndPhase1_valueChosen_inp: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and b :: "nat"   and bk :: "DiskBlock"   and r :: "Proc"   and v :: "InputsOrNi" assumes "HEndPhase1 s s' q"   and "Inv2a s"   and "b \<in> \<Union> (range Ballot)"   and "bk \<in> blocksOf s r"   and "bk \<in> blocksSeen s q"   and "b \<le> bal bk"   and "maxBalInp s b v"   and "Inv1 s" shows "inp (dblock s' q) = v"
ex_less_of_nat_mult: fixes x :: "'a"   and y :: "'a" assumes "(0::'a) < x" shows "\<exists>n. y < of_nat n * x"
simps_8: fixes f :: "'c \<Rightarrow> 'b" shows "map_option f None = None"
gso_connect: fixes us :: "rat vec list" shows "snd (gs.main us) = gram_schmidt n us"
distinct_1: fixes x1 :: "int" shows "Const x1 \<noteq> Any"
tm_comp_fetch_second_zero: fixes B :: "(action \<times> nat) list"   and s :: "nat"   and x :: "cell"   and a :: "action"   and A :: "(action \<times> nat) list" assumes "fetch B s x = (a, 0)"   and "tm_wf0 A"   and "s \<noteq> 0" shows "fetch (A |+| B) (s + length A div 2) x = (a, 0)"
dim_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "dim_vec (Matrix.vec n f) = n"
mUnionMT_def: fixes uu :: "'b itself" shows "mUnionMT uu = fold_mset (munionMT TYPE('b)) (Monad_Overloading.return {#})"
w: shows "wuntil \<phi> \<psi> \<pi>l"
intros_11: shows "String \<sqsubset>\<^sub>B OclAny"
select_convs_1: fixes carrier :: "'c set"   and pop :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mop :: "'c \<Rightarrow> 'c"   and zero :: "'c"   and tp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and un :: "'c"   and more :: "'d" shows "(\<cdot>\<^sub>r\<^bsub>\<lparr>carrier = carrier, pop = pop, mop = mop, zero = zero, tp = tp, un = un, \<dots> = more\<rparr>\<^esub>) = tp"
ir_While_False: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com" assumes "ir_hoare (\<lambda>s s'. P s s' \<and> \<not> bval b s) SKIP c' Q" shows "ir_hoare P (WHILE b DO c) c' Q"
type_definition_Tangle_Diagram: shows "type_definition Rep_Tangle_Diagram Abs_Tangle_Diagram {x. is_tangle_diagram x}"