hcju/isabelleps huggingface dataset:
l_new_character_data_get_disconnected_nodes_def: fixes get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_character_data_get_disconnected_nodes get_disconnected_nodes_locs \<equiv> \<forall>h new_character_data_ptr h' r ptr'. h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr \<longrightarrow> h \<turnstile> new_character_data \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_disconnected_nodes_locs ptr' \<longrightarrow> r h h'"
vs_fst: shows "hd vs = hd (fst (splitAt ram1 vs))"
height_Unbal_r: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat"   and d :: "nat"   and x :: "'b"   and r :: "'b tree"   and r' :: "'b tree"   and l :: "'b tree"   and y :: "'b" assumes "RBTi1 TYPE('b) bal_i"   and "RBTi1.ins bal_i n (d + 1) x r = Unbal r'"   and "bal_i n (height \<langle>l, y, r\<rangle> + d)" shows "height l < height r'"
to_gc_conf_append: fixes r :: "Vars.var list"   and r' :: "Vars.var list"   and c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" shows "to_gc_conf (r @ r') c = to_gc_conf r (to_gc_conf r' c)"
rep_eq: fixes x :: "'b pmf" shows "pmf x = (\<lambda>xa. measure_pmf.prob x {xa})"
is_valid_problem_sas_plus_then_strips_transformation_too: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "is_valid_problem_strips (\<phi> \<Psi> )"
SUP_cong: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d \<Rightarrow> 'e"   and D :: "'d \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
tl_set: fixes q :: "'c list" assumes "distinct q" shows "set (tl q) = set q - {hd q}"
Quotient_equiv_abs1: fixes R :: "'j \<Rightarrow> 'j \<Rightarrow> bool"   and Abs :: "'j \<Rightarrow> 'k"   and Rep :: "'k \<Rightarrow> 'j"   and T :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'j"   and y :: "'j" assumes "Quotient R Abs Rep T"   and "R x y" shows "T x (Abs y)"
that_1: assumes "l \<^bold>\<in> x" shows "thesis"
F: shows "Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<lfloor>Some ` (H ` ran (heap (pre_post \<tau>)) - {None})\<rfloor>\<rfloor>) = \<lfloor>\<lfloor>Some ` (H ` ran (heap (pre_post \<tau>)) - {None})\<rfloor>\<rfloor>"
widens_trans: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and Ss :: "ty list"   and Ts :: "ty list"   and Us :: "ty list" assumes "P \<turnstile> Ss [\<le>] Ts"   and "P \<turnstile> Ts [\<le>] Us" shows "P \<turnstile> Ss [\<le>] Us"
elem_set: fixes x :: "'d"   and xo :: "\<langle>'d\<rangle>\<^sub>\<bottom>" shows "(x \<in> set_option xo) = (xo = \<lfloor>x\<rfloor>)"
dis_log_alt_def: fixes \<G> :: "'b cyclic_group" shows "dis_log_alt \<G> \<equiv> 0 < order \<G>"
semiring_normalization_rules_7: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.comm_semiring_1 times one plus zero" shows "times a b = times b a"
sample_vars_union: fixes A :: "'v set"   and B :: "'v set"   and X :: "'v \<Rightarrow> int pmf"   and m :: "'m" assumes "finite A"   and "finite B" shows "sample_vars X (A \<union> B) m = sample_vars X A (sample_vars X B m)"
EndPhase0_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc" shows "EndPhase0 s s' p = (phase s p = 0 \<and> IsMajority {d. hasRead s p d p} \<and> (\<exists>b\<in>Ballot p. (\<forall>r\<in>allBlocksRead s p. mbal r < b) \<and> dblock s' = (dblock s)(p := (SOME r. r \<in> allBlocksRead s p \<and> (\<forall>s\<in>allBlocksRead s p. bal s \<le> bal r))\<lparr>mbal := b\<rparr>)) \<and> InitializePhase s s' p \<and> phase s' = (phase s)(p := 1) \<and> inpt s' = inpt s \<and> outpt s' = outpt s \<and> disk s' = disk s)"
Comp_in_Hom: fixes A :: "'o"   and B :: "'o"   and C :: "'o"   and f :: "'a"   and g :: "'a" assumes "A \<in> Obj"   and "B \<in> Obj"   and "C \<in> Obj"   and "f \<in> Hom A B"   and "g \<in> Hom B C" shows "Comp C B A g f \<in> Hom A C"
rel_pmf_return_pmf2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'a pmf"   and x :: "'b" shows "rel_pmf R M (return_pmf x) = (\<forall>a\<in>set_pmf M. R a x)"
alphabet_swapSyms: fixes a :: "'a"   and t :: "'a tree"   and b :: "'a" assumes "a \<in> alphabet t"   and "b \<in> alphabet t" shows "alphabet (swapSyms t a b) = alphabet t"
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 <#> H <#> K = M <#> (H <#> K)"
radical_sqrt_normal_form_lemma: fixes e :: "expr"   and r :: "expr" assumes "radicals e \<noteq> {}"   and "\<forall>s\<in>radicals e. r \<notin> radicals s"   and "r \<in> radicals e" shows "\<exists>a b. 0 \<le> \<lbrace>r\<rbrace> \<and> \<lbrace>e\<rbrace> = \<lbrace>a\<rbrace> + \<lbrace>b\<rbrace> * sqrt \<lbrace>r\<rbrace> \<and> radicals a \<union> radicals b \<union> radicals r \<subseteq> radicals e \<and> r \<notin> radicals a \<union> radicals b"
fac: shows "factors G (bs[0 := bs ! 0 \<otimes> u]) (b \<otimes> u)"
non_sync_interleaving: fixes t :: "'a event list"   and u :: "'a event list"   and C :: "'a event set" assumes "(set t \<union> set u) \<inter> C = {}" shows "setinterleaving (t, C, u) \<noteq> {}"
rational_approximation: fixes e :: "real"   and x :: "real" assumes "0 < e" obtains r :: "real" where "r \<in> \<rat>" and "\<bar>r - x\<bar> < e"
measurable_restrict_space1: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and \<Omega> :: "'b set" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "f \<in> restrict_space M \<Omega> \<rightarrow>\<^sub>M N"
sp_from_subsumee: fixes v1 :: "'a \<times> nat"   and v2 :: "'a \<times> nat"   and es :: "('a \<times> nat) edge list"   and v :: "'a \<times> nat" assumes "(v1, v2) \<in> subs"   and "SubRel.subpath g v1 es v subs"   and "out_edges g v1 = {}" shows "es = [] \<or> SubRel.subpath g v2 es v subs"
termi_mn: fixes xs :: "nat list"   and n :: "nat"   and f :: "recf"   and r :: "nat" assumes "length xs = n"   and "terminate f (xs @ [r])"   and "rec_exec f (xs @ [r]) = 0"   and "\<forall>i<r. terminate f (xs @ [i]) \<and> 0 < rec_exec f (xs @ [i])" shows "terminate (Mn n f) xs"
padd_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and p\<^sub>1 :: "'b \<times> 'b \<times> 'b"   and p\<^sub>2 :: "'b \<times> 'b \<times> 'b" assumes "class.ell_field divide inverse times one plus zero minus uminus" shows "ell_field.padd times one plus zero minus a p\<^sub>1 p\<^sub>2 = (let (x\<^sub>1, y\<^sub>1, z\<^sub>1) = p\<^sub>1; (x\<^sub>2, y\<^sub>2, z\<^sub>2) = p\<^sub>2 in if z\<^sub>1 = zero then p\<^sub>2 else if z\<^sub>2 = zero then p\<^sub>1 else let d\<^sub>1 = times x\<^sub>2 z\<^sub>1; d\<^sub>2 = times x\<^sub>1 z\<^sub>2; l = minus d\<^sub>1 d\<^sub>2; m = minus (times y\<^sub>2 z\<^sub>1) (times y\<^sub>1 z\<^sub>2) in if l = zero then if m = zero then ell_field.pdouble times one plus zero minus a p\<^sub>1 else (zero, zero, zero) else let h = minus (times (times (power.power one times m 2) z\<^sub>1) z\<^sub>2) (times (plus d\<^sub>1 d\<^sub>2) (power.power one times l 2)) in (times l h, minus (times (minus (times d\<^sub>2 (power.power one times l 2)) h) m) (times (times (power.power one times l 3) y\<^sub>1) z\<^sub>2), times (times (power.power one times l 3) z\<^sub>1) z\<^sub>2))"
mat_of_row_dim_2: fixes y :: "'a Matrix.vec" shows "dim_col (mat_of_row y) = dim_vec y"
IH_3: shows "num_syms s = num_syms (Fun g ts)"
l_set_shadow_root_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_shadow_root type_wf set_shadow_root set_shadow_root_locs \<equiv> ((\<forall>element_ptr disc_nodes h h'. writes (set_shadow_root_locs element_ptr) (set_shadow_root element_ptr disc_nodes) h h') \<and> (\<forall>h element_ptr shadow_root. type_wf h \<longrightarrow> element_ptr |\<in>| element_ptr_kinds h \<longrightarrow> h \<turnstile> ok set_shadow_root element_ptr shadow_root)) \<and> (\<forall>h element_ptr shadow_root. h \<turnstile> ok set_shadow_root element_ptr shadow_root \<longrightarrow> element_ptr |\<in>| element_ptr_kinds h) \<and> (\<forall>w element_ptr h h'. w \<in> set_shadow_root_locs element_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> object_ptr_kinds h = object_ptr_kinds h') \<and> (\<forall>w element_ptr h h'. w \<in> set_shadow_root_locs element_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> type_wf h = type_wf h')"
cases: fixes a1 :: "('a \<times> real) msllist"   and a2 :: "real \<Rightarrow> real"   and a3 :: "(real \<Rightarrow> real) list" assumes "is_expansion_aux a1 a2 a3" obtains f :: "real \<Rightarrow> real" and basis :: "(real \<Rightarrow> real) list" where "a1 = MSLNil" and "a2 = f__" and "a3 = basis__" and "\<forall>\<^sub>F x in at_top. f__ x = 0" and "length basis__ = Suc (expansion_level TYPE('a))"   | xs :: "('a \<times> real) msllist" and f :: "real \<Rightarrow> real" and C :: "'a" and b :: "real \<Rightarrow> real" and e :: "real" and basis :: "(real \<Rightarrow> real) list" where "a1 = MSLCons (C, e) xs" and "a2 = f" and "a3 = b # basis" and "is_expansion_aux xs (\<lambda>x. f x - eval C x * b x powr e) (b # basis)" and "is_expansion C basis" and "\<And>e'. e < e' \<Longrightarrow> f \<in> o(\<lambda>x. b x powr e')" and "ms_exp_gt e (ms_aux_hd_exp xs)"
hyps_2: assumes "NF t" shows "r[s/0] \<degree>\<degree> ss \<rightarrow>\<^sub>l t"
positive_smult: fixes A :: "complex mat"   and n :: "nat"   and c :: "complex" assumes "A \<in> carrier_mat n n"   and "positive A"   and "0 \<le> c" shows "positive (c \<cdot>\<^sub>m A)"
DisjAssoc2: fixes H :: "fm set"   and A :: "fm"   and B :: "fm"   and C :: "fm" assumes "H \<turnstile> (A OR B) OR C" shows "H \<turnstile> A OR B OR C"
zero_less_limit_ordinal: fixes x :: "ordinal" assumes "limit_ordinal x" shows "0 < x"
base: fixes x :: "'a"   and xs :: "'a list" shows "P (x # xs) []"
card_le_inj: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B"   and "card A \<le> card B" shows "\<exists>f. f ` A \<subseteq> B \<and> inj_on f A"
prems: shows "n'' = hd (n' # ns) \<and> m = last (n' # ns)"
bdd_sane_def: fixes bdd :: "bdd" shows "bdd_sane bdd \<equiv> pointermap_sane (dpm bdd) \<and> mi.map_invar_impl (dcl bdd) bdd"
not_choice_simps_3: fixes p1 :: "('a, 'b, 'c, 'd) seqp"   and p2 :: "('a, 'b, 'c, 'd) seqp" shows "\<not> not_choice (p1 \<oplus> p2)"
execd_hext: fixes s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr option \<times> 'addr frame list) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and t :: "'thread_id"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and s' :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr option \<times> 'addr frame list) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" assumes "P \<turnstile> s -t\<triangleright>ta\<rightarrow>\<^bsub>jvmd\<^esub> s'" shows "shr s \<unlhd> shr s'"
0: shows "padic_val p f = int (LEAST k. f (Suc k) \<noteq> \<zero>\<^bsub>residue_ring (p ^ Suc k)\<^esub>)"
fNext_not_member: fixes S :: "'a fset" shows "fNext S |\<notin>| S"
eq: fixes U :: "'b set" shows "{x \<in> topspace X. f x \<in> U} = {x \<in> topspace X. g x \<in> U}"
homotopic_with_imp_continuous_maps: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "homotopic_with P X Y f g" shows "continuous_map X Y f \<and> continuous_map X Y g"
Branching_Leaf_Leaf_Tree: fixes T :: "tree"   and T1 :: "tree"   and T2 :: "tree" assumes "T = Branching T1 T2" shows "\<exists>B. branch (B @ [True]) T \<and> branch (B @ [False]) T"
brw_step_abs: fixes \<Sigma> :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)"   and \<Sigma>' :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)"   and \<delta> :: "('d, 'e) ta_rule set" assumes "(\<Sigma>, \<Sigma>') \<in> brw_step \<delta>" shows "(brw_\<alpha> \<Sigma>, brw_\<alpha> \<Sigma>') \<in> br'_step \<delta>"
dghm_ArrMap_app_in_HomCod_Arr: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
prj_simps_1: shows "C.seq \<mu>\<nu>.prj\<^sub>1 \<p>\<^sub>1[\<nu>.leg0 \<cdot> \<mu>\<nu>.prj\<^sub>0, \<pi>.leg1]"
BackwardSlice_axioms: shows "BackwardSlice sourcenode targetnode kind (valid_edge prog) (_Entry_) (Defs prog) (Uses prog) id (While_CFG_wf.wod_backward_slice prog)"
UN_constant: fixes c :: "'b set"   and A :: "'c set" shows "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
irc: fixes X :: "'a set" shows "irc_on X linear_cf"
coprime_1_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one one a"
cards: shows "cards = map (\<lambda>k. (k, local.compute_nr_of_jordan_blocks A ev k)) [1..<Suc kk]"
nhds_prod: fixes a :: "'b"   and b :: "'c" shows "nhds (a, b) = nhds a \<times>\<^sub>F nhds b"
integer_mod_group_0: fixes n :: "nat" shows "0 \<in> carrier (Z n)"
intros_4: fixes p :: "'b list"   and G :: "'b list set"   and q :: "'b list"   and w :: "'b list" assumes "p \<in> \<langle>G\<rangle>\<^sub>F"   and "q \<in> \<langle>G\<rangle>\<^sub>F"   and "p \<cdot> w \<in> \<langle>G\<rangle>\<^sub>F"   and "w \<cdot> q \<in> \<langle>G\<rangle>\<^sub>F" shows "w \<in> \<langle>G\<rangle>\<^sub>F"
rep_nat_term_of_exps: fixes a :: "nat"   and b :: "nat"   and i :: "nat" shows "rep_nat_term (of_exps a b i) = (pp_of_fun (\<lambda>x. if x = 0 then a else if x = 1 then b else 0), if \<exists>y. snd (rep_nat_term y) = i then i else 0)"
block_set_nempty_imp_block_ex: assumes "arcs_blocks \<noteq> {#}" shows "\<exists>bl. bl \<in># arcs_blocks"
hyps: fixes ys :: "'a list" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"   and "mset ys = mset xs" shows "fold f xs = fold f ys"
eq_swap_lproject_visible: fixes u :: "'action list"   and v :: "'action list" assumes "u =\<^sub>S v" shows "lproject visible (llist_of u) = lproject visible (llist_of v)"
strategyproofness_1: shows "(pmf (sds R27) a < pmf (sds R2) a \<or> pmf (sds R27) a + pmf (sds R27) c < pmf (sds R2) a + pmf (sds R2) b) \<or> pmf (sds R27) a = pmf (sds R2) a \<and> pmf (sds R27) a + pmf (sds R27) c = pmf (sds R2) a + pmf (sds R2) b"
b_def: shows "b = wo.max2 (wo.max2 x y) (wo.max2 z t)"
Px_def: shows "Px \<equiv> {y \<in> P. y \<sqsubset> x} \<union> {x}"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" shows "a1 \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L a2 = ((\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2) \<or> (\<exists>S. S \<in>S a1 \<and> S \<in>S a2) \<or> (\<exists>T. T \<in>T a1 \<and> T \<in>T a2) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L Q \<and> Q \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L R))"
pp_sup_p: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "uminus (uminus (sup x (uminus x))) = top"
hyps: fixes tu :: "'a event list"   and t :: "'a event list"   and u :: "'a event list"   and v :: "'a event list" assumes "tu setinterleaves ((t, u), A)"   and "tuv setinterleaves ((tu, v), A)" shows "\<exists>uv. uv setinterleaves ((u, v), A) \<and> tuv setinterleaves ((t, uv), A)"
u: shows "u \<turnstile>\<^bsub>v,n\<^esub> \<lambda>i' j'. if i' = v c1 \<and> j' = 0 then Le r else if i' = 0 \<and> j' = v c1 then Le (- r) else M i' j'"
create_claim_length_stable: fixes ts :: "traffic"   and d :: "cars"   and n :: "nat"   and ts' :: "traffic"   and v :: "view"   and c :: "cars" shows "ts \<^bold>\<midarrow>c( d, n ) \<^bold>\<rightarrow> ts' \<longrightarrow> len v ts c = len v ts' c"
commute_rtrancl: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "commute R S" shows "commute R\<^sup>*\<^sup>* S\<^sup>*\<^sup>*"
coboundedI2: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b \<le> c" shows "a \<sqinter> b \<le> c"
reachable_weakenE: fixes s :: "'d"   and A :: "('d, 'e) automaton"   and P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" assumes "s \<in> reachable A P"   and "\<And>a. P a \<Longrightarrow> Q a" shows "s \<in> reachable A Q"
A: fixes w :: "'a vec" shows "A w \<in> carrier_mat n n"
cases: fixes a1 :: "int"   and a2 :: "(int \<times> Instr) list"   and a3 :: "Prim"   and a4 :: "(int \<times> Instr) list \<times> int" assumes "(a1, a2, a3, a4) \<in> compilePrim" obtains l :: "int" and code :: "(int \<times> Instr) list" and i :: "int" where "a1 = l__" and "a2 = code__" and "a3 = IntPrim i" and "a4 = (code__[l__\<mapsto>const (IVal i)], l__ + 1)"   | l :: "int" and code :: "(int \<times> Instr) list" and f :: "Val \<Rightarrow> Val" and x :: "Var" where "a1 = la__" and "a2 = codea__" and "a3 = UnPrim f__ x__" and "a4 = (codea__[la__\<mapsto>load x__][la__ + 1\<mapsto>unop f__], la__ + 2)"   | l :: "int" and code :: "(int \<times> Instr) list" and f :: "Val \<Rightarrow> Val \<Rightarrow> Val" and x :: "Var" and y :: "Var" where "a1 = lb__" and "a2 = codeb__" and "a3 = BinPrim fa__ xa__ y__" and "a4 = (codeb__[lb__\<mapsto>load xa__][lb__ + 1\<mapsto>load y__][lb__ + 2\<mapsto>binop fa__], lb__ + 3)"   | l :: "int" and code :: "(int \<times> Instr) list" where "a1 = lc__" and "a2 = codec__" and "a3 = NilPrim" and "a4 = (codec__[lc__\<mapsto>const (RVal Nullref)], lc__ + 1)"   | l :: "int" and code :: "(int \<times> Instr) list" and x :: "Var" and y :: "Var" where "a1 = ld__" and "a2 = coded__" and "a3 = ConsPrim xb__ y" and "a4 = (coded__[ld__\<mapsto>load y][ld__ + 1\<mapsto>load xb__][ld__ + 2\<mapsto>new LIST][ld__ + 3\<mapsto>store tmp][ld__ + 4\<mapsto>load tmp][ld__ + 5\<mapsto>putfield LIST HD][ld__ + 6\<mapsto>load tmp][ld__ + 7\<mapsto>putfield LIST TL][ld__ + 8\<mapsto>load tmp], ld__ + 9)"   | l :: "int" and code :: "(int \<times> Instr) list" and f :: "Class \<times> Method" where "a1 = le__" and "a2 = codee__" and "a3 = CallPrim fb__ []" and "a4 = (codee__[le__\<mapsto>invokeS (fst fb__) (snd fb__)], le__ + 1)"   | l :: "int" and code :: "(int \<times> Instr) list" and x :: "Var" and f :: "Class \<times> Method" and args :: "Var list" and OUT :: "(int \<times> Instr) list \<times> int" where "a1 = l" and "a2 = code" and "a3 = CallPrim f (x # args)" and "a4 = OUT" and "(l + 1, code[l\<mapsto>load x], CallPrim f args, OUT) \<in> compilePrim"
summable_moebius_over_square: shows "summable (\<lambda>n. moebius_mu n / (real n)\<^sup>2)"
prems_5: shows "length lm3 = ba - aa - Suc n"
select_convs_3: fixes prg :: "prog"   and cls :: "qtname"   and lcl :: "lenv"   and more :: "'a" shows "lcl \<lparr>prg = prg, cls = cls, lcl = lcl, \<dots> = more\<rparr> = lcl"
open_chart_last_domain: shows "open (Collect chart_last_domainP)"
cong_altdef_nat: fixes b :: "nat"   and a :: "nat"   and m :: "nat" assumes "b \<le> a" shows "[a = b] (mod m) = (m dvd a - b)"
lens_equiv_pres_indep: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and Z :: "'g \<Longrightarrow> 'e" assumes "X \<approx>\<^sub>L Y"   and "Y \<bowtie> Z" shows "X \<bowtie> Z"
l_one: fixes x :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F" shows "\<one>\<^bsub>F\<^esub> \<otimes>\<^bsub>F\<^esub> x = x"
openin_topspace: fixes U :: "'a topology" shows "openin U (topspace U)"
no_solution_0_fst_nat: fixes b :: "nat" shows "\<not> solution (0, b)"
assms_5: fixes A :: "'a"   and B :: "'a" assumes "equalish A B" shows "equalish B A"
inj_single: fixes k :: "'b" shows "inj (Poly_Mapping.single k)"
hyps: shows "a<x>.Q \<longmapsto> \<alpha> \<prec> Q'"
meld_def: fixes bq1 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and bq2 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.meld bq1 bq2 = SkewBinomialHeapStruc.meldUniq (SkewBinomialHeapStruc.uniqify bq1) (SkewBinomialHeapStruc.uniqify bq2)"
Int_atLeastAtMostR1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.linorder less_eq less" shows "ord.atMost less_eq b \<inter> ord.atLeastAtMost less_eq c d = ord.atLeastAtMost less_eq c (ord.min less_eq b d)"
two_comparisons_into_case_order_3: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b"   and P :: "'c"   and Q :: "'c"   and R :: "'c" assumes "comparator comp" shows "(if le_of_comp comp x y then if le_of_comp comp y x then P else Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
simps_3: fixes xs :: "'a msstream" shows "un_MSLCons2 (msllist_of_msstream xs) = (case xs of MSSCons x xs \<Rightarrow> msllist_of_msstream xs)"
spanning_forest_eq: fixes E' :: "(nat \<times> int \<times> nat) set" shows "s.SpanningForest E' = spanning_forest \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E'\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>"
ins_edge_def: fixes e :: "'b \<times> 'b"   and g :: "'b ugraph" shows "ins_edge e g \<equiv> graph (nodes g) (insert e (edges g))"
RECT_def: fixes B :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and x :: "'c" shows "REC\<^sub>T B x \<equiv> if trimono B then flatf_gfp B x else top"
2: shows "class.nontriv TYPE('b) = (2 \<le> CARD('b))"
spr_simObs: fixes t :: "'s Trace"   and ec :: "'s odlist"   and a :: "'a" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRsingle.sim_equiv_class a t" shows "local.spr_simObs a ec = envObs a (tLast t)"
C_greater_zero: fixes y :: "int" assumes "y \<in> C" shows "0 < y"
preserves_iso: fixes f :: "'a" assumes "local.iso f" shows "local.iso (trg f)"
run_nondet_def: fixes nondetT :: "('b, 'a) nondetT" shows "run_nondet nondetT = (case nondetT of NondetT x \<Rightarrow> x)"
PO_m1_inv1r_cache: shows "reach m1 \<subseteq> m1_inv1r_cache"
intro: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and S :: "'c set" assumes "v0_v1_Digraph G v0 v1"   and "Separation_axioms G v0 v1 S" shows "Separation G v0 v1 S"
ev_cong: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream"   and Q1 :: "'a stream \<Rightarrow> bool"   and Q2 :: "'a stream \<Rightarrow> bool" assumes "alw P \<omega>"   and "\<And>\<omega>. P \<omega> \<Longrightarrow> Q1 \<omega> = Q2 \<omega>" shows "ev Q1 \<omega> = ev Q2 \<omega>"
emeasure_measure_pmf_not_zero: shows "emeasure (measure_pmf p) s \<noteq> 0"