hcju/isabelleps huggingface dataset:
l7_25: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and C :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong C A C B" shows "\<exists>X. Tarski_neutral_dimensionless.Midpoint Bet Cong X A B"
can_syncI: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and LT :: "('l + 't + 't) set" assumes "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "LT = collect_waits ta" shows "t \<turnstile> \<langle>x, m\<rangle> LT \<wrong>"
less_eq_imp_Interp_subseteq_Interp: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "C \<le> D" shows "Interp C \<subseteq> Interp D"
read_only_unowned: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> 'e) list"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'c"   and \<D> :: "'d"   and \<O> :: "nat set"   and \<R> :: "'e" assumes "read_only_unowned \<S> ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "\<O> \<inter> read_only \<S> = {}"
dbm_entry_dbm_lt: fixes u :: "'c \<Rightarrow> 'd"   and c1 :: "'c"   and c2 :: "'c"   and a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "dbm_entry_val u (Some c1) (Some c2) a"   and "a \<prec> b" shows "dbm_entry_val u (Some c1) (Some c2) b"
Ints_power: 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 a :: "'a"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus"   and "a \<in> ring_1.Ints one plus zero minus" shows "power.power one times a n \<in> ring_1.Ints one plus zero minus"
exec_Normal_elim_cases_3: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "'b"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,Abrupt s\<rangle> \<Rightarrow> t" obtains "t = Abrupt s"
lemma1a': fixes tp :: "bool list"   and tq :: "bool list"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and p :: "bool list" assumes "(tp, tq) \<in> (eps (RegExp2NAe.conc L R))\<^sup>*"   and "tp = True # p" shows "(\<exists>q. tq = True # q \<and> (p, q) \<in> (eps L)\<^sup>*) \<or> (\<exists>q r. tq = False # q \<and> (p, r) \<in> (eps L)\<^sup>* \<and> fin L r \<and> (start R, q) \<in> (eps R)\<^sup>*)"
inject: fixes x2 :: "'g"   and y2 :: "'g" shows "(\<lfloor>x2\<rfloor> = \<lfloor>y2\<rfloor>) = (x2 = y2)"
pfa_empty_def: fixes cap :: "nat" shows "pfa_empty cap \<equiv> pfa_init cap default (0::'c)"
case_prodI: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d" assumes "f a b" shows "case (a, b) of (c, d) \<Rightarrow> f c d"
iwlsFSbSw_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "iwlsFSbSw MOD \<equiv> iwlsFSb MOD \<and> igSwapAllIPresIGWlsAll MOD \<and> igSwapCls MOD"
mono_mndet_F_process: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process"   and Q :: "'a \<Rightarrow> 'a process" assumes "\<forall>x\<in>A. P x \<sqsubseteq>\<^sub>F Q x" shows "mndet A P \<sqsubseteq>\<^sub>F mndet A Q"
lemmaD: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "w \<Turnstile> \<phi>"   and "\<And>\<psi>. G \<psi> \<in> \<G>\<^sub>F\<^sub>G \<phi> w \<Longrightarrow> threshold \<psi> w (\<G>\<^sub>F\<^sub>G \<phi> w) = Some 0" shows "accept\<^sub>M \<phi> (\<G>\<^sub>F\<^sub>G \<phi> w) w"
elim_dest: fixes \<sigma> :: "('b \<times> 'b trm) list"   and v :: "'b"   and t :: "'b trm" assumes "elim \<sigma> v" shows "v \<notin> vars_of (t \<lhd> \<sigma>)"
orecvmsg_recvmsg: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and a :: "'e seq_action" assumes "orecvmsg P \<sigma> a" shows "recvmsg (P \<sigma>) a"
B_A_step: fixes a' :: "'b"   and b' :: "'b"   and a :: "'a" assumes "B a' b'"   and "sim1 a a'"   and "PA a"   and "PB a'" shows "\<exists>b. A a b \<and> sim1 b b'"
K'_def: shows "K' \<equiv> {i \<in> K. r i \<le> 1}"
integer_hull_integer_cone: fixes C :: "'a vec set" assumes "C \<subseteq> carrier_vec n"   and "C \<subseteq> \<int>\<^sub>v" shows "integer_hull (cone C) = cone C"
exec_serial_then_path_to: fixes s :: "nat \<Rightarrow> nat option"   and ops :: "(nat, nat) sas_plus_operator list" assumes "s \<in> valid_states"   and "\<forall>op\<in>set ops. \<exists>ast_op\<in>set ast\<delta>. op = abs_ast_operator ast_op" shows "path_to s ((map (fst \<circ> the \<circ> lookup_action) \<circ> rem_condless_ops s) ops) (execute_serial_plan_sas_plus s ops)"
winding_number_zero: fixes S :: "complex set" assumes "SC_Chain S"   and "simply_connected S" shows "connected S \<and> (\<forall>\<gamma> z. path \<gamma> \<and> path_image \<gamma> \<subseteq> S \<and> pathfinish \<gamma> = pathstart \<gamma> \<and> z \<notin> S \<longrightarrow> winding_number \<gamma> z = 0)"
id_comp: fixes g :: "'a \<Rightarrow> 'b" shows "id \<cdot> g = g"
zcount_eq_zero_iff: fixes M :: "'b zmultiset"   and x :: "'b" shows "(zcount M x = 0) = (x \<notin>#\<^sub>z M)"
relcompE: fixes xz :: "'e \<times> 'f"   and r :: "('e \<times> 'g) set"   and s :: "('g \<times> 'f) set" assumes "xz \<in> r O s" obtains x :: "'e" and y :: "'g" and z :: "'f" where "xz = (x, z)" and "(x, y) \<in> r" and "(y, z) \<in> s"
\<theta>\<theta>': shows "\<guillemotleft>\<theta> \<cdot> (\<theta>' \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[f, w', w] : f \<star> w' \<star> w \<Rightarrow> f\<guillemotright>"
vars_st': shows "vars_mset (App s t) \<subseteq># vars_mset (App s t')"
power_by_squaring: shows "efficient_funpow (*) (1::'a) = (^)"
Sigma_insert: fixes x :: "'c"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" shows "Sigma (insert x A) B = Pair x ` B x \<union> Sigma A B"
M5exist_var3: fixes k :: "'a"   and l :: "'a"   and q :: "'a"   and t :: "'a"   and r :: "'a" assumes "k \<parallel> l"   and "l \<parallel> q"   and "q \<parallel> t"   and "t \<parallel> r" shows "\<exists>lqt. k \<parallel> lqt \<and> lqt \<parallel> r"
a_consistentI: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "stack_elem list"   and a :: "Arity"   and e :: "exp"   and as :: "Arity list" assumes "edom ae \<subseteq> domA \<Gamma> \<union> upds S"   and "Astack S \<sqsubseteq> a"   and "(ABinds \<Gamma>\<cdot>ae \<squnion> \<A>\<^bsub>a\<^esub> e \<squnion> AEstack as S) f|` (domA \<Gamma> \<union> upds S) \<sqsubseteq> ae"   and "stack_consistent as S" shows "a_consistent (ae, a, as) (\<Gamma>, e, S)"
is_measure_trivial: fixes f :: "'c \<Rightarrow> nat" shows "is_measure f"
measurable_lebesgue_cong: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and M :: "'b measure" assumes "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "(f \<in> lebesgue_on S \<rightarrow>\<^sub>M M) = (g \<in> lebesgue_on S \<rightarrow>\<^sub>M M)"
4: shows "t \<le> v * v\<^sup>T \<sqinter> - - g"
injf_max_order_preserving: fixes E :: "'a set"   and n :: "nat" assumes "\<forall>x. \<exists>y\<in>E. x < y" shows "injf_max n E < injf_max (Suc n) E"
realistic_executions_def: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and output_f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and s0 :: "'a"   and current :: "'a \<Rightarrow> 'd"   and cswitch :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and kprecondition :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and realistic_execution :: "'b list list \<Rightarrow> bool"   and control :: "'a \<Rightarrow> 'd \<Rightarrow> 'b list list \<Rightarrow> 'b option \<times> 'b list list \<times> 'a"   and kinvolved :: "'b \<Rightarrow> 'd set"   and ifp :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and vpeq :: "'d \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and AS_set :: "'b list set"   and invariant :: "'a \<Rightarrow> bool"   and AS_precondition :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and aborting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and waiting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and execs :: "'d \<Rightarrow> 'b list list" assumes "Interruptible_Separation_Kernel kstep output_f s0 current cswitch kprecondition realistic_execution control kinvolved ifp vpeq AS_set invariant AS_precondition aborting waiting" shows "Interruptible_Separation_Kernel.realistic_executions realistic_execution execs \<equiv> \<forall>d. realistic_execution (execs d)"
subgraph_in_degree: fixes G :: "('b, 'c) pre_digraph"   and H :: "('b, 'c) pre_digraph"   and v :: "'b" assumes "fin_digraph G"   and "subgraph H G" shows "in_degree H v \<le> in_degree G v"
fresh_at_def: fixes t :: "('form, 'rule, 'c, 'd) itnode rose_tree"   and xs :: "nat list" shows "fresh_at t xs = (case rev xs of [] \<Rightarrow> {} | i # is' \<Rightarrow> freshenLC (iAnnot (tree_at t (rev is'))) ` a_fresh (inPorts' (iNodeOf (tree_at t (rev is'))) ! i))"
fail_option_def: shows "local.fail_option = OptionT (return None)"
has_field_derivative_def: fixes f :: "'e \<Rightarrow> 'e"   and D :: "'e"   and F :: "'e filter" shows "(f has_field_derivative D) F = (f has_derivative (*) D) F"
minarc_min: fixes y :: "'a"   and x :: "'a" shows "(bijective (y * top) \<and> bijective (y[[top]])) \<and> y \<sqinter> x \<noteq> bot \<longrightarrow> sum_class.sum (minarc x \<sqinter> x) \<le> sum_class.sum (y \<sqinter> x)"
bound_on_all_plans_bounds_MPLS_charles_i: fixes P :: "(('c, bool) fmap \<times> ('c, bool) fmap) set \<Rightarrow> bool"   and f :: "(('c, bool) fmap \<times> ('c, bool) fmap) set \<Rightarrow> nat" assumes "\<forall>PROB s as. P PROB \<and> finite PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)" shows "\<forall>PROB s as. P PROB \<and> finite PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> Inf {n. n \<in> PLS_charles s as PROB} < f PROB"
binary_meet_exists: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.Finite_Meet_cpo below" shows "\<exists>l. below l x \<and> below l y \<and> (\<forall>z. below z x \<longrightarrow> below z y \<longrightarrow> below z l)"
simps_10: fixes i :: "int"   and c :: "int"   and e :: "MIR.num" shows "iszlfm (NDvd i (CN 0 c e)) = (\<lambda>bs. 0 < c \<and> 0 < i \<and> numbound0 e \<and> isint e bs)"
uses'_codem_correct: fixes g :: "'g"   and nodes_of_uses :: "('val, 'node set) mapping"   and "next" :: "'node \<times> 'val" assumes "uninst_code.phi_equiv_mapping (p_g g s1_phis) g nodes_of_uses (uninst_code.ssa.useNodes_of (u_g g s1_uses) g)"   and "next \<in> Mapping.keys (p_g g s1_phis g)" shows "usesOf (uninst_code.uses'_codem (u_g g s1_uses) g next (uninst_code.substitution_code (p_g g s1_phis) g next) nodes_of_uses) = uninst_code.uses'_code (u_g g s1_uses) (p_g g s1_phis) g next"
g0: shows "\<guillemotleft>ECC.prod f0 (local.dom f0) \<cdot> ECC.dup (local.dom f0) : local.dom f0 \<rightarrow> ECC.prod (cod f0) (local.dom f0)\<guillemotright>"
wp1_alternative_not_mt: fixes p :: "('b, 'c) Combinators list" assumes "wellformed_policy1_strong p" shows "p \<noteq> []"
InitObjectRed: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and sh' :: "sheap"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "sh C = \<lfloor>(sfs, Prepared)\<rfloor>"   and "C = Object"   and "sh' = sh(C \<mapsto> (sfs, Processing))" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>INIT C' (C # Cs,True) \<leftarrow> e,(h, l, sh'),b\<rangle>"
Vinv2D: fixes x :: "v_state"   and r :: "nat" assumes "x \<in> Vinv2" shows "no_defection x (votes x r) r"
rel_funE: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and x :: "'b"   and y :: "'c" assumes "(A ===> B) f g"   and "A x y" obtains "B (f x) (g y)"
Top: fixes \<Psi> :: "'b" shows "\<Psi> \<turnstile> \<top>"
OT_14_P1_sec: fixes R1_12_0T :: "nat \<Rightarrow> nat \<Rightarrow> bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'v_OT121 spmf"   and n :: "nat"   and m1 :: "bool \<times> bool \<times> bool \<times> bool"   and m2 :: "bool \<times> bool"   and D :: "(bool \<times> bool \<times> bool \<times> bool) \<times> (bool \<times> bool \<times> bool \<times> bool \<times> bool \<times> bool) \<times> 'v_OT121 \<times> 'v_OT121 \<times> 'v_OT121 \<Rightarrow> bool spmf" shows "OT_14.adv_P1 (R1_12_0T n) n m1 m2 D \<le> 3 * adv_OT12 n"
map_gpv'_comp: fixes f :: "'t \<Rightarrow> 'q"   and g :: "'u \<Rightarrow> 'r"   and h :: "'s \<Rightarrow> 'v"   and f' :: "'w \<Rightarrow> 't"   and g' :: "'x \<Rightarrow> 'u"   and h' :: "'v \<Rightarrow> 'y"   and gpv :: "('w, 'x, 'y) gpv" shows "map_gpv' f g h (map_gpv' f' g' h' gpv) = map_gpv' (f \<circ> f') (g \<circ> g') (h' \<circ> h) gpv"
no_overlaps_not_unefined: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and ft :: "('c, 'e) flow_entry_match list"   and p :: "'d" assumes "no_overlaps \<gamma> ft" shows "OF_priority_match \<gamma> ft p \<noteq> Undefined"
x_c_par_idem: fixes x :: "'a" shows "x \<cdot> 1\<^sub>\<pi> \<parallel> (x \<cdot> 1\<^sub>\<pi>) = x \<cdot> 1\<^sub>\<pi>"
RP_ground_derive: fixes St :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set"   and St' :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set" assumes "St \<leadsto> St'" shows "sr_ext.derive (grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) (grounding_of_clss (N_of_state St' \<union> P_of_state St' \<union> Q_of_state St'))"
total_onI: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y\<rbrakk> \<Longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r" shows "total_on A r"
dir1: shows "\<forall>s\<^sub>1 s\<^sub>2. (P s\<^sub>1|||Q s\<^sub>2) \<sqsubseteq> S (s\<^sub>1, s\<^sub>2)"
set_mode_get_child_nodes: fixes ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" shows "\<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
final_bury_correct: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int" assumes "(c, s) \<Rightarrow> s'" shows "(bury c UNIV, s) \<Rightarrow> s'"
comp_ide_self: fixes a :: "'b" assumes "B.ide a" shows "a \<cdot>\<^sub>B a = a"
strict_boundedE: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subset># b \<inter># c" obtains "a \<subset># b" and "a \<subset># c"
CIId: fixes A :: "ZF"   and T :: "(ZF, ZF) Axioms" assumes "A \<in> Ty\<^bsub>aS\<^bsub>T\<^esub>\<^esub>" shows "id\<^bsub>CanonicalCat T\<^esub> A = [A,Vx,A]\<^bsub>T\<^esub>"
reflexive: fixes M :: "'a trm" shows "M \<rightarrow>\<beta>\<^sup>* M"
inf_aci_1: fixes x :: "'a multiset"   and y :: "'a multiset" shows "x \<inter># y = y \<inter># x"
measurable_emeasure_Pair1: fixes Q :: "('a \<times> 'b) set" assumes "Q \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1"
ring_prime_def: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" shows "ring_prime\<^bsub>R\<^esub> a = (a \<noteq> \<zero>\<^bsub>R\<^esub> \<and> prime R a)"
u'_def: fixes x :: "'a" shows "u' x = lim (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else (0::'b))"
induct: fixes P :: "'a list \<Rightarrow> bool"   and list :: "'a list" assumes "P []"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (x1 # x2)" shows "P list"
assms: shows "low_equal s1 s2 \<and> t1 = mem_mod 10 addr val s1 \<and> t2 = mem_mod 10 addr val s2"
wr1: shows "w1 \<equiv>\<^sub>E r'\<^sub>1"
empty_fv_not_var: fixes t :: "('d, 'c) Term.term"   and x :: "'c" assumes "fv t = {}" shows "t \<noteq> Var x"
count_append: fixes a :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "PermutationLemmas.count a (xs @ ys) = PermutationLemmas.count a xs + PermutationLemmas.count a ys"
i_in_Red: shows "\<iota> \<in> Red_I (N \<union> Red_F N)"
poly_add_coeff: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "local.coeff (poly_add p1 p2) = (\<lambda>i. local.coeff p1 i \<oplus> local.coeff p2 i)"
Src_Inv: fixes t :: "'a local.term" assumes "Can t" shows "Src (Inv t) = Src t"
sel_2: fixes x21 :: "('a, 'b) tm"   and x22 :: "('a, 'b) tm" shows "head (App x21 x22) = head0 x21"
prod_pOp_assoc: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and a :: "'c \<Rightarrow> 'd"   and b :: "'c \<Rightarrow> 'd"   and c :: "'c \<Rightarrow> 'd" assumes "\<forall>k\<in>I. aGroup (A k)"   and "a \<in> carr_prodag I A"   and "b \<in> carr_prodag I A"   and "c \<in> carr_prodag I A" shows "prod_pOp I A (prod_pOp I A a b) c = prod_pOp I A a (prod_pOp I A b c)"
cap_def: fixes xoyo :: "'v vertex \<times> 'v vertex" shows "cap xoyo = (case xoyo of (SOURCE, yo) \<Rightarrow> if yo = \<langle>b\<rangle> then weight \<Gamma> b - d_IN (h 0) b else 0 | (vertex.SINK, \<langle>y\<rangle>) \<Rightarrow> if y \<in> A \<Gamma> then weight \<Gamma> y else 0 | (vertex.SINK, _) \<Rightarrow> 0 | (\<langle>x\<rangle>, SOURCE) \<Rightarrow> 0 | (\<langle>x\<rangle>, vertex.SINK) \<Rightarrow> if x \<in> A \<Gamma> then weight \<Gamma> x - d_OUT (h 0) x else 0 | (\<langle>x\<rangle>, \<langle>y\<rangle>) \<Rightarrow> if edge \<Gamma> x y then h 0 (x, y) else if edge \<Gamma> y x then max (weight \<Gamma> x) (weight \<Gamma> y) else 0)"
simps_5: fixes V :: "'a set"   and uu :: "('b, 'a) Term.term"   and uv :: "('b, 'a) Term.term"   and S :: "('b, 'a) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (\<langle>uu == uv\<rangle> # S) = wf'\<^sub>s\<^sub>s\<^sub>t V S"
comp_ring_morphisms: fixes \<eta> :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and addA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zeroA :: "'d"   and oneA :: "'d"   and B :: "'e set"   and addB :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and multB :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zeroB :: "'e"   and oneB :: "'e"   and \<theta> :: "'e \<Rightarrow> 'f"   and C :: "'f set"   and addC :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and multC :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zeroC :: "'f"   and oneC :: "'f" assumes "ring_homomorphism \<eta> A addA multA zeroA oneA B addB multB zeroB oneB"   and "ring_homomorphism \<theta> B addB multB zeroB oneB C addC multC zeroC oneC" shows "ring_homomorphism (\<theta> \<circ> \<eta> \<down> A) A addA multA zeroA oneA C addC multC zeroC oneC"
update_convs_8: fixes dests' :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> nat option"   and ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool"   and more :: "'a" shows "dests_update dests' \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests' dests, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr>"
isomorphic_tuple_intro: fixes repr :: "'a \<Rightarrow> 'b \<times> 'c"   and abst :: "'b \<times> 'c \<Rightarrow> 'a"   and v :: "('a, 'b, 'c) tuple_isomorphism" assumes "\<And>x y. (repr x = repr y) = (x = y)"   and "\<And>z. repr (abst z) = z"   and "v \<equiv> Tuple_Isomorphism repr abst" shows "isomorphic_tuple v"
precision_spec_def: shows "precision_spec = SPEC (\<lambda>prec. True)"
Im_complex_of_real: fixes z :: "real" shows "Im (complex_of_real z) = 0"
unfold_thesis: shows "vec_space.rank (length signs) (mat_of_cols (dim_row (M_mat signs subsets)) (take_indices (cols (M_mat signs subsets)) (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets))))) = length (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))"
transpE: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "transp r"   and "r x y"   and "r y z" obtains "r x z"
first_summand_not_universe: fixes x :: "i"   and y :: "i" assumes "CEMU"   and "x \<oplus> y \<noteq> u" shows "x \<noteq> u"
add_le_mono: fixes i :: "nat"   and j :: "nat"   and k :: "nat"   and l :: "nat" assumes "i \<le> j"   and "k \<le> l" shows "i + k \<le> j + l"
step_invariant_mono: fixes ws :: "('a list \<times> 's \<times> 's) list"   and ps :: "('s \<times> 's) list"   and N :: "('s \<times> 's) set"   and ws' :: "('a list \<times> 's \<times> 's) list"   and ps' :: "('s \<times> 's) list"   and N' :: "('s \<times> 's) set" assumes "step_invariant (ws, ps, N) = (ws', ps', N')" shows "snd ` set ws \<union> set ps \<subseteq> snd ` set ws' \<union> set ps'"
higher_order: fixes A :: "'b tensor"   and B :: "'b tensor" assumes "order A = 1"   and "cprank_max1 B" shows "cprank_max1 (A \<otimes> B)"
oddE: fixes a :: "'a" assumes "odd a" obtains b :: "'a" where "a = (2::'a) * b + (1::'a)"
insertI: fixes A :: "'c set"   and a :: "'c" assumes "finite A" shows "finite (Set.insert a A)"
bindET_fmapET_unitET: fixes f :: "'h \<rightarrow> 'e"   and m :: "'h\<cdot>('f, 'g) errorT" shows "bindET\<cdot>(fmapET\<cdot>f\<cdot>m)\<cdot>unitET = fmapET\<cdot>f\<cdot>m"
dependent_zero: fixes A :: "('a, 'b) vec set" assumes "0 \<in> A" shows "vec.dependent A"
allowed_delete_deliver: fixes i :: "'a"   and x :: "'a"   and j :: "nat" assumes "Deliver (i, Delete x) \<in> set (history j)" shows "\<exists>i' n' v b. Deliver (i', Insert (x, v, b) n') \<sqsubset>\<^sup>j Deliver (i, Delete x)"
gh: fixes y :: "(real, 'm) vec" assumes "y \<in> T" shows "h y \<in> S \<and> g (h y) = y"
dom_subid: fixes x :: "'a" shows "d x \<le> (1::'a)"
prfOf_def: fixes n :: "'trm"   and \<phi> :: "'fmla" shows "prfOf n \<phi> \<equiv> bprv (PPf n \<langle>\<phi>\<rangle>)"
abs_def: fixes implem :: "chan \<Rightarrow> msg"   and S :: "msg set" assumes "semivalid_implem implem" shows "semivalid_implem.abs implem S \<equiv> {Chan x A B M |x A B M. M \<in> payload \<and> implem (Chan x A B M) \<in> S}"
\<sigma>1_eq: shows "lst (s, [\<beta>_step]) = fst \<sigma>1"
choose_affine_subset: fixes S :: "'c set"   and d :: "int" assumes "affine S"   and "- 1 \<le> d"   and "d \<le> aff_dim S" obtains T :: "'c set" where "affine T" and "T \<subseteq> S" and "aff_dim T = d"
b: shows "(SUP n. Inf (u ` {n + 1..})) = (SUP n\<in>{1..}. Inf (u ` {n..}))"
simps_49: fixes x101 :: "stmt"   and x102 :: "stmt"   and x2 :: "expr" shows "x101 Finally x102 \<noteq> Expr x2"