hcju/isabelleps huggingface dataset:
sees_method_mono2: fixes C' :: "char list"   and C :: "char list"   and P :: "'a prog"   and wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list"   and b' :: "staticb"   and Ts' :: "ty list"   and T' :: "ty"   and m' :: "'a"   and D' :: "char list" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D"   and "P \<turnstile> C' sees M, b' :  Ts'\<rightarrow>T' = m' in D'" shows "b = b' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> P \<turnstile> T' \<le> T"
artanh_real_at_left_1: shows "filterlim artanh at_top (at_left 1)"
ToTC_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "T.ToTC f \<equiv> if CCC.arr f then (fst f \<otimes> fst (snd f)) \<otimes> snd (snd f) else null"
alpha_res: fixes bs :: "atom set"   and x :: "'b"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom set"   and y :: "'b" shows "(bs, x) \<approx>res R f p (cs, y) = (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y)"
karatsuba_mult_poly_code_unfold: shows "(*) = karatsuba_mult_poly"
no_matching_Goto_append2: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs1 :: "'c rule list"   and rs2 :: "'c rule list" assumes "no_matching_Goto \<gamma> p (rs1 @ rs2)" shows "no_matching_Goto \<gamma> p rs2"
chain_def: fixes A :: "'c set"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and C :: "'c set" shows "pred_on.chain A P C = (C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. P\<^sup>=\<^sup>= x y \<or> P\<^sup>=\<^sup>= y x))"
simps: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and a1 :: "('a, 'd) L_state"   and a2 :: "('a, 'd) L_state" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.L_bisimilar satisfies transition effect_apply L a1 a2 = ((\<exists>P F Q. a1 = EF (F, P) \<and> a2 = EF (F, Q) \<and> effect_nominal_ts.FL_bisimilar satisfies transition effect_apply L F P Q) \<or> (\<exists>P F Q f. a1 = AC (f, F, effect_apply f P) \<and> a2 = AC (f, F, effect_apply f Q) \<and> effect_nominal_ts.FL_bisimilar satisfies transition effect_apply L F P Q \<and> f \<in>\<^sub>f\<^sub>s F))"
is_aligned_iff_dvd_nat: fixes ptr :: "'b word"   and n :: "nat" shows "is_aligned ptr n = (2 ^ n dvd unat ptr)"
image_mset_mset_set: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" assumes "inj_on f A" shows "image_mset f (mset_set A) = mset_set (f ` A)"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>G' f : G' a \<rightarrow>\<^sub>D G' b\<guillemotright>"
ih_1: shows "{$$} \<turnstile> e\<^sub>2, eP\<^sub>2, eV\<^sub>2 : \<tau>"
arcsin_less_arcsin: assumes "- 1 \<le> 0"   and "0 < x"   and "x \<le> 1" shows "arcsin 0 < arcsin x"
foldl_Nil: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and a :: "'a" shows "foldl f a [] = a"
many_reducible_lm_1: fixes A :: "nat set" assumes "many_reducible_to univ_ce A" shows "\<not> computable A"
simps: fixes x :: "'a"   and y :: "'a" shows "dirvec (x, y) = y - x"
mult_nonneg_nonpos: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq b zero" shows "less_eq (times a b) zero"
c: shows "h \<turnstile> local.a_get_component ptr \<rightarrow>\<^sub>r c"
FiniteBroadcastEnvironmentIndependentInit_axioms: shows "FiniteBroadcastEnvironmentIndependentInit jkbp agents"
span_def: fixes S :: "'a vec set" shows "NC.span S = {NC.lincomb a A |a A. finite A \<and> A \<subseteq> S}"
list_all_append: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'d list" shows "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
S'V: fixes s' :: "nat \<Rightarrow> nat option" assumes "simulate_plan \<pi>s I = Some s'" shows "s' \<in> valid_states"
bind_returns_result_E2: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and y :: "'c" assumes "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>r y"   and "pure f h" obtains x :: "'d" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>r y"
safeAborts_def: fixes s :: "('a, 'b, 'c) SLin_state" shows "safeAborts s \<equiv> if initialized s then initAborts s else uninitAborts s"
hyps: shows "MPoly_Type.degree (Var x ^ i) x = i"
hseq_char: fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "B.hseq \<nu> \<mu> = (B.arr \<mu> \<and> B.arr \<nu> \<and> src \<nu> = trg \<mu>)"
find_base_vectors_6: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "row_echelon_form A"   and "A \<in> carrier_mat nr nc" shows "vectorspace.dim class_ring (module_vec TYPE('b) nc\<lparr>carrier := mat_kernel A\<rparr>) = nc - card {i. i < nr \<and> row A i \<noteq> 0\<^sub>v nc}"
A: shows "set vs = {x, y} \<union> set (filter (\<lambda>z. z \<noteq> x \<and> z \<noteq> y) vs)"
goal_fulfilled_M: assumes "c_offending_flows m \<lparr>nodes = V, edges = E\<rparr> \<noteq> {}" shows "all_security_requirements_fulfilled M (delete_edges (generate_valid_topology_some M Es \<lparr>nodes = V, edges = E\<rparr>) (set (minimalize_offending_overapprox (c_sinvar m) Es [] \<lparr>nodes = V, edges = E\<rparr>)))"
inner_is_valid: fixes n :: "'b LDCFG_node" assumes "local.CFGExit.inner_node n" shows "CFGExit.valid_node n"
module_mat_def: fixes ty :: "'b itself"   and nr :: "nat"   and nc :: "nat" shows "module_mat ty nr nc \<equiv> \<lparr>carrier = carrier_mat nr nc, mult = (*), one = 1\<^sub>m nr, zero = 0\<^sub>m nr nc, add = (+), smult = (\<cdot>\<^sub>m)\<rparr>"
bounded_iff: fixes b :: "'a multiset"   and c :: "'a multiset"   and a :: "'a multiset" shows "(b \<union># c \<subseteq># a) = (b \<subseteq># a \<and> c \<subseteq># a)"
induct: fixes x1 :: "trm"   and x2 :: "trm"   and P :: "trm \<Rightarrow> trm \<Rightarrow> bool" assumes "x1 \<leadsto> x2"   and "\<And>s s' t. \<lbrakk>s \<leadsto> s'; P s s'\<rbrakk> \<Longrightarrow> P (App s t) (App s' t)"   and "\<And>t t' s. \<lbrakk>t \<leadsto> t'; P t t'\<rbrakk> \<Longrightarrow> P (App s t) (App s t')"   and "\<And>x t s. P (App (\<Lambda> x . t) s) (t[x::=s])"   and "\<And>t t' x. \<lbrakk>t \<leadsto> t'; P t t'\<rbrakk> \<Longrightarrow> P (\<Lambda> x . t) (\<Lambda> x . t')"   and "\<And>x t. x \<sharp> t \<Longrightarrow> P (\<Lambda> x . App t (Var x)) t"   and "\<And>s s' x t. \<lbrakk>s \<leadsto> s'; P s s'\<rbrakk> \<Longrightarrow> P (s to x in t) (s' to x in t)"   and "\<And>t t' s x. \<lbrakk>t \<leadsto> t'; P t t'\<rbrakk> \<Longrightarrow> P (s to x in t) (s to x in t')"   and "\<And>s x t. P ([s] to x in t) (t[x::=s])"   and "\<And>x s. x \<sharp> s \<Longrightarrow> P (s to x in [Var x]) s"   and "\<And>x y u s t. \<lbrakk>x \<sharp> y; x \<sharp> u\<rbrakk> \<Longrightarrow> P ((s to x in t) to y in u) (s to x in t to y in u)"   and "\<And>s s'. \<lbrakk>s \<leadsto> s'; P s s'\<rbrakk> \<Longrightarrow> P [s] [s']" shows "P x1 x2"
emeasure_density_const: fixes A :: "'b set"   and M :: "'b measure"   and c :: "ennreal" assumes "A \<in> sets M" shows "emeasure (density M (\<lambda>_. c)) A = c * emeasure M A"
\<Delta>_expectation_X: shows "\<Delta>\<^sub>a \<le> \<mu>\<^sup>2 + \<Delta>\<^sub>d"
head_add_arc: fixes G :: "('d, 'c) pre_digraph"   and a :: "'c" shows "head (pre_digraph.add_arc G a) = head G"
simps: fixes ctf :: "String.literal \<Rightarrow> typ option"   and uu :: "String.literal \<Rightarrow> nat option"   and uv :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" shows "const_type (ctf, uu, uv) = ctf"
Rep_trm: fixes x :: "trm" shows "Rep_trm x \<in> Collect trm_Rep_set"
bisimCasePushRes: fixes x :: "name"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and \<Psi> :: "'b" assumes "x \<sharp> map fst Cs" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(Cases Cs) \<sim> Cases map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>x\<rparr>P)) Cs"
mem_mod_w32_def: fixes asi :: "word8"   and addr :: "word36"   and byte_mask :: "word4"   and data_w32 :: "word32"   and state :: "'b sparc_state" shows "mem_mod_w32 asi addr byte_mask data_w32 state \<equiv> let addr' = addr AND 68719476732; addr0 = addr' OR 0; addr1 = addr' OR 1; addr2 = addr' OR 2; addr3 = addr' OR 3; byte0 = ucast (data_w32 >> 24); byte1 = ucast (data_w32 >> 16); byte2 = ucast (data_w32 >> 8); byte3 = ucast data_w32; s0 = if byte_mask AND 8 >> 3 = 1 then mem_mod asi addr0 byte0 state else state; s1 = if byte_mask AND 4 >> 2 = 1 then mem_mod asi addr1 byte1 s0 else s0; s2 = if byte_mask AND 2 >> 1 = 1 then mem_mod asi addr2 byte2 s1 else s1; s3 = if byte_mask AND 1 = 1 then mem_mod asi addr3 byte3 s2 else s2 in s3"
1: shows "r\<^sup>T \<le> r\<^sup>T * ((r\<^sup>T * g\<^sup>\<star>)\<^sup>T * (r\<^sup>T * g\<^sup>\<star>) \<sqinter> g)\<^sup>\<star>"
upd_rows_insert: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and I :: "'d set"   and f :: "'d \<Rightarrow> ('c, 'd) vec" shows "upd_rows M (insert i I) f = upd_row (upd_rows M I f) i (f i)"
weight_from_list_correct: fixes l :: "(('a \<times> 'a) \<times> nat) list"   and u :: "'a"   and v :: "'a"   and d :: "nat" assumes "valid_weight_repr l"   and "((u, v), d) \<in> set l" shows "weight_from_list l {u, v} = d"
simps_32: fixes vd :: "'a acom"   and ve :: "'a acom"   and v :: "bexp"   and va :: "'a acom"   and vb :: "'a acom"   and vc :: "'a" shows "(vd;; ve \<sqsubseteq> IF v THEN va ELSE vb {vc}) = False"
set_mode_ptr_in_heap: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root_mode :: "shadow_root_mode" assumes "h \<turnstile> ok set_mode shadow_root_ptr shadow_root_mode" shows "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h"
sup_least: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and z :: "'b" assumes "class.semilattice_sup sup less_eq less"   and "less_eq y x"   and "less_eq z x" shows "less_eq (sup y z) x"
perm_mem_right: fixes pi :: "(name \<times> name) list"   and x :: "name"   and A :: "name set" assumes "rev pi \<bullet> x \<in> A" shows "x \<in> pi \<bullet> A"
outer_measure_le: fixes f :: "'a set \<Rightarrow> ennreal"   and A :: "nat \<Rightarrow> 'a set"   and X :: "'a set" assumes "positive M f"   and "increasing M f"   and "range A \<subseteq> M"   and "X \<subseteq> \<Union> (range A)" shows "outer_measure M f X \<le> (\<Sum>i. f (A i))"
leadsTo_Join: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set"   and T :: "'b set"   and G :: "'b program" assumes "F \<in> A \<longmapsto> B"   and "T - B \<subseteq> awp F T"   and "\<And>X. X \<in> wens_set F B \<Longrightarrow> T \<inter> X - B \<subseteq> awp G (T \<inter> X)" shows "F \<squnion> G \<in> T \<inter> A \<longmapsto> B"
assms_1: shows "\<Gamma> \<turnstile> e : Low"
sourcesD_2: fixes a :: "'a"   and \<mu> :: "'a" assumes "a \<in> sources \<mu>" shows "weak_unit a"
ldistinct_lfilterD: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist"   and n :: "nat"   and m :: "nat"   and a :: "'b" assumes "ldistinct (lfilter P xs)"   and "enat n < llength xs"   and "enat m < llength xs"   and "P a"   and "xs $ n = a"   and "xs $ m = a" shows "m = n"
carrier_submanifold: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and S :: "'c set" assumes "submanifold charts k S" shows "manifold.carrier (manifold.charts_submanifold charts S) = S \<inter> manifold.carrier charts"
cases: fixes a1 :: "nat"   and a2 :: "nat"   and a3 :: "nat list"   and a4 :: "nat list"   and a5 :: "nat list" assumes "Form_Body a1 a2 a3 a4 a5" obtains xs :: "nat list" and ys :: "nat list" and a :: "nat list" and as :: "nat list list" and b :: "nat list" and bs :: "nat list list" and ka :: "nat" and kb :: "nat" and c :: "nat list" and d :: "nat list" and zs :: "nat list" where "a1 = ka" and "a2 = kb" and "a3 = xs" and "a4 = ys" and "a5 = zs" and "length xs < length ys" and "xs = concat (a # as)" and "ys = concat (b # bs)" and "a # as \<in> lists (- {[]})" and "b # bs \<in> lists (- {[]})" and "length (a # as) = ka" and "length (b # bs) = kb" and "c = acc_lengths 0 (a # as)" and "d = acc_lengths 0 (b # bs)" and "zs = concat [c, a, d, b] @ interact as bs" and "strict_sorted zs"
that_9: fixes b\<^sub>2 :: "bool" assumes "x = \<bottom>"   and "y = CB\<cdot>(Discr b\<^sub>2)" shows "thesis"
checkI_OKI: fixes inp :: "'a \<Rightarrow> 'b option" shows "checkI (OKI inp) = inp"
that_1: assumes "r = x"   and "\<sigma>' = \<sigma>"   and "\<not> is_exn \<sigma>" shows "thesis"
LinkMark_def: shows "LinkMark = (\<lambda>(i, j). case (i, j) of (init, loop) \<Rightarrow> Q1''_a \<sqinter> Q2''_a | (init, _) \<Rightarrow> \<top> | (loop, init) \<Rightarrow> \<top> | (loop, loop) \<Rightarrow> Q3''_a \<sqinter> (Q4''_a \<sqinter> Q5''_a) | (loop, final) \<Rightarrow> Q6''_a | (final, b) \<Rightarrow> \<top>)"
bind_gpv_sel': fixes r :: "('f, 'd, 'e) gpv"   and f :: "'f \<Rightarrow> ('b, 'd, 'e) gpv" shows "the_gpv (r \<bind> f) = the_gpv r \<bind> (\<lambda>generat. if is_Pure generat then the_gpv (f (generat.result generat)) else return_spmf (IO (generat.output generat) (\<lambda>input. continuation generat input \<bind> f)))"
reduced_GB_subset_monic_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]"   and "is_monomial_set F" shows "punit.reduced_GB F \<subseteq> punit.monic ` F"
elims: fixes x :: "'b \<Rightarrow> 'a"   and xa :: "'b \<times> 'b"   and y :: "'a \<times> 'a" assumes "pairself x xa = y" obtains f :: "'b \<Rightarrow> 'a" and x :: "'b" and y :: "'b" where "xa__ = f" and "xa = (x, y)" and "ya__ = (f x, f y)"
comp_assoc: fixes f :: "'h \<Rightarrow> 'f"   and g :: "'g \<Rightarrow> 'h"   and h :: "'e \<Rightarrow> 'g" shows "f \<circ> g \<circ> h = f \<circ> (g \<circ> h)"
eval_evaln: fixes G :: "prog"   and s0 :: "state"   and t :: "Term.term"   and v :: "vals"   and s1 :: "state" assumes "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)" shows "\<exists>n. G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s1)"
\<I>\<^sub>\<tau>_4: shows "\<I>\<^sub>\<tau> \<Turnstile> \<langle>unlabel \<A>'\<rangle>"
dist_add_cancel2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "dist (plus b a) (plus c a) = dist b c"
foundation2: fixes \<tau> :: "'b state \<times> 'b state" shows "\<tau> |\<noteq> false"
wfp_onI_min: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x Q. \<lbrakk>x \<in> Q; Q \<subseteq> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>Q. \<forall>y\<in>A. r y z \<longrightarrow> y \<notin> Q" shows "wfp_on r A"
seq_cons: fixes \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b"   and r :: "'a rule"   and t :: "state"   and rs :: "'a rule list"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[r], Undecided\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, t\<rangle> \<Rightarrow> t'" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r # rs, Undecided\<rangle> \<Rightarrow> t'"
r: fixes s :: "'s" assumes "s \<in> S\<^sub>r" shows "\<exists>n. s \<in> r n"
store_sops_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "bool"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>\<Union> (fst ` store_sops is). i < t"
S_imp_NS: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term" assumes "S s t" shows "NS s t"
r_def: shows "r = poly_add_sign (degree p) (degree q) * poly_add p q"
inf_le2: fixes x :: "'g multiset"   and y :: "'g multiset" shows "x \<inter># y \<subseteq># y"
a: fixes x :: "'a"   and y :: "'a" shows "boa x y \<longrightarrow> boa' x y"
sums_finite: fixes N :: "nat set"   and f :: "nat \<Rightarrow> 'b" assumes "finite N"   and "\<And>n. n \<notin> N \<Longrightarrow> f n = (0::'b)" shows "f sums sum f N"
set_scf_list: fixes xs :: "'c list"   and scf :: "nat \<Rightarrow> nat" assumes "\<forall>i<length xs. 0 < scf i" shows "set (scf_list scf xs) = set xs"
proj_zero: fixes k :: "nat" shows "proj_poly k 0 = 0"
position_def: fixes e :: "'a list" shows "position e \<equiv> enat (length e) \<le> 2 * N"
Union_transfer: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "(rel_set (rel_set A) ===> rel_set A) \<Union> \<Union>"
lm032: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G = injectionsUniverse \<inter> (Range -` all_partitions G \<inter> Domain -` Pow N)"
assms_1: shows "\<forall>m>(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows). A $ m $ mod_type_class.from_nat k = (0::'a)"
unfold: shows "the_or = foldr\<cdot>tror\<cdot>FF"
subst_heap_def: shows "subst_heap \<equiv> \<lambda>x0 x1 x2. projr (subst_subst_heap_sumC (Inr (x0, x1, x2)))"
is_Zp_cauchyI: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "\<And>n. \<exists>N. \<forall>n0 n1. N < n0 \<and> N < n1 \<longrightarrow> s n0 n = s n1 n" shows "is_Zp_cauchy s"
jkbpCSt_jkbpCS_subset: fixes t :: "'s Trace" shows "spr_jkbpCSt t \<subseteq> SPRsingle.jkbpCS"
mod_mult_self4: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and b :: "'c"   and c :: "'c"   and a :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (times b c) a) b = modulo a b"
zeta_trivial_zero: fixes n :: "nat" assumes "even n"   and "n \<noteq> 0" shows "zeta (- of_nat n) = 0"
loc_reach_subset: fixes c :: "'Com"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "'Com"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "loc_reach \<langle>c', mds', mem'\<rangle> \<subseteq> loc_reach \<langle>c, mds, mem\<rangle>"
eq_OO: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "(=) OO R = R"
eq_iff: fixes x :: "'b list"   and y :: "'b list" shows "(x = y) = (suffix x y \<and> suffix y x)"
boundedE: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "semilattice_order f less_eq less"   and "less_eq a (f b c)" obtains "less_eq a b" and "less_eq a c"
igWlsAllDisj_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gWlsAllDisj MOD" shows "igWlsAllDisj (fromMOD MOD)"
possible_steps_alt_aux: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and d :: "nat"   and t :: "transition" assumes "possible_steps e s r l i = {|(d, t)|}" shows "ffilter (\<lambda>((origin, dest), t). origin = s \<and> Label t = l \<and> length i = Arity t \<and> apply_guards (Guards t) (join_ir i r)) e = {|((s, d), t)|}"
strict_trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<prec> a"   and "c \<prec> b" shows "c \<prec> a"
induct: fixes P :: "Ferrack.num \<Rightarrow> bool"   and a0 :: "Ferrack.num" assumes "P (Bound 0)"   and "\<And>a b. \<lbrakk>P a; \<And>x xa y. \<lbrakk>x = rsplit0 a; (xa, y) = x\<rbrakk> \<Longrightarrow> P b\<rbrakk> \<Longrightarrow> P (Add a b)"   and "\<And>a b. P (Add a (Neg b)) \<Longrightarrow> P (Sub a b)"   and "\<And>a. P a \<Longrightarrow> P (Neg a)"   and "\<And>c a. P a \<Longrightarrow> P (Mul c a)"   and "\<And>c a. P a \<Longrightarrow> P (CN 0 c a)"   and "\<And>v c a. P a \<Longrightarrow> P (CN (Suc v) c a)"   and "\<And>v. P (C v)"   and "\<And>va. P (Bound (Suc va))" shows "P a0"
rel_witness_option_2: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option" assumes "rel_option A x y" shows "map_option fst (rel_witness_option (x, y)) = x"
Pair_inject: fixes a :: "'k"   and b :: "'l"   and a' :: "'k"   and b' :: "'l" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
rename_leadsTo: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "bij h" shows "(rename h F \<in> h ` A \<longmapsto> h ` B) = (F \<in> A \<longmapsto> B)"
degree_primitive_part: fixes p :: "'d poly" shows "degree (primitive_part p) = degree p"
sums_single: fixes i :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "(\<lambda>r. if r = i then f r else (0::'b)) sums f i"
\<Gamma>\<^sub>T\<^sub>O\<^sub>Y_simps: shows "\<Gamma>\<^sub>T\<^sub>O\<^sub>Y PToy = {PToy-:0}receive(\<lambda>msg'. msg_update (\<lambda>_. msg')) . {PToy-:1}\<lbrakk>\<lambda>\<xi>. \<xi>\<lparr>nhid := state.id \<xi>\<rparr>\<rbrakk> ({PToy-:2}\<langle>is_newpkt\<rangle> {PToy-:3}\<lbrakk>\<lambda>\<xi>. \<xi>\<lparr>no := max (no \<xi>) (num \<xi>)\<rparr>\<rbrakk> {PToy-:4}broadcast(\<lambda>\<xi>. Pkt (no \<xi>) (state.id \<xi>)) . {PToy-:5}\<lbrakk>clear_locals\<rbrakk> call(PToy) \<oplus> {PToy-:2}\<langle>is_pkt\<rangle> ({PToy-:6}\<langle>\<lambda>\<xi>. if no \<xi> < num \<xi> then {\<xi>} else {}\<rangle> {PToy-:7}\<lbrakk>\<lambda>\<xi>. \<xi>\<lparr>no := num \<xi>\<rparr>\<rbrakk> {PToy-:8}\<lbrakk>\<lambda>\<xi>. \<xi>\<lparr>nhid := sid \<xi>\<rparr>\<rbrakk> {PToy-:9}broadcast(\<lambda>\<xi>. Pkt (no \<xi>) (state.id \<xi>)) . {PToy-:10}\<lbrakk>clear_locals\<rbrakk> call(PToy) \<oplus> {PToy-:6}\<langle>\<lambda>\<xi>. if num \<xi> \<le> no \<xi> then {\<xi>} else {}\<rangle> {PToy-:11}\<lbrakk>clear_locals\<rbrakk> call(PToy)))"
allowed_delete_deliver_in_set: fixes es :: "('a \<times> ('a, 'b) operation) event list"   and i :: "'a"   and m :: "'a"   and j :: "nat" assumes "es @ [Deliver (i, Delete m)] prefix of j" shows "\<exists>i' n v b. Deliver (i', Insert (m, v, b) n) \<in> set es"
eqvtChainI: fixes P :: "pi"   and P' :: "pi"   and perm :: "(name \<times> name) list" assumes "P \<Longrightarrow>\<^sub>\<tau> P'" shows "perm \<bullet> P \<Longrightarrow>\<^sub>\<tau> perm \<bullet> P'"
FAILURES_def: fixes P :: "('c event list \<times> 'c event set) set \<times> 'c event list set" shows "FAILURES P = fst P"