hcju/isabelleps huggingface dataset:
norm_beta_complete_multi: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and D :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and l' :: "'d"   and Z :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, [D]\<^bsub>v,n\<^esub>\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>l', Z\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "valid_dbm D" obtains D' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" where "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^sub>\<N>* \<langle>l', D'\<rangle>" and "[D']\<^bsub>v,n\<^esub> = Z" and "valid_dbm D'"
F_anal_valid: shows "\<forall>twoC\<in>two_chain. analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> i) j"
generateI: fixes G :: "('e, 'f) monoid_scheme"   and E :: "'e set"   and H :: "'e set" assumes "Group.group G"   and "subgroup E G"   and "H \<subseteq> E"   and "\<And>K. \<lbrakk>subgroup K G; H \<subseteq> K\<rbrakk> \<Longrightarrow> E \<subseteq> K" shows "E = generate G H"
ide_in_hom: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "ide a = \<guillemotleft>a : a \<rightarrow> a\<guillemotright>"
simps_3: fixes c :: "int"   and e :: "MIR.num" shows "\<beta> (Eq (CN 0 c e)) = [Sub (C (- 1)) e]"
simps_1: fixes p :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "repeat 0 p = Skip"
eq_abs_iff': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "(a = abs b) = (less_eq zero a \<and> (b = a \<or> b = uminus a))"
Exec_Equal_State_def: fixes localState :: "'d \<Rightarrow> 'e"   and trans_fun :: "'f message_af \<Rightarrow> 'd \<Rightarrow> 'd" shows "Exec_Equal_State localState trans_fun \<equiv> \<forall>c1 c2 m. localState c1 = localState c2 \<longrightarrow> trans_fun m c1 = trans_fun m c2"
E_wf_1: shows "fst ` flows_fix \<T> \<subseteq> hosts \<T>"
comp_in_homE: fixes g :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and c :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.VV.in_hom (S.VV.comp g f) a c" obtains b :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" where "S.VV.in_hom f a b" and "S.VV.in_hom g b c"
igFreshIGAbs2_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igFreshIGAbs2 MOD \<equiv> \<forall>ys y xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igFresh MOD ys y X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD xs x X)"
tuple: shows "\<guillemotleft>tuple (local.comp (pr1 a b) h) (local.comp (pr0 a b) h) : local.dom h \<rightarrow> local.prod a b\<guillemotright>"
Union_eqpoll_Times: fixes A :: "'c set"   and F :: "'c \<Rightarrow> 'd set"   and B :: "'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> F x \<approx> B"   and "pairwise (\<lambda>x y. disjnt (F x) (F y)) A" shows "\<Union> (F ` A) \<approx> A \<times> B"
has_bochner_integral_integrable: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "integrable M f" shows "has_bochner_integral M f (integral\<^sup>L M f)"
statelist'_exec_plan: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and p :: "('c, 'd) fmap list" assumes "statelist' s as = p" shows "exec_plan s as = last p"
app\<^sub>i_Return: fixes P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and T :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (Return, P, pc, mxs, T\<^sub>r, T # ST, LT) = subtype P T T\<^sub>r"
inconsistent_fm: fixes S :: "('a, 'b) fm set"   and p :: "('a, 'b) fm" assumes "consistent S"   and "\<not> consistent ({p} \<union> S)" obtains S' :: "('a, 'b) fm list" where "set S' \<subseteq> S" and "p # S' \<turnstile> \<^bold>\<bottom>"
obj_at_def: fixes P :: "('b, 'c, 'd) object \<Rightarrow> bool"   and r :: "'d" shows "obj_at P r \<equiv> \<lambda>s. case sys_heap s r of None \<Rightarrow> False | Some obj \<Rightarrow> P obj"
ee_e: shows "arc_to_ends G e = (u, head G e)"
distinct_15: fixes x21 :: "nat"   and x22 :: "nat"   and x41 :: "nat"   and x42 :: "nat" shows "x21:deliver(x22) \<noteq> connect(x41, x42)"
T: fixes f :: "'a"   and g :: "'a" shows "N (f + g) \<le> ennreal (max (defect N1) (defect N2)) * N f + ennreal (max (defect N1) (defect N2)) * N g"
integ_exist_b: shows "\<forall>(k1, \<gamma>1)\<in>one_chain1. \<forall>b\<in>basis. line_integral_exists F {b} (snd (f (k1, \<gamma>1)))"
reconstruct_def: fixes shares :: "bool \<times> bool" shows "reconstruct shares = (let (a, b) = shares in return_spmf (a \<oplus> b))"
nz: shows "coeff f 0 \<noteq> (0::'a)"
Rep_FingerTree: fixes x :: "('c, 'd) FingerTree" shows "Rep_FingerTree x \<in> {t. FingerTreeStruc.ft_invar t}"
DynPDG_ddep_edge_CFG_path_2: fixes n :: "'node"   and V :: "'var"   and as :: "'edge list"   and n' :: "'node" assumes "n -{V}as\<rightarrow>\<^sub>d\<^sub>d n'" shows "as \<noteq> []"
inner_loop2_def: fixes dst :: "'a"   and succl :: "'a list"   and u :: "'b"   and PRED :: "'a \<Rightarrow> 'b option"   and N :: "'a set" shows "Graph.inner_loop2 dst succl u PRED N \<equiv> nfoldli succl (\<lambda>(f, uu_, uu_). \<not> f) (\<lambda>v (f, PRED, N). if PRED v \<noteq> None then return (f, PRED, N) else let PRED = PRED(v \<mapsto> u) in assert (v \<notin> N) \<bind> (\<lambda>_. let N = insert v N in return (v = dst, PRED, N))) (False, PRED, N)"
c4_2: shows "set (labels_conv (fst \<gamma>1''', map (Pair True) (snd \<gamma>1'''))) = set (labels \<gamma>1''')"
mult_ac_2: fixes a :: "'m"   and b :: "'m" shows "a * b = b * a"
ring_homomorphism_to_poly: shows "ring_homomorphism (\<lambda>i. [:i:])"
cis_2pi_nat: fixes n :: "nat" shows "cis (2 * pi * real n) = 1"
minGraphProps8: fixes g :: "graph"   and v :: "nat" assumes "minGraphProps g"   and "v \<in> \<V> g" shows "distinct (normFaces (facesAt g v))"
set_eqvt: fixes p :: "perm"   and xs :: "'a list" shows "p \<bullet> set xs = set (p \<bullet> xs)"
t_disj: shows "t = REAL \<or> t = INTEG"
redundancy_implies_entailment: fixes C :: "'at Literal set"   and S :: "'at Literal set set" assumes "redundant C S" shows "entails S C"
steps_is_node: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat"   and q :: "bool list"   and w :: "bool list list" assumes "aut_nfa A n"   and "nfa_is_node A q"   and "list_all (is_alph n) w" shows "nfa_is_node A (nfa_steps A q w)"
cases: fixes a1 :: "'node"   and a2 :: "'node"   and a3 :: "nat" assumes "distance a1 a2 a3" obtains n :: "'node" and as :: "'edge list" and n' :: "'node" and x :: "nat" where "a1 = n" and "a2 = n'" and "a3 = x" and "n -as\<rightarrow>\<^sub>\<iota>* n'" and "length as = x" and "\<forall>as'. n -as'\<rightarrow>\<^sub>\<iota>* n' \<longrightarrow> x \<le> length as'"
induct: fixes x1 :: "BExpr"   and x2 :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and x3 :: "Var \<Rightarrow> Val"   and x4 :: "Var \<Rightarrow> Val"   and P :: "BExpr \<Rightarrow> ((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "(x1, x2, x3, x4) \<in> varD"   and "\<And>b s A t. \<lbrakk>\<not> evalB b s; A s t\<rbrakk> \<Longrightarrow> P b A s t"   and "\<And>b s A w t. \<lbrakk>evalB b s; A s w; (b, A, w, t) \<in> varD; P b A w t\<rbrakk> \<Longrightarrow> P b A s t" shows "P x1 x2 x3 x4"
singular_chain_chain_map: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and X' :: "'d topology"   and g :: "'c \<Rightarrow> 'd" assumes "singular_chain p X c"   and "continuous_map X X' g" shows "singular_chain p X' (chain_map p g c)"
mult_cancel_right2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(times a c = c) = (c = zero \<or> a = one)"
f4: shows "vec (\<alpha> \<cdot> A) = vec_smult \<alpha> (vec A)"
sets_eq_imp_space_eq: fixes M :: "'c measure"   and M' :: "'c measure" assumes "sets M = sets M'" shows "space M = space M'"
af_letter_alt_def_2: fixes \<phi> :: "'b ltl"   and \<nu> :: "'b set" shows "af_G_letter \<phi> \<nu> = step (Unf\<^sub>G \<phi>) \<nu>"
vec_lambda_beta: fixes g :: "'e \<Rightarrow> 'd"   and i :: "'e" shows "vec_lambda g $h i = g i"
strip_widen_acom: fixes C1 :: "'a acom"   and C2 :: "'a acom" assumes "strip C1 = strip C2" shows "strip (C1 \<nabla> C2) = strip C1"
weakStatImpI: fixes Q :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<Psi>'. \<exists>Q' Q''. \<Psi> \<rhd> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q' \<and> insertAssertion (extractFrame P) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q') \<Psi> \<and> \<Psi> \<otimes> \<Psi>' \<rhd> Q' \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q'' \<and> (\<Psi> \<otimes> \<Psi>', P, Q'') \<in> Rel" shows "\<Psi> \<rhd> P \<lessapprox><Rel> Q"
cSUP_upper2: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<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 f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and x :: "'d"   and u :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_above less_eq (f ` A)"   and "x \<in> A"   and "less_eq u (f x)" shows "less_eq u (Sup (f ` A))"
rep_inverse: fixes x :: "('c, 'd) mapping" shows "Mapping (mapping.rep x) = x"
quotient_group_lift_to_quotient_set: fixes H :: "'g set"   and g :: "'g" assumes "Group H \<and> order.greater_eq G H"   and "normal H"   and "g \<in> G" shows "BinOpSetGroup.Abs_G_perm (G // LCoset_rel H) (+) (LCoset_rel H `` {g}) \<rightarrow> H = LCoset_rel H `` {g}"
AE_integrable_fst: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x, y). f x y)" shows "AE x in M2. integrable M1 (f x)"
diff_delta_lnexp_cf7_def: shows "diff_delta_lnexp_cf7 \<equiv> \<lambda>x. x ^ 14 / (numer_cf7 (- x) * numer_cf7 x)"
map_minsky_def: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and M :: "('c \<times> ('c, 'd) Op) set" shows "map_minsky f g M = map_prod f (map_Op f g) ` M"
Suc_llength: fixes n :: "nat"   and xs :: "'c llist" assumes "enat (Suc n) < llength xs" shows "enat n < llength xs"
unique_half_chamber_system_f: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and f' :: "'b \<Rightarrow> 'b"   and g' :: "'b \<Rightarrow> 'b" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "OpposedThinChamberComplexFoldings X f' g' C0"   and "g' ` C0 = g ` C0" shows "f' \<turnstile> ChamberComplex.chamber_system X = f \<turnstile> ChamberComplex.chamber_system X"
internal_iface_name_match_refl: fixes i :: "char list" shows "internal_iface_name_match i i"
cod_simp: fixes f :: "'a \<times> 'a" assumes "C.VV.arr f" shows "C.VV.cod f = C.VxV.cod f"
rep_eq: fixes x :: "'b" shows "finfun_apply (finfun_const x) = (\<lambda>xa. x)"
similarity_eye: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "similarity eye M = M"
inducts: fixes x1 :: "('a, 'b) Proc"   and x2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and P :: "('a, 'b) Proc \<Rightarrow> ('a, 'b) Proc \<Rightarrow> bool" assumes "(x1, x2) \<in> encoding.indRelL Enc"   and "\<And>S. P (TargetTerm (Enc S)) (SourceTerm S)" shows "P x1 x2"
analytic_on_analytic_at: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set" shows "(f analytic_on s) = (\<forall>z\<in>s. f analytic_on {z})"
simps: fixes line1 :: "'a Line"   and line2 :: "'a Line" shows "meets line1 line2 = (\<exists>p. inLine p line1 \<and> inLine p line2)"
sem: shows "sterm_sem (local.adjoint I \<sigma> \<nu>) \<theta> = sterm_sem (local.adjoint I \<sigma> \<omega>) \<theta>"
mult_zero_left: fixes times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and a :: "'g" assumes "class.mult_zero times zero" shows "times zero a = zero"
verts_subdivide: fixes G :: "'b pair_pre_digraph"   and e :: "'b \<times> 'b"   and w :: "'b" shows "pverts (subdivide G e w) = pverts G \<union> {w}"
set_iterator_filter_def: fixes P :: "'b \<Rightarrow> bool" shows "set_iterator_filter P \<equiv> set_iterator_image_filter (\<lambda>x. if P x then Some x else None)"
gbinomial_series_def: fixes abort :: "bool"   and x :: "real" shows "gbinomial_series abort x = gbinomial_series_aux abort x 0 1"
free_tv_alpha: fixes A :: "'a"   and n :: "nat"   and t1 :: "typ" shows "free_tv ($ (\<lambda>x. TVar (if x \<in> free_tv A then x else n + x)) t1) - free_tv A \<subseteq> {x. \<exists>y. x = n + y}"
Ord_sup: fixes k :: "hf"   and l :: "hf" assumes "Ord k"   and "Ord l" shows "Ord (k \<squnion> l)"
sup2: fixes B :: "'g"   and k :: "'g"   and b :: "'g"   and a :: "'g" assumes "B \<equiv> sup k b" shows "sup a B \<equiv> sup k (sup a b)"
initedD: fixes C :: "qtname"   and g :: "oref \<Rightarrow> obj option" assumes "inited C g" shows "\<exists>obj. g (Inr C) = Some obj"
step: fixes s2 :: "nat"   and f :: "nat \<Rightarrow> nat"   and s1 :: "nat"   and e1 :: "((nat \<times> nat) \<times> transition) fset"   and r1 :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and r2 :: "nat \<Rightarrow>f value option"   and es :: "(String.literal \<times> value list) list" assumes "s2 = f s1"   and "\<forall>(s1', t1)|\<in>|possible_steps e1 s1 r1 l i. \<exists>(s2', t2)|\<in>|possible_steps e2 s2 r2 l i. evaluate_outputs t1 i r1 = evaluate_outputs t2 i r2 \<and> execution_simulation f e1 s1' (evaluate_updates t1 i r1) e2 s2' (evaluate_updates t2 i r2) es" shows "execution_simulation f e1 s1 r1 e2 s2 r2 ((l, i) # es)"
increasing: fixes x :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "x \<in> Field leq" shows "(x, trim1 x) \<in> leq"
subst_is_fresh_2: fixes y :: "Vars.var"   and z :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list" assumes "atom y \<sharp> z"   and "atom ` domA \<Gamma> \<sharp>* y" shows "atom y \<sharp> \<Gamma>[y::h=z]"
term_variants_pred_inv_1: fixes P :: "'a \<Rightarrow> 'a list"   and f :: "'a"   and T :: "('a, 'b) Term.term list"   and h :: "'a"   and S :: "('a, 'b) Term.term list" assumes "term_variants_pred P (Fun f T) (Fun h S)" shows "length T = length S"
cfg'': shows "cfg'' = (c\<^sub>1'', s'')"
no_grey_refs_not_reachable: fixes s :: "'mut process_name \<Rightarrow> ('a, 'mut, 'b, 'c) Model.local_state"   and r :: "'c" assumes "no_grey_refs s"   and "reachable_snapshot_inv s"   and "white r s" shows "\<not> reachable r s"
intro: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice" assumes "sequential (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)"   and "seq_distrib_left_axioms (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)" shows "seq_distrib_left (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)"
scc_of_head_inter: shows "head G a \<in> G.scc_of (tail G a) \<inter> H.scc_of (head G a)"
cases: fixes x :: "fterm literal" obtains p :: "char list" and ts :: "fterm list" where "x = Pos p__ ts__"   | p :: "char list" and ts :: "fterm list" where "x = Neg p ts"
simps_5: fixes p :: "('a, 'b) form"   and q :: "('a, 'b) form"   and s :: "'a FOL_Fitting.term"   and k :: "nat" shows "Or p q[s/k] = Or (p[s/k]) (q[s/k])"
all_eq: shows "\<forall>v\<in>B. v mod p_i = v mod p_j"
kcomp_distl: fixes f :: "'b \<Rightarrow> 'd set"   and g :: "'d \<Rightarrow> 'c set"   and h :: "'d \<Rightarrow> 'c set" shows "f \<circ>\<^sub>K (g \<squnion> h) = f \<circ>\<^sub>K g \<squnion> f \<circ>\<^sub>K h"
simps_2: fixes m :: "'m"   and ms :: "'m list" shows "RSpan (m # ms) = {r \<cdot> m |r. r \<in> R} + RSpan ms"
l2_inv6_def: shows "l2_inv6 \<equiv> {s. \<forall>R. in_progress (progress s R) xsk \<longrightarrow> (\<exists>X N. guessed_frame R xsk = Some (Exp X (NonceF (R $ N))))}"
sel_10: fixes x31 :: "'c"   and x32 :: "'b"   and x33 :: "'a" shows "pac_res (Extension x31 x32 x33) = x33"
is_PairVal_def: fixes val :: "val" shows "is_PairVal val = (\<exists>x51 x52. val = <|x51, x52|>)"
WhileStepInduct: fixes b :: "expr"   and c :: "com"   and s1 :: "char list \<Rightarrow> val option"   and s2 :: "char list \<Rightarrow> val option"   and \<Gamma> :: "char list \<Rightarrow> secLevel option" assumes "\<langle>while (b) c,s1\<rangle> \<rightarrow>* \<langle>Skip,s2\<rangle>"   and "\<And>s1 s2. \<langle>c,s1\<rangle> \<rightarrow>* \<langle>Skip,s2\<rangle> \<Longrightarrow> \<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2"   and "\<Gamma>,High \<turnstile> c" shows "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2"
sel_2: fixes x1 :: "int"   and x2 :: "int" shows "b (X x1 x2) = x2"
square_free_factorizationD_5: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list" assumes "local.square_free_factorization p (c, bs)" shows "distinct bs"
simps_2: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'d" shows "(\<oplus>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub>) = add"
insert_eq_cut_less_cut_greater: fixes n :: "'b"   and I :: "'b set" shows "insert n I = I \<down>< n \<union> {n} \<union> I \<down>> n"
zero_aezfun_apply: fixes x :: "'b" shows "aezfun 0 x = (0::'a)"
intuitionistic_sep_conj_sep_true_simp: fixes P :: "'a \<Rightarrow> bool" assumes "intuitionistic P" shows "(P \<and>* (\<lambda>s. True)) = P"
lower_rank_of_eq_upper_rank_of: shows "rank_of \<E> = upper_rank_of \<E>"
add_cf_commute: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c"   and "pol_coeff S d" shows "\<forall>j\<le>max (fst c) (fst d). snd (add_cf S c d) j = snd (add_cf S d c) j"
AE_add: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and f' :: "'c \<Rightarrow> 'd"   and g' :: "'c \<Rightarrow> 'd" assumes "AEeq M f g"   and "AEeq M f' g'" shows "AEeq M (\<lambda>w. f w + f' w) (\<lambda>w. g w + g' w)"
siso_ParT: fixes cl :: "('test, 'atom, 'choice) cmd list" assumes "properL cl"   and "sisoL cl" shows "siso (ParT cl)"
IH: fixes cs :: "'edge list" shows "upd_rev_cs cs (a # as') = upd_rev_cs cs as'"
R_2: shows "u \<in> [u]\<^sub>\<R>"
split_list_last_propE: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" obtains ys :: "'g list" and x :: "'g" and zs :: "'g list" where "xs = ys @ x # zs" and "P x" and "\<forall>z\<in>set zs. \<not> P z"
h2: shows "p = isolate_variable_sparse p x 0 * Var x ^ 0"
tab_is_tabulation: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg r tab (tab\<^sub>0 r) (tab\<^sub>1 r)"
tape_of_nl_app_Suc: fixes list :: "nat list"   and ab :: "nat" shows "<list @ [Suc ab]> = <list @ [ab]> @ [Oc]"
nonpos_Ints_mult: fixes x :: "'a"   and y :: "'a" assumes "x \<in> \<int>\<^sub>\<le>\<^sub>0"   and "y \<in> \<int>\<^sub>\<le>\<^sub>0" shows "x * y \<in> \<nat>"
smult_l_minus: fixes a :: "'a"   and x :: "'c" assumes "a \<in> carrier R"   and "x \<in> carrier M" shows "\<ominus> a \<odot>\<^bsub>M\<^esub> x = \<ominus>\<^bsub>M\<^esub> (a \<odot>\<^bsub>M\<^esub> x)"
not_mem_frontier_diff: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and t :: "'c"   and M :: "'c zmultiset"   and N :: "'c zmultiset" assumes "dataflow_topology summary results_in"   and "t \<notin>\<^sub>A frontier M - frontier N"   and "t \<notin>\<^sub>A frontier N - frontier M" shows "zcount (zmset_of (mset_set (set_antichain (frontier M))) - zmset_of (mset_set (set_antichain (frontier N)))) t = 0"
denote_partial_density_operator: fixes S :: "com"   and \<rho> :: "complex mat" assumes "well_com S"   and "partial_density_operator \<rho>"   and "\<rho> \<in> carrier_mat d d" shows "partial_density_operator (denote S \<rho>)"
castmsg_simps_2: fixes P :: "'d \<Rightarrow> bool"   and d :: "nat"   and m :: "nat" shows "castmsg P (d:deliver(m)) = True"
defs_uses_disjoint: fixes n :: "'a"   and g :: "('a, 'b, 'c) gen_cfg_wf" assumes "n \<in> set (gen_wf_\<alpha>n g)" shows "gen_wf_defs g n \<inter> gen_wf_uses g n = {}"
filter_eq_nths: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "filter P xs = nths xs {i. i < length xs \<and> P (xs ! i)}"
intro: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool" assumes "\<And>P \<phi> p. satisfies P \<phi> \<Longrightarrow> satisfies (p \<bullet> P) (p \<bullet> \<phi>)"   and "\<And>P \<alpha>Q p. transition P \<alpha>Q \<Longrightarrow> transition (p \<bullet> P) (p \<bullet> \<alpha>Q)" shows "nominal_ts satisfies transition"
simple_matches_andD: fixes m1 :: "'a simple_match"   and p :: "('a, 'b) simple_packet_scheme"   and m2 :: "'a simple_match" assumes "simple_matches m1 p"   and "simple_matches m2 p" shows "\<exists>m. simple_match_and m1 m2 = Some m \<and> simple_matches m p"
Resolution_unnecessary: fixes T :: "'b literal set set"   and S :: "'b literal set set"   and D :: "'b literal set" assumes "\<forall>C\<in>T. S \<turnstile> C" shows "(T \<union> S \<turnstile> D) = (S \<turnstile> D)"
singleton_sup_single_single: fixes default :: "unit \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" shows "singleton default (single x \<squnion> single y) = (if x = y then x else default ())"
image_eq_imp_comp: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and g :: "'d \<Rightarrow> 'b"   and B :: "'d set"   and h :: "'b \<Rightarrow> 'e" assumes "f ` A = g ` B" shows "(h \<circ> f) ` A = (h \<circ> g) ` B"
real_root_less_iff: fixes n :: "nat"   and x :: "real"   and y :: "real" assumes "0 < n" shows "(root n x < root n y) = (x < y)"
less_1: fixes ls :: "'a list"   and ls' :: "'a list" assumes "length ls < length lsa__"   and "\<And>xs x ys x' ys'. \<lbrakk>ls = xs @ [x] @ ys; ls' = xs @ [x'] @ ys'; x \<noteq> x'\<rbrakk> \<Longrightarrow> thesis"   and "\<And>xs. \<lbrakk>ls = ls' @ xs; xs \<noteq> []\<rbrakk> \<Longrightarrow> thesis"   and "\<And>xs. \<lbrakk>ls @ xs = ls'; xs \<noteq> []\<rbrakk> \<Longrightarrow> thesis"   and "ls \<noteq> ls'"   and "ls \<noteq> []"   and "ls' \<noteq> []" shows "thesis"
trapprop: shows "stable_on (fst ` (CX \<union> P)) trap"