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vendor/scrivo/highlight.php/test/detect/mizar/default.txt

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+::: ## Lambda calculus
+
+environ
+
+  vocabularies LAMBDA,
+      NUMBERS,
+      NAT_1, XBOOLE_0, SUBSET_1, FINSEQ_1, XXREAL_0, CARD_1,
+      ARYTM_1, ARYTM_3, TARSKI, RELAT_1, ORDINAL4, FUNCOP_1;
+
+  :: etc...
+
+begin
+
+reserve D for DecoratedTree,
+        p,q,r for FinSequence of NAT,
+        x for set;
+
+definition
+  let D;
+
+  attr D is LambdaTerm-like means
+    (dom D qua Tree) is finite &
+::>                          *143,306
+    for r st r in dom D holds
+      r is FinSequence of {0,1} &
+      r^<*0*> in dom D implies D.r = 0;
+end;
+
+registration
+  cluster LambdaTerm-like for DecoratedTree of NAT;
+  existence;
+::>       *4
+end;
+
+definition
+  mode LambdaTerm is LambdaTerm-like DecoratedTree of NAT;
+end;
+
+::: Then we extend this ordinary one-step beta reduction, that is,
+:::  any subterm is also allowed to reduce.
+definition
+  let M,N;
+
+  pred M beta N means
+    ex p st
+      M|p beta_shallow N|p &
+      for q st not p is_a_prefix_of q holds
+        [r,x] in M iff [r,x] in N;
+end;
+
+theorem Th4:
+  ProperPrefixes (v^<*x*>) = ProperPrefixes v \/ {v}
+proof
+  thus ProperPrefixes (v^<*x*>) c= ProperPrefixes v \/ {v}
+  proof
+    let y;
+    assume y in ProperPrefixes (v^<*x*>);
+    then consider v1 such that
+A1: y = v1 and
+A2: v1 is_a_proper_prefix_of v^<*x*> by TREES_1:def 2;
+ v1 is_a_prefix_of v & v1 <> v or v1 = v by A2,TREES_1:9;
+then
+ v1 is_a_proper_prefix_of v or v1 in {v} by TARSKI:def 1,XBOOLE_0:def 8;
+then  y in ProperPrefixes v or y in {v} by A1,TREES_1:def 2;
+    hence thesis by XBOOLE_0:def 3;
+  end;
+  let y;
+  assume y in ProperPrefixes v \/ {v};
+then A3: y in ProperPrefixes v or y in {v} by XBOOLE_0:def 3;
+A4: now
+    assume y in ProperPrefixes v;
+    then consider v1 such that
+A5: y = v1 and
+A6: v1 is_a_proper_prefix_of v by TREES_1:def 2;
+ v is_a_prefix_of v^<*x*> by TREES_1:1;
+then  v1 is_a_proper_prefix_of v^<*x*> by A6,XBOOLE_1:58;
+    hence thesis by A5,TREES_1:def 2;
+  end;
+ v^{} = v by FINSEQ_1:34;
+  then
+ v is_a_prefix_of v^<*x*> & v <> v^<*x*> by FINSEQ_1:33,TREES_1:1;
+then  v is_a_proper_prefix_of v^<*x*> by XBOOLE_0:def 8;
+then  y in ProperPrefixes v or y = v & v in ProperPrefixes (v^<*x*>)
+  by A3,TARSKI:def 1,TREES_1:def 2;
+  hence thesis by A4;
+end;